CHARACTERIZATION OF SPHERICAL METAMATERIALS BY HUANG NINGYUN B.ENG. COMMUNICATION ENGINEERING XIAN JIAOTONG UNIVERSITY, 2000 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 indeed feel great pleasure to express my sincere gratitude to my supervisor Professor Li Le-Wei, for his valuable guidance, advice, strong support and many useful opinions and suggestions on many parts of the research work in the process of my postgraduate program. I would like to particularly acknowledge and thank Prof. Li who has given me continual help and great encouragement, which have always inspired me to work harder and have given me confidence. I am also very grateful to Professor Leong Mook Seng and Professor Yeo Tat Soon for their kind encouragement and helpful suggestions. My gratitude is also extended to my fellow laboratory members and many friends in Microwave Division, for their kind help and advice when I encountered some difficulties in the project. Last but not least, I take this opportunity to express my deep thanks to my beloved parents and younger sister for their encouragement and support. Huang Ningyun 15 March 2004 Table of Contents 1 TABLE OF CONTENTS ABSTRACT ……………………………………………………………………………..I ACKNOWLEDGEMENT ……………………………………..………………………II 1 2 INTRODUCTION....................................................................................................... 1 1.1 PROBLEM DESCRIPTION ....................................................................................... 1 1.2 MOTIVATION AND OBJECTIVES ............................................................................ 2 1.3 OUTLINE OF THE THESIS ...................................................................................... 2 1.4 ORIGINAL CONTRIBUTION.................................................................................... 3 BACKGROUND INTRODUCTION OF METAMATERIAL............................. 5 2.1 NEGATIVE REFRACTIVE INDEX ............................................................................ 5 2.2 NEGATIVE REFRACTION MAKES A PERFECT LENS ............................................... 8 2.3 LATERAL DISPLACEMENT OF A GAUSSIAN BEAM REFLECTED FROM A GROUNDED METAMATERIAL ........................................................................................... 9 3 2.3.1 Numerical Results ....................................................................................... 9 2.3.2 Simulation Results..................................................................................... 14 SPHERICAL DYADIC GREEN’S FUNCTIONS............................................... 17 3.1 INTRODUCTION .................................................................................................. 17 3.2 FUNDAMENTAL FORMULATION .......................................................................... 19 3.2.1 Eigenfunction Expansion of DGF in Unbounded Media.......................... 19 Table of Contents 2 3.2.2 DGFs for Spherical Multilayered Media.................................................. 23 3.2.3 The Recurrence Matrix Equations for the Coefficients of Scattering DGFs 24 3.3 4 SCATTERING OF SPHERE ................................................................................ 30 4.1 Rayleigh Scattering................................................................................... 30 4.1.2 Mie Scattering........................................................................................... 32 SCATTERING BY TWO SPHERES .......................................................................... 36 METAMATERIAL SPHERE................................................................................ 44 5.1 INTRODUCTION .................................................................................................. 44 5.2 CURRENT DISTRIBUTION LOCATED OUTSIDE THE SPHERE ................................. 44 5.2.1 Introduction............................................................................................... 44 5.2.2 Using Two Methods to Obtain the Coefficients ........................................ 45 5.2.3 Coefficients in the Special Case of Metamaterial..................................... 48 5.2.4 Calculation of the Electrical Field ........................................................... 51 5.3 6 INTRODUCTION .................................................................................................. 30 4.1.1 4.2 5 CONCLUSIONS .................................................................................................... 29 CURRENT DISTRIBUTION LOCATED INSIDE THE SPHERE .................................... 54 5.3.1 Introduction............................................................................................... 54 5.3.2 The Calculation of the Coefficients........................................................... 55 5.3.3 Coefficients in the Special Case of Metamaterial..................................... 57 5.3.4 Calculation of the Electrical Field ........................................................... 61 METAMATERIAL SPHERICAL SHELL.......................................................... 65 Table of Contents 6.1 INTRODUCTION .................................................................................................. 65 6.2 CURRENT DISTRIBUTION LOCATED OUTSIDE THE SPHERICAL SHELL ................. 65 6.2.1 Introduction............................................................................................... 65 6.2.2 The Calculation of the Coefficients........................................................... 66 6.2.3 Coefficients in the Special Case of Metamaterial..................................... 70 6.2.4 Calculation of the Electrical Field ........................................................... 73 6.3 7 3 CURRENT DISTRIBUTION LOCATED INSIDE THE SPHERICAL SHELL .................... 76 6.3.1 Introduction............................................................................................... 76 6.3.2 The Calculation of the Coefficients........................................................... 77 6.3.3 Coefficients in the Special Case of Metamaterial..................................... 80 6.3.4 Calculation of the Electrical Field ........................................................... 83 MULTI-SPHERICAL LAYERS OF METAMATERIALS ............................... 87 7.1 INTRODUCTION .................................................................................................. 87 7.2 CURRENT DISTRIBUTION LOCATED OUTSIDE THE SPHERICAL MULTILAYERS .... 87 7.2.1 Introduction............................................................................................... 87 7.2.2 The Calculation of the Coefficients........................................................... 88 7.2.3 Coefficients in the Special Case of Metamaterial..................................... 93 7.2.4 Calculation of the Electrical Field ........................................................... 95 7.3 CURRENT DISTRIBUTION LOCATED INSIDE THE SPHERICAL MULTILAYERS ....... 99 7.3.1 Introduction............................................................................................... 99 7.3.2 The Calculation of the Coefficients......................................................... 100 7.3.3 Coefficients in the Special Case of Metamaterial................................... 104 4 Table of Contents 7.3.4 8 Calculation of the Electrical Field ......................................................... 106 SIMULATION RESULTS ................................................................................... 110 8.1 SIMULATION RESULTS AND DISCUSSION OF METAMATERIAL SPHERE ............. 110 8.2 SIMULATION RESULTS AND DISCUSSION OF METAMATERIAL SPHERICAL SHELL 114 8.3 SIMULATION RESULTS AND DISCUSSION OF METAMATERIAL SPHERICAL MULTILAYERS ............................................................................................................. 118 9 SUMMARY ........................................................................................................... 125 REFERENCES.............................................................................................................. 