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... procedure of T -matrix of dielectric cylinders(s) Analysis of Finite- sized EBG Materials and Devices by Scattering Matrix Method Chapter T -matrix of Conducting and Dielectric Cylinders and Scattering Matrix. .. the RCS of the same cylinder Analysis of Finite- sized EBG Materials and Devices by Scattering Matrix Method Chapter T -matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method. .. a N Analysis of Finite- sized EBG Materials and Devices by Scattering Matrix Method (3.58) Chapter T -matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 30 Define matrix
ANALYSIS OF F INITE- SIZED ELECTROMAGNETIC B ANDGAP M ATERIALS AND D EVICES BY SCATTERING M ATRIX M ETHOD BY WANG QUANXIN B.ENG. BIOMEDICAL ENGINEERING SHANGHAI JIAOTONG UNIVERSITY, 2001 A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATINAL UNIVERSITY OF SINGAPORE 2004 @ National University of Singapore, All Right Reserved 2004 i ACKNOWLEDGEMENT I would like to express my sincere gratitude to Dr. Zhang Yaojiang, not only for the clear and valuable guidance and support to this project, which has led me get into the gate of research, but also for the encouragement and positive comments, which have always given me confidence in the last two years. I am very grateful to Dr. Li Er Ping, who gives the expert advice, which has helped me a lot in completing the project successfully. His guidance and comments have aided me in solving difficulties and creating new ideas. I must also thank Prof. Ooi Ban Leong, for helping me a lot in understanding basic knowledge of Electromagnetics. Acknowledgements should also go to other IHPC faculties, students and all my friends. All of them created a happy and inspiring environment for me and greatly helped me when I was doing my project. I must take this opportunity to express my deep love and gratefulness to my parents. They are always my source of encouragement. Author 10 December 2003 i ABSTRACT Keywords: Electromagnetic bandgap (EBG), photonic crystals, scattering matrix method, addition theorem, ferrite or chiral cylinder, tunable EBG devices Besides conventional applications in frequency selective surfaces or gratings, periodic structures regain extensive studies nowadays due to potential utilizations in either novel patch antenna design or photonic crystal waveguides. The property of electromagnetic bandgap (EBG), shown in photonic crystals, makes it very promising in future high density optical interconnects. In this thesis, scattering matrix method is used to simulate the transmission properties of the finite-sized two dimensional EBG materials. By the implementation of addition theorem, multiple scatterings for different dielectric rods are accurately modeled and the forbidden frequency or wavelength bands are efficiently predicted. The results are validated by comparison with other methods as well as some reference data. The method is extended to study some novel EBG structures which contain ferrite or chiral cylinders as defects, where several tunable pho tonic crystal devices are proposed including ferrite defect filters, couplers and Y-branches. The study provid es a new approach to control the flow of light in photonic crystals. ii Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Table of Contents 3 TABLE OF CONTENTS ABSTRACT… … … … … … … … … … … … … … … … … … … … … … … … … … … . . … … . I ACKNOWLEDGEMENT … … … … … … … … … … … … . .… … … … … … … … … … … II SUMMARY … … … … … … … … … … … … . .… … … … … … … … … … … … … … … … ...III 1 INTRODUCTION....................................................................................................... 1 2 3 1.1 PROBLEM D ESCRIPTION ....................................................................................... 1 1.2 SCOPE OF WORK .................................................................................................. 2 1.3 O RIGINAL C ONTRIBUTION .................................................................................... 3 BACKGROUND KNOWLEDGE ........................................................................... 4 2.1 E LECTROMAGNETIC BANDGAP STRUCTURE......................................................... 4 2.2 SCATTERING MATRIX METHOD.......................................................................... 7 T-MATRIX OF CONDUCTING AND DIELECTRIC CYLINDERS AND SCATTERING MATRIX METHOD .................................................................... 9 3.1 SCATTERING OF METAL C YLINDER ...................................................................... 9 3.1.1 Scattering Matrix of a Metal Cylinder....................................................... 9 3.1.2 Scattering Matrix Method for Metal Cylinder Array................................ 13 3.2 SCATTERING OF DIELECTRIC C YLINDER ............................................................ 21 3.2.1 Scattering Matrix of a Dielectric Cylinder ............................................... 22 3.2.2 Scattering Matrix Method for Parallel Dielectric Cylinder Array ........... 25 3 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Table of Contents 3.2.3 Parameters of Dielectric Cylinder EBG ................................................... 29 3.2.4 The Field Calculation Inside the Dielectric Cylinders ............................. 37 3.3 4 4 HARD WARE IMPLEMENTATION.......................................................................... 39 3.3.1 The EBG Structure .................................................................................... 39 3.3.2 Experiment Facilities ................................................................................ 40 3.3.3 Experiment Results and Discussion .......................................................... 43 3.3.4 Conclusion ................................................................................................ 45 ELECTROMAGNETIC BANDGAP STRUCTURES COMPOSED BY MULTI-LAYERED CYLINDERS, FERRITE AND CHIRAL CYLINDERS 46 4.1 SCATTERING MATRIX OF MULTILAYERED DIELECTRIC C YLINDERS .................. 46 4.1.1 S-parameter Method ................................................................................. 47 4.1.2 Scattering of Coated Cylinder EBG .......................................................... 51 4.2 SCATTERING MATRIX OF F ERRITE C YLINDERS .................................................. 53 4.2.1 T-matrix of Ferrite Cylinder ..................................................................... 54 4.2.2 Scattering of Ferrite Cylinder EBG .......................................................... 58 4.3 4.3.1 Scattering Matrix of Chiral Cylinder........................................................ 62 4.3.2 Scattering of Chiral Cylinder .................................................................... 73 4.4 5 SCATTERING MATRIX OF C HIRAL C YLINDERS ................................................... 62 A GGREGATED T- MATRIX OF MULTIPLE C YLINDERS .......................................... 74 NOVEL ELECTROMAGNETI C BANDGAP DEVICES CONSTRUCTED WITH COATED CYLINDERS OR FERRITE CYLINDERS AS DEFECTS 81 5.1 T- JUNCTION FILTERS C OMPOSED OF C OATED D IELECTRIC EBG ....................... 81 4 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Table of Contents 5.1.1 A T-junction Filter .................................................................................... 81 5.2 6 5 TUNABLE EBG D EVICES WITH FERRITE D EFECTS ............................................. 90 5.2.1 EBG Filter Tuned by Ferrite Defects........................................................ 90 5.2.2 Tunable EBG Coupler............................................................................... 93 5.2.3 Y-branch Filters ........................................................................................ 97 5.2.4 Conclusions ............................................................................................. 100 EXCITATION OF ELECTROMAGNETIC BANDGAP STRUCTURES BY GAUSSIAN BEAM AND WIRE LINE SOURCES ......................................... 101 6.1 6.1.1 Gaussian Beam ....................................................................................... 101 6.1.2 Scattering Matrix .................................................................................... 103 6.2 7 EBG ANALYSIS USING GAUSSIAN B EAM ILLUMINATION ................................ 101 WIRE L INE EXCITATION OF EBG STRUCTURES ................................................ 105 SUMMARY............................................................................................................. VI REFERENCES .............................................................................................................. 108 5 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Summary SUMMARY In this project, two-dimensional finite cylinder Electromagnetic Bandgap structures are studied by using the Scattering matrix method. Basic theory of the scattering matrix for cylinder array is described. Some useful devices based on 2-D EBG structures, including the coated cylinder EBG and ferrite cylinder EBG, are examined. Scattering matrix method is a semi-analytic method that takes advantage of the analytical solution of circular cylinders and addition theorem of harmonic waves. It is efficient in the calculation of transmission and field distribution of two dimensional cylinder EBG structures. The detailed process and examples of metal and dielectric cylinder EBG are given and discussed. Coated cylinder will alternate the EM properties of a cylinder, and consequently, modify the EM properties of EBG structures. A new T-junction filter device with coated cylinder is described in chapter 5 and it is discussed under conditions of different rod radius and ring radius. Ferrite cylinder is a good controller for EBG structure due to its unique characteristic that its EM property changes with the applied DC magnetic field. Its scattering matrix is derived in chapter 4 and some devices based on ferrite cylinder EBG are described and discussed. Those EBG devices with ferrite cylinder are tunable of its transmission property owing to the EM property of ferrite material. vi Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Summary Chiral cylinder EBG is also briefly described and its scattering matrix is derived. Similar to ferrite cylinder EBG, it can also be used in tunable EBG device design. However, due to limited time, no device based on chiral cylinder has been given in our project, and future work will focus on this area. This project can be extended to three-dimensional EBG case, which is a very promising area. vii Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method List of Figures List of Figures Fig 2.1: Model of interconnect.............................................................................................5 Fig 2.2: An example of electromagnetic bangap structurte .................................................6 Fig 3.1: Calculation modle of single metal c ylinder ............................................................9 Fig 3.2: Electric field distribution of single metal cylinder as Radius= λ .........................12 Fig 3.3: RCS of single metal cylinder with Radius= λ .....................................................12 Fig 3.4: Calculation model of two dimens ional cylinder array .........................................13 Fig 3.5: Translation model in the cylindrical coordinate system.......................................14 Fig 3.6: Geometry of triangular lattice metal EBG structure .............................................19 Fig 3.7: Transmission versus wavelength for the crystal in Fig 3.6 ..................................19 Fig 3.8: Electric field distributions of TM-polarized wave in EBG of Fig 3.6 For (a) λ = 7.35 (b) λ = 9 ...........................................................................................19 Fig 3.9: Scattering pattern by three cylinders ( ka = 0.75 , kd = 2π , θ =90o ). (a) is from [13], (b) is our result. ................................................................................................20 Fig 3.10: Scattering pattern of metal cylinder array in Fig 3.9 computed with different truncation numbers of the expansion.........................................................................21 viii Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method List of Figures Fig 3.11: Outside cylinder electric field distribution of dielectric cylinder with radius= λ , ε r = 8.41................................................................................................... … 24 Fig 3.12: RCS of single dielectric cylinder with radius= λ , ε r = 8.41 .............................24 Fig 3.13: Geometry of triangular lattice EBG with dielectric cylinders. ...........................26 Fig 3.14: Transmission spectra from a EBG structure.......................................................27 Fig 3.15: Magnetic field distributions of TE-polarized wave around EBG in Fig 3.12 (a) ω a / 2π c = 0.96 (b) ω a / 2π c = 0.40 .....................................................................27 Fig 3.16: Electric field distributions of TE-polarized wave around EBG of Fig 3.12 (a) ω a / 2π c = 0.96 (b) ω a / 2π c = 0.40 .....................................................................28 Fig 3.17: Transmission of versus wavelength compared with [1], nc = 2.9 ε r = 8. 41 ..........28 Fig 3.18: Geometry of Triangular lattice dielectric cylinder EBG with one defect. .........30 Fig 3.19: Transmission spectrum versus wavelength for the crystal in Fig 3.16 compared with the same crystal but without defect....................................................................30 Fig 3.20: Electric field distribution of EBG in Fig 3.16 with the resonant mode for λ = 9.0572 ± i0.00092 . ..............................................................................................31 Fig 3.21: EBG with two defects (a) Distant defects (b) Near defects................................32 Fig 3.22: Transmission versus wavelength for the EBGs in Fig 3.19 ...............................33 ix Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method List of Figures Fig 3.23: The finite size EBG structure .............................................................................33 Fig 3.24 : Tuning of PBG properties by additional rods. d x = 4, d y = 4 , r = 0.6 , ε r = 8.41, a = 4 ...........................................................................................................33 Fig 3.25 Electric field distribution ωa / 2πc = 0.401 (a) without additional rows (b) d=2.0 ..........................................................................................................................34 Fig 3.26: Finite size EBG with different rod radius ε r = 8.41 ..........................................35 Fig 3.27: Transmission spectra of PBG structures shown in Fig.3 with different rod radius R at r = 0.6 , Spacing=4 ............................................................................................35 Fig 3.28: Finite size EBG with different rod permittivity at Radius = 0.6 ,Spacing=4..... 