127 Abstract i Abstract Materials possessing both negative permittivity and permeability simultaneously, i.e., left-handed materials or metamaterials were firstly introduced in 1968 by V.G.Veselago. This special kind of materials demonstrates very different characteristics from the conventional materials. In this thesis, the electromagnetic fields produced by a dipole in the presence of metamaterial spheres are analyzed. Firstly, some properties of metamaterial, including negative refractive index and lateral displacement of a Gaussian beam reflected from a grounded metamaterial, are introduced and discussed. Secondly, spherical dyadic Green’s functions in such a material are analyzed, and the eigenfunction expansion of DGF and the recurrence matrix equations for the coefficients of scattering DGF are emphasized. Thirdly, the scattering of normal material sphere is introduced. Scattering by two separate spheres is analyzed in particular. Finally, the spherical metamaterial objects are analyzed. Three cases are considered respectively: the single sphere, the spherical shell and the multi-layer sphere. Finally, simulation results are given and discussed. List of Figure 1 List of Figures Fig 2.1: A medium with negative refractive index bend light to a negative angle with the surface normal. Light formerly diverges from a point source and then set in reverse and converges back into a point. The light reaches a focus for a second time after it released from the medium........................................................................................... 8 Fig 2.2: Configuration of a Gaussian beam incident upon a slab with thickness d = d 2 − d1 .................................................................................................................. .10 Fig 2.3: Time-averaged power density on the xz plane for a 30° incidence of a Gaussian beam upon a grounded slab of thickness d = 6λ with ε1 = ε 0 and µ = µ0 . This is the simulation result by using the formula developed in this thesis ......................... 15 Fig 2.4: Reference time-averaged power density on the xz plane for a 30° incidence of a Gaussian beam with ε1 = ε 0 and µ = µ0 . This is the simulated result by using the formula developed in reference [6]........................................................................... 15 Fig 2.5: Time-averaged power density on the xz plane for a 30° incidence of a Gaussian beam upon a grounded slab of thickness d = 6λ with ε1 = −ε 0 and µ = − µ0 . This is the simulation result by using the formula developed in this thesis ......................... 16 Fig 2.6: Time-averaged power density on the xz plane for a 30° incidence of a Gaussian beam upon a grounded slab of thickness d = 6λ with ε1 = −ε 0 and µ = − µ0 . This is the simulated result by using the formula developed in reference [6] ...................... 16 List of Figure 2 Fig 3.1: Geometry of a spherically multilayered medium ................................................ 18 Fig 8.1: Structure of the metamaterial sphere. ................................................................ 110 Fig 8.2: Near field of metamaterial sphere when k2 = k1 = k0 , a = 1λ , b = 2λ ............ 112 Fig 8.3: Near field of metamaterial sphere when k2 = 2k1 , a = 1λ , b = 2λ . ................. 112 Fig 8.4: Near field of metamaterial sphere when k2 = 4k1 , a = 1λ , b = 2λ .................. 113 Fig 8.5: Near field of metamaterial sphere when k2 = − k1 , a = 1λ , b = 2λ ............. …113 Fig 8.6: Near field of metamaterial sphere when k2 = −2k1 , a = 1λ , b = 2λ . ............... 114 Fig 8.7: Structure of the metamaterial spherical shell. ................................................... 115 Fig 8.8: Near field of metamaterial spherical shell when k2 = k1 , a1 = 4λ , a2 = 2λ ... 116 Fig 8.9: Near field of metamaterial spherical shell when k2 = 2k1 . a1 = 4λ , a2 = 2λ .. 116 Fig 8.10: Near field of metamaterial spherical shell when k2 = − k1 , a1 = 4λ , a2 = 2λ .117 Fig 8.11: Near field of metamaterial spherical shell when k2 = −2k1 , a1 = 4λ , a2 = 2λ 117 Fig 8.12: Structure of the spherical mutilayers ( m = 5 ). ................................................ 118 Fig 8.13: Near field of 5-layer metamaterial spheres when k2 = k1 , a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ . ................................................................................... 119 List of Figure 3 Fig 8.14: Near field of 5-layer metamaterial spheres when k2 = 2k1 , a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ ................................................................................... 119 Fig 8.15: Near field of 5-layer metamaterial spheres when k2 = −k1 , a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 10 . ....................................................................... 120 Fig 8.16: Near field of 5-layer metamaterial spheres when k2 = − k1 , a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 20 ....................................................................... 120 Fig 8.17: Near field of 5-layer metamaterial spheres when k2 = − k1 , a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ ., n = 30 ...................................................................... 121 Fig 8.18: Near field of 5-layer metamaterial spheres when k2 = − k1 , a1 = 5λ , a2 = 4λ , a3 = 3λ , a4 = 2λ , b = 5.5λ , n = 30 ....................................................................... 121 Fig 8.19: Near field of 5-layer metamaterial spheres when k2 = −2k1 , a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 20 ..................................................... 122 Fig 8.20: Near field of 5-layer metamaterial spheres when k2 = −2k1 , a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 30 ..................................................... 122 Fig 8.21: Near field of 5-layer metamaterial spheres when k2 = −2k1 , a1 = 5λ , a2 = 4λ , a3 = 3λ , a4 = 2λ , b = 5.5λ , n = 30 ........................................................................ 123 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 Chapter1 1 Introduction 1 INTRODUCTION 1.1 PROBLEM DESCRIPTION In 1968, metamaterial, a kind of artificial materials with the electric permittivity and magnetic permeability simultaneously negative was firstly introduced by V.G. Veselago, who predicted that if such a material could be found, it would exhibit very unusual electromagnetic scattering phenomena. In recent years, the electromagnetic properties of this kind of special materials have attracted much attention. Such metamaterial has optical properties that would be impossible to be found in a conventional material and could make novel antennas or perfect optical lenses theoretically. Metamaterials can be engineered to have specific electromagnetic behaviors that are physically impossible for natural materials. Now, in this thesis, we focus on the electromagnetic properties of a spherically multilayered medium, which is metamaterial. The spectral-domain electromagnetic Dyadic Green’s Function is constructed for defining the electromagnetic fields in the multilayered media. The scattering dyadic Green’s function in each layer is constructed and the coefficients of the function are derived. The general solution can be applied to specific geometries, e.g., two-, three- and four-layered media which are frequently employed in practical cases. Boundary conditions are also used to solve this problem. Results and discussions are given in succession. 1 Characterization of Spherical Metamaterials Chapter1 Introduction 2 1.2 MOTIVATION AND OBJECTIVES The research aims to obtain electromagnetic radiation characteristics of a dipole in each layer of the spherically multilayered media. The metamaterial exhibits different properties from other conventional materials, such as opposite electromagnetic wave phase velocity. Accordingly, the media made of metamaterial will show a different electromagnetic radiation pattern. This is what we are interested in. Metamaterial’s special properties are also very beneficial in potential practical uses. The simulation results of models with different layers are given and discussed. 1.3 OUTLINE OF THE THESIS The thesis is divided into 8 chapters, as follows: Chapter 2 gives the basic information about this thesis by providing an overview of background knowledge for the metamaterials. The Chapter 3 gives thorough description about spherical dyadic Green’s functions, which are used in this project. Some necessary implementations of the theories used in the thesis are also provided here. In Chapter 4, detailed description regarding the scattering by spheres is provided. The scattering of two spheres is emphasized. 2 Characterization of Spherical Metamaterials Chapter1 Introduction 3 Chapter 5 analyzes the fields associated with a single sphere, using the spherical dyadic Green’s function. Two circumstances are considered: the source locates outside the sphere and inside the sphere, respectively. The specific case, i.e., the metamaterial sphere case, is also studied. Chapter 6 analyzes the fields associated with a spherical shell. Similar to those in Chapter 5, two cases are considered where the source locates outside the sphere shell and inside the sphere shell, respectively, and also examined is the specific case of metamaterial sphere. In Chapter 7, the general case is considered where spherical multilayer is involved, with the source set to be outside the layers and inside the layers, respectively. The metamaterial case is considered. In Chapter 8, the simulation results of the single sphere and spherical multi layers are given. Chapter 9 gives a summary of the whole thesis. 