35 Fig 3.29: Transmission spectra of EBG with different rod permittivity............................36 Fig 3.30: Transmission spectra of EBG structure with different odd a nd even column rods. ....................................................................................................................................36 Fig 3.31: The 2D EBG structure M x = 9, M y = 9, a = 4, r0 = 0.6, d = 4, l = 6 . .............37 Fig 3.32: Effects of mixed metal(even row) and dielectric EBG. .....................................37 Fig 3.33: Electric field distributions inside cylinders ........................................................38 Fig 3.34: Geometry of 2D metal cylinder EBG structure. .................................................39 x Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method List of Figures Fig 3.35: Manufactured 2D metal-cylinder EBG. The board used in the structure is a wooden board .............................................................................................................40 Fig 3.36: Horn antenna: 1.5-18 GHz. ................................................................................41 Fig 3.37: Sweep oscillator: Hewlett Packard 8350A .........................................................42 Fig 3.38: Frequency converter: Hewlett Packard 8511B 45MHz~50GHz. .......................42 Fig 3.39: Microwave receiver: Hewlett Packard 8530A. ..................................................43 Fig 3.40: RCS of the 2D metal-cylinder EBG structure ....................................................44 Fig 3.41: Transmission of the 2D metal EBG ....................................................................44 Fig 4.1: Calculation model of multilayer cylinder .............................................................47 Fig 4.2: S- matrix network ..................................................................................................47 Fig 4.3: Calculation model 1 for S-parameter ...................................................................48 Fig 4.4: Calculation model 2 for S-parameter ...................................................................49 Fig 4.5: (a) Geometry of triangular lattice metal EBG structure. The segment above the structure is the one used for the computation of the transmission. (b) Transmission spectra of ring rod EBG structure ..............................................................................52 Fig 4.6: Comparison of transmission spectra of coated dielectric rod...............................53 xi Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method List of Figures Fig 4.7: Electric field distribution r1 = 0.7, r2 = 0.3, nc1 = 5.0, nc 2 = 2.9 , (a) λ = 9.6 passband (b) λ = 8.0 stop-band ........................................................................................53 Fig 4.8: Scattering Patterns of BSC for (a) for single ferrite cylinders compared with [15] (b) an array of circular ferrite cylinders ( d = 1.5λ ) compared with [15].................59 Fig 4.9: Geometry of the ferrite cylinder EBG. .................................................................60 Fig 4.10: Transmission spectra of EBG with different added DC magnetic field . ............60 Fig 4.11: Field distribution of EBG in Fig 4.9 with different added DC magnetic field intensity at λ = 10.3663 . (a) M z = 1× 1013 A/m (b) M z = 2 × 1013 A/m......................60 Fig 4.12: Transmission spectra of EBG in Fig 4.9 with different magnetic susceptibility.61 Fig 4.13: Transmission spectra of EBG in Fig 4.9 with different gyro magnetic ratio .....61 Fig 4.14: Calculation model of 2-D chiral cylinder...........................................................62 Fig 4.15: Echo width of one chiral cylinder compared with paper (a) our simulation result (b) from paper[24] ......................................................................................................73 Fig 4.16: Test of the rightness of equation by comparing special case with dielectric cylinders .....................................................................................................................73 Fig 4.17: Calculation model for two cylinder combined aggregate T-matrix ...................75 Fig 4.18: Calculation model of aggregate T- matrix. ..........................................................75 xii Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method List of Figures Fig 4.19: EBG structure model (a) real structure (b) combined structure .........................79 Fig 4.20: Comparison of transmission spectra of usual scattering matrix method and aggregate T-matrix method ........................................................................................79 Fig 5.1: 2-D T-junction EBG structure ..............................................................................82 Fig 5.2: Transmission spectra of dielectric EBG in Fig 5.1 a = 1 , radius = 0.15 , ε r = 8.41.....................................................................................................................82 Fig 5.3: Electric field distribution of EBG in Fig 5.1 (a) λ = 2.09 (b) λ = 2.13 (c) λ = 2.29 ................................................................................................................83 Fig 5.4: Transmission spectra of hollow EBG in Fig 9 (inner radius=0.05) .....................84 Fig 5.5: Electric field distribution of EBG in Fig 12 (a) λ = 2.02 (b) λ = 2.20 ...........85 Fig 5.6: Transmission spectra of hollow EBG in Fig 9 (inner radius=0.1) .......................85 Fig 5.7: Electric field distribution of EBG in Fig 14 (a) λ = 1.81 (b) λ = 1.99 .................85 Fig 5.8: Transmission spectra of metal inner layer EBG in Fig 9(inner radius=0.05) .87 Fig 5.9: Electric field distribution of EBG in Fig 14 (a) λ = 4.20 (b) λ = 3.39 . ...........87 Fig 5.10: Transmission spectra of meta l inner layer EBG in Fig 9 (inner radius=0.1) ....88 Fig 5.11 Electric field distribution of EBG in Fig 14 (a) λ = 2.45 (b) λ = 2.53 (c) λ = 2.90 (d) λ = 4 ....................................................................................................89 xiii Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method List of Figures Fig 5.12: EBG with one ferrite cylinder. ...........................................................................90 Fig 5.13 Transmission spectra of EBG in Fig 5.12 with different added DC magnetic field ............................................................................................................................90 Fig 5.14: Electric field distribution for EBG in Fig 5 with different added DC magnetic field intensity. (a) mz = 1× 10 14 A/m λ = 9 .2367 (b) mz = 1 .5 × 10 14 A/m, mz = 1× 1014 A/m, λ = 8.9611 (d) mz = 1 .5 × 10 14 A/m, λ = 9 .2367 (c) λ = 8.9611 .................................91 Fig 5.15: Transmission spectra with (a) different H0/Mz (b) different gyromagnetic ratio ............................................................................................................................92 Fig 5.16: Transmission spectra with different added DC magnetic field. .........................93 Fig 5.17 The geometry of alternate Coupler......................................................................94 Fig 5.18: Transmission characteristics of coupler with varying added magnetic field intensities. ..................................................................................................................94 Fig 5.19: Electric field distribution with different added DC magnetic field intensity at λ = 9.0936 ， γ = 10 （ a ） M z =0 (dielectric cylinder) (b) M z =0.2E14 (c) M z =5E14...................................................................................................................95 Fig 5.20: Geometry of coupler with ferrite defects............................................................95 Fig 5.21: Transmission of coupler with ferrite defects with different added magnetic field intensities ...................................................................................................................95 xiv Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method List of Figures Fig 5.22: Electric field distribution with different added DC magnetic field intensity at， γ = 10 （a） λ = 9.0198 , M z =0.3E14 (b) λ = 9.0198 , M z =1E14 (c) λ = 10.1300, M z =0.3E14 (d) λ = 10 .1300 , M z =1E14. .................................................................96 Fig 5.23 Geometry of ferrite the Y-branch filter ...............................................................98 Fig 5.24: Transmission spectra of the Y-branch structure when left discontinuity M z =0.8E14, right discontinuity M z =3E14. .............................................................98 Fig 5.25: Electric field distribution of Y-branch filter with different wavelength at γ = 10 when left discontinuity M z =0.8E14, right discontinuity M z =3E14 （ a ） λ = 9.0000 (b) λ = 9.3023 (c) λ = 10 .1695 ............................................................99 Fig 5.26: Transmission spectra of the Y-branch structure when (a) left discontinuity M z =0.5E14, right discontinuity M z =3E14 (b) left discontinuity M z =6E14, right discontinuity M z =3E14.............................................................................................99 Fig 6.1: Incidence model of Gaussian beam....................................................................101 Fig 6.2: Geometry of EBG structure for Gaussian incidence calculation........................105 Fig 6.3: Electric field distributions of Gaussian beam incidence (a) λ = 39.5 , stop-band (b) λ = 0.2 , pass-band ..............................................................................................105 Fig 6.4: Translation model in the cylindrical coordinate system for Hankel function....106 xv Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method List of Figures Fig 6.5: EBG structure with wire source .........................................................................107 Fig 6.6: Electric field distributions at resonant mode λ = 9.0575 ...................................107 xvi Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method List of Symbols List of Symbols E: Electric field H: Magnetic field k: Wave number k0 : Wave number in free space λ: Wave length λ0 : Wave length in free space f : Frequency ω: Angular frequency ε: Permittivity nc : Refractive index µ Permeability : εr : Relative permittivity µr : Relative permeability ε0 : Permittivity in free space µ0 : Permeability in free space xiiii Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method List of Symbols T: T-matrix f: Scattering matrix a: Incident matrix a: Mutual effect matrix γ: Gyro magnetic ratio ξc : Chiral admittance of the medium εc : Effective permittivity xiiii Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 1 1 Introduction 1 INTRODUCTION 1.1 PROBLEM DESCRIPTION Electromagnetic bandgap (EBG) structures are typically a class of periodic refractive materials which exhibit useful band rejection behavior, which means that, in some specific frequency band, electromagnetic wave propagation is totally prohibited for any polarization. The discovery of EBG structure makes it possible to control EM wave propagations and leads to numerous novel applications in the optical or microwave technologies. 2D cylinder array EBG structure is widely investigated at present due to the fact that it is easy to fabricate using nowadays microelectronics technologies, and this project will focus on it. Full- wave solvers, such as Finite Difference Time Domain (FDTD) Method and Finite Element Method (FEM) as well as Beam Propagation Method (BPM), are accurate and flexible in modeling EBG structures, but they are time consuming and thus, inefficient. Recently, scattering matrix method has been used to analyze EBG structures [1] [2]. It takes advantage of the addition theorem of cylindrical harmonics to compute multiple scattering among parallel cylinders. Therefore, transmission property of finite-sized 2D EBG materials could be simulated as a scattering problem. Either near field distribution or far radiation pattern could be obtained conveniently by using this method. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 1 Introduction 2 Instead of conventional metal or dielectric EBG structures, in this thesis, scattering matrix method is extended to study several novel media EBGs including ferrite and chiral cylinder EBGs. Furthermore, some new EBG devices such as tunable ferrite filters, couplers and Y-branches, are proposed. This research project provides a new approach to design novel EBG circuits. 1.2 SCOPE OF WORK The thesis is divided into the following chapters: Chapter 2 overviews the background knowledge of EBG and Scattering matrix method. Chapter 3 describes the underlying theory, and scattering matrix method for metal cylinder and dielectric cylinder EBG structures which is used in this project. In Chapter 4, detailed description of scattering matrix method for many special cylindrical EBGs, including coated cylinder, ferrite cylinder and chiral cylinder EBG, is provided. In addition, a useful aggregated T- matrix method is presented. Chapter 5 provides some useful devices based on coated cylinder EBG Structure and ferrite cylinder EBG structure. Their EM property and potential use are discussed. Chapter 6 studies the scattering matrix of EBG structures under two special incidences, namely Gaussian beam and wire source. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 1 Introduction 3 Chapter 7 serves as a brief summary of the entire project. 1.3 ORIGINAL CONTRIBUTION (I) Conference paper Zhang Yaojiang, Wang Quanxin, and Li Erping, “Analysis of finite-size 2D coated electromagnetic bandgap structures by scattering matrix method”, ISAPE2003, Beijing, 17-19 Nov, 2003 (II) Journal Papers Wang Quanxin, Zhang Yaojiang, Li Erping and Ooi Ban Leong, “Modeling of electromagnetic band gap structure devices tuned by ferrite cylinders”, Microwave and Optical Technology Letters. (Accepted) Wang Quanxin, Zhang Yaojiang, Li Erping and Ooi Ban Leong, “Analysis of finite-sized 2D coated electromagnetic band gap structures by scattering matrix,” IEEE Tran on Selected Topics in quantum Electronics. (Submitted) Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 2 Background Knowledge 4 2 BACKGROUND KNOWLEDGE 2.1 ELECTROMAGNETIC BANDGAP STRUCTURE With the increase of operating frequencies of electronic circuits, the integrated circuits suffer more electromagnetic radiation problems than ever. Accordingly, the usual electrical interconnects are facing more and more challenges to keep signal integrity. Therefore, it has been predicted that the optical interconnects will become the dominant interconnects for integrated circuits, which are correspondingly named optical integrated circuit. In the extreme high-frequency band, wave will propagate through the wall at the sharp bend (Fig 2.1 (a)). Consequently, the circular arc bend is designed to reduce the energy loss (Fig 2.1 (b)) at the waveguide band. However, the arc bend can only reduce energy loss partially. But partial energy can penetrate through the waveguide wall not at the bend position. In contrast, the optical interconnect are able to reduce the scattering loss caused by the sharp bend made of usua l materials. Photonic bandgap (PBG, also called Electromagnetic bandgap) structure is used to achieve this purpose because of its perfect ability to prohibit the wave propagation in some specified frequency band. The concept of PBG appeared in 1987 [6] for the first time. It was borrowed from semiconductor crystals in analogy to their electronic bandgap. The work of Yablonovitch makes it possible to produce a full three dimensional photonic bang gap structure lately. The fabrication and design of photonic bandgap structures or devices and the simulation and modeling of such kind of materials are gaining importance recently Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 2 Background Knowledge (a) 5 (b) Fig 2.1 Model of interconnect Electromagnetic bandgap structure is a refractive periodic structure which can effectively prevent the propagation of electromagnetic waves in a specified band of frequency. Usually, this forbidden band is held for all incident angles and for all polarizations of electromagnetic waves if the lattice potential is strong enough. The special period of the stack is called the lattice constant. Fig 2.2 gives a simple example of Electromagnetic bandgap structure with 2D parallel dielectric cylinders. There are three main types of EBGs, that is, one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) according to the dimensionality of the stack [3]. EBGs that work in the microwave and far-infrared regions are relatively easy to be fabricated. On the other hand, those that work in the optical region, especially 3D ones are difficult to fabricate due to their small lattice constant. However, with the development of various Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 2 Background Knowledge 6 process technologies, the fabrication of such EBG structures has become possible in the last ten years, and many good EBG structures with a lattice constant less than 1 millimeter are now available. 1.3 1.2 1.1 Fig 2.2 Example of 2D electromagnetic bandgap structures EBG structure can be used in various microwave and millimeter wave devices and antennas due to the wide-bandwidth of the forbidden frequency band. One application is in the metallic waveguides, which does not allow electromagnetic waves to propagate below a certain threshold frequency, and the wave can only propagate along its axis. Another application is the dielectric mirror. Waves in a certain frequency range, when incident on it, is completely reflected. This property makes it a potential candidate in various applications, such as dielectric mirrors, dielectric Far-Perot filters [5], and distributed feedback lasers. Other potential utilizations include ground plane, waveguide structures, antenna gain enhancement, cavities and miniaturization of antenna dimensions. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 2 Background Knowledge 7 2.2 SCATTERING MATRIX M ETHOD Up to now, various methods, including the plane wave expansion (PWE) [7]-[9], the Transfer- matrix method and finite difference time domain (FDTD) method [10], have been used to analyze the EM properties of Electromagnetic bandgap structures. Among these methods, scattering matrix method is a semi-analytic method, which takes the advantage of the analytical solution of circular cylinders and the addition theorem of harmonic waves. It is, therefore, an efficient approach. This method links the incident field with the scattered field in the form of Fourier-Bessel expansions when applied to cylinders, and each rod is characterized by its scattering matrix. Then the scattering problem is reduced to the resolution of a linear system due to the translation properties of Bessel functions. The incident field for scattering matrix method can be arbitrary. It can be a plane wave, a Gaussian beam, a thin wire source or some other kind of source. The theory will be detailed in the following chapter. Compared with other methods used to calculate EBG structures, scattering matrix method is an efficient and attractive approach. Generally, FDTD and PWE method are time consuming, and they also need much computer resource. Besides, the accuracy of the results when they are applied to calculate round structures is a problem. Scattering matrix method only takes one-tenth of the computing time in comparison to the FDTD method [1] [2]; moreover it is able to guarantee the accuracy. According to [2], the computation time for a field distribution in a finite 2-D crystal by this method is shorter than one -tenth of that by the FDTD method. Transfer-matrix method can only be used to calculate the Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 2 Background Knowledge 8 transmission properties whereas scattering matrix method can give both field distributions and transmission spectra [2]. Another disadvantage of the Transfer-matrix method is that the transfer matrix is singular for structures with dimensions larger than the electron Fermi wavelength, which can also be removed by scattering matrix method [11]. However, we must point out one limitation of this approach: the circle that contains one cylinder cannot interact with the boundary of another circle [1], and this will cause accuracy problem when it is applied to noncircular cylinders. Moreover, the scattering matrix method will become inefficient when the number of cylinder become large or the cylinder radius become large because in these cases, the dimensions of the matrices become very large. In this thesis, we extend the application of scattering matrix method into analysis of EBG structures composed by coated dielectric cylinders, ferrite or chiral rods. Cascaded Sparameter approach is used to obtain the T- matrix of coated dielectric cylinders. Moreover, T-matrix of ferrite and chiral cylinders are also derived rigorously. Based on an extensive study of different EBG materials, several tunable EB G devices are proposed. These include the filters, the couplers and the Y-branch junction constructed by ferrite defects. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 9 3 T-MATRIX OF CONDUCTING AND DIELECTRIC CYLINDERS AND S CATTERING M ATRIX METHOD This chapter mainly introduces the underlying theories of the scattering matrix method used for the simulation of metal cylinder array EBG structures and the dielectric cylinder array EBG structures [31]-[35]. Some simulation examples are conducted and the results of electric field distribution, magnetic field distribution and transmission spectra are given. 3.1 SCATTERING OF METAL CYLINDER 3.1.1 SCATTERING M ATRIX OF A M ETAL CYLINDER Y ρj φj P θ X Fig 3.1 Calculation model of one metal cylinder Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 10 Let us consider a plane wave (TM polarization) with an angular θ (in Fig 3.1). The model is two dimensional, which means the cylinder is infinite in length in the z direction, and the incident field is also z- invariant. The total electric field at point P, which is located outside the cylinder, can be expressed as follows E z = E zinc + E zsca , (3.1) where Ε inc is the electric field of an incident wave from excitation points, and Ε sca z z is the scattering field from the cylinder. Because of the properties of the Helmholtz equation, the incident field can be written as a Fourier-Bessel expansion [1] [2] E zinc = ∞ ∑a J n = −∞ n n ( kρ)e inφ , (3.2) where ρ and φ are the distance and angle of the point P from the cylinder, k = 2π / λ = ω / c is the wave number outside the cylinder, J n is Bessel function of the first kind, a n is the incident coefficient. Let us assume that the incide nt wave is expressed as v v − ik • r E inc = e− ik0 ( x cosθ + y sinθ ) = e− ik 0 ρ (cosθ cosφ + sinθ z =e sinφ ) (3.3) where k 0 represents the wave number in free space and θ is the incident angle with respect to the X − axis . Based on the equation e ikρ cosφ = +∞ ∑J n (kρ )( −i ) n e inφ , n =−∞ the incident wave can be expressed as Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (3.4) Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method E zinc = e −ik0 ρ (cosφ cosθ +sinθ sin φ ) = e −iko ρ cos(φ −θ ) = +∞ ∑J n ( k 0 ρ )(− i ) n e in(φ −θ ) , 11 (3.5) n =−∞ ( x = ρ cosφ ; y = ρ sin φ ). Because − π2 ni ( −i ) n = e , (3.6) the incident wave for one cylinder can be expressed as follows Einc = ∞ ∑e − in ( π2 +θ ) J n ( kρ ) einφ . (3.7) n =−∞ Similar to the incident field, the scattering field of one cylinder can be expressed as Esca = ∞ ∑bH n =−∞ n (2) n ( k ρ )e inφ , (3.8) where bn is the scattering coefficient, which characterizes the scattered field of the cylinder, and it is linked to the incident field through bn = Tn an , (3.9) where Tn is a known squa re matrix element and can be obtained by applying the Boundary condition: Ea = 0 = E( inc) a + E( sca)a to the cylinder surface. The parameters Tn is obtained as Tn = bn J (ka) = − (n2 ) , an H n (ka) Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (3.10) Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method J − M ( ka) − H ( 2 ) ( ka) −M 0 T = . . . 0 0 − . . J − M +1 ( ka) . . H −(2M) +1 ( ka) . . . . . . 0 . . 0 , . . J M ( ka) − ( 2) H M ( ka) 12 0 (3.11) where a is the cylinder radius. Based on the above theory, the electric field distribution outside a two dimensional metal cylinder and its RCS are given in Figs 3.2 and 3.3. As an example, the cylinder radius is set to equal to the wavelength for the two Figs. Radius=λ (λ ) 1 0 -1 -2 -2 -1 0 1 2 (λ) 1.900 -- 2.000 1.800 -- 1.900 1.700 -- 1.800 1.600 -- 1.700 1.500 -- 1.600 1.400 -- 1.500 1.300 -- 1.400 1.200 -- 1.300 1.100 -- 1.200 1.000 -- 1.100 0.9000 -- 1.000 0.8000 -- 0.9000 0.7000 -- 0.8000 0.6000 -- 0.7000 0.5000 -- 0.6000 0.4000 -- 0.5000 0.3000 -- 0.4000 0.2000 -- 0.3000 0.1000 -- 0.2000 0 -- 0.1000 6 Radius=λ 5 4 RCS 2 3 2 1 0 50 100 150 200 250 300 350 Angle(θ) Fig 3.2 Electric field distribution of single Fig 3.3 RCS of single metal cylinder metal cylinder with Radius= λ with Radius= λ Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 13 3.1.2 SCATTERING MATRIX M ETHOD FOR M ETAL CYLINDER ARRAY ρj Y φj P θ X Fig 3.4 Calculation model of two dimensional Cylinder Consider an EBG consisting of a set of N parallel cylinders (Fig 3.4). The incident wave for cylinder j can be written as ( EZ ) j = ∞ ∑a jn Jn ( k ρ j )e inφ j , (3.12) n =−∞ where a j is a column matrix that represents the actual incident field of the cylinder. When the incident field is a plane wave, borrowing the concept derived earlier for a single cylinder, we easily achieve −ik0rj −in( π2 +θ ) a jn = e e =e −ik0 rj (cosθ cosϕj +sinθ sinϕ j ) −in( π2 +θ ) e , e −ik0 rj (cosθ cosϕ j +sinθ sinϕ j ) e −i ( −M )( 2 +θ ) −ik0 rj (cosθ cosϕ j +sinθ sinϕ j ) −i ( − M +1)( π +θ ) 2 e e . a= , . . π e −ik0 rj (cosθ cosϕ j +sinθ sinϕ j) e −iM ( 2 +θ ) (3.13) π Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (3.14) Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 14 where r j is the position of cylinder j , and ϕ j is the angle of the line from start point to the cylinder. The total scattering field from all the cylinders can be expressed as follows N ∞ _ sca E total =∑∑ f z j =1 n =−∞ (inφ j ) jn H n(2) ( k ρ j ) e (3.15) where f jn is the actual scattered field from the cylinder j , H n( 2 ) is the n th order Hankel function of the second kind. The total incident wave to cylinder j consists of two parts. One is directly from the incident wave, while the other is the scattering waves from the other cylinder s. O φ φ' y ρ ? − ?' ?' o' φ '' y '' x x '' Fig 3.5 Translation Model in the cylindrical coordinate system Correspondingly, the scattering wave can be divided into two parts. Therefore, the scattering coefficient of cylinder j can be expressed as [1] [2] [31] Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method f j = Tja j + ∑ Tja ji f i , 15 (3.16) i≠ j where T i stands for the transmission matrix coefficient from j th cylinder, and a ij is the coupling of i th cylinder to j th cylinder, whose elements can be obtained from the addition theorem of cylindrical harmonics [12]. The addition theorem for Hankel function is '' r r H m(2) ( k ρ − ρ ' )eimφ = ∞ ∑H n =−∞ (2) n −m (k ρ ' ) e− i ( n−m )φ J n ( kρ ) einφ . ' (3.17) It means that the scattering wave from one cylinder can be expressed as the incident wave of another cylinder. In our equation, however, we use the second kind of Bessel function instead. The elements of α ij matrix simply contain exponential and Hankel functions. H −( 2Mi) −( −Mj ) (kρ ' )e −i [ − Mi −( −Mj )]φ . a ij = . . . H ( 2) ' _ i [ Mi− (Mj )]φ ' Mi −( − Mj ) ( kρ )e ' . . . . . . . . . H −(2Mi) −Mj ( kρ ' )e − i[_ Mi −Mj ]φ . . . . . . . . . (2 ) ' _ i [ Mi −Mj ]φ H Mi − Mj (kρ ) e ' ' , (3.18) where the matrix is limited to (2 M i +1)*(2 M j +1), as in the computation we cannot use infinity. Therefore, n = [− M i , M i ] m − [ −M j , M j ] (3.19) then α ij ( n, m ) = H n( 2−)m (kρ ' ) e −i (n− m)φ ' Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (3.20) Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 16 For the convenience of coding, we let P = n + M i +1 Q = m + M j +1 n = [1,2M i + 1] m = [1,2 M j + 1] , (3.21) , (3.22) then α ij can be rewritten as α ij ( p, q ) = H (p2−)q + Mj −Mi (kρ ' )e −i ( p −q + Mj − Mi)φ . ' (3.23) Equation (3.16) can be written as f1 f2 . = . . fN T1 T2 . . . a 1 T1 a 2 . + . . TN a N T2 . . . 0 a 12 a 0 21 . . . TN a N1 a N2 . . . . . . . a 1N f1 a 2N f 2 . . . . . . . . . 0 f N (3.24) Here, a ij is a known squared matrix. The scattering matrix can be written compactly as f = T a + Taf , (3.25) f = (I − Ta ) −1 Ta , (3.26) Together with the f matrix the scattering wave can be re-written as follows N Ez = Ezinc + ∑ j =1 ∞ ∑ n =−∞ inφ j f jn H n(2) ( k ρ j ) e , f = (I − Ta ) −1 Ta , a jn = e − ik0 r j (cosθ cos ϕ j +sin θ sin ϕ j ) (3.27) (3.28) e −in ( π2 +θ ) , Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (3.29) Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method T jn = − 17 J n ( kaj ) , H n( 2 ) ( kaj ) (3.30) and α ij ( n, m ) = H n( 2−)m (kρ ' ) e −i (n − m )φ . ' (3.31) Equation (3.27) can be rewritten as Ε z = Ε inc z + [H 1 H2 f 1 f 2 . . . . H N ] , . . f N (3.32) where [ H j = H −(2M) j ( kρ j )e i ( − M j )φ j H −( 2M) j +1 (kρ j )e i ( − M j +1 )φ j . . . H M(2 )j (kρ j )e iM jφ j ] (3.33) Transmission curve is very useful when analyzing the properties of EBG structures, and it is usually represented by the Poynting vector. It can be derived from the Maxwell equation, namely, r r r ∇ × Ε = iω Β = iωµ Η . For our case, µ = µ 0 ( 4π ×10 −7 (3.34) henry / m ) , is the permeability of free space. In our case, electric field has only the Z direction component. The resulting magnetic field becomes r ∇ × ( Ε zˆ) z Η= iωµ 0 = 1 ∂Ε z r ∂Ε z r ( Χ+ Υ ) ( ω = 2πf ). iωµ0 ∂y ∂x Therefore magnetic field in the x − direction can be written as Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (3.35) Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method r Η= 1 ∂Ε z r 1 ∂Ε z r Χ= Χ. iωµ 0 ∂ y 2π if µ 0 ∂ y 18 (3.36) By using equation (3.36), the magnetic field can be obtained. For the calculation of transmission spectra, the Poynting power is obtained by averaging the Poynting power on a line behind the PBG. The line is selected to be two times of the cylinder periodicity in length, and the distance to PBG is the same as the periodicity, i.e. s=∫ line _ end line _ start r s .dl . (3.37) The transmission is defined as the ratio of the so-obtained Poynting power to that of incident wave. By varying the frequency, we can recover the relation between transmissions (reflection) versus frequency. The expected frequency bandgap can be observed from the Fig of transmission (reflection) versus frequency. Fig 3.6 shows the geometry of a two dimensional metal cylinder array. M x = 9, M y = 9, a = 4, r0 = 0.6, d = 4, l = 6 . Fig 3.7 gives the transmission of this array versus wavelength. The incident wave propagates from the bottom of this array, and the segment should be short enough to ensure that no power flowing around the EBG will be collected. Stop-band appears when wavelength is lager than 7.8. Fig 3.9 gives the electric field distribution of this array in the pass-band and the stop-band respectively, which verifies the transmission curve. The transmission can assume values greater than 0dB at some wavelength owing to the focusing effect. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 19 10 35 Metal Cylinder Radius=0.6 30 0 Transmission(dB) 25 20 15 10 5 -10 -20 -30 -40 0 -50 -5 -5 0 5 10 15 20 25 30 6 35 7 8 9 10 11 12 Wavelength(λ) Fig 3.6 Geometry of triangular lattice Fig 3.7 Transmission versus metal EBG structure. The segment wavelength for the crystal in Fig 3.6 below the structure is the one used for λ=9 λ=7.35 1.891 -- 4.000 0.8944 -- 1.891 0.4229 -- 0.8944 0.2000 -- 0.4229 0.0946 -- 0.2000 0.0447 -- 0.0946 0.02115 -- 0.0447 0.01000 -- 0.02115 30 20 0.5712 -- 2.200 0.1483 -- 0.5712 0.0385 -- 0.1483 0.00999 -- 0.0385 0.002595 -- 0.00999 6.736E-4 -- 0.002595 1.749E-4 -- 6.736E-4 4.54E-5 -- 1.749E-4 30 20 10 10 0 0 -10 -10 0 10 20 30 (a) 40 0 10 20 30 40 (b) Fig 3.8 Electric field distributions of TM-polarized wave in EBG of Fig 3.6. For (a) λ = 7.35 (b) λ = 9 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method 13 Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 20 3.0 2.5 Transmission 2.0 1.5 1.0 0.5 0.0 0 60 120 180 240 300 360 θ (a) Fig 3.9 Scattering pattern by three cylinders (ka (b) = 0.75 , kd = 2π , θ =90o ). (a) is from [13 ], (b) is our result. Fig 3.9 shows the RCS of a simple metal cylinder array to verify the method described in this project. It can be seen that the results obtained from our method agree well with those obtained from paper [13] As we have seen from these expansion equations, the expansion series of the field is infinity. However, in our computation and codes, it is necessary to give the specific truncation numbers of the series. In fact, this truncation number varies with the radius of the cylinder. The truncation number becomes larger with the increase of the cylinder radius. In specific case, we should decide the number through our experience. Fig. 3.10 gives an example of the comparison of scattering pattern of the metal cylinder array in Fig. 3.9 with different truncation numbers. We can see that there is a little difference Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 21 between the two curves with the truncation numbers of 3 and 5, while the curves with truncations numbers of 5 and 7 are nearly no difference. Electric field N=3(-1 to 1) N=5(-2 to 2) N=7(-3 to 3) Angle ( θ) Fig 3.10 Scattering pattern of metal cylinder array in Fig 3.9 computed with different truncation numbers of the expansion 3.2 SCATTERING OF DIELECTRIC CYLINDER Compared with the metal cylinder EBG, the scattering matrix expression of dielectric cylinder is the same except for the T- matrix, which is due to the different boundary conditions are applied to the cylinder surface, Therefore, in this part we will focus on the derivation procedure of T-matrix of dielectric cylinders(s). Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 22 3.2.1 SCATTERING MATRIX OF A DIELECTRIC CYLINDER According to the boundary condition, the electric field along the z-direction and magnetic field along the φ -direction are both continuous at ρ = a ( radius ) E( inside) a = E( inc) a + E( sca)a (Along z direction), (3.38a) H ( inside) a = H (inc) a + H ( sca) a (Along φ direction). (3.38b) and We have known from the metal cylinder part that the electric field can be written as Fourier-Bessel expansion and the derivation procedure of the magnetic field along the φ -direction is given below. Suppose the incident pla ne wave assumes the form E zinc = e − jk0 ( x cosθ + y sin θ ) = e − jk0 ρ (cosθ cosφ +sin θ sinφ ) , (3.39) according to Maxwell’s equation, r r ∇ × Ε = − iωΒ = − iωµ Η , (3.40) r r ∇Ε z × Z = −i ωµΗ , (3.41) r ∂Ε z r r ρ × Z = −i ωµΗ φ φ , ∂ρ (3.42) ∂Ε z = iωµΗ φ , ∂ρ (3.43) and the magnetic field along the φ -direction can be obtained as Ηφ = 1 ∂Ε z (µ = µ r µ 0 ) . iωµ ∂ρ (3.44) Hence the incident electric field ( z direction) and magnetic field ( φ direction) can be written as Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method ∞ E inc = z ∑aJ n n ( k1 ρ ) einφ , 23 3.45a) n =−∞ H φinc = k1 ∞ an J n ' (k1 ρ )einφ , ∑ µ1 n =−∞ (3.45b) where k1 is the wave number outside the cylinder, J n' is the derivative of Bessel function of the first kind. Similarly, the scattering electric field ( z direction) and magnetic field ( φ direction) can be written as follows E zsca = ∞ ∑bH n (2) n (k1 ρ )e inφ , (3.46a) n =−∞ Hφsca = k1 ∞ ∑ bn H n( 2 ) ' (k 1ρ )e inφ , µ1 n =−∞ (3.46b) where H n(2)' is the derivative of Hankel function of the second kind. Inside the cylinder, the electric field ( z direction) and magnetic field ( φ direction) can be written as follows E inside = z Hφinside = ∞ ∑cJ n =−∞ k2 µ2 n n (k 2 ρ ) einφ , ∞ ∑c J n =−∞ n ' n ( k2 ρ )einφ , (3.47a) (3.47b) where k2 is the wave number inside the cylinder. Combining (3.45), (3.46),(3.47) with the boundary condition (3.38), we obtain J n (k1 a ) + Tn H n( 2 ) (k1 a ) = C n J n ( k 2 a) , (3.48a) k1 ' k k J n (k1a ) + Tn 1 H n(2)' (k1 a) = Cn 2 J n ' ( k2 a ) . µ1 µ1 µ2 (3.48b) Thus the T- matrix of dielectric cylinder can be obtained by solving (3.48) Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method Tn = − 24 nc J n' (k1a ) J n ( k2 a ) − η J n ( k1 a ) Jn' (k 2 a ) nc H n(2)' (k1a ) J n ( k2 a ) − η H n(2) (k1a ) Jn ' ( k2 a ) , (3.49) η = 1(TE ), n (TM ) 2 c and cn = J n (k1a ) + Tn H n( 2) ( k1a ) J n ( k 2a ) , (3.50) where nc is the refractive index of the cylinder. The total electric field can written as E z = E zinc + E zsca = E zinc + = E zinc + ∞ ∑bH n =−∞ n (2) n (k 1 ρ )einφ ∞ ∑T a H n =−∞ n n (2) n (k 1 ρ)e inφ . Dielectric cylinder ｅr=8.41) 8 1.750 -- 2.000 1.500 -- 1.750 1.250 -- 1.500 1.000 -- 1.250 0.7500 -- 1.000 0.5000 -- 0.7500 0.2500 -- 0.5000 0 -- 0.2500 2 1 7 Single dielctric cylinder εr=8.41, Rdius=1 6 5 4 0 RCS (ｌ) (3.51) 3 -1 2 1 -2 -2 -1 0 1 2 (ｌ) 0 -1 0 50 100 150 200 250 300 350 Angle(θ) Fig 3.11 Outside cylinder electric field Fig 3.12 RCS of single dielectric distribution of dielectric cylinder with cylinder with radius= λ , ε r = 8.41 Fig 3.11 gives the electric field distribution outside a 2-D dielectric cylinder with radius r = λ , dielectric constant ε r = 8.41. The incident field is excited form the bottom. Fig 3.12 shows the RCS of the same cylinder. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 25 3.2.2 SCATTERING M ATRIX M ETHOD FOR PARALLEL DIELECTRIC CYLINDER ARRAY The scattering coefficient matrix of the dielectric cylinder array has the same expression as that of the metal cylinder array. The total wave outside the cylinder is expressed as N E z = E zinc + ∑ j =1 =Ε inc z + [ H1 ∞ ∑ n =−∞ inφ j f jn H n(2) ( k ρ j )e H2 f1 f2 . . . . HN ] . . fN , f = (I − Ta ) −1 Ta , a jn = e Tn = − (3.52) (3.53) − ik0 r j (cosθ cos ϕ j + sinθ sin ϕ j ) e − in( π2 +θ ) , nc J n' (k1a ) J n ( k2a ) −η J n ( k1a ) Jn' (k 2 a ) nc H n(2)' (k1a ) J n ( k2a ) −η H n(2) (k1a ) Jn ' ( k2 a ) , (3.54) (3.55) η = 1(TE), n (TM ), 2 c α ij ( n, m ) = H n( 2−)m (kρ ' ) e −i (n − m )φ , ' (3.56) and Ε z = Ε inc z + [H 1 H2 f 1 f 2 . . . . H N ] . . . f N Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (3.57) Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 26 Dielectric cylinder EBG has been widely studied and used [1], [2]. EBG showed in Fig 3.13 is a triangular lattice EBG with M = 9 , M = 5 , a = 0.45µ m, r = 0.075µ m , 0 x y d = 0.45 µ m, l = 1.0 µ m , ε r = 12.25 , which is typical for semiconductor columns in a vacuum[2]. Fig 3.14 shows the transmission spectra for TM and TE polarization. It shows that the EBG in Fig 3.13 exhibits different transmission spectra for different polarizations. Figs 3.15 and 3.16 give, respectively, the magnetic and electric field distributions around this EBG for TE polarization. The incident wave is strongly reflected and the transmission is sup pressed by the EBG when the frequency lies inside the EBG. Due to the finite width of the EBG structure, the incident wave is scattered so that radial patterns appear at the side edges. When the frequency is outside the EBG, the incident wave is almost transmitted through the structure. It contributes a phase delay, which in turn causes the focusing effect mentioned before. Dielectric cylinder ε r=12.25 Radius=0.075 µm, spacing=0.45 µm Fig 3.13 Geometry of triangular lattice EBG with dielectric cylinders. The segment above the structure is the one for the computation of transmission Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 27 10 Transmissison(dB) 0 -10 -20 -30 -40 TM-polarization TE-polarization -50 0.2 0.4 0.6 0.8 1.0 1.2 Normalized Frequency ω a/2 πc (a) (b) Fig 3.14 Transmission spectra from a EBG structure, solid and dashed curves indicate TM and TE polarization （ a） from[2] (b) our simulation result ωa/2πc=0.96 ωa/2πc=0.40 0.00394 -- 0.00450 0.00337 -- 0.00394 0.002812 -- 0.00337 0.002250 -- 0.002812 0.001687 -- 0.002250 0.001125 -- 0.001687 5.625E-4 -- 0.001125 0 -- 5.625E-4 (a) 0.00437 -- 0.00500 0.00375 -- 0.00437 0.003125 -- 0.00375 0.002500 -- 0.003125 0.001875 -- 0.002500 0.001250 -- 0.001875 6.25E-4 -- 0.001250 0 -- 6.25E-4 (b) Fig 3.15 Magnetic field distributions of TE-polarized wave around EBG in Fig 3.12 (a) ω a / 2π c = 0.96 (b) ω a / 2π c = 0.40 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method ωa/2πc=0.96 28 ωa/2πc=0.40 1.260 -- 1.400 1.120 -- 1.260 0.9800 -- 1.120 0.8400 -- 0.9800 0.7000 -- 0.8400 0.5600 -- 0.7000 0.4200 -- 0.5600 0.2800 -- 0.4200 0.1400 -- 0.2800 0 -- 0.1400 2.100 -- 2.400 1.800 -- 2.100 1.500 -- 1.800 1.200 -- 1.500 0.9000 -- 1.200 0.6000 -- 0.9000 0.3000 -- 0.6000 0 -- 0.3000 (a) (b) Fig 3.16 Electric field distributions of TE-polarized wave around EBG of Fig 3.12 (a) ω a / 2π c = 0.96 (b) ω a / 2π c = 0.40 35 1 a 30 Q L d from [4] our simualtion result 0 Transmission(10dB) 25 My rods 20 15 10 5 -2 -3 -4 0 -5 -5 -1 0 5 10 15 20 Mx rods (a) 25 30 35 -5 6 7 8 9 10 11 12 13 wavelength(λ) (b) Fig.3.17 Transmission of versus wavelength compared with [1], nc = 2.9 ε r = 8.41 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 29 A 2D triangle lattice structure is shown in Fig.3.17 (a). Fictitious line Q is for evaluating the Poynting vector to approximate the transmission coefficient. Plane wave is assumed to incident the lattice normally from - y direction in this paper although it is easy to include the excitation from monopole antennas. Here, M x = 9 , M y = 9 , a = 4 , r0 = 0.6 , d = 4 , l = 6 . The rod in each cell can be dielectric, metal or multilayered cylinder. Fig.3.17 (b) gives the transmission of EBG in Fig.3.17 (a), and it is compared with the curves in [1]. The two curves agree very well, which verifies our equations and code. 3.2.3 PARAMETERS OF DIELECTRIC CYLINDER EBG By studying the parameters of the EBG structure, we can see that by modifying the rod distance, radius and permittivity, the bandgap property of the PBG str ucture could be tuned. Consequently, a novel device could be designed by varying these parameters locally to change the direction or frequency band of waves. 3.2.3.1 EBG with Defects It is worth to note that equation (3.16) can be rewritten as: f1 I −T1α12 L − T1α1 N f1 f − T α I L −T2α 2 N f2 = [ A] 2 = 2 2 1 M M M O M M I fN fN − TN α N 1 − TN α N 2 L Ta 1 1 . T 2a 2 M TN a N Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (3.58) Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 30 Define matrix A as: I −T α A = 2 21 M −TN α N 1 − T1α1 2 I M −TN α N 2 L −T1α1N L −T2α2 N O M L I (3.59) For equation (3.58), the frequencies satisfying det( A ) = 0 are the resonant frequencies of the set of cylinders and the corresponding eigenvectors stand for the resonant modes. Equation (3.58) can be solved by the Newton iterative method. EBGs with defects usually have resonant modes due to the fact that, at this frequency, a resonance occurs in the microcavity made by the defect(s). This microcavity plays the role of a relay for photons. Some examples are given in this section to illustrate the influences of defects to EBGs. 10 εr=8.41 Dielectric Cylinder εr=8.41, Radius=0.6, Spacing=4 35 with one defect without defect Radius=0.6,Spacing=4 30 0 Transmission(dB) 25 20 15 10 -10 -20 -30 5 0 -40 -5 -5 0 5 10 15 20 25 30 35 6 7 8 9 10 11 12 13 Wavelength(λ) Fig 3.18 Geometry of triangular lattice Fig 3.19 Transmission spectrum versus dielectric cylinder EBG with one defect. The wavelength for the crystal in Fig 3.16 segment above the structure is the one for the computation of transmission Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 31 Fig 3.18 shows the calculated models of a densely-packed EBG structure with one defect. M x = 9, M y = 9, a = 4, r0 = 0.6, d = 4, l = 6 , and ε r = 8.41 . The TM polarization is adopted. Fig 3.19 gives the transmission spectrum of this structure compared with that of the same structure but without defect. The bandgap lies between 7.5 and 10.8 (-30dB). As we have expected, a sharp transmission peak appears at the wavelength λ = 9.0572 ± i 0.00092 (this value can also be obtained by using Newton iterative method). Fig 3.20 gives the electric field distribution for this mode. The imaginary part of λ p is due to the fact that the mode has certain losses, which means that some energy radiates out of the structure. λ=9.057 EBG with 66.50 -- 70.00 63.00 -- 66.50 59.50 -- 63.00 56.00 -- 59.50 52.50 -- 56.00 49.00 -- 52.50 45.50 -- 49.00 42.00 -- 45.50 38.50 -- 42.00 35.00 -- 38.50 31.50 -- 35.00 28.00 -- 31.50 24.50 -- 28.00 21.00 -- 24.50 17.50 -- 21.00 14.00 -- 17.50 10.50 -- 14.00 7.000 -- 10.50 3.500 -- 7.000 0 -- 3.500 Fig 3.20 Electric field distribution of EBG in Fig 3.16 with the resonant mode for λ = 9.0572 ± i 0.00092 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 32 Next consider an EBG structure with two defects. The EBG structures in Fig 3.21 have the same parameters as the one in Fig 3.18, but with two defects. The distance between two defects of EBG in Fig 3.21(a) and Fig 3.21(b) is different. Fig 3.22 gives the transmission spectra of them. We can see that two resonant modes appeared. And we can also see that the resonant wavelengths are more split with the increasing of the distance between defects. 50 50 Dielectric Cylinder Radius=0.6, Spacing=4 εr=8.41 45 40 40 35 35 30 30 25 25 20 20 15 15 10 10 5 5 0 0 -5 -5 -10 -10 0 10 20 30 Dielectric Cylinder Radius=0.6, Spacing=4 εr=8.41 45 40 50 -10 -10 0 10 (a) 20 30 40 50 (b) Fig 3.21 EBG with two defects (a) Distant defects (b) Near defects It can be seen that the defect traps the EM energy coupled from incident plane wave. This effect enables EBG structure potential candidates of various components such as filters, couplers and efficient semiconductor light emitters, etc. Further research will focus on the simulation of photonic crystal devices formed by the defects. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 33 10 Fig 3.19(a) Fig 3.19(b) Transmission(dB) 0 -10 -20 -30 -40 -50 6 8 10 12 Wavelength(λ) Fig 3.22 Transmission versus wavelength for the EBGs in Fig 3.19 3.2.3.2 Effects of rod distance on transmission spectra of EBG d 10 without additional rods d=1.2 d=2.0 Tranmission(dB) 0 -10 -20 -30 -40 dy 2r -50 0.25 dx 0.30 0.35 0.40 0.45 0.50 0.55 0.60 ωa/2πc Fig 3.23 The finite size EBG structure Fig 3.24 Tuning of PBG properties by additional rods. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 34 Consider an EBG structure shown in Fig 3.23. It is an 8-row by 11-column rod array. The horizontal and vertical periodicities are dx and dy respectively. For each column rod, an additional row with distance d is designed to tune the bandgap property of the structure. Fig 3.24 shows that additional rods could convert the previous bandgap into a pass-band, and the band width decreases with an increase in d . Fig 3.25 shows the field distribution of the two cases when ωa / 2πc = 0.401 . It is clear that additional rods change the stop frequency band to a pass-band. 1.926 -- 2.200 1.653 -- 1.926 1.379 -- 1.653 1.105 -- 1.379 0.8313 -- 1.105 0.5575 -- 0.8313 0.2838 -- 0.5575 0.01000 -- 0.2838 30 y 20 10 40 20 10 0 0 -10 -10 -10 0 10 20 30 40 50 1.751 -- 2.000 1.502 -- 1.751 1.254 -- 1.502 1.005 -- 1.254 0.7562 -- 1.005 0.5075 -- 0.7562 0.2587 -- 0.5075 0.01000 -- 0.2587 30 Y Axis 40 -20 -10 0 10 x X Axis (a) (b) Fig 3.25 Electric field distribution 20 30 40 ωa / 2πc = 0.401 (a) without additional rows (b) d=2.0 3.2.3.3 Effects of rod radius on the tra nsmission spectra of PBG We will study the effects of rod radius on the transmission properties of the EBG structure in this part. The finite EBG structure investigated is shown in Fig 3.26. The transmission spectra of three kinds of rod radius R are given in Fig 3.27. It is obvious that the rod radius of even column could be used to tune the stop or pass-band. By using different size rod, potential novel devices such as filters and couplers could be designed. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 35 10 2R dy 2r dx Transmission Coefficients (dB) R=1.5 R=2.5 R=3.0 0 -10 -20 -30 -40 -50 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 ωa/2πc Fig 3.26 Finite -sized EBG Fig 3.27 Transmission spectra of PBG structures with different rod radi i. shown in Fig.3 with different rod radiii ε = 3.2.3.4 Effects of rod permittivity on transmission spectra of PBG ε1 ε2 dy 2r dx Fig 3.28 Finite-sized EBG with different rod permittivity values = Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 36 Fig 3.28 shows the proposed EBG structure. By varying the permittivity of the rods, different scattering characteristics are obtained. Fig 3.29 gives the comparison of the transmission spectra of different permittivity but with same odd and even rods. It can be seen that different permittivity could vary greatly on the bandgap properties of PBG. Fig 3.30 gives the spectra of different odd and even column rods. Obviously, the pass-band of one structure is the stop-band of another structure. 5 0 -10 -20 -30 -40 ε1=8.0,ε2=8.0 ε1=13.8,ε2=13.8 -50 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Transmission Coefficients (dB) Transmission Coefficients (dB) 10 0 -5 -10 -15 -20 ε1=3.8 ε2 =8.0 ε1=3.8 ε2 =13.8 -25 -30 -35 0.1 0.2 ωa/2πc Fig 3.29 Transmission spectra of EBG with different rod permittivity. 0.3 ωα/2πc 0.4 0.5 Fig 3.30 Transmission spectra of EBG structure with different odd and even rods. 3.2.3.5 EBG with both metal and dielectric cylinders Fig 3.31 shows the geometry of the mixed EBG structure. Cylinders in even rows are metal cylinders while in the odd rows are the dielectric cylinders with dielectric constant ε r = 8.41 . Fig 3.32 compares the spectra of EBG consisted of metal, dielectric Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 37 and mixed rods. It can be seen that there is no much influence of mixed rods on the passband compared with metal rods. This is because metal rod disturbs electric field much more than the dielectric rod. 20 35 a 30 Q 10 d 25 Transmission(dB) 0 20 M y rods metal rods dielectric rods εr=8.41 mixed rods (even row: metal) L 15 10 5 -10 -20 -30 -40 0 -50 -5 -5 0 5 10 15 20 25 30 35 6 7 8 Mx rods 9 10 11 12 13 wavelength Fig 3.31 The 2D EBG structure Fig 3.32 Effects of mixed metal(even M x = 9, M y = 9, a = 4, r0 = 0.6, d = 4, l = 6 row) and dielectric EBG 3.2.4 THE FIELD CALCULATION I NSIDE THE DIELECTRIC CYLINDERS Although we only compute the field outside the dielectric cylinders, the field inside the cylinders needs to consider in some cases. The electric field inside cylinder can be written as follows E inside = z ∞ ∑ p jn H n( 2 ) ( k ρ j )e inφ j . (3.60) n =−∞ For the cylinder array, similar to the derivation of the scattering coefficient f j , the inside field coefficient can be written as Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method p j = c j (a j + ∑ a ji f i ) , 38 (3.61) i ≠j where cj can be derived from equation (3.48), it can be written as cn = J n (k1a ) + Tn H n( 2) ( k1a ) J n ( k 2a ) . (3.62) p j can be simplified Given that f j = T(a j j + ∑ a ji fi ) , we combine it with equation (3.61). i≠ j −1 as p j = c jTj fj , where cj , pj , f j , a j , are all column matrix containing x jn ( x represent c, p, f, a) elements, and a ji is a square matrix as mentioned before. 