1.4 ORIGINAL CONTRIBUTION Conference paper: 3 Characterization of Spherical Metamaterials Chapter1 Introduction 4 Le-Wei Li, Ningyun Huang, Qun Wu and Zhong-Cheng Li, "Macroscopic Characteristics of Electromagnetic Waves Radiated by a Dipole in the Presence of Metamaterial Sphere ({Invited})", Proc. of 5th Asia-Pacific Engineering Research Forum on Microwaves and Electromagnetic Theory, Kyushu University, Fukuoka, Japan, July 29-30, 2004 4 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 5 2 BACKGROUND INTRODUCTION OF METAMATERIAL After metamaterial was first introduced in 1968 by Veselago[3], many scientists have tried to realize it in the real world. A composite medium, based on a periodic array of interspaced conducting, nonmagnetic copper split ring resonators and continuous wires, which exhibits a frequency region in the microwave regime with simultaneously negative values of effective permeability µ and permittivity ε , forms a metamaterial. It has been predicted that such phenomena as the Doppler Effect, the Cerenkov radiation, and the Snell’s law are inverted. The shape of the ring can be both circular and square. The rings and wires are on opposite sides of the boards. The structure is improved for better material characteristics in the following years. In this chapter some basic properties of the metamaterial are presented. The negative refractive index of metamaterials is introduced firstly; then the lateral displacement of Gaussian beam reflected from grounded metamaterial is introduced afterwards. 2.1 NEGATIVE REFRACTIVE INDEX The real part of the refractive index of a nearly transparent and passive medium is usually taken to have only positive values. Through an analysis of a current source radiating into a metamaterial, it can be determined that the sign of the real part of the refractive index is actually negative. The regime of negative index leads to unusual electromagnetic wave 5 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 6 propagation and merits further exploration. The negative refractive index can be derived from the analysis of the equation of the electric field and the equation of the work. The general solution to the one-dimensional wave equation with an arbitrary current density, in frequency domain, can be written as [7]: E (ω ) = − z (ω ) j (ω ) , c (2.1) where the generalized impedance can be defined [7] as: z (ω ) = µ (ω ) , n(ω ) (2.2) where n is refractive index. We require that the source on average do positive work on the fields. Thus we define the quantity P (ω ) = j (ω ) 1 z (ω ) 2 c 2 , (2.3) which should be greater than zero [7]. In order to make sure the positive work, we get z (ω ) > 0. In a metamaterial, since µ < 0 , we conclude that the solution with n < 0 leads to the correct interpretation that the currents perform work on the fields. The existence of negative refractive index is consistent with causality, which introduces the constraints: d [ε (ω )ω ] dω > 1 and d [ µ (ω )ω ] dω > 1, (2.4) valid for nearly transparent media. Wave propagation and wave interaction with current sources in metamaterial are therefore necessarily complicated by the implicit frequency dependence of the material parameters, and even simple geometries can lead to 6 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 7 mathematical or numerical complexity. For nearly transparent media, Im [ z (ω )] can be neglected since it is obtained from the reference [7]: d (nω ) 1 d ωµ 1  d (ωε ) 1 d (ωµ )  = + (ωε z + ) = z . 2 dω 2  dω dω z z dω  Applying Eq. (2.4) to Eq. (2.5), we have: velocity of a wave, defined as vg = (2.5) d (nω ) z + z −1 > > 1. Furthermore, the group dω 2 c , must therefore always be positive and less d (nω ) dω than c in either normal media or metamaterial. At last, we can find that at some frequency range, the wave propagates with a negative index of refraction. We can also get the same result from analyzing the composite medium which is made use of an array of metal posts to create a frequency region with ε < 0 . The structure is interspersed with an array of split ring resonators (SRRs) having a frequency region with µ < 0, which is introduced in the beginning of this chapter. The thin wire medium can be described by the dielectric function [7]: ε (ω ) = 1 − ω 2p , ω2 (2.6) where the plasma frequency ω p is related to the geometry of the wire array. 7 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 8 We can assume that the wire and the SRR arrays do not interact directly, the index of the refraction of this structure can be presented as: n(ω ) = ε (ω ) µ (ω ). Thus, it can be obtained that µ and ε are simultaneously negative in the region between ω 0 and ω b , and there exist propagating modes in this region. 2.2 NEGATIVE REFRACTION MAKES A PERFECT LENS The sharpness of the image is determined by the wavelength of the light for a conventional lens. A slab of material with negative refractive index, i.e., a metamaterial slab, can focus all the Fourier components of a 2D image, even including those do not propagate in a radiative manner. With the current technology, this kind of super-lenses can be realized in the microwave band provided that the material has no transmission loss. Figure 2.1 A medium with negative refractive index bend light to a negative angle with the surface normal. Light formerly diverges from a point source and then set in reverse and converges back to a point. The light reaches a focus for a second time after it released from the medium. 8 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 9 Assuming that n = −1, we can seen from the figure that it obeys Snell’s laws of refraction at the surface as light inside the medium makes a negative angle with the surface normal. Another characteristic we should notice of this structure is the double focusing effect revealed. Evanescent waves emerge from the far side of the medium enhanced in amplitude by the transmission process, so the medium can also cancel the decay of evanescent waves. This does not violate energy conservation, because evanescent waves do not transport energy. Several developments in technology make such a lens a practical possibility, at least in some regions of the spectrum. 2.3 LATERAL DISPLACEMENT OF A GAUSSIAN BEAM REFLECTED FROM A GROUNDED METAMATERIAL A dramatic negative lateral shift can be observed when a Gaussian beam reflected from a grounded metamaterial, which is distinctly different from a shift made by a conventional grounded slab [6]. 2.3.1 NUMERICAL RESULTS The waves inside and outside the material can be solved analytically from Maxwell’s equations and the boundary conditions at the interfaces, then, the field values in all regions can be clearly determined. A system as shown in Fig. 2.2 has the following incident wave in region “I”: ∞ Eiy = ∫ dk x ei ( kx x + k0 z z )ψ (k x ), −∞ (2.7) 9 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 10 where ψ (k x ) = g 2 π e− g 2 ( k x − kix )2 / 4 . (2.8) Y I ( ε 0 , µ0 ) 0 II ( ε1 , µ1 ) d1 θi III ( ε 2 , µ2 ) d2 X Figure 2.2 Configuration of a Gaussian beam incident upon a slab with thickness d = d 2 − d1 ˆ ix + zk ˆ 0 sin θ i + zk ˆ iz = xk ˆ 0 cos θ i , where θ i is The Gaussian beam is centered about k i = xk the incident angle. In region “I”, we can obtain the expression of the E 0 y , which is the sum of the incident wave and reflection wave [6]: E ∞ 0y = ∫ dkxψ (kx )(eik0 z z + Re−ik0 z z )eikx x . −∞ (2.9) Because 10 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial (2.10) ∇ × E = jωµ H Hx = − 11 1 ∂E y , jωµ ∂z (2.11) and 1 ∂E y , jωµ ∂x Hz = (2.12) we can obtain: ∞ − k0z −∞ ωµ 0 H 0 x = ∫ dk xψ (k x ) ∞ kx −∞ ωµ 0 H 0 z = ∫ dk xψ (k x ) (e ik0 z z − Re −ik0 z z )e ik x x , (2.13a) (e ik0 z z + Re −ik0 z z )e ik x x . (2.13b) In region “II”, we have ∞ E1 y = ∫ dk xψ (k x )( Ae ik1 z z + Be −ik1 z z )e ik x x , −∞ ∞ − k1 z −∞ ωµ1 H 1x = ∫ dk xψ (k x ) ( Ae ik1 z z − Be −ik1 z z )e ik x x , (2.14) (2.15a) and ∞ kx −∞ ωµ1 H 1z = ∫ dk xψ (k x ) ( Ae ik1 z z + Be −ik1 z z )e ik x x . (2.15b) In region “III”, we have ∞ E 2 y = ∫ dk xψ (k x )Te ik 2 z z +ik x x , −∞ (2.16) thus we can obtain: ∞ − k2z −∞ ωµ 2 H 2 x = ∫ dk xψ (k x ) Te ik 2 z z +ik x x , (2.17a) 11 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial ∞ kx −∞ ωµ 2 H 2 z = ∫ dk xψ (k x ) Te ik 2 z z + ik x x . 