3 Frequency=30Mhz cylinder Radius=1, spacing=3 εr=8.41 2 1 0 -1 -2 -3 -1 0 1 2 3 4 1.520 -- 1.600 1.440 -- 1.520 1.360 -- 1.440 1.280 -- 1.360 1.200 -- 1.280 1.120 -- 1.200 1.040 -- 1.120 0.9600 -- 1.040 0.8800 -- 0.9600 0.8000 -- 0.8800 0.7200 -- 0.8000 0.6400 -- 0.7200 0.5600 -- 0.6400 0.4800 -- 0.5600 0.4000 -- 0.4800 0.3200 -- 0.4000 0.2400 -- 0.3200 0.1600 -- 0.2400 0.0800 -- 0.1600 0 -- 0.0800 Fig 3.33 Electric field distributions inside cylinders Fig 3.33 illustrates the electric field distribution inside the dielectric cylinders of a simple array with two cylinders. The adapted frequency of the incident wave is 30MHz , the cylinder radius of r = 1m , spacing of d = 3m , and dielectric constant of ε r = 8.41. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 39 3.3 HARD WARE IMPLEMENTATION In this chapter, a simple 2D metal-cylinder EBG structure is presented. An experiment is carried out to test the EM properties of the fabricated model, and the experiment results are compared with simulation results. 3.3.1 THE EBG S TRUCTURE The geometry of the 2D metal EBG is given in Fig 3.34. The cylinder Radius = 0.0025m , spacing = 0.03m . The incident wave propagates from the bottom of the structure and TM polarization is adopted. The fabricated structure is presented in Fig 3.35. 0.22 0.20 Metal Cylinder 0.18 0.16 0.14 m 0.12 0.005 0.10 0.08 0.06 0.04 0.02 0.00 0.00 0.05 0.10 0.15 0.20 m Fig 3.34 Geometry of 2D metal cylinder EBG structure Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 40 Fig 3.35 Manufactured 2D metal -c ylinder EBG. The board used in the structure is a wooden board 3.3.2 EXPERIMENT FACILITIES (1) Horn antenna : 1.5-18 GHz (Fig 3.36) (2) Sweep Oscillator: Hewlett Packard 8350A (Fig 3.37): The HP 8350A is an earlier version of the HP 8350B; Single slot; designed to hold the 86200 series (with the 11869A adapter) and the 83500 series of plug in modules. Enhanced features make measurements easier and more precise. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method (3) Frequency Converter: Hewlett Packard 8511B 45MHz~50GHz (Fig 3.38): 41 The 8511B Four Channel Frequency Converter is a 45MHz to 50GHz frequency converter which combined with the 8510C Network Analyzer (not included) results in a general purpose four-channel magnitude/phase receiver. (4) Microwave Receiver: Hewlett Packard 8530A (Fig 3.39): 8530A is a fast and accurate microwave receiver designed for both manual and automated antenna measurement and radar cross-section measurement applications. It offers fast data acquisition speeds, excellent sensitivity, wide dynamic range, multiple test channels, and frequency agility without compromising measurement accuracy. The distance between source antenna and receiver is 10 meter. Fig 3.36 Horn antenna: 1.5-18 GHz Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 42 Fig 3.37 Sweep oscillator: Hewlett Packard 8350A Fig 3.38 Frequency converter: Hewlett Packard 8511B 45MHz~50GHz The experiment was conducted in a chamber in which the reflection wave was minimized because the wall and ground was covered with special materials. In the process of experiment, the horn antenna is put at a bracket as the source of wave, while another antenna is put at another bracket at the same height with the source to receive the wave. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 43 The transmission is obtained by varying the source frequency which was achieved by adjusting the frequency converter. The pattern (RCS) was measured by turn the turn the antenna which was used as receiver to form a angle against the source horn antenna. Fig 3.39 Microwave receiver: Hewlett Packard 8530A 3.3.3 EXPERIMENT R ESULTS AND DISCUSSION Fig 3.40 shows the RCS of this metal EBG Structure from both experiment and simulation result at f = 3GHz . The peak appears around θ = 90 o . The peak value from simulation result is 0.88dB, while the experimental value is 0.25dB, the error is 0.63dB. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 44 1.0 Experiment Result simulation Result 0.5 RCS(dB) 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 0 50 100 150 200 250 300 350 Angle(θ ) Fig 3.40 RCS of the 2D metal-cylinder EBG structure 5 0 Transmission(dB) -5 -10 -15 -20 -25 -30 Expiriment Result Simulation Result -35 -40 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Frequency(GHz) Fig 3.41 Transmission of the 2D metal EBG Fig 3.41 gives the experimental result of transmission spectrum of the 2D metal EBG structure, and the simulation result is also presented in this figure to make comparison. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T-matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 45 The characteristic that it can forbid the wave propagating is verified. At low frequency, the transmission is lower than -25dB, and at f = 2.42GHz , the transmission even down to -30dB. In the pass-band, the transmission slightly changes in the range from -11.92 to 3.38 dB (-3.38~1.18dB in the simulation curve). The experimental result verifies the simulation result. There is a threshold frequency at f = 3.4GHz (-25dB). Below this frequency, the wave can not penetrate the EBG structure; while above f = 3.7GHz , the wave can almost totally penetrate through it. In some frequency, the transmission is over 0dB, which is owing to the focusing effect. However, the focusing effect cannot be seen from the experiment result, which is due to the experimental errors. The error percentage at f = 2.42Ghz , f = 2.82GHz are 23.7 % and 24.3%, respectively. The errors may be owing to the reason that the cylinder length is not infinity. Due to the lack of materials, our cylinder length is set to be 5cm instead of infinity. 3.3.4 CONCLUSION 2D cylinder EBG structure is easy to manufacture, therefore, we have this 2D metal EBG fabricated. This experiment provides me a good opportunity to learn several important instruments frequently used in electromagnetic experiments. It also verifies the rightness of my equations and codes. However, because of the limitation of time, materials, techniques used in the EBG- manufacture process and influence of testing environment, there is some error in the experiment result. Some more accurate tests will be carried out to learn the EM properties of the EBG Structures. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 46 4 ELECTROMAGNETIC BANDGAP S TRUCTURES COMPOSED BY MULTI- LAYERED CYLINDERS, FERRITE AND CHIRAL CYLINDERS In this chapter, we mainly introduce three special kinds of EBG structures, including coated cylinder EBG, ferrite cylinder EBG and chiral cylinder EBG, and their EM properties are described. It begins with the coated cylinder EBG, and the S-parameter is used to derive its scattering matrix. The subsequent section leads to a discussion about ferrite cylinder EBG. Thirdly, the chiral cylinder EBG is briefly introduced. Finally, the aggregated T- matrix algorithm, which is not only useful in reducing the matrix dimension of scattering matrix but also necessary in some special cases, is described. 4.1 SCATTERING MATRIX OF M ULTILAYERED DIELECTRIC CYLINDERS In this section, we use the scattering matrix method to analyze the finite coated EBG rods. The S-parameter method is described and T- matrix of coated circular cylinders is derived by the S-parameter method. The coated EBG shows novel characteristics and provides additional freedom to tune the band gaps. Some changes in the dielectric constant and thickness of the coated layer or inner layer will change the electromagnetic properties of EBG. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 47 4.1.1 S- PARAMETER M ETHOD Consider a two layer coated cylinder as an example. Fig 4.1 shows the general calculation model. Fig 4.2 gives the s- matrix network chart. I II a1 a2 Rb III Ra b1 b1 Fig 4.1 Calculation model of multilayer cylinder a1 s21 b2 T s11 b1 s22 s12 T0 a2 Fig 4.2 S-matrix network Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 48 From the S- matrix chart, the relations between parameters can be written as b1 s11 s12 a 1 = , b 2 s21 s22 a 2 (4.1) a2 = Tob 2 , (4.2) and where T 0 is the T- matrix of single cylinder, which has been derived and its elements has the following form Tn = − nc J n' (k 2 Ra ) J n ( k3 Ra ) − η J n ( k2 Ra ) J n' (k3 Ra ) , nc H n(2)' (k2 Ra ) J n ( k3 Ra ) − η H n(2) (k2 Ra ) J n' ( k3 Ra ) (4.3) η = 1(TE ), n (TM ). 2 c Consider the relation between b 1 and a1 , which is the T- matrix of the coated cylinder. Let us consider two special cases: (in this part, a = Rb ) (I) Case one, a1 = 1, a 2 = 0 a1 b2 b1 Fig 4.3 Calculation model for S-parameter The inner cylinder is removed due to the fact a 2 = 0 , which means that no scattering from the third dielectric in the second area. In this case, from the equation of the S-matrix (4.1), b1 = s11, b2 = s21 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 49 Since a1 = 1 , b1 is the T-matrix of the cylinder and its element can be written as Tn = − nc J n' ( k1 a ) Jn (k2 a ) −η J n (k1 a ) J n' ( k2 a ) , nc H n(2)' (k1a )H n(2) (k 2 a ) −η H n(2) ( k1a ) H n(2)'( k2 a ) (4.4) η = 1(TE), n (TM ). 2 c According to the boundary condition, E I = E Iinc + E Isca = E II . (4.5) Due to the fact that no scattering field exists in area 2, we obtained J n ( k1 a ) e i φn + T n H n( 2 ) ( k1 a ) e iφ n = b 2 n J n ( k 2 a ) e iφ n , (4.6) with b2 n = J n (k1a ) + Tn H n( 2) ( k1a ) J n ( k 2a ) . b1 , b2 , namely s11 , s21 , are derived (II) Case two, a1 = 0 , a 2 = 1 a2 b2 b1 Fig 4.4 Calculation model 2 for S-parameter Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (4.7) Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 50 In this case, from equation of the S- matrix (4.1), b1 = s12 , b 2 = s 22 a 2 = 1 . Therefore, b2 is the T- matrix of the cylinder, and its element can be written as Tn = − nc H n(2)' (k2 a ) H n(2) ( k1 a ) − η H n(2) (k 2 a )H n(2)'( k1 a ) , nc J n' (k2 a ) H n(2) ( k1 a ) − η J n ( k2 a ) H n( 2 )(' k1a ) η = 1(TE ), nc2 (TM ), nc = nc2 1 , (4.8) According to the boundary condition, E inc = E inc + E Isca = E II . I (4.9) Due to the fact that no scattering field exists in area 2 J n (k 2 a )e iφn + Tn H n(2) (k 2 a )e iφn = b1n J n ( k1a ) ei φn , (4.10) with b1n = J n ( k2a ) + Tn H n( 2) (k2 a ) J n ( k1a ) . (4.11) Up to now, we have derived S parameter, T0 . Take them into S- matrix equation. Τ = [ S11 + S12 Τ0 ( Ι − S 22 Τ0 ) −1 S 21 ] , (4.12) where S11 n nc J n' ( k1 a ) Jn (k 2 a ) − η J n (k1a ) J n' ( k2 a ) =− , nc H n(2)' ( k1 a ) J n (k 2 a ) − η H n(2) ( k1 a ) Jn ' (k2 a ) η = 1(TE ), nc2 (TM ), nc = nc12 , S 21n = J n ( k1a ) + Tn H n( 2) ( k1a ) J n ( k 2a ) (4.13a) , Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (4.13b) Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders S12 n = H n( 2 ) ( k 2 a ) + T n J n (k 2 a ) S22 n = − H n( 2 ) ( k1 a ) , 51 (4.13c) nc H n(2)' ( k2 a ) H n(2)( k1 a ) − η H n(2) ( k2 a ) H n( 2 )(' k1a ) , nc J n' ( k2 a ) H n(2)( k1a ) − η J n ( k2 a ) H n(2)'( k1 a ) η = 1(TE ), nc2 (TM ), nc = nc2 1 , , (4.13d) nc J n' (k 2 Ra ) J n ( k3 Ra ) − η J n ( k2 Ra ) Jn' (k3 Ra ) T0 n = − , nc H n(2)' (k 2 Ra ) J n ( k3 Ra ) − η H n(2) ( k2 Ra ) Jn ' ( k3 Ra ) η = 1(TE ), nc2 (TM ), nc = nc2 3 , (4.13e) and Τn = [Τ0 n S12 n S 21n + S11n (Ι − Τ0 n S 22n )] /( Ι − Τ0 n S 22n ) . (4.13f) The above derivation procedure is for two layer cylinder. For multilayer cylinder with more than 2 layers, we can use S-parameter method repeatedly from the inner most layer to obtain the T- matrix of the outermost layer 4.1.2 SCATTERING OF COATED CYLINDER EBG We have known that, for multiple - layer cylinders, the ir T- matrix can be derived by using microwave equivalent network method in which every dielectric interface is viewed as the S-parameters. Based on the equations given in the above section 4.1.1, the scattering of coated cylinder EBG structure can be calculated. The T-matrix of two layer coated cylinder can be written as Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders ? =[S 11 + S 12 ? 0 (?- S 22 ? 0 ) -1 S21 ] , 52 (4.14) where s11,s12,s21,s22 and T0 are square matrices. Fig 4.5 (a) shows the geometry of a coated EBG structure. M x = 9, M y = 9 , a = 4, r0 = 0.6 d = 4, l = 6 . Fig 4.5 (b) gives an example of transmission spectra of ring rods. It can be seen that enlarging the air core will reduce the stop-band gradually. Instead of the air core, Fig 4.6 gives an example of the spectra of EBG consisting of two layered of rods with different dielectric constants. It shows that the pass-band can be shifted or tuned by varying the geometric parameters of different dielectrics. Fig 4.7 gives the electric field distribution in pass-band and stop-band ( r1 = 0.7, r2 = 0.3, nc1 = 5.0, nc 2 = 2.9 ) 35 a Q 20 L 25 20 My rods :r=0.6,εr=8.41 :r1=0.6,r2=0.1,εr1=8.41,εr2=1 :r1=0.6,r2=0.3,εr1=8.41,εr2=1 :r1=0.6,r2=0.55,εr1=8.41,εr2=1 d Transmission(dB) 30 15 10 0 -10 10 -20 5 -30 0 -5 -5 -40 0 5 10 15 20 Mx rods 25 30 35 6 7 8 9 10 11 12 13 wavelength( λ) Fig 4.5 (a) Geometry of triangular lattice metal EBG structure. The segment above the structure is the one used for the computation of the transmission. (b) Transmission spectra of ring rod EBG structure Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 53 10 Transmission(dB) 0 -10 -20 -30 -40 s olid:r1=0.7,r2=0.6,ε r1=25, εr 2=8.41 dot:r=0.7, εr =25 dash:r1=0.7,r2=0.3,ε r1=25, εr 2=8.41 dash_dot:r=0.6,ε r=8.41 -50 6 8 10 12 wavelengthλ Fig 4.6Comparison of transmission spectra of coated dielectric rod （a） Fig 4.7 Electric field distribution (b) r1 = 0.7, r2 = 0.3, nc1 = 5.0, nc 2 = 2.9 , (a) λ = 9.6 pass-band (b) λ = 8.0 stop-band 4.2 SCATTERING MATRIX OF FERRITE CYLINDERS Scattering problems of ferrite cylinder have be en studied by many authors [14], [15]. When a Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 54 ferrite cylinder is magnetized along its axis of incidence, the scattering pattern becomes asymmetrical about the direction of incidence. The permeability of cylinder can be controlled by adjusting the DC magnetic field intensity, and accordingly, the EM property can also be modified by adjusting the physical properties (gyromagnetic ratio and susceptibility) [14]-[16]. Due to the fact that the added DC magnetic field intensity can be adjusted easily, we can get the corresponding scattering properties easily. Ferrite cylinder EBG is seldom investigated although there are many papers deal with EM properties of ferrite cylinders. By mixing dielectric cylinders with ferrite cylinders, some useful devices can be easily obtained. 4.2.1 T- MATRIX OF FERRITE CYLINDER Consider an infinitely long ferrite cylinder; a dc magnetic field is applied along the z − direction . The incident wave is a plane wave along the x − direction . The field consists two separate parts: one is polarized normal to the cylinder axis, i.e. E = E y , the other is parallel to the axis, namely E = E z . For the former, there is no nonreciprocal interaction between the field and the magnetization of ferrite due to the fact that the magnetic field of the wave is parallel to the axis and to the applied magnetic DC field. For the latter, the AC field interacts with the processing magnetic dipoles of the ferrite because the magnetic DC and AC fields are normal to each other. Thus according to Maxwell’s equations, we have ∇ × H = jω D = jωε E , (4.15a) r ∇ × E = − jω B = − jωµ ⋅ H . (4.15b) and Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 55 Because there is no variation along the z − direction , equation(4.15b) can be rewritten as ∂ 1 0 1 1 0 1 ∂x B=− • ∇ E = − z • Ez . jω 1 0 jω 1 0 ∂ ∂y (4.16) The magnetic field in free space is thus given as H= 1 −1 0 1 µ0 • • ∇E z . jω −1 0 (4.17) In the ferrite cylinder, we have H =− 1 0 1 ( µ ) −1 • • ∇E z jω −1 0 (4.18) A ferrite cylinder has the tensor permeability given as[14] µ − ju ' 0 µ = ju ' µ 0 , 0 0 1 (4.19) where µ = µ0 γ 2 µ 0 H 0 Bz − ω 2 , γ 2 µ 0 2H 0 2 − ω2 (4.20a) k = µ0 ωγµ0 M z . γ µ 0 2H 0 2 − ω 2 (4.20b) 2 µ0 is the permeability in the air, and ω is the angular frequency of electromagnetic waves, H 0 is the DC magnetic field, and M z is the DC magnetization, and the value of H 0 / M z , which is usually constant for specific material known as magnetic susceptibility. Bz = µ 0 ( H 0 + M z ) , γ is the gyromagnetic ratio. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 56 From equation (4.19), we have ( µ ) −1 = 1 µ −k2 2 µ − jk . jk µ (4.21) By combining equation (4.18) with equation (4.21), we obtained H =− 1 1 − jk 2 jω µ − k 2 −µ µ • ∇E z . − jk (4.22) The magnetic field in Cartesian coordinates is thus Hx = ∂E z ∂E jk −µ z j ω( µ − k ) ∂x ∂y , (4.23a) Hy = ∂E z ∂E µ + jk z jω ( µ − k ) ∂x ∂y . (4.23b) 1 2 2 1 2 2 In cylindrical coordinates, it is rewritten as Hr = 1 ∂E z 1 ∂E z k + jµ 2 ω (µ − k ) ∂r r ∂φ , (4.24a) Hφ = 1 ∂E z 1 ∂E z jµ −k . 2 ω( µ − k ) ∂r r ∂φ (4.24b) 2 and 2 Based on the equations above, the entire field inside and outside cylinder can be written as follows : (I) Incident field ( µ = µ 0 , k = 0 for free space) E z inc = ∞ ∑aJ n n ( β r ) e− jnφ , n =−∞ Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (4.25a) Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders H r inc = 1 ωµ0 r = jβ ωµ 0 Hφ (II) inc ∞ ∑aJ n n =−∞ ∑a n ( β r ) ne− jnφ , J n ( β r )e − jnφ . ' n 57 (4.25b) (4.25c) n Scattering Field ∞ ∑bH ( β r )e− jnφ , (4.26a) H r sca = 1 ∞ bn H n(2) ( β r ) ne− jnφ , ∑ ωµ 0r n=−∞ (4.26b) H φ sca = jβ ωµ0 (4.26c) E z sca = (III) n =−∞ n (2) n ∞ ∑nH n =−∞ n (2)' n ( β r ) e− jnφ . Inside cylinder field E z inside = ∞ ∑c J n =−∞ n n ( β 2 r )e− jnφ , (4.27a) H r inside = ∞ 1 µ ∞ ' − jnφ k β c J ( β r ) e + cn J n (β 2 r ) ne − jnφ , 2 ∑ n n 2 ∑ 2 2 ω( µ − k ) r n =−∞ n =−∞ (4.27b) H r inside = ∞ j k ∞ ' − jnφ µβ c J ( β r ) e + cn J n (β 2 r ) ne − jnφ . 2 ∑ n n 2 ∑ 2 2 ω( µ − k ) r n =−∞ n =−∞ (4.27c) Then, the boundary condition can be applied at r = a (cylinder radius) E zinc + E zsca = E zinside, (4.28a) H φinc + H φsca = H φinside . (4.28b) Letting bn = Tn an , combining equation (4.25) (4.26) and (4.27) with equation (4.28), we have Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders Tn = − D n ( β 2 a) J n ' ( β a) − J n (β 2 a) J n ( β a) Dn ( β 2 a) H − J n ( β 2 a) H ( 2) ' n ( 2) n J n (β a ) , (2 ) ( β a) H n ( β a) (β a ) 58 (4.29) where Dn ( β 2 a) = µ0 β 2 k n ' [ J n ( β 2 a) + J n ( β 2 a)] , µ eff β µ β 2a (4.30a) β 2 = ω 2 µ eff ⋅ ε , (4.30b) µ2 − k2 . µ (4.30c) and µ eff = When M z = 0 , the cylinder becomes a dielectric cylinder. 4.2.2 SCATTERING OF FERRITE CYLINDER EBG Let us consider a plane-wave scattering with two parallel linear arrays of circular ferrite cylinders as shown in Fig.4.8 (b) where the parameters are Radius = 0.12λ , ε r = 10 , H0 / M z = 9.3 , ω ( µ0 r M z ) = 10 (inset). The BCS (Bistatic scattering cross section) can then be expressed as [15] σ b (θ ,φ ) = σ b ,0 M ∑e 2 j k0 di [cos(α i −φ ) +cos(α i −θ )] (4.31) i =1 σ b ,0 is the BCS when the single cylinder is located at origin o in seclusion from the others which is expressed as Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 59 2 σ b ,0 π jn ( −φ ) 4 ∞ = ∑ fne 2 . k0 n =−∞ (4.32) Fig.4.8 (a) shows the scattering pattern BCS of single ferrite cylinder. The patterns of Figs Fig.4.8 (a) and (b) agree well with the results given in [15], which verifies the rightness of our equations and code. BCS of single ferrite cylinder 90 k0 σb,o 120 14 Our simulation result Result from paper[2] 60 150 30 λ /4 60 d 150 300 200 6 4 2 30 100 0 0 θ =0o 180 -100 180 0 0 2 100 4 200 6 330 210 10 12 14 120 400 8 8 k0σb 500 10 0 700 600 12 0 Results from [2] Our simulaiton results9 0 300 400 210 330 Radius=0.12λ 240 300 270 εr=10 H0/ M0 =9.3 600 700 H0/M0 =9.3 Radius=0.12λ εr=10 500 240 300 ω/(µ 0 )IrIM0 =10 270 ω/( µ0)IrIM0 =10 (a) (b) Fig. 4.8 Scattering Patterns of BSC for (a) for single ferrite cylinders compared with [15] (b) an array of circular ferrite cylinders ( d = 1.5λ ) compared with [15] Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 20 35 a 30 L Q d dielectric cylinder mz=1E13 h0=0.01mz γ =10 mz=2E13 h0=0.01mz γ =10 mz=5E13 h0=0.01mz γ =10 10 Transmission(dB) 25 20 M y rods 60 15 10 5 0 -10 -20 -30 0 -40 -5 -5 0 5 10 15 20 25 30 35 6 7 8 9 10 11 12 13 wavelength(λ) Mx rods Fig 4.10 Transmission spectra of EBG Fig 4.9 Geometry of the ferrite cylinder with different added DC magnetic field EBG mz=1E13 H0=1E11 λ=10.3663 mz=2E13 H0=2E11 λ=10.3663 1.925 -- 2.200 1.650 -- 1.925 1.375 -- 1.650 1.100 -- 1.375 0.8250 -- 1.100 0.5500 -- 0.8250 0.2750 -- 0.5500 0 -- 0.2750 30 20 2.625 -- 3.000 2.250 -- 2.625 1.875 -- 2.250 1.500 -- 1.875 1.125 -- 1.500 0.7500 -- 1.125 0.3750 -- 0.7500 0 -- 0.3750 30 20 10 (Figure 6) 10 0 0 -10 -10 0 10 20 30 40 0 (a) 10 20 30 40 (b) Fig 4.11 Field distribution of EBG in Fig 4.9 with different added DC magnetic field intensity at λ = 10.3663 . (a) M z = 1× 1013 A/m (b) M z = 2 × 1013 A/m Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 20 20 mz=1E13,h0=0.01mz,γ =10 mz=1E13,h0=0.5mz, γ =10 mz=1E13,h0=1mz, γ =10 mz=1E13,h0=2mz, γ =10 10 0 -10 -20 0 -10 -20 -30 -30 -40 -40 6 7 8 9 10 11 12 13 mz=1E13 h0=0.01mz γ =10 mz=1E13 h0=0.01mz γ =1 mz=1E13 h0=0.01mz γ =30 10 Transmission(dB) Transmission(dB) 61 14 Wavelength( λ) 6 7 8 9 10 11 12 13 Wavelength(λ ) Fig 4.12 Transmission spectra of EBG in Fig Fig 4.13 Transmission spectra of EBG in 4.9 with different magnetic susceptibility. Fig 4.9 with different gyro magnetic ratio. In this section, the geometry of EBG structure used in Fig 4.9 is given as M x =9, M y =9, r0 =0.6, a =4, d = 4.3 , l = 7 , and ε r = 8.41 . All the cylinders are ferrite cylinders. A DC magnetic field is added. In this section, the TM polarization is adopted. Fig 4.10 gives the transmissions of a ferrite cylinder EBG structures when different DC magnetic fields are added. It shows that for certain ferrite cylinders, with the increasing of added magnetic field, the stop-band becomes narrower and narrower. It means that we can control the stop-band in a certain range by adjusting the added magnetic field intensity. It is a useful property for device design. Fig 4.11 shows the electric field distribution in different added DC magnetic field intensity. It verifies the curves in Figs 4.10. Figs 4.12 and 4.13 show the influence of cylinder material on scattering properties. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 4.3 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 62 SCATTERING MATRIX OF CHIRAL CYLINDERS Chiral media have been studied over many years and it has been widely used in areas such as antenna, microstrip substrate and waveguide [17]-[22]. Unlike dielectric or metal cylinders, chiral scatterers produce both co-polarized and cross-polarized scattered fields which make it more difficult to analyze. We will use the scattering matrix method to analyze it in this section. 4.3.1 SCATTERING MATRIX OF CHIRAL CYLINDER 4.3.1.1 The derivation of scattering matrix Y ρj φj θ X Fig 4.14 Calculation model of 2-D chiral c ylinder Assume a plane wave has the following form E zinc = e − jk0 ( x cosθ + y sin θ ) = e − jk0 ρ (cosθ cosφ +sin θ sinφ ) , (4.33a) H zinc = e − jk0 ( x cosθ + y sinθ ) = e − jk0 ρ (cosθ cosφ +sinθ sinφ ) (4.33b) and Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 63 Because of the properties of the Helmholtz equation, they can be rewritten as ∞ E zinc = ∑a J n n (kρ ) e inφ , (4.34a) n =−∞ and H zinc = ∞ ∑b J n =−∞ n n (kρ) e in φ . (4.34b) Similarly , the scattering field around chiral cylinder can be written as E zsca = ∞ ∑c n =−∞ n H n( 2 ) ( kρ)e inφ , (4.35a) and H zsca = ∞ ∑d n =−∞ n H n( 2 ) ( kρ) e inφ , (4.35b) where c n Tnee = he d n Tn Tneh an . Tnhh bn (4.36) Unlike that of usual materials, its T- matrix has the following form T−eeM he T− M 0 0 . T= . . . 0 0 T−ehM T−hhM 0 . . . . . . 0 . . . . 0 0 0 0 0 ee T− M +1 T−ehM +1 T−heM +1 T−hhM +1 . . . . . . . . . . 0 . . . . . . . . . . . . . . . . 0 0 0 0 . . . . . . . . . . . . . . . . . 0 0 . . 0 0 ee 0 0 TM TMeh 0 0 TMhe TMhh . . . . . . (4.37) As discussed before, for multiple cylinders, the scattering coefficient can be written as Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders f j = Tjaj + ∑ Tja ji fi , 64 (4.38) i≠ j ( f j ) en (T j ) ee n = ( f ) h (T ) he j n j n f1 f2 . = . . fN T1 ( a j ) n (T j ) een (T j ) eh n + (b j ) n ∑ (T ) he (T j ) hh i ≠ j n j n T2 . . . a1 T1 a2 . + . . TN aN T2 . . . (α ji ) en (T j ) eh n 0 (T j ) hh n 0 a 21 . . . TN a N1 a 12 0 . . ( f i ) en , (4.39) (α ji ) hn ( f i ) hn 0 . . . a N2 . . . . a1N f1 a 2 N f2 . . , . . . . 0 fN (4.40) where ( f i ) e−M h ( f i ) −M ( f i ) e− M +1 h ( f i ) − M +1 . , fi = . . . e ( fi ) M ( f i ) hM (4.41) (a i ) e− M h (a i ) − M ( ai ) e−M +1 h ( ai ) −M +1 . , ai = . . . e (ai ) M (ai ) hM (4.42) Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders (Ti ) ee −M he (Ti ) − M 0 0 . Ti = . . . 0 0 (Ti ) eh −M 0 . . . . 0 0 0 0 0 ee (Ti ) − M +1 (Ti ) eh − M +1 he (Ti ) − M +1 (Ti ) hh − M +1 . . . . . . . . . . 0 . . (Ti ) 0 (α ji ) −M , − M 0 . . . a ji = . . . (α ji ) M , − M 0 hh −M 0 (α ji ) −M ,− M . . . . . . 0 (α ji ) M , − M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . 0 0 eh (Ti ) M (Ti ) hh M (4.43) (α ji ) − M , M . . . , . . . 0 (α ji ) M , M (4.44) 0 0 0 0 . . . . . . . . . 0 . 0 ee 0 (Ti ) M . . 0 0 (Ti ) he M . (α ji ) − M , M 0 . . . . . . . . . . . . . ( α ) . ji M , M 0 . . . . . . . . . . . . . . . . . . . . . . . . 65 0 where ((α ji ) n , m ) = H n( 2−)m ( kρ ' )e −i ( n − m )φ . ' (4.45) 4.3.1.2 Derivation of T-matrix Elements The constitutive relationships for a chiral medium can be written as [18], [23] D = ε E − jξc B , H= 1 B − jξc E , µ Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (4.46a) (4.46b) Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 66 where ξc is chiral admittance of the medium. If ξ c , µ or ε are complex, the medium is lossy. They can be simplified as D = ε c E − j µξc H , (4.47a) B = µ H + j µξc E , (4.47b) where ε c = ε + µξ c2 is the effective permittivity. The Maxwell’s equation is re-written in a matrix form as E E ∇ × = [K ] H H 0 + , J (4.48) and E ρ 1 ∇• = , H ε − jξ c (4.49) where ωµξ c jωε c − jωµ . ωµξ c [K ] = (4.50) Combining equatio n (4.48) with equation (4.49), the source- free wave equation in chiral media can be written as E 2 E ∇2 + [ K ] = 0 . H H (4.51) We can remove the coupling of [ K ] through [A]−1[K ][A] = kR . − k L [A] can be written as Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (4.52) Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 1 [A] = j η c 1 − j , η c 67 (4.53) where ηc = µ , εc (4.54) and the chiral wavenumbers are k R = ω µε c + ωµξc , k L = ω µε c − ωµξc . (4.55) and we define HR E = [ A] , H HL (4.56) ER , EL are the electric fields of the right and left circularly polarized waves with propagation constant k R , k L , respectively [23]. Substituting equation (4.56) into equation (4.48) and equation (4.49), we obtain E K E jη −1 ∇ × R = R R + J c , 2 1 EL K L E L (4.57) E R ρ 1 − η cξ c . ∇ • = EL 2ε 1 + η c ξc (4.58) and The uncoupled source-free wave equation in chiral media is given as E k2E ∇ 2 R + R2 R = 0 . E L k L EL Based on this equation, E R , and E L can be written as Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (4.59) Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders ∞ ∑w ER = J n (k R ρ )e inφ n J n ( k L ρ )e inφ n 68 , (4.60a) . (4.60b) n = −∞ ∞ ∑p EL = n =−∞ The field distributions in Fourier-Bessel expansion can be given as (1) Along the Z-direction E zinc = ∞ ∑a J n =−∞ H zinc = E zsca = H zsca = n n (kρ) e ∞ ∑b J n =−∞ n ∞ ∑c n =−∞ n n =−∞ (kρ) e , in φ H n( 2 ) ( kρ)e ∞ ∑d n inφ n (4.61a) , inφ H n( 2 ) ( kρ) e (4.61b) , inφ (4.62a) . (4.62b) According to equation (4.56), the field inside the ferrite cylinder can be written as E zinside = E R z + E L z = ∞ ∑ [w J n =−∞ H zinside = H R z + H L z = (2) j ηc n n (k R ρ ) + pn J n (k L ρ ) ]e ∞ ∑ [w J n =−∞ n n inφ , (k R ρ ) − pn J n ( k L ρ )]e (4.63a) inφ . (4.63b) Along φ − direction For a source free region, we have E E ∇ × = [K ] , H H where Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (4.64) Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders ωµξ c jωε c [K ] = − jωµ . ωµξ c 69 (4.65) Because r r r E = Et + Ez eˆz , (4.66a) r r r H = H t + H z eˆz , (4.66b) and ∇ = ∇t + ∂ eˆz , ∂z (4.67) we achieve r r r ∂ ∇ × E = (∇ t + eˆz ) × ( E t + E z eˆz ) . ∂z For normal incidence i.e. (4.68) ∂ = 0 , then ∂z r r r r r r r ∇ × E = ∇ t × ( E t + E zeˆz ) = ∇ t × E t + ∇ t × ( E z eˆz ) = ∇ t × E t + ∇ t E z × eˆz . (4.69) r r We know ∇t × Et is along the Z-direction , and ∇ t × ( E zeˆz ) is in the transverse. Similarly , the magnetic field is expressed as r r r r r r r ∇ × H = ∇ t × ( H t + H zeˆz ) = ∇ t × H t + ∇ t × ( H z eˆz ) = ∇ t × H t + ∇ t H z × eˆz (4.70) resulting in r r Et ˆz −1 ∇ t E z × e r = [K ] r H ∇ H × eˆ t t z z Since Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (4.71) Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders r ∂E ∂E 1 ∂E z ∇ t E z × eˆz = ( z eˆ? + eˆφ + z eˆz ) × eˆz ∂ρ ρ ∂φ ∂z 70 (4.72) similarly to the earlier deriva tion procedure for electric field, the magnetic field can be written as r ∂H 1 ∂H z ∂H z ∇ t H z × eˆz = ( z eˆ? + eˆφ + eˆz ) × eˆz . ∂ρ ρ ∂φ ∂z (4.73) In the φ -direction , ∂E z ∂E , eˆρ × eˆz = − z θˆ φ ∂ρ ∂ρ (4.74a) ∂H z ∂H eˆρ × eˆz = − z θˆ φ , ∂ρ ∂ρ (4.74b) then ∂E z r − Eφ r = [K ]−1 ∂ρ H ∂H z φ − ∂ρ . (4.75) In fact, for the conventional dielectric, [K ] = 0 j ωε − jωµ , 0 (4.76) The field expressions outside the cylinder and along the φ -direction can be obtained as Eφ = − Hφ = 1 ∂H z , j ωε ∂ρ 1 ∂E z . jωµ ∂ρ whereas for the field inside the cylinder, we have Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (4.77a) (4.77b) Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders ∂E ∂H z − µξ c z + j µ ∂ρ ∂ρ Eφ = ∂E z ∂H z − µξc jε c ∂ρ ∂ρ Hφ = 71 [ ], (4.78a) [ ]. (4.78b) ω ( µξc ) 2 − µε c ω ( µξ c ) 2 − µε c We can use the boundary conditions to derive the T- matrix E zinc + E zsca = E zinisde / ρ = a , (4.79a) Eφinc + Eφsca = Eφinside / ρ = a , (4.79b) H zinc + H zsca = H zinisde / ρ =a . (4.79c) H φinc + H φsca = H φinside / ρ =a . (4.79d) Finally, combining equations (4.61), (4.62), (4.63), (4.77) , (4.78) and (4.79), the relation between incident and scattering is given as X 3 Y2 − Y3 X 2 X Y − Y1 X 2 = 1 2 X 3 Y1 − Y3 X 1 dn X 2 Y1 − Y2 X 1 cn X 4 Y2 − Y4 X 2 X 1Y2 − Y1 X 2 an . X 4 Y1 − Y4 X 1 bn X 2 Y1 − Y2 X 1 (4.80) Therefore, the T- matrix for chiral cylinder is X 3Y2 − Y3 X 2 X Y −Y X Tn = 1 2 1 2 X 3Y1 − Y3 X1 X 2 Y1 − Y2 X 1 X 4Y2 − Y4 X 2 X 1Y2 − Y1 X 2 , X 4Y1 − Y4 X1 X 2Y1 − Y2 X1 where Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (4.81) Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders X 1 = −[ AJ n (k L a) + BJ n (k R a)]H n(2 ) (k1 a) , 72 (4.82a) ' X 2 = 2J n ( k R a) J n ( k L a) H n( 2 ) (k1 a) + jη c [ AJ n ( k L a ) − BJ n ( k R a)] H n( 2 ) ( k1 a) , (4.82b) X 3 = [ AJ n ( k L a ) + BJ n ( k R a )]J n (k1 a) , (4.82c) X 4 = −2 J n ( k R a ) J n ( k L a) J n ( k1a) − jη c [ AJ n ( k L a ) − BJ n ( k R a)] J n ( k1a) , (4.82e) ' ' Y1 = 2 J n (k R a) J n ( k L a ) H n( 2 ) (k1 a ) − [CJ n ( k L a) + DJ n ( k R a )]H n( 2 ) ( k1 a) , (4.82f) Y 2 = jη c [CJ n ( k L a) − DJ n ( k R a)] H n( 2 ) ( k1 a) , (4.82g) Y3 = −2 J n ( k R a ) J n ( k L a) J n (k1a) + [CJ n (k L a) + DJ n ( k R a )]J n (k1 a) , (4.82h) Y 4 = −[CJ n ( k L a ) − DJ n (k R a)] jη c J n ( k1 a) , (4.82i) ' and jε 1 (ξc − A= k1 ( µξ c − ε c ) 2 jε 1 (ξ c + B= 1 ' ) k R J n ( k R a) ηc 1 ' )k L J n ( k L a ) ηc k1 ( µξ c − ε c ) 2 − µ1 (ε c − C= , (4.83a) , µξ c ' )k R J n ( k R a ) ηc k1 ( µ 2ξ c − µε c ) 2 (4.83b) , (4.83c) . (4.83d) and − µ1 (ε c + D= µξc ' ) k L J n ( k L a) ηc k1 ( µ 2ξ c − µε c ) 2 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 73 4.3.2 SCATTERING OF CHIRAL CYLINDER 0 echo width(dB/m) -5 Ez -10 chiral cylinder, radius=0.1 meter Frequency=300MHz,µr=4.0, ε r=1.5, ξ c=0.0005 -15 -20 0 30 60 90 120 150 180 φ(deg) (a) (b) Fig 4.15 Echo width of one chiral cylinder compared with paper (a) our simulation result (b) from paper[24] Due to its property that both permittivity and permeability will change with the frequency, like ferrite cylinder, chiral cylinder EBG has various potential applications such as filter. 15 10 echo width(dB/m) 5 0 -5 -10 -15 -20 dielectric cylinder ε=3 µ=2 chiral cylinder ε=3 µ=2 ξc=0 -25 -30 0 60 120 180 240 300 360 φ (degree) Fig 4.16 Test of the rightness of equation by comparing special case with dielectric cylinders. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 74 Fig 4.15 shows the Echo Width (BSCS) of a single chiral cylinder. Our result agrees well with that from paper [24]. Fig 4.16 gives the comparison of the Echo Width of a special chiral cylinder, namely a dielectric cylinder by using equations of this part and equations in previous part (for dielectric cylinder). The two curves agree very well, which further verifies the rightness of our equations. 4.4 AGGREGATED T-MATRIX OF MULTIPLE CYLINDERS In this section, we will represent an effective algorithm, which can combine the T-matrix of several cylinders into one T- matrix, and accordingly, these cylinders can be viewed as one cylinder. It reduces the dimension of T-matrix, which not only reduces a large amount of computer resource needed but also saves much computation time. Moreover, in some cases, such as investigating some special inhomogeneous cylinders by using the scattering matrix method, it is necessary to view several cylinders as a single object. Fig 4.17 shows the geometry of two cylinders. Assume a plane wave with incident angle θ . Fig 4.18 gives the calculation model of combining two T-matrixes. Let a 1 , a 2 , f 1 and f 2 respectively be the incident coefficient matrices and scattering matrices of these two cylinders. a and f are respectively the incident coefficient matrix and scattering matrix of the combined cylinder. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders P α α1 α2 r1 r2 R θ Fig 4.17 Calculation model for two cylinder combined aggregate T-matrix a1 a2 f2 f1 a f θ Fig 4.18 Calculation model of aggregate T-matrix Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method 75 Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 76 we obtain a1 = ß10 a , (4.84a) a2 = ß20a , (4.84b) ß10 and ß20 can be obtained by using the Addition theorem (for the first kind) J m (k ρ − ρ ' )e imφ = ∑ J n −m ( kρ ' ) J n (kρ )e inφ −i (n −m )φ , '' ' (4.85) n J − Mi −( − Mj ) (kρ ' )e − i[ − Mi −( − Mj )]φ . ßij = . . . J ' _ i [ Mi− ( Mj )]φ ' ( k ρ ) e Mi − ( − Mj ) ' . . . . . . . . . J Mi− (− Mj ) (kρ ' ) e −i [_ Mi− Mj ]φ . . . . . . J Mi − Mj ( kρ ' ) e _ i [ Mi −Mj ]φ . . . ' ' . (4.86) here, the matrix is limited to (2 M i +1)*(2 M j +1), let n = [− M i , M i ], m − [ − M j , M j ], (4.87) then β ij ( n, m ) = J n −m ( kρ ' )e −i ( n − m )φ . ' (4.88) For the convenience of coding, suppose P = n + M i + 1, Q = m + M j + 1, (4.89) n = [1,2M i + 1], m = [1,2 M j + 1], (4.90) then Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders β ij ( p, q ) = J p −q +Mj − Mi ( kρ ' )e −i ( p − q + Mj −Mi )φ , ' (4.92a) f 2 = T2 a 2 + T2 a 21f 1 , (4.92b) a 1 T1 + T2 a 2 0 T2 a 21 −1 f1 0 f = I − T a 2 2 21 T1 a 12 T1 0 0 f1 0 f = I − T a 2 2 21 T1 a 12 T1 0 0 −1 a 12 f 1 , 0 f 2 (4.93) 0 a 1 , T2 a 2 (4.94) 0 ß10 a, T2 ß20 (4.95) f = ß01f 1 + ß02f 2 , f = [ß01 (4.91) f 1 = T1 a1 + T1 a f 2 , 12 f1 T1 = f 2 77 (4.96) 0 ß02 ] I − T2 a 21 f ß02 ] 1 = [ß01 f 2 −1 T1a 12 T1 0 0 0 ß10 a, T2 ß20 (4.97) and T = f/a = [ß01 0 ß02 ] I − T2 a 21 −1 T1 a 12 T1 0 0 0 ß10 . T2 ß20 (4.98) Finally, the combined T- matrix is obtained as T = f/a = [ß01 I ß02 ] − T2 a 21 −1 − T1 a 12 T1 I 0 0 ß10 , T2 ß20 (4.99) where ß01 and ß02 can be obtained by using the Addition theorem (for Ha nkel functions) H m(1) ( k ρ − ρ ' )e imφ = ∑ J n− m ( kρ ' ) e −i( n− m)φ H n( 2) ( kρ )e inφ , '' ' n Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (4.100) Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 78 and J − Mi −( − Mj ) (kρ ' )e − i[ − Mi −( − Mj )]φ . ßij = . . . J ' _ i [ Mi− ( Mj )]φ ' Mi − (− Mj ) ( kρ ) e ' . . . . . . . . . J Mi− (− Mj ) (kρ ' ) e −i [_ Mi− Mj ]φ . . . . . . J Mi − Mj ( kρ ' ) e _ i [ Mi −Mj ]φ . . . ' ' . (4.101) Please note, here, the matrix is limited to (2 M i +1)*(2 M j +1). Let n = [− M i , M i ], (4.102) m − [ −M j , M j ], then β ij ( n, m ) = J n −m ( kρ ' )e −i ( n − m )φ . ' (4.103) For the convenience of developing code, let us suppose: P = n + M i + 1, (4.104) Q = m + M j + 1, n = [1,2M i +1], (4.105) m = [1,2 M j + 1], then β ij ( p, q ) = J p −q +Mj − Mi ( kρ ' )e −i ( p − q + Mj −Mi )φ . ' Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (4.106) Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 79 25 25 left cylinder before being combined right cylinder before being combined combined cylinder 20 20 15 15 10 10 5 5 0 0 -5 -5 0 5 10 15 20 0 25 5 10 (a) 15 20 25 (b) Fig 4.19 EBG structure model (a) real structure (b) combined structure 10 Transmission(dB) 0 -10 -20 -30 aggregated T-mate=rix method usual scattering matrix method -40 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ωa/2πc Fig 4.20 Comparison of transmission spectra of usual scattering matrix method and aggregate T-matrix me thod. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 4 EBG Structures Composed by Multi-layered Cylinders, Ferrite and Chiral Cylinders 80 Fig 4.19 shows the EBG structure and the virtual structure combined. The cylinder radius a = 4 , and spacing r = 0.6 , dielectric constant ε r = 8.41. Fig 4.20 gives the comparison of transmission spectra by using aggregated T- matrix and usual scattering matrix method. They agree well, which verifies the rightness of our theory. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects 81 5 N OVEL ELECTROMAGNETIC BANDGAP DEVICES CONSTRUCTED WITH COATED CYLINDERS OR FERRITE CYLINDERS AS DEFECTS EBG can be used in various microwave and millimeter wave devices and antennas due to the wide-bandwidth of the forbidden frequency band. In this chapter, some kinds of devices based on coated cylinder EBG and ferrite cylinder EBG are described. 5.1 T-JUNCTION FILTERS COMPOSED OF COATED DIELECTRIC EBG The coated EBG shows novel characteristics and provides additional freedom to tune the bandgaps. Some changes in the dielectric constant and thickness of the coated layer or inner layer will change the electromagnetic properties of EBG. This property will lead to a good candidate for novel device design. 5.1.1 A T-JUNCTION FILTER In this section, we will analyze a T-junction EBG structure, which is a triangular lattice of dielectric cylinders in air. The dielectric constant of the cylinders is ε r = 8.41 . Due to the discontinuities, inside the bandgap of EBG, the wave can effectively be guided through the channel, which in turn gives the filtering behavior [27]. When we replace the cylinder Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects 82 with coated cylinder, the transmission will change and some of these changes have potential application for T-junction filter design. Fig 5.1 gives the geometry of the T-junction structure. A TM plane wave propagating from top of the structure is assumed as the incident wave. Short lines put in both ports are for the calculation of the transmission spectra in port2 and port 3. The Poynting power is obtained by averaging the power (along the channel direction) on the line. Both of the two lines have length 3.5. Line in port 2 lies apart from both of the two neighbor rows of cylinders(cylinder centre) by the distance 0.25; Line in port 3 lies apart from both of its neighbor rows of cylinders by the distance 0.5. We also put a short line behind the EBG to calculate the transmission of it in vertical direction. a port 1 port 3 a/4 -------------- ---------------- a/2 port 2 Transmission(dB) 0 -10 -20 -30 ---------------line 1 line 2 line 1 line 2 line 3 -40 line 3 -50 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 wavelength (λ) Fig 5.1 2-D structure T-junction EBG Fig 5.2 Transmission spectra of dielectric EBG in Fig 5.1 a = 1 , radius = 0.15 , ε r = 8.41 Fig 5.2 shows the transmission spectra of port 2 and port 3 of EBG in Fig 5.1. In the stopband of the EBG, at the lower wavelength range, port2 is in its pass-band, while port3 is in stop-band. For low wavelength, the wave cannot propagate through both ports. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects (a) (b) (c) Fig 5.3 Electric field distribution of EBG in Fig 5.1 (a) (c) λ λ = 2.09 (b) λ = 2.13 = 2.29 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method 83 Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects 84 With an increase of wavelength, port3 in its pass-band, port 2 in its stop-band. This property makes it as a good filter. Fig 5.3 shows the electric field distribution of the EBG structure adopted in Fig 5.1 for different frequenc ies. The filtering property of this Tjunction filter is verified. 5 0 Transmission(dB) -5 -10 -15 -20 -25 -30 -35 line 1 -40 line 2 line 3 -45 -50 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 wavelength(λ) Fig 5.4 Transmission spectra of hollow EBG in Fig 9 (inner radius=0.05) The geometry of EBG of Fig 5.4 has the same geometry as that in Fig 5.1 except that its cylinders are hollow (with air core, the inner radius is 1/3 of the outer radius ). Fig 5.4 shows the transmission spectra of port 2 and port 3 of this structure. Compared with those curves in Fig 5.2, the spectra have changed in that pass-band for port2 shifts to highe r frequency, and the bandwidth for port 3 has increased. Fig 5.5 shows the electric field distribution in different wavelength. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects (a) (b) Fig 5.5 Electric field distribution of EBG in Fig 12 (a) λ = 2.02 (b) λ = 2.20 0 Transmission(dB) -5 -10 -15 -20 -25 -30 line 1 line 2 line 3 -35 -40 1.4 1.6 1.8 2.0 2.2 2.4 wavelength(λ) Fig 5.6 Transmission spectra of hollow EBG in Fig 9 (inner radius=0.1) Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method 85 Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects 86 The geometry of EBG in Fig 5.6 has the same geometry as Fig 9, but its cylinders are hollow (air core), and the air cylinder radius is 0.1. Fig 5.6 shows the transmission spectra of port 2 and port 3 of this coated EBG structure. Compared with the curves in Figs 5.2 and 5.4, it is obvious that the stop-band pass-band have been shifted. Fig 5.7 shows the electric field distribution in different wavelength for this coated cylinder EBG filter. (a) Fig 5.7 Electric field distribution of EBG in Fig 14 (a) (b) λ = 1.81 (b) λ = 1.99 The geometry of EBG in Fig 5.8 has the same geometry as Fig 5.1 except that its cylinders have metal inne r layer, and the inner radius is 1/3 of the outer radius. Fig 5.8 shows the transmission of port 2 and port 3 of this structure. Compared with the curves in Fig 10, in no frequency band I which wave will be guided through port 2, however, it can always be guided through port 3. Higher transmission is achieved in some frequency band. Fig 5.9 shows the electric field distribution of this T-junction filter at different wavelengths. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects 10 Transmission(dB) 0 -10 -20 -30 -40 line 1 line 2 line 3 -50 2.0 2.5 3.0 3.5 4.0 4.5 wavelength (λ) Fig 5.8 Transmission spectra of metal inner layer EBG in Fig 9 (inner radius=0.05) (a) Fig 5.9 Electric field distribution of EBG in Fig 14 (a) (b) λ = 4.20 (b) λ = 3.39 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method 87 Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects 88 10 Transmission(dB) 0 -10 line 1 line 2 line 3 -20 -30 -40 -50 -60 2.0 2.5 3.0 3.5 4.0 4.5 wavelength (λ) Fig 5.10 Transmission spectra of metal inner layer EBG in Fig 9 (the inner radius is 2/3 of the outer radius) The geometry of EBG in Fig 5.10 has the same geometry as shown in Fig 5.1, but its cylinders have metal inner layer, and the inner radius is 0.1. Fig 5.10 shows the transmission of port 2 and port 3 of this structure. Compared with curves in Figs 5.2 and 5.4, whereas port 2 has a narrow pass-band, port 3 has a wide pass-band and a high transmission in some frequency bands. Fig 5.11 shows the electric field distribution with different wavelength. The T junction structure shows the filtering behavior. By adding air or metal core, the transmission properties have been changed, which can be used to modify the performance of photonic crystal devices. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects (a) (b) (c) (d) Fig 5.11 Electric field distribution of EBG in Fig 14 (a) λ = 2.45 (b) λ = 2.53 (c) λ = 2.90 (d) λ = 4 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method 89 Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects 90 5.2 TUNABLE EBG DEVICES WITH FERRITE DEFECTS By generating ferrite defects in dielectric EBG structures, some useful devices can be designed to achieve some desired functions. We can adjust the EM properties of the ferrite cylinder EBG by adjusting the applied DC magnetic field intensity. Compared with devices consisting purely of dielectric cylinders or conducting cylinders, those with ferrite defects can be adjusted to the desired scattering properties. EBG FILTER TUNED BY FERRITE D EFECTS 5.2.1 35 mz=1E14 h0=0.01mz gama=10 mz=1.5E14 h0=0.01mz gama=10 mz=0.8E14 h0=0.01mz gama=10 10 30 0 Transmission(dB) 25 20 15 10 -10 -20 -30 5 -40 0 -5 0 5 10 15 20 25 30 35 6 7 8 9 10 11 12 13 wavelngth( λ) Fig 5.12 EBG with one ferrite cylinder Fig 5.13 Transmission spectra of EBG in Fig 5.12 with different added DC magnetic In this section, the filtering behavior of an EBG structure is analyzed. The EBG geometry is showed in Fig 5.12. One cylinder in this dielectric array is replaced by a ferrite cylinder. A DC magnetic field is applied along the axis of this ferrite cylinder. The dielectric Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects 91 constant of all cylinders in this EBG is ε r = 8.41 . In this part, TM polarization incident wave is adopted. (a) (b) (c) (d) Fig 5.14 Electric field distributions for EBG in Fig 5 with different added DC magnetic field intensity. (a) mz = 1× 10 14 A/m mz = 1×1014 A/m, λ = 8 .9611 (d) λ = 9 .2367 (b) mz = 1 .5 × 10 14 A/m, mz = 1 .5 × 10 14 A/m, λ = 9 .2367 λ = 8 .9611 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (c) Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects 92 Fig 5.13 is the transmission spectra of this EBG structure with different applied DC magnetic field s. It shows that a sharp transmission peak appears in the stop-band, which is due to the fact that a resonance occurs in the microcavity created by the ferrite defect. From Fig 5.13, we can also see that when the applied magnetic field intensity changes, the transmission peak will change its position. This property makes it a frequency selection filter device. Fig 5.14 gives the electric field distributions with different applied magnetic field s. As predicted, resonance occurs and a filtering behavior is presented. 20 mz=1E15 h0=0.01mz gama=10 mz=1E15 h0=0.005mz gama=10 mz=1E15 h0=0.02mz gama=10 10 mz=1E13 h0=0.01mz γ=10 mz=1E13 h0=0.01mz γ=1 mz=1E13 h0=0.01mz γ=100 mz=1E13 h0=0.01mz γ=30 10 Transmission(dB) Tranbsmission(dB) 0 -10 -20 -30 0 -10 -20 -30 -40 -40 6 7 8 9 10 Wavelength( λ) (a) 11 12 13 6 7 8 9 10 11 12 13 Wavelength(λ) (b) Fig 5.15 Transmission spectra with (a) different H0/Mz (b) different gyromagnetic ratio Fig 5.15 is the transmission spectra with different susceptibilities and gyromagnetic ratios. The curves show how the ferrite cylinder properties affect the scattering properties. From Fig 5.16, it is noticed that when the applied magnetic field intensity becomes much larger Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects 93 or much smaller, more than one transmission peaks appear, which also provides potential uses in device design. 5 mz=1E14 h0=1E13 gama=10 mz=0.5E14 h0=0.5E13 gama=10 mz=8E14 h0=8E13 gama=10 0 Tranbsmission(dB) -5 -10 -15 -20 -25 -30 -35 -40 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.512.012.5 13.0 13.5 Wavelength( λ ) Fig 5.16 Transmission spectra with different added DC magnetic field This ferrite defect EBG structure shows the filtering behavior and shows the frequency selection property through adjusting the added magnetic field intensity, which can be used as frequency selective device. 5.2.2 TUNABLE EBG COUPLER 5.2.2.1 A Tunable Coupler The EBG structure of this tunable coupler with ferrite cylinders is shown in Fig 5.17, where the cylinder radius r0 =0.6, a =4, d = 2 , l = 7 , and ε r = 8.41 . A DC magnetic field is applied along z direction. In this section, the TM polarization incident wave propagates from the bottom of this structure. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects 94 We put some ferrite cylinders between the two discontinuities (Fig 5.17) while other cylinders are dielectric cylinders. Then, when a wave propagates from bottom, the structure becomes an adjustable coupler. Simulatio n results are shown in Fig 5.18 and it is obvious that compared with the dielectric cylinder coupler, coupler with ferrite cylinder has narrower pass-band. Moreover, the pass-band becomes narrower and narrower with an increase of added magnetic field intensity. 