12 (2.17b) The coefficients R , A , B and T can be obtained by matching the boundary conditions for the tangential components of electric and magnetic fields at x = d1 and x = d 2 . We apply the boundary conditions: nˆ × E1 = nˆ × E2 , (2.18a) nˆ × H1 = nˆ × H 2 ; (2.18b) Because nˆ = zˆ , then: zˆ × H 0 z = zˆ × H1z = zˆ × H 2 z = 0 . (2.19) Thus we have [6] when z = d1 , zˆ × E0 y = zˆ × E1 y , (2.20a) zˆ × H 0 x = zˆ × H1x ; (2.20b) zˆ × E1 y = zˆ × E2 y , (2.21a) zˆ × H1x = zˆ × H 2 x . (2.21b) when z = d 2 , From (2.20) and (2.21), four equations are derived [6]: e ik0 z d1 + Re −ik0 z d1 = Ae ik1 z d1 + Be −ik1 z d1 , − k0z ωµ 0 (e ik0 z d1 − Re −ik0 z d1 ) = − k1 z ωµ1 ( Ae ik1 z d1 − Be −ik1 z d1 ) , (2.22a) (2.22b) 12 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial Ae ik1 z d 2 + Be −ik1 z d 2 = Te ik 2 z d 2 , − k1 z ( Ae ik1 z d 2 − Be −ik1 z d 2 ) = ωµ1 − k2z ωµ 2 13 (2.22c) Te ik 2 z d 2 . (2.22d) By solving (2.22a-d), we have: R01 + R12 e i 2 k1 z ( d 2 − d1 ) i 2 k0 z d1 e , 1 + R01 R12 e i 2 k1 z ( d 2 − d 1) (2.23a) A= 2e − i ( k1 z − k0 z ) d1 , (1 + p 01 ) 1 + R01 R12 e i 2 k1 z ( d 2 − d1 ) (2.23b) B= 2 R12 e −i ( k1 z − k0 z ) d1 e i 2 k1 z d 2 , (1 + p 01 ) 1 + R01 R12 e i 2 k1 z ( d 2 − d1 ) (2.23c) R= [ ] [ ] 4e ik0 z d1 e ik1 z ( d 2 − d1 ) e −ik 2 z d 2 T= , (1 + p 01 )(1 + p12 ) 1 + R01 R12 e i 2 k1 z ( d 2 − d1 ) [ ] (2.23d) where R01 = 1 − p 01 1 − p12 , R12 = , 1 + p12 1 + p12 p 01 = µ 0 k1 z µk , p12 = 1 2 z . µ1 k 0 z µ 2 k1 z (2.24a) (2.24b) Now we consider the special case that ε 1 = −ε 0 , µ1 = − µ 0 , d 1 = 0 and d 2 = d . Because region 2 is perfectly conducting and ε 2 is close to infinity, we can simplify the coefficients R , A , B and T in this special case k = ω εµ , (2.25) so k1z = −k0 z , (2.26) 13 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 14 and p01 = 1 , R01 = 0 . (2.27) Because k2 z = ω ε 2 µ 2 is close to infinity, p12 is close to infinity, and R12 = −1 . (2.28) We have [6] R = R12 e i 2 k1 z d = R12 e − i 2 k0 z d = e −i 2 k0 z d , A=1, B = R12 e i 2 k1 z d = R12 e −i 2 k0 z d = e − i 2 k0 z d , T=0. 2.3.2 (2.29a) (2.29b) (2.29c) (2.29d) SIMULATION RESULTS The time averaged power density for the TE case is given as: Sn = 2 2 1  Re ( Eny H nz* )  +  Re ( Eny H nx* )  ,    2  (2.30) where n =0,1,2 denotes the regions. Fig. 2.3 and Fig. 2.5 are the simulation results obtained in this thesis, Fig. 2.4 and Fig. 2.6 are the simulation results in [6]. Fig 2.3 and Fig. 2.4 are the time-averaged power density for the normal material. Fig. 2.5 and Fig. 2.6 are the time-averaged power density for the metamaterial. 14 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 15 10 8 x/λ 6 4 2 0 -2 -2 -1 0 1 2 z/λ 3 4 5 6 7 8 9 10 ° Figure 2.3 Time-averaged power density on the xz plane for a 30 incidence of a Gaussian beam upon a grounded slab of thickness d = 6λ with ε1 = ε 0 and µ = µ0 . This is the simulated result by using the formula developed in this thesis Figure 2.4 Reference time-averaged power density on the xz plane for a beam with ε1 = ε 0 and µ = µ0 . 30° incidence of a Gaussian This is the simulated result by using the formula developed in reference [6] 15 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 16 2 0 x/λ -2 -4 -6 -8 -10 -2 -1 0 1 2 3 4 5 6 7 8 9 10 z/λ Figure 2.5 Time-averaged power density on the xz plane for a upon a grounded slab of thickness d = 6λ with ε1 = −ε 0 30° incidence of a Gaussian beam and µ = − µ0 . This is the simulated result by using the formula developed in this thesis ° Figure 2.6 Time-averaged power density on the xz plane for a 30 incidence of a Gaussian beam upon a grounded slab of thickness d = 6λ with ε1 = −ε 0 and µ = − µ0 . This is the simulated result by using the formula developed in reference [6] 16 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 17 3 SPHERICAL DYADIC GREEN’S FUNCTIONS 3.1 INTRODUCTION Green’s function is named in honor of English mathematician and physicist George Green (1793-1841). Green’s function is basically distribution due to a point source, a solution to a linear differential equation and a building block that can be used to construct many useful solutions. The linear differential equation is derived from the boundary value problem due to a unit excitation source .The exact form of the Green’s function depends on the differential equation, the geometrical shape and the type of the boundary conditions present. The solution involves all the electromagnetic field phenomena. The quasi-static solution is a special case of the general Dyadic Green’s Function at the zero frequency. In general, there are two typical applications when the Dyadic Green’s Function (DGF) is applied to solve boundary-value problems. One of them is: for a particular structure due to an arbitrarily assumed distributed source, a DGF is employed as an electromagnetic (EM) response of the dielectric medium involved, and then the EM field can be formulated. The other application is: we can use a DGF to derive the parameters matrices for the unknown source current distribution. Except for a few simple geometries, the Green’s functions are difficult to be obtained, with each new geometry requiring a new formulation. This is the main limitation of Green’s function technique. 17 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 18 This chapter will present a detailed description of electromagnetic dyadic Green’s function in spherically multilayered media. Figure 3.1 shows the spherically N -layered geometry. The transmitter with an arbitrary electric current distribution J s or an arbitrary magnetic current distribution M s is located in the s th (source) layer ( s = 1, 2,..., N ). The receiver lies in the f th (field) layer ( f = 1, 2,..., N ) of the spherically N -layered system. Figure 3.1 Geometry of a spherically multilayered medium 18 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 19 3.2 FUNDAMENTAL FORMULATION 3.2.1 EIGENFUNCTION EXPANSION OF DGF IN UNBOUNDED MEDIA Electromagnetic problems can always be solved by beginning with the analysis of Maxwell’s equations. The Maxwell’s functions in dyadic form can be given as: ∇ × E = iωµ0 H, (3.1a) ∇ × H = J − iωε 0 E, (3.1b) ∇ ⋅ J = iωρ , (3.1c) ∇ × (ε 0E ) = ρ , (3.1d) ∇ × ( µ0 H ) = 0. (3.1e) From (3.1), the vector field equations can be found from, ∇ × ∇ × E ( R ) − k 2 E ( R ) = iωµ J ( R ) , (3.2a) ∇ × ∇ × H ( R ) − k 2H ( R ) = ∇ × J ( R ) . (3.2b) We consider a general case, that is, a structure of a spherically N -layered geometry which is shown in Figure 3.1. Throughout the analysis, a time dependence exp ( −iω t ) for the fields is assumed. The electric and magnetic current sources J s and M s lie in the s th layer ( s = 1, 2,...N ) of the medium. From (3.2), we can get that the expressions of the electromagnetic radiation fields- E f and H f in the f th layer ( f = 1, 2,...N ) can be found from: ∇ × ∇ × E f − k 2f E f = iωµ f J f δ fs − ( ∇ × M ) f δ fs , (3.3a) 19 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function ∇ × ∇ × H f − k 2f H f = iωε f M f δ fs + ( ∇ × J ) f δ fs , 20 (3.3b) where δ fs is the Kronecker delta function, and ε f , µ f are the permittivity, permeability of the medium, respectively. The propagation constant in the f th layer of the medium is:  iσ k f = ω µ f ε f 1 + f  ωε f    ,  (3.4) where σ f denotes the conductivity of the medium. Moreover, the electric and magnetic fields, E f and H f , due to the electric and magnetic current sources J s and M s , can be expressed by the electric type and magnetic type of dyadic Green’s functions G (e fs ) ( R, R′ ) and G (mfs ) ( R, R′ ) , respectively: E f ( R ) = iωµ f ∫∫∫ G (e fs ) ( R, R′ ) ⋅ J s ( R′ ) dV ′, (3.5a) H f ( R ) = iωµ f ∫∫∫ G (mfs ) ( R, R′ ) ⋅ M s ( R′ ) dV ′, (3.5b) Vs Vs where Vs indicates the volume occupied by the sources in the s th layer, R is the position vector of the field point and R′ is the position vector of the source point. The relationship between the dyadic Green’s function of the electric and the dyadic Green’s function of the magnetic type is subject to the reciprocity theorem. The magnetic DGF can be obtained from the electric DGF by simple replacements of E → H, H → −E, J → M, M → −J , µ → ε and ε → µ . Only DGF of the electric type will be considered to avoid unnecessary repetition. 20 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 21 Substituting (3.5a) into (3.3a), we have: ∇ × ∇ × G (e fs ) ( R, R′ ) − k 2f G (e fs ) ( R, R′ ) = Iδ ( R , R ′ ) δ fs , (3.6) where I is the unit dyadic, δ ( R, R′ ) is the Dirac delta function. The electric type of dyadic Green’s function G (e fs ) ( R, R′ ) satisfies the boundary conditions at the spherical interfaces r = a j ( j = 1, 2,....N − 1 ): ( f +1) s  fs , rˆ × G (e ) = rˆ × G e 1 µf rˆ × ∇ × G (e fs ) = 1 µ f +1 ( f +1) s  rˆ × ∇ × G e . (3.7a) (3.7b) Based on the scattering superposition method, the dyadic Green’s function can be separated into two parts: the unbounded dyadic Green’s function and the scattering dyadic Green’s function, as follows: G (e fs ) ( R, R′ ) = G 0 e ( R, R′ ) δ fs + G (esfs ) ( R, R′ ) , (3.8) where the superscript ( fs ) is the layers where the field point and source point locate, respectively, while the subscript e indicates the electric type of the dyadic Green’s function, and the subscript s identifies the scattering dyadic Green’s function. The unbounded dyadic Green’s function G 0 e ( R, R′ ) corresponds to the contribution of the direct waves from an unbounded medium, i.e., an infinite homogeneous space. The scattering dyadic Green’s function G (esfs ) ( R, R′ ) represents an additional contribution of 21 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 22 the multiple reflected and transmitted waves due to the presence of the spherical boundary of the dielectric layered media. Using the contour integration method in the complex h -plane, the dyadic Green’s function in the unbounded medium can be represented in terms of the normalized spherical vector wave functions as a result of the residue theorem. The unbounded DGF > under the spherical coordinate system is expressed for r r ′ as: < G 0e ( R, R′) = ˆˆ ik rr δ ( R − R′ ) + s 2 ks 4π ∞ n 2n + 1 (n − m)! ∑∑ ( 2 − δ ) n(n + 1) (n + m)! 0 m n =0 m=0 M e ( ks ) M′e ( ks ) + N e ( k s ) N′e ( ks ) , r ≥ r ′,  o mn o mn o mn o mn × (1) (1) M eo mn ( ks ) M′oe mn ( ks ) + N oe mn ( k s ) N′oe mn ( ks ) , r ≤ r ′, (1) (1) (3.9) where the prime indicates the coordinates ( r ′,θ ′, φ ′ ) of the current source J f , m and n is the eigen-value parameters, M e mn is the ΤΕ -wave spherical vector wave function that o represents the electric field of the ΤΕ mn modes, and N e mn is the ΤΜ -wave spherical o vector wave function that represents the electric field of the ΤΜ mn modes. The superscript (1) of the vector wave functions denotes the third-type spherical Bessel function or the first-type spherical Hankel function. M e mn and N e mn can be expressed as: o o sin m jn ( kr ) Ρ mn ( cos θ ) mφθˆ M e mn ( k ) = ∇ × φ e mn ( k ) r  = ∓ o  o  cos sin θ d Ρ m ( cos θ ) cos mφφˆ, − jn ( kr ) n sin dθ (3.10a) 22 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 23 and N e mn ( k ) = o cos 1 n(n + 1) jn ( kr ) Ρ mn ( cos θ ) mφ rˆ ∇ × ∇ × φ e mn ( k ) r  =  o  sin k kr m sin  m m 1 d  rjn ( kr )   d Ρ n ( cos θ ) cos mφθˆ ∓ mφφˆ  , + Ρ n ( cos θ )  sin cos sin θ kr dr dθ   (3.10b) where r = rrˆ φ e o mn (k ) = jn (kr )Ρ mn ( cos θ ) (3.11a) cos ( nφ ) , sin (3.11b) with the subscripts e and o denoting the even and odd modes, respectively. 3.2.2 DGFS FOR SPHERICAL MULTILAYERED MEDIA The scattering dyadic Green’s function should have the similar form of the unbounded dyadic Green’s function, and the multiple reflection and transmission effects should be considered as well. The electromagnetic fields always consist of the radial wave modes that propagate outwards and inwards under the spherical coordinate. Assuming that the current source is located in the s th layer, we can obtain the scattering dyadic Green’s function in the f th layer among the multi layers. Now we can express the scattering DGF utilizing the spherical Bessel and Hankel function and Kroncker delta as follows: 23 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function G es( fs ) (R, R′) = {(1 − δ N f iks 4π ∞ n ∑∑ (2 − δ n=0 m=0 0 ) 24 2n + 1 (n − m)! n(n + 1) (n + m)! )M (1) (k f ) (1 − δ s1 ) AMfs M′e mn (ks ) + (1 − δ sN ) BMfs M′e(1)mn (ks )  e o mn o o   + (1 − δ fN )N (1) (k f ) (1 − δ s1 ) ANfs N′e mn (ks ) + (1 − δ sN ) BNfs N′e(1)mn (ks )  e o mn o o   1 1 fs N fs (1) + (1 − δ f )M e mn (k f ) (1 − δ s )CM M′e mn (ks ) + (1 − δ s ) DM M′e mn (ks )  o o o   (3.12) } + (1 − δ 1f )N e mn (k f ) (1 − δ s1 )CNfs N′e mn (ks ) + (1 − δ sN ) DNfs N′e(1)mn (ks )  , o o o   where AMfs, N , BMfs, N , CMfs, N and DMfs , N ( s, f = 1, 2,..., N ) are the coefficients of the scattering DGF to be solved, and the superscript N is the number of the layers of the multi-layer medium. Same as the expression of the unbounded dyadic Green’s function, the superscript (1) indicates that the third-type spherical Bessel function or the first-type spherical Hankel function should be chosen in the function of the spherical wave vector functions: M e mn and N e mn . Because the normal first-type spherical Bessel function can be o o used to represent both outgoing and incoming waves, it should also be chosen for the rest of the vector wave functions. 3.2.3 THE RECURRENCE MATRIX EQUATIONS FOR THE COEFFICIENTS OF SCATTERING DGFS We should derive the coefficients of the scattering dyadic Green’s function from the boundary conditions. The boundary conditions can be expressed in terms of the dyadic Green’s function shown in (3.7a) and (3.7b). We can rewrite them in the form of linear equation system, which can also be replaced by the coefficient matrix equations as: 24 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function   ℑ ff CMfs  ff AMfs      fs s  fs  δ D ℑ + ( ) B    ff M f  ff M  + =   fs fs ∂ ff AN  C ∂ℑ ff N   ∂  fs  fs s  ∂ ff BN  ∂ℑ ff ( DN + δ f )   25 ( f +1) f + δ fs +1 )   ℑ  ( f +1) f CM   ( f +1) s ( f +1) f    ℑ( f +1) f DM ( f +1) f BM  , (3.13a) +  ( f +1) f  ( f +1) s s  + δ f +1 )  ∂ℑ( f +1) f CN ( f +1) f ( AN    ∂ℑ( f +1) f DM( f +1) f  ∂ ( f +1) f BN( f +1) s    ( f +1) f (A ( f +1) s M and ∂ ff AMfs    k f ∂ ff BMfs  k f + µ f  ff ANfs  µ f  fs   ff BN  ∂  ∂ℑ ff CMfs     fs s ∂ℑ ff ( DM + δ f )  k f +1    = fs C ℑ ff N   µ f +1    ℑ D fs + δ s  f )   ff ( N  + δ fs +1 )   ∂ ( f +1) f BM( f +1) s   ( f +1) s + δ fs +1 )  ( f +1) f ( AN  ( f +1) s ( f +1) f BN  ( f +1) f (A ( f +1) s M (3.13b)  ∂ℑ( f +1) f CM( f +1) f    k f +1 ∂ℑ( f +1) f DM( f +1) f  , + µ f +1  ℑ( f +1) f C N( f +1) f   ( f +1) f   ℑ( f +1) f DM  where ℑil = jn ( ki al ) , (3.14a) = hn(1) ( ki al ) , (3.14b) il ∂ℑil = 1 d [ ρ jn ( ρ ) ] , ρ dρ ρ =k a (3.14c) i l ∂ il (1) 1 d  ρ hn ( ρ )  = ρ dρ . (3.14d) ρ = ki al The coefficient matrix equations are the results of simplifying the complicated algebraic calculation of the linear equation system. Equations (3.13a) and (3.13b) can be simplified as: 25 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function  AM( f,+N1) s + δ fs +1  1   = H ,V ( f +1) s  BM , N  TFf  AMfs , N  RFfH ,V  fs  + H ,V  BM , N  TFf 26  CMfs, N  ,  fs s  DM , N + δ f  (3.15a) and  CM( f,+N1) s  RPfH ,V  ( f +1) s  = H ,V  DM , N  TPf  AMfs , N  1  fs  + H ,V  BM , N  TPf  CMfs, N  ,  fs s  DM , N + δ f  (3.15b) where RPfH = µ f k f +1∂ ( f +1) f ff − µ f +1k f ∂ ff µ f k f +1ℑ ff ∂ ( f +1) f − µ f +1k f ∂ℑ ff ( f +1) f , (3.16a) RFfH = µ f k f +1∂ℑ( f +1) f ℑ ff − µ f +1k f ∂ℑ ff ℑ( f +1) f , µ f k f +1∂ℑ( f +1) f ff − µ f +1k f ℑ( f +1) f ∂ ff (3.16b) RPfV = µ f k f +1 ( f +1) f ∂ ff − µ f +1k f ff ∂ µ f k f +1∂ℑ ff ( f +1) f − µ f +1k f ℑ ff ∂ , (3.16c) RFfV = µ f k f +1ℑ( f +1) f ∂ℑ ff − µ f +1k f ℑ ff ∂ℑ( f +1) f , µ f k f +1ℑ( f +1) f ∂ ff − µ f +1k f ∂ℑ( f +1) f ff (3.16d) TPfH = µ f k f +1 (ℑ( f +1) f ∂ ( f +1) f − ∂ℑ( f +1) f µ f k f +1ℑ ff ∂ ( f +1) f − µ f +1k f ∂ℑ ff TFfH = TPfV = TFfV = µ f k f +1 (∂ℑ( f +1) f µ f k f +1∂ℑ( f +1) f ( f +1) f ff ( f +1) f ( f +1) f − ∂ℑ( f +1) f ) , (3.16e) , (3.16f) , (3.16g) , (3.16h) ( f +1) f ( f +1) f − µ f +1k f ℑ( f +1) f ∂ ( f +1) f ff ( f +1) f − ℑ( f +1) f ∂ µ f k f +1 (∂ℑ( f +1) f ( f +1) f − ℑ( f +1) f ∂ µ f k f +1∂ℑ ff ( f +1) f − µ f +1k f ℑ ff ∂ µ f k f +1 (ℑ( f +1) f ∂ µ f k f +1ℑ( f +1) f ∂ ( f +1) f ) ff ( f +1) f ) ( f +1) f ( f +1) f − µ f +1k f ∂ℑ( f +1) f  AMNs, N   CM1s , N   Ns  =  1s  = 0.  BM , N   DM , N  ) ff (3.16i) 26 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 27 The subscripts F and P indicate the centrifugal and centripetal waves, respectively. T( HP, F ) f and R(HP , F ) f represent the centripetal and centrifugal transmission (corresponding to T ) and reflection (corresponding to F ) contributions from ΤΕ waves (corresponding to the superscript H). T(VP , F ) f and R(VP , F ) f represent the centripetal and centrifugal transmission and reflection contributions from ΤΜ waves (corresponding to the superscript V ). Now a method of formulating the coefficients of the scattering dyadic Green’s function for spherical arbitrary multilayered medium will be provided. However, the matrix equations for the coefficients are coupled. Hence, the de-coupling of these equations is needed to obtain the general expression of the coefficients of the scattering dyadic Green’s function. From (3.15a) and (3.15b), we have:  AM( f,+N1) s + δ fs +1  ( f +1) s  CM , N  1  H ,V BM( f,+N1) s   TFf = DM( f,+N1) s   RPfH ,V   TPfH ,V RFfH ,V   TFfH ,V   AMfs, N  ⋅  fs 1  CM , N TPfH ,V   , + δ  BMfs , N fs M ,N D s f (3.17) where the symbols T( HP ,,FV ) f and R(HP,,VF ) f represent the equivalent transmission and reflection coefficients given by (3.15a) and (3.15b), respectively. Given the following coefficient matrix, parameter matrix and the operators are:  A fs CMfs , N =  Mfs, N CM , N BMfs , N  , DMfs, N  (3.18a) 27 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function f M ,N T  1  TFfH ,V =  H ,V RPf  TPfH ,V  28   , 1 TPfH ,V   H ,V RFf TFfH ,V 1 0  A11 =   , and 0 0 (3.18b) 0 0 A 22 =  . 