10 Dielctric cylinder Ferritecylinder l mz=0(dielectric cylinder) mz=0.8E14 mz=5E14 5 d 0 Transmission(dB) -5 -10 -15 -20 -25 -30 -35 2r0 a -40 -45 5 6 7 8 9 10 11 12 13 14 wavelength(λ) Fig 5.17 The geometry of Fig 5.18 Transmission characteristics of alternate Coupler coupler with varying added magnetic field Fig 5.19 shows the Electric field distribution at λ = 9.0936 , which verifies the rightness of transmission characteristics showed in Fig 5.18. The most important advantage of this coupler is that its pass-band can be adjusted to become narrower whereas coupler with only dielectric cylinders can only achieve a fixed pass-band. More desired properties can be achieved by changing the positions or the number of these ferrite cylinders. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects (a) (b) (c) Fig 5.19 Electric field distribution with different added DC magnetic field intensity at λ = 9.0936 ， γ = 10 （ a） M z =0 (dielectric cylinder) (b) M z =0.2E14 (c) 5.2.2.2 A Coupler with Ferrite Defects 20 Dielectric cylinder Ferrite cylinder mz=0.3E14 mz=0.5E14 mz=1.0E14 10 Transmission(dB) 0 -10 -20 -30 -40 -50 6 Fig 5.20 Geometry of coupler with ferrite defects 7 8 9 10 11 12 13 Fig 5.21 Transmission of coupler with ferrite defects with different added magnetic field intensities. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method 95 Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects (a) (c) 96 (b) (d) Fig 5.22 Electric field distribution with different added DC magnetic field intensity at γ = 10 （ a ） λ = 9.0198 , M z =0.3E14 (b) λ = 9.0198 , M z =1E14 (c) λ = 10 .1300 , M z =0.3E14 (d) λ = 10.1300 , M z =1E14 The geometry of this coupler structure is shown in Fig 5.20, where r0 =0.6, a =4, d = 2 , l = 7 (the length unit is the same as wavelength), and ε r = 8.41 .Ferrite discontinuities are created by replacing some dielectric cylinders with ferrite cylinders. A DC magnetic field is applied along the axis of ferrite cylinders. When the same DC magnetic field is applied to both discontinuities, due to the same permeability and Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects 97 permittivity of the two discontinuities, coupling will arise in certain frequency ranges. In this section, the TM polarization incident wave is from the bottom of this structure. Owing to the fact that DC magnetic field can be changed easily, the frequency band in which coupling happens is shown in Fig 5.21 proves our prediction and is adjustable. The transmission band shifts to higher frequency range (low wavelength value) with an increase of the DC magnetic field. Fig 5.22 gives the comparison of electric field distribution with different added DC magnetic field intensities at different wavelengths, which verifies the results of Fig 5.21 and shows the adjustability of our coupler. Compared with the conventional discontinuity couplers [2], this coupler is more flexible. Compared with the coupler in Fig 3.1, it changed the frequency range, which provides potential use for coupler design. By varying the shape of discontinuities or the number of ferrite cylinders, devices with different pass-band s can be achieved. 5.2.3 Y- BRANCH FILTERS Fig 5.23 gives the geometry of the Y-branch filter. r0 =0.6, a =4, d = 2 , l = 7 (the length unit is the same as wavelength) , and ε r = 8.41 . Ferrite discontinuities are created by replacing some dielectric cylinders with ferrite cylinders. A DC magnetic field is applied along the axis of ferrite cylinder s. Incident wave propagates from the bottom of this structure. The TM polarization is adopted for the incident wave . Owing to the existence of ferrite cylinders, the wave can go though the discontinuities in some frequency band s. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects 98 Fig 5.24 shows the transmission spectra when the applied DC magne tic field in left discontinuity, M z =0.8E14, whereas that in right discontinuity, M z =3E14. The transmissions band for left and right port are different, this is due to the different applied DC magnetic field intensity. By adjusting the DC magnetic field intensity, the filter becomes an adjustable filter, which is more flexible compared with the conventional filters [2]. Fig 5.25 shows the electric field distribution of the Y-branch coupler against different wavelengths when the added DC magnetic field in the left discontinuity, M z =0.8E14, and in right discontinuity is, M z =3E14. As predicted in Fig 5.24, it shows the filtering behavior. dielectric cylinder ferrite cylinder left port right port 0 Transmission(dB) -10 -20 -30 -40 left discontinuity M z=0.8E14 right discontinuity Mz =3E14 -50 7 8 9 10 11 12 13 Wavelength(λ ) Fig 5.23 Geometry of the Y- Fig 5.24 Transmission spectra of the Y-branch branch filter structure when left discontinuity discontinuity M z =0.8E14, right M z =3E14 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects (a) (b) 99 (c) Fig 5.25 Electric field distribution of Y-branch filter with different wavelengths at γ = 10 when left discontinuity M z =0.8E14, right discontinuity M z =3E14 （ a） λ = 9.0000 (b) λ = 9.3023 (c) λ = 10 .1695 10 10 left ferrite discontinuity Mz=0.5E14 right ferrite discontinuity Mz=3E14 left ferrite discontinuity Mz=6E14 right ferrite discontinuity Mz=3E14 0 Transmission(dB) Transmission(dB) 0 -10 -20 -30 left port right port -10 -20 -30 left port right port -40 -40 -50 7 8 9 10 11 12 13 Wavelength(λ) (a) 7 8 9 10 11 13 (b) Fig 5.26 Transmission spectra of the Y-branch structure when discontinuity 12 Wavelength(λ) (a) left M z =0.5E14, right discontinuity M z =3E14 (b) left discontinuity M z =6E14, right discontinuity M z =3E14 To achieve a different stop-band, the added DC magnetic intensity is changed. Fig 5.26 give the transmission spectra with the added DC magnetic field in left discontinuity, Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 5 Novel EBG Devices Constructed with Coated Cylinders or Ferrite Cylinders as Defects 100 M z =0.5E14, while in right discontinuity, M z =3E14, and the applied DC magnetic field in left discontinuity, M z =6E14, in right discontinuity, M z =3E14 respectively. 5.2.4 CONCLUSIONS By replacing some dielectric cylinders with ferrite cylinders, the EM characteristics become adjustable. Compared with those devices with only dielectric cylinders, they have much more flexibility. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 6 Excitation of EBG Structures by Gaussian Beam and Wire Line Source 101 6 EXCITATION OF ELECTROMAGNETIC BANDGAP STRUCTURES BY G AUSSIAN BEAM AND WIRE LINE SOURCES We have investigated cases of plane wave as the incidence. However, in some case some special waves are needed as the incidence. In this chapter, two kinds of incident waves, Gaussian beam and wire source, are described and the ir applications to EBG structures are studied by using the scattering matrix method. 6.1 EBG ANALYSIS USING GAUSSIAN BEAM ILLUMINATION 6.1.1 GAUSSIAN B EAM Y P( ρ , θ ) X (0, y0 ) Fig 6.1 Incidence model of Gaussian beam Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 6 Excitation of EBG Structures by Gaussian Beam and Wire Line Source 102 Some sources such as horn antenna give Gaussian beam wave. Suppose a Gaussian beam whose source located at y = − y0 (Fig 6.1). The far field produced by the beam source can be written as −β E inc z ( x, − y0 ) = E0 e 2 2 x , (6.1) where β 2 = a 2 + jb 2 and 1/ β is related to the beam width. Using the Fourier integral [28], i.e. E inc = z 1 2π ∞ ∫ ∫ ∞ −∞ −∞ Ezinc (x , − y0 )e jα x d x.e − j ( y+ y0 ) k02 −α 2 − jα x dα , (6.2) it can be rewritten as E inc z = 1 2 πβ ∫ ∞ −∞ a −α 2 /4 β 2 − j( y+ y0 ) k02 −α 2 − jα x dα . (6.3) Equation (6.3) can be approximately integrated. Suppose βλ is smaller than unity, so the energy is concentrated in a narrow band at α ≅ 0 . Under this condition the phase function (k 02 − α 2 ) can be expanded in a Taylor’s series about α = 0 , k0 2 − α 2 ≅ k0 − 12 αk0 . 2 (6.4) Then equation (6.3) can be rewritten as 2 E xinc 1 ≅ e −( β x ) /(1− jY )− jk0 Y( +Y0 ) , E0 1 − jY where Y = 2β 2 (1/ k0 )(Y + Y0 ) Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (6.5) Chapter 6 Excitation of EBG Structures by Gaussian Beam and Wire Line Source 103 6.1.2 SCATTERING MATRIX The difference between scattering matrices of plane wave scattering and Gaussian beam scattering is the incident coefficients, therefore, in this part the incident coefficients for Gaussian beam incidence is derived. Under polar coordinate system, x = ρ sin(θ ) , y = ρ cos(θ ),α = k 0 sin γ (α ) then the factor of the exponential function in the integrand now becomes y k02 − α 2 + xα = k0ρ cos[θ − γ (α )] , (6.6) then it is expressed as e − jkρ cos[θ −γ (α )] = ∞ ∑ j −n e jn [θ −γ (α )] J n ( k0 ρ ) . (6.7) n =−∞ Combine equation (6.3) with equation (6.7) ∞ Eziinc = ∑ j − ne jnθ J n (k0 ρ ) Anh , E0 n=−∞ (6.8) where A = h n 1 2 πβ ∫ ∞ −∞ − e 2 α − jy0 k 02 −α2 − jnγ (α ) 4β 2 dα . (6.9) The scattered field is given as ∞ E zsca = ∑ j − n e jnθ H n(2) (k0 ρ ) Bnh , E0 n =−∞ (6.10) Bnh = Tn Anh . (6.11) where Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 6 Excitation of EBG Structures by Gaussian Beam and Wire Line Source 104 Because βλ is smaller than unity, so the energy is concentrated on in a narrow band at α ≅ 0 . Under this condition the phase function γ (α ) can be expanded in a Taylor’s series about α = 0 , α 1 α 3 α + ( ) + ... ≅ . k0 6 k0 k0 γ (α ) = sin −1 (α / k0 ) = (6.12) Substituting equation (6.4) and (6.9) into equation (6.12) and using the well-known integral formula [29] Anh ≅ Anh1 = − j k0 y0 n 2 β) k0 (1− jY0 ) ( e e 1 − jY0 − , (6.13) where Y0 = 2β 2 y0 / k0 , the incident and scattered field can be written as E inc z = E zsca = − jk0 y0 e 1 − jY0 n =−∞ − j k0 y0 ∞ e 1 − jY0 ∑aJ n n ( kρ ) e n 2 β) k0 (1 − jY0 ) (6.14a) ( ∑bH n =−∞ n 2 β) k0 − (1− jY0 ) ( ∞ n (2) n (k ρ )e − (6.14b) Fig 6.2 shows an EBG structure with M x = 9, M y = 9, Spacing = 4, Radius = 0.6 , ε r = 8.41. The incident Gaussian beam β = 0.000041+j0.00007 , y0 = − 5 . Fig 6.3 gives the electric field distribution when the frequency is in stop-band and pass-band, respectively. Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 6 Excitation of EBG Structures by Gaussian Beam and Wire Line Source 105 Radius=3, Spacing=20 Fig 6.2 Geometry of EBG structure for Gaussian incidence calculation λ=0.2 λ=39.5 250 2.101 -- 2.400 1.803 -- 2.101 1.504 -- 1.803 1.205 -- 1.504 0.9063 -- 1.205 0.6075 -- 0.9063 0.3088 -- 0.6075 0.01000 -- 0.3088 200 150 100 1.225 -- 1.400 1.050 -- 1.225 0.8750 -- 1.050 0.7000 -- 0.8750 0.5250 -- 0.7000 0.3500 -- 0.5250 0.1750 -- 0.3500 0 -- 0.1750 50 0 -50 -100 -100 -50 0 50 100 150 (a) 200 250 (b) Fig 6.3 Electric field distributions of Gaussian beam incidence (a) λ (b) λ = 39.5 , stop-band = 0.2 , pass-band 6.2 WIRE LINE EXCITATION OF EBG STRUCTURES An infinitely thin wire parallel to the z − axis , acts as an antenna. It can be expressed as Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Chapter 6 Excitation of EBG Structures by Gaussian Beam and Wire Line Source (2) E inc z ( r) = H 0 (k r − r0 ) , 106 (6.15) where r0 is the position of the source in x − y plane. According to the addition theorem (Fig 6.4) H m(1) ( k ρ − ρ ' )e imφ = ' ∞ ∑H n =−∞ (2) n− m ( k ρ ' )e −i( n− m) φ J n ( kρ ) einφ . ' (6.16) O ρ Φ f ' ? Y ? −?' ' '' f o '' Y '' X X '' Fig 6.4 Translation model in the cylindrical coordinate system for Hankel function Because in this case m = 0 , it can be rewritten as H 0(1) ( k ρ − ρ ' ) = ∞ ∑H n =−∞ (2) n (k ρ ' ) e−inφ J n ( kρ ) einφ . ' (6.17) Similar to plane wave incidence, the incident field can be expressed as ∞ jnφ . E inc z = ∑ an J n (k ρ )e n =∞ Comparing it with equation (6.17), the incident coefficient can be expressed as Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method (6.18) Chapter 6 Excitation of EBG Structures by Gaussian Beam and Wire Line Source an = H n(2) (k ρ ' )e − jnφ . ' 107 (6.19) Fig 6.5 shows an example of EBG with wire source in it and Fig 6.6 gives the electric field distribution when the structure is in its resonant mode for λ = 9.0575 . The resonant frequency can be obtained by solving the scattering matrix equation. Chapter 3 has described the procedure in detail. %(1) Radius=0.6,Spacing=4,εr=8.41 10 0 -10 -20 wire source -30 -20 Fig 6.5 EBG structure with wire source -10 0 10 20 Fig 6.6 Electric field distributions at resonant mode λ = 9.0575 Analysis of Finite-sized EBG Materials and Devices by Scattering Matrix Method Reference 108 REFERENCES [1] G. Tayeb and D. 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Finite- sized EBG Materials and Devices by Scattering Matrix Method List of Symbols T: T -matrix f: Scattering matrix a: Incident matrix a: Mutual effect matrix γ: Gyro magnetic ratio ξc : Chiral admittance of the medium εc : Effective permittivity xiiii Analysis of Finite- sized EBG Materials and Devices by Scattering Matrix Method Chapter 1 1 Introduction 1 INTRODUCTION 1.1 PROBLEM DESCRIPTION Electromagnetic. .. EBG materials, several tunable EB G devices are proposed These include the filters, the couplers and the Y-branch junction constructed by ferrite defects Analysis of Finite- sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T -matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 9 3 T -MATRIX OF CONDUCTING AND DIELECTRIC CYLINDERS AND S CATTERING M ATRIX METHOD. .. photonic bang gap structure lately The fabrication and design of photonic bandgap structures or devices and the simulation and modeling of such kind of materials are gaining importance recently Analysis of Finite- sized EBG Materials and Devices by Scattering Matrix Method Chapter 2 Background Knowledge (a) 5 (b) Fig 2.1 Model of interconnect Electromagnetic bandgap structure is a refractive periodic structure... cylinder combined aggregate T -matrix 75 Fig 4.18: Calculation model of aggregate T- matrix 75 xii Analysis of Finite- sized EBG Materials and Devices by Scattering Matrix Method List of Figures Fig 4.19: EBG structure model (a) real structure (b) combined structure 79 Fig 4.20: Comparison of transmission spectra of usual scattering matrix method and aggregate T -matrix method 79 Fig... Microwave and Optical Technology Letters (Accepted) Wang Quanxin, Zhang Yaojiang, Li Erping and Ooi Ban Leong, Analysis of finite- sized 2D coated electromagnetic band gap structures by scattering matrix, ” IEEE Tran on Selected Topics in quantum Electronics (Submitted) Analysis of Finite- sized EBG Materials and Devices by Scattering Matrix Method Chapter 2 Background Knowledge 4 2 BACKGROUND KNOWLEDGE 2.1 ELECTROMAGNETIC. .. Electric field distribution of single Fig 3.3 RCS of single metal cylinder metal cylinder with Radius= λ with Radius= λ Analysis of Finite- sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T -matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 13 3.1.2 SCATTERING MATRIX M ETHOD FOR M ETAL CYLINDER ARRAY ρj Y φj P θ X Fig 3.4 Calculation model of two dimensional Cylinder... pass-band 105 Fig 6.4: Translation model in the cylindrical coordinate system for Hankel function 106 xv Analysis of Finite- sized EBG Materials and Devices by Scattering Matrix Method List of Figures Fig 6.5: EBG structure with wire source 107 Fig 6.6: Electric field distributions at resonant mode λ = 9.0575 107 xvi Analysis of Finite- sized EBG Materials and Devices by Scattering. .. (cosθ cosϕ j +sinθ sinϕ j) e −iM ( 2 +θ ) (3.13) π Analysis of Finite- sized EBG Materials and Devices by Scattering Matrix Method (3.14) Chapter 3 T -matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 14 where r j is the position of cylinder j , and ϕ j is the angle of the line from start point to the cylinder The total scattering field from all the cylinders can be expressed... cylinder j can be expressed as [1] [2] [31] Analysis of Finite- sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T -matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method f j = Tja j + ∑ Tja ji f i , 15 (3.16) i≠ j where T i stands for the transmission matrix coefficient from j th cylinder, and a ij is the coupling of i th cylinder to j th cylinder, whose elements... Calculation model of one metal cylinder Analysis of Finite- sized EBG Materials and Devices by Scattering Matrix Method Chapter 3 T -matrix of Conducting and Dielectric Cylinders and Scattering Matrix Method 10 Let us consider a plane wave (TM polarization) with an angular θ (in Fig 3.1) The model is two dimensional, which means the cylinder is infinite in length in the z direction, and the incident field