0 1  (3.18c) CMf 1, N = TMf −, N1 ⋅⋅⋅ TM1 , N ( C11 M , N + A 22 ) , (3.19a) Now we can obtain the relations for s = 1: for s ≠ 1 and N : { } CMfs , N = TMf −, N1 ⋅⋅⋅ TMs , N TMs −,1N ⋅⋅⋅ TM1 , N C1Ms , N +  H ( f − s − 1) A 22 − H ( f − s ) A11  , (3.19b) and for s = N : CMfN, N = TMf −, N1 ⋅⋅⋅ TM1 , N C1MN, N − H ( f − N ) A11 , (3.19c) where the step function H ( x − x0 ) is given as: 1, x ≥ x0 . H ( x − x0 ) =  0, x < x0 (3.20) To obtain the results of the coefficients from equations (3.19a), (3.19b) and (3.19c), we assume that the field point locates in the last layer, i.e. f = N . The coefficients in the first and last layers are easy to obtain from the equation (3.16i). Then, the rest of the coefficients can be found directly by using the recurrence relations in (3.19) once again. So far, the general expressions of the scattering dyadic Green’s function for the spherical multilayered media have been provided. 28 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 29 3.3 CONCLUSIONS In this chapter, we presented the general expression of the dyadic Green’s function for spherical multilayered medium, including the unbounded dyadic Green’s function and the scattering dyadic Green’s function. The complete set of general coefficients of the scattering dyadic Green’s function is also provided analytically. Furthermore, only the electric type of dyadic Green’s functions is presented in this thesis in order to avoid unnecessary repetition. The magnetic type of the dyadic Green’s functions can be easily obtained by making the simple change of E → H, H → −E, J → M, M → −J , µ → ε and ε → µ . The dyadic Green’s function for cylindrically layered media has also been derived in [21]. 29 Characterization of Spherical Metamaterials Chapter 4 Scattering of Sphere 30 4 SCATTERING OF SPHERE 4.1 INTRODUCTION 4.1.1 RAYLEIGH SCATTERING Rayleigh scattering represents the scattering property of the electromagnetic wave due to a particle much smaller than the wavelength. We can assume a spherical particle with radius a , permittivity ε and permeability µ locates at the origin of the coordinate. The incident electric field is polarized in the z direction: ˆ 0 eikx . Ein = zE (4.1) The solution of the scattered field can be given as: −iωµ Ileikr E sc = 4π r 2  i  2 i  i    i   ˆ ˆ r 2 cos 1 sin + + + + θ θ θ        ,    kr  kr    kr  kr   (4.2a) and −ikIleikr  i  ˆ H sc = φ  + 1 sin θ , 4π r  kr  (4.2b) where Il (polar distance) will be decided by the value of E0 and ε . Because of kr > 1. From (4.2a) and (4.2b), we have:  ε − ε  2 2 a ikr Eθ = −  s  k a E0 e sin θ , r  ε s + 2ε  (4.9a) and Hφ = ε E. µ θ (4.9b) For the perfectly conducting sphere, the inner electric field of the sphere Ein is zero. From (4.4) and (4.6a), we have: Il = −i 4π ka 3 ε E. µ 0 (4.10) 4.1.2 MIE SCATTERING Mie scattering is brought forward when the radius of the sphere is bigger than the Rayleigh scattering case. The solutions of the sphere scattering problem due to planar waves can be obtained precisely by applying the boundary conditions. For convenience, we should introduce Debye potentials: π e and π m , then the spherical wave can be divided to ΤΜ wave and ΤΕ wave due to rˆ . The ΤΜ wave due to rˆ includes [25] A = rπ e , (4.11a) 32 Characterization of Spherical Metamaterials Chapter 4 Scattering of Sphere 1 ∂ ∂ H = ∇ × A = θˆ π e − φˆ π e . sin θ ∂φ ∂θ 33 (4.11b) The ΤΕ wave due to rˆ includes: Z = rπ m , 1 ∂ ∂ E = ∇ × Z = θˆ π m − φˆ π m . sin θ ∂φ ∂θ (4.12a) (4.12b) Debye potentials satisfy the following Helmholtz function under the spherical coordinate: π  + k 2 )  e  = 0, π m  (4.13) 1 ∂2 1 ∂ ∂ 1 ∂2 ∇ = + sin θ + . r ∂r 2 r 2 sin θ ∂θ ∂θ r 2 sin 2 θ ∂φ 2 (4.14) (∇ 2 where 2 Applying the Maxwell’s equations and (4.13), we have [25] Er = Eθ =  i  ∂2 2  2 rπ e + k rπ e  , ωε  ∂r  (4.15a) i 1 ∂2 1 ∂ πm, rπ e + ωε r ∂r ∂θ sin θ ∂φ (4.15b) 1 ∂2 ∂ rπ e − πm, ∂θ ωε r sin θ ∂r ∂φ (4.15c)  i  ∂2 2  2 rπ m + k rπ m  , ωµ  ∂r  (4.15d) i 1 ∂2 1 ∂ πe, rπ m + ωµ r ∂r ∂θ sin θ ∂φ (4.15e) Eφ = i Hr = − Hθ = − 33 Characterization of Spherical Metamaterials Chapter 4 Scattering of Sphere Hφ = − 34 1 ∂2 ∂ π e. rπ m − ωµ r sin θ ∂r ∂φ ∂θ i (4.15f) Assume a sphere located in the origin of the coordinate has a radius of a , permittivity of ε and permeability of µ . The incident planar wave is given as: ˆ 0 eikz = xE ˆ 0 eik cosθ , E = xE (4.16a) ε E0 eik cosθ . µ H = yˆ (4.16b) The incident wave propagates in the z direction. We have ∞ eikr cosθ = ∑ ( −i ) n=0 −n ( 2n + 1) jn ( kr ) Pn ( cosθ ), (4.17) Thus the incident wave can be expanded as: Er = E0 sin θ cos φ eikr cosθ =− iE0 cos φ ( kr ) 2 ∞ ∑ ( −i ) ( 2n + 1) Jˆ ( kr ) P ′ ( cosθ ), −n n n =1 (4.18) n where Jˆn ( kr ) = krjn ( kr ) . (4.19) From (4.4), π e and π m can be obtained as follows [25]: E cos φ πe = − 0 ωµ r ( −i ) ( 2n + 1) Jˆ kr P ′ cosθ , ) n( ) ∑ n( n ( n + 1) n =1 ∞ −n (4.20a) 34 Characterization of Spherical Metamaterials Chapter 4 Scattering of Sphere E sin φ πm = 0 kr 35 ( −i ) ( 2n + 1) Jˆ kr P ′ cosθ . ) n( ) ∑ n( n ( n + 1) n =1 −n ∞ (4.20b) The Debye potentials of scattered field can be expressed as: π es = − π ms = E0 cos φ ωµ r ∞ ∑ a Hˆ ( kr ) P ′ ( cosθ ), (4.21a) E0 sin φ ∞ ′ bn Hˆ (1) ∑ n ( kr ) Pn ( cos θ ). kr n =1 (4.21b) n =1 (1) n n n where (1) Hˆ (1) n ( kr ) = krhn ( kr ) . (4.22) The inner field of the sphere can also be expressed by Debye potentials as: E0 cos φ ∞ ˆ ∑ c J ( k r ) P ′ ( cosθ ), ωµ s r n =1 n n s n (4.23a) E0 sin φ ks r (4.23b) π ei = − π mi = ∞ ∑ d Jˆ ( k r ) P ′ ( cosθ ). n =1 n n s n The unknown coefficients can be obtained by applying the boundary conditions at the surface of r = a, we have −n −i ) ( 2n + 1) − ε s µ Jˆn′ ( ka ) Jˆn ( ks a ) + εµ s Jˆn ( ka ) Jˆn′ ( k s a ) ( an = i , n ( n + 1) ε s µ Hˆ n(1)′ ( ka ) Jˆn ( ks a ) − εµ s Hˆ n(1) ( ka ) Jˆn′ ( k s a ) (4.24a) −n −i ) ( 2n + 1) − ε s µ Jˆn ( ka ) Jˆn′ ( k s a ) + εµ s Jˆn′ ( ka ) Jˆn ( ks a ) ( bn = , i n ( n + 1) ε s µ Hˆ n(1) ( ka ) Jˆn′ ( ks a ) − εµ s Hˆ n(1)′ ( ka ) Jˆn ( ks a ) (4.24b) i εsµ ( −i ) ( 2n + 1) i cn = , n ( n + 1) ε s µ Hˆ n(1)′ ( ka ) Jˆn ( ks a ) − εµ s Hˆ n(1) ( ka ) Jˆn′ ( ks a ) (4.24c) −n 35 Characterization of Spherical Metamaterials Chapter 4 Scattering of Sphere 36 − εµ s ( −i ) ( 2n + 1) i dn = . n ( n + 1) ε s µ Hˆ n(1) ( ka ) Jˆn′ ( ks a ) − εµ s Hˆ n(1)′ ( ka ) Jˆn ( ks a ) −n (4.24d) When the sphere satisfies ka b G e 0 (R, R′) = − ik 1 ˆˆ RRδ (R − R′) + 1 2 k1 4π ∑C mn M (1) (k1 )M′(k1 ) + N (1) (k1 )N′(k1 ) , (5.5a) ik 1 ˆˆ RRδ (R − R′) + 1 2 k1 4π ∑C mn M (k1 )M′(1) (k1 ) + N(k1 )N′(1) (k1 ) , (5.5b) m,n when R < b G e 0 (R, R′) = − m,n 45 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere 46 with M e mn and N e mn given in equations (3.10a) and (3.10b). o o The boundary conditions at the surface of the sphere ( R = a ) of electric dyadic Green functions are expressed as follows: Rˆ × G e1 (R, R′) = Rˆ × G e 2 (R, R′), (5.6a) 1 ˆ 1 ˆ R × ∇ × G e1 (R, R′) = R × ∇ × G e 2 (R, R′). µ1 µ2 (5.6b) Because R = a < b , the electric free space DGF should use the equation (5.5b). According to the functions of the boundary conditions, we may obtain that the 21 21 coefficients BM11 , B11 N , DM , DN must satisfy the following equations: jn ( ρ1 ) + BM11 hn(1) ( ρ1 ) = DM21 jn ( ρ 2 ), (5.7a) (1) k1  [ ρ1 jn ( ρ1 )]′ k2  21 [ ρ 2 jn ( ρ 2 )]′  11 [ ρ1hn ( ρ1 )]′  + BM  =  DM , µ1  ρ1 ρ1 ρ2  µ2   [ ρ1 jn ( ρ1 )]′ ρ1 + B11 N [ ρ1hn(1) ( ρ1 )]′ ρ1 = DN21 [ ρ 2 jn ( ρ 2 )]′ ρ2 , (1) 21  k 2  [ jn ( ρ1 ) + B11  jn ( ρ 2 ), N hn ( ρ1 )] = DN  µ1  µ2  k1 (5.7b) (5.7c) (5.7d) where ρ1 = k1a, ρ 2 = k2 a. (5.8) By solving the equations from (5.7a) to (5.7d), the expressions of the four coefficients are obtained as: BM11 = k1µ 2 ρ 2 jn ( ρ 2 )[ ρ1 jn ( ρ1 )]′ − k2 µ1 ρ1 jn ( ρ1 )[ ρ 2 jn ( ρ 2 )]′ , k2 µ1 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k1µ 2 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ (5.9a) 46 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere 47 B11 N = k2 µ1 ρ 2 jn ( ρ 2 )[ ρ1 jn ( ρ1 )]′ − k1µ 2 ρ1 jn ( ρ1 )[ ρ 2 jn ( ρ 2 )]′ , k1µ 2 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k2 µ1 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ (5.9b) DM21 = k1µ 2 ρ 2 hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ 2 ρ 2 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ , k2 µ1 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k1µ 2 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ (5.9c) DN21 = k1µ 2 ρ 2 hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ 2 ρ 2 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ . k1µ 2 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k2 µ1 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ (5.9d) The recurrence matrix equations for the coefficients of scattering dyadic Green’s function introduced in Chapter 3 will be used to obtain the coefficients. Applying N = 2 and s = 1 to equation (3.16i) and equation (3.17), we have: 21 11 BMN = DMN = 0,  1 ( f +1)1   T HV  BMN  Ff  ( f +1)1  =  HV  DMN   RPf  TPfHV RFfHV   f1 TFfHV   BMN  ⋅  , 1 1 f  1   DMN + δ f  TPfHV  (5.10a) (5.10b) where TFfHV , TPfHV , RFfHV , RPfHV are determined from equation (3.16a) to equation (3.16h). For f = 1, the equation (5.10b) can be rewritten as follows: 21 MN B and = 21 DMN = RFHV1 11 + HV ( DMN + 1) , TF 1 (5.11a) RPHV 1 11 11 1 BMN + HV ( DMN + 1) . HV TP1 TP1 (5.11b) 1 TFHV 1 11 MN B From the above two equations together with equation (5.10a), the following equations will be derived: 11 BMN = − RFHV1 , (5.12a) 47 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere 21 DMN = and 1 TPHV 1 (1 − R HV F1 48 RPHV 1 ). (5.12b) By applying f = 1 to the expressions of TFfHV , TPfHV , RFfHV , and RPfHV , the coefficients are obtained: BM11 = − RFH1 = k1µ 2 ρ 2 jn ( ρ 2 )[ ρ1 jn ( ρ1 )]′ − k2 µ1 ρ1 jn ( ρ1 )[ ρ 2 jn ( ρ 2 )]′ , k2 µ1 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k1µ 2 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ (5.13a) V B11 N = − RF 1 = k2 µ1 ρ 2 jn ( ρ 2 )[ ρ1 jn ( ρ1 )]′ − k1µ 2 ρ1 jn ( ρ1 )[ ρ 2 jn ( ρ 2 )]′ , k1µ 2 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k2 µ1 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ (5.13b) DM21 = k1µ 2 ρ 2 hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ 2 ρ 2 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ 1 H H − = 1 R R ( F1 P1 ) k µ ρ h(1) ( ρ )[ ρ j ( ρ )]′ − k µ ρ j ( ρ )[ ρ h(1) ( ρ )]′ , (5.13c) TPH1 2 1 1 n 1 2 n 2 1 2 2 n 2 1 n 1 DN21 = k1µ 2 ρ 2 hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ 2 ρ 2 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ 1 V V − = 1 R R ( F1 P1 ) k µ ρ h(1) ( ρ )[ ρ j ( ρ )]′ − k µ ρ j ( ρ )[ ρ h(1) ( ρ )]′ . (5.13d) TPV1 1 2 1 n 1 2 n 2 2 1 2 n 2 1 n 1 It is obvious that the above four equations are the same as the results derived from the boundary conditions. 5.2.3 COEFFICIENTS IN THE SPECIAL CASE OF METAMATERIAL Now let us consider the metamaterial case, i.e., k2 = − k1 and µ 2 = − µ1 . We know (−1) k z jn ( z ) = ∑   k = 0 k !Γ ( n + k + 1)  2  ∞ n+2k , (5.14) thus we have  j (− z ), When n is even, jn ( z ) =  n  − jn (− z ), When n is odd. (5.15) Based on the properties of the Bessel functions, we have the following equations: 48 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere 49 1  [ ρ1 jn ( ρ1 )]′ = jn ( ρ1 ) + ρ1[ jn ( ρ1 )]′ = jn ( ρ1 ) + ρ1  [ jn −1 ( ρ1 ) − jn +1 ( ρ1 )] , 2   (5.16a) 1  [ ρ 2 jn ( ρ 2 )]′ = jn ( ρ 2 ) + ρ 2 [ jn ( ρ 2 )]′ = jn ( ρ 2 ) + ρ 2  [ jn −1 ( ρ 2 ) − jn +1 ( ρ 2 )] . 2  (5.16b) When n is even, n + 1 and n − 1 are both odd, we have: [ ρ 2 jn ( ρ 2 )]′ = [ ρ1 jn ( ρ1 )]′. (5.17a) When n is odd, then n + 1 and n − 1 are both even, we have: [ ρ 2 j n ( ρ 2 )]′ = −[ ρ1 j n ( ρ1 )]′ . (5.17b) 21 21 After applying above two equations to the expressions of BM11 , B11 N , DM , DN , the results of the coefficients in the special case can be obtained. For BM11 , when n is even: k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ 2 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ = − (1) , hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ BM11 = (5.18a) when n is odd: − k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ 2 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ , = − (1) hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ BM11 = (5.18b) therefore BM11 = − 2 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ , h ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ (1) n (5.19) 49 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere 50 where n can be either odd or even. . For B11 N , when n is even: k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ −k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ 2 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ = − (1) , hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ BN11 = (5.20a) when n is odd: −k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ 2 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ = − (1) , hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ BN11 = (5.20b) therefore B11 N = − 2 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ = An , h ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ (1) n (5.21) whenever n is even or odd. For DM21 , when n is even: DM21 = k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ − k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ h(1) ( ρ )[ ρ j ( ρ )]′ − jn ( ρ1 )[hn(1) ( ρ1 )]′ = − n(1) 1 1 n 1 , hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[hn(1) ( ρ1 )]′ (5.22a) when n is odd: 50 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere DM21 = k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ h(1) ( ρ )[ ρ j ( ρ )]′ − jn ( ρ1 )[hn(1) ( ρ1 )]′ = n(1) 1 1 n 1 . hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[hn(1) ( ρ1 )]′ 51 (5.22b) For DN21 , when n is even: DN21 = k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ − k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ h(1) ( ρ )[ ρ j ( ρ )]′ − jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ = − n(1) 1 1 n 1 , hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ (5.23a) when n is odd: DN21 = k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ h(1) ( ρ )[ ρ j ( ρ )]′ − jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ = n(1) 1 1 n 1 . hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ DM21 = DN21 . It is obvious that (5.23b) (5.24) 5.2.4 CALCULATION OF THE ELECTRICAL FIELD Rectangular coordinate can be transformed to spherical coordinate as: xˆ = sin θ cos φ rˆ + cos θ cos φθˆ − sin φφˆ. (5.25) For the infinitesimal horizontal electric dipole introduced in the beginning of section 5.2, because θ ′ = 0, φ ′ = 0 , we have: xˆ = θˆ . (5.26) Therefore, we have 51 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere 52  n(n + 1) , m =1 mpnm (cosθ )  = 2 , lim θ →0 sin θ  0, otherwise (5.27a)  n(n + 1) , m =1 dpnm (cosθ )  . lim = 2 θ →0 dθ  0, otherwise (5.27b) Applying m = 1 to equation (5.3), we have: Cmn = 2 × 2n + 1 1 . × n(n + 1) n(n + 1) (5.28) From the above three equations, the following equation can be obtained: mpnn (cos θ ) dp m (cosθ ) 2n + 1 = Cmn × lim n = . θ →0 θ →0 sin θ dθ n(n + 1) Cmn × lim (5.29) Additionally, we have: dV ′ = R′2 dR′ sin θ ′dθ ′dφ ′. (5.30) Now the electrical fields will be calculated in three cases. For R > b : E1 (R ) = iωµ1 ∫∫∫ G e1 (R, R′)iJ (R′)dV ′ = iωµ1cG e1 (R, R′)i xˆ = iωµ1cG e1 (R, R′)iθˆ, (5.31) then derived from equations (3.10a), (3.10b) and (5.29), the result can be obtained as: E1 (R ) = − k1ωµ1c ∞ 2n + 1 i  jn ( ρb ) + An hn(1) ( ρb )  M o(1)1n (k1 ) ∑ 4π n =1 n(n + 1)  { } (5.32) + ( ρb jn ( ρb ))′ + Bn ( ρ h ( ρb ))′ N (k1 ) ρ b , (1) b n (1) e1n where ρb = k1b. For a < R < b : 52 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere 53 E 2 (R ) = iωµ1 ∫∫∫ G e1 (R, R′)iJ (R′)dV ′ = iωµ1cG e1 (R, R′)i xˆ = iωµ1cG e1 (R, R′)iθˆ =− k1ωµ1c ∞ 2n + 1 i{hn(1) ( ρb )[M o1n (k1 ) + An M o(1)1n (k1 )] ∑ 4π n =1 n(n + 1) (5.33)   +  ρb hn(1) ( ρb ) ′  Ν e1n (k1 ) + Bn N (1) e1n ( k1 )  ρ b  .  For R < a : E3 (R ) = iωµ2 ∫∫∫ G e 2 (R, R′)iJ (R′)dV ′ = iωµ2 cG e 2 (R, R′)i xˆ = iωµ2 cG e 2 (R, R′)iθˆ =− (5.34)  [ ρb hn(1) ( ρb )]′ k1ωµ2 c ∞ 2n + 1  (1) • Dn N e1n (k2 )  . hn ( ρb ) • Cn M o1n (k2 ) + ∑ 4π n =1 n(n + 1)  ρb  In the metamaterial case, the above expression can be written as: E3 (R ) = iωµ2 ∫∫∫ G e 2 (R , R ′)iJ (R ′)dV ′ = iωµ2 cG e 2 (R, R′)i xˆ = iωµ2 cG e 2 (R, R′)iθˆ =  [ ρb hn(1) ( ρb )]′ k1ωµ1c ∞ 2n + 1  (1) ( ) ( ) • − + • Dn N e1n (− k1 )  . h ρ C M k  ∑ n b n o1n 1 4π n =1 n(n + 1)  ρb  (5.35) We use the expressions of M e mn ( k ) and N e mn ( k ) , i.e., equations (3.10a) and (3.10b), o o together with the properties of the Bessel function shown in equations (5.15), (5.17a) and (5.17b), we have: when n is even, M e mn ( k ) = M e mn ( −k ) , (5.36a) N e mn ( k ) = − N e mn ( −k ) , (5.36b) M e mn ( k ) = −M e mn ( −k ) , (5.36c) N e mn ( k ) = N e mn ( −k ) . (5.36d) o o when n is odd, o o o o o o 53 Characterization of Spherical Metamaterials Chapter 5 5.3 Metamaterial Sphere 54 CURRENT DISTRIBUTION LOCATED INSIDE THE SPHERE 5.3.1 INTRODUCTION The single layered sphere is centered at (0, 0, 0). An infinitesimal horizontal electric dipole with current moment c points in the x -direction and locates at R′ = 0,θ ′ = 0, φ ′ = 0 (the center of the sphere). J (R′) = c δ ( R′ − 0)δ (θ ′ − 0)δ (φ ′ − 0) xˆ. R′2 sin θ ′ (5.37) Scattering dyadic Green’s functions can be expressed generally as in equation (3.12). Applying this equation to special case where N = 2 , s = 2, we can get the scattering DGFs of region 1 (outside the sphere) and region 2 (inside the sphere): ′ G12 es ( R , R ) = ik2 4π G es22 (R, R′) = ∑C mn m,n ik2 4π ∑C m,n  AM12 M (1) (k1 )M′(k2 ) + AN12 N (1) (k1 )N′(k2 )  , (5.38a) CM22 M (k2 )M′(k2 ) + CN22 N(k2 )N′(k2 )  , (5.38b) mn where Cmn = (2 − δ 0 ) 2n + 1 (n − m)! , n(n + 1) (n + m)! (5.39) 22 22 and the coefficients AM12 , A12 N , CM , C N need to be solved for. 54 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere 55 5.3.2 THE CALCULATION OF THE COEFFICIENTS The recurrence matrix equations for the coefficients of scattering dyadic Green’s function introduced in the chapter 3 will be used to obtain the coefficients. Green functions of region 1 and region 2 are presented as follows respectively: ′ G e1 (R, R′) = G12 es ( R , R ), (5.40a) Ge2 (R, R′) = Ge0 (R, R′) +Ges22 (R, R′), (5.40b) where the electric free space DGF is: because R > 0 G e 0 (R, R′) = − 1 垐 ik RRδ (R − R ′) + 2 2 k2 4π ∑C m,n mn M (1) (k2 )M′(k2 ) + N (1) (k2 )N′(k2 ) , (5.41) where M e mn and N e mn are given in equations (3.10a) and (3.10b), and Cmn is shown in o o equation (5.39). Applying N = 2 and s = 2 to equation (3.16i) and equation (3.17), we have: 22 12 AMN = CMN = 0,  1  f +1) 2 ( 2  AMN + δ f +1   TFfHV   =  HV ( f +1) 2  CMN   RPf  TPfHV (5.42a) RFfHV   f2 TFfHV   AMN   ⋅ C f 2  , 1   MN  TPfHV  (5.42b) where TFfHV , TPfHV , RFfHV , RPfHV are determined from equation (3.16a) to equation (3.16h). For f = 1, the equation (5.40b) can be rewritten as follows: 55 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere 22 AMN +1 = 22 CMN = and 1 TFHV 1 12 AMN + RFHV1 12 CMN , TFHV 1 RPHV 1 12 12 1 AMN + HV CMN . HV TP1 TP1 56 (5.43a) (5.43b) From the above two equations together with equation (5.42a), the following coefficients are derived: 12 AMN = TFHV 1 , 22 CMN = and HV TFHV 1 RP1 . TPHV 1 (5.44a) (5.44b) Apply f = 1 to the expressions of TFfHV , TPfHV , RFfHV , and RPfHV . Then the results of the coefficients are obtained as follows: AM12 = TFH1 = k2 µ1 ρ1hn(1) ( ρ 2 )[ ρ 2 jn ( ρ 2 )]′ − k2 µ1 ρ1 jn ( ρ 2 )[ ρ 2 hn(1) ( ρ 2 )]′ , k2 µ1 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k1µ 2 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ (5.45a) V A12 N = TF 1 = k2 µ1 ρ1 jn ( ρ 2 )[ ρ 2 hn(1) ( ρ 2 )]′ − k2 µ1 ρ1hn(1) ( ρ 2 )[ ρ 2 jn ( ρ 2 )]′ , k2 µ1 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ − k1µ 2 ρ1hn(1) ( ρ1 )[ ρ 2 jn ( ρ 2 )]′ (5.45b) CM22 = TFH1 RPH1 k2 µ1 ρ1hn(1) ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′ − k1µ 2 ρ 2 hn(1) ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ , (5.45c) = TPH1 k1µ 2 ρ 2 j n ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ − k2 µ1 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ C N22 = TFV1 RPV1 k2 µ1 ρ 2 hn(1) ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ − k1µ 2 ρ1hn(1) ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′ = . (5.45d) TPV1 k1µ 2 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k2 µ1 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ 56 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere 57 5.3.3 COEFFICIENTS IN THE SPECIAL CASE OF METAMATERIAL Now let us consider the case that the sphere is metamaterial, i.e., k2 = − k1 and µ 2 = − µ1 . We consider the same case as before where current distribution is located outside the sphere. Because: (−1) k z jn ( z ) = ∑   k = 0 k !Γ ( n + k + 1)  2  ∞ n+2k , (5.46) we have  j (− z ), When n is even, jn ( z ) =  n  − jn (− z ), When n is odd. (5.47) Together with the properties of the Bessel functions, we have the following equations: 1  [ ρ1 jn ( ρ1 )]′ = jn ( ρ1 ) + ρ1[ jn ( ρ1 )]′ = jn ( ρ1 ) + ρ1  [ jn −1 ( ρ1 ) − jn +1 ( ρ1 )] , 2  (5.48a) 1  [ ρ 2 jn ( ρ 2 )]′ = jn ( ρ 2 ) + ρ 2 [ jn ( ρ 2 )]′ = jn ( ρ 2 ) + ρ 2  [ jn −1 ( ρ 2 ) − jn +1 ( ρ 2 )] , 2  (5.48b) where ρ1 and ρ 2 have been given in equation (5.8). Then, it is easy to get the results shown below. When n is even, n + 1 and n − 1 are both odd, we have: [ ρ 2 jn ( ρ 2 )]′ = [ ρ1 jn ( ρ1 )]′. (5.49a) When n is odd, then n + 1 and n − 1 are both even, we have: [ ρ 2 j n ( ρ 2 )]′ = −[ ρ1 j n ( ρ1 )]′ . (5.49b) Based on the properties of the Hankel function. 57 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere 58 We have: hn(1) ( z ) = j− n ( z ) − jn ( z ) e − iπ n i sin ( nπ ) ( z < ∞, arg z < π ). (5.50) Because j− n ( z ) = ( −1) jn ( z ) , n (5.51) the following can be obtained: hn(1) ( ρ 2 ) = hn(1) ( ρ1 ) , when n is even, (5.52a)  j ( ρ ) − jn ( ρ 2 ) e− iπ n ′  j− n ( ρ 2 ) ′ −  jn ( ρ 2 ) ′ e − iπ n [hn(1) ( ρ 2 )]′ =  − n 2  = i sin ( nπ ) i sin ( nπ )    j ( ρ ) − jn ( ρ1 ) e = − −n 1 i sin ( nπ )  − iπ n ′ (1)  = −[hn ( ρ1 )]′,  [ ρ 2 hn(1) ( ρ 2 )]′ = hn(1) ( ρ 2 ) + ρ 2 [hn(1) ( ρ 2 )]′ = hn(1) ( ρ1 ) + ρ1[hn(1) ( ρ1 )]′ = [ ρ1hn(1) ( ρ1 )]′; hn( ) ( ρ 2 ) = 1 when n is odd, j− n ( ρ1 ) + jn ( ρ1 ) e −iπ n i sin ( nπ ) , (5.52b) (5.52c) (5.53a)  j− n ( ρ 2 ) − jn ( ρ 2 ) e− iπ n ′  j− n ( ρ 2 ) ′ −  jn ( ρ 2 ) ′ e − iπ n ′ [hn ( ρ 2 )] =   = i sin ( nπ ) i sin ( nπ )   (5.53b) − iπ n ′  − j ( ρ ) − jn ( ρ1 ) e  =  −n 1  , i sin ( nπ )   (1) (1) [ ρ 2 hn ( ρ 2 )]′ = hn(1) ( ρ 2 ) + ρ 2 [hn(1) ( ρ 2 )]′ = j− n ( ρ1 ) + jn ( ρ1 ) e − iπ n i sin ( nπ )  j− n ( ρ1 ) + jn ( ρ1 ) e− iπ n ′ + ρ1   . i sin ( nπ )   (5.53c) 58 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere 59 22 22 After applying above equations to the expressions of AM12 , A12 N , CM , C N , the results of the coefficients in the special case can be obtained. For AM12 , when n is even: − k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ A = − k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ 12 M hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ , = − (1) hn ( ρ1 )[ ρ1 j n ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ (5.54a) when n is odd: AM12 = k1µ1 ρ1hn(1) ( ρ 2 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′ k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ h (1) ( ρ )[ ρ j ( ρ )]′ − jn ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′ , = n(1) 2 1 n 1 hn ( ρ1 )[ ρ1 j n ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ (5.54b) where [ ρ 2 hn(1) ( ρ 2 )]′ and hn(1) ( ρ 2 ) are expressed in equations (5.53c) and (5.53a). For A12 N , when n is even: AN12 = − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ + k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ + k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ j ( ρ )[ ρ h (1) ( ρ )]′ − hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ , = − n 1 1 n(1) 1 jn ( ρ1 )[ ρ1hn ( ρ1 )]′ + hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ (5.55a) when n is odd: A12 N = k1µ1 ρ1 jn ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′ − k1µ1 ρ1hn(1) ( ρ 2 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ − k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ j ( ρ )[ ρ h(1) ( ρ )]′ − hn(1) ( ρ 2 )[ ρ1 jn ( ρ1 )]′ , = − n 1 2 n(1) 2 jn ( ρ1 )[ ρ1hn ( ρ1 )]′ + hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ (5.55b) 59 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere 60 where [ ρ 2 hn(1) ( ρ 2 )]′ and hn(1) ( ρ 2 ) are expressed in equations (5.53c) and (5.53a). It is obvious that AM12 = A12 N , (5.56) wherever n is even or odd. For CM22 , when n is even: CM22 = −k1µ1 ρ1hn(1) ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ − k1µ1 ρ1hn(1) ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ k1µ1 ρ1 j n ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ + k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ 2hn(1) ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ ; =− j n ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ + hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ (5.57a) when n is odd: CM22 = −k1µ1 ρ1hn(1) ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′ − k1µ1 ρ1hn(1) ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ −k1µ1 ρ1 j n ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ − k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ h(1) ( ρ )[ ρ h(1) ( ρ )]′ + hn(1) ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ = n 1 2 n(1) 2 , j n ( ρ1 )[ ρ1hn ( ρ1 )]′ + hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ (5.57b) where [ ρ 2 hn(1) ( ρ 2 )]′ and hn(1) ( ρ 2 ) are expressed in equations (5.53c) and (5.53a). For C N22 , when n is even: C 22 N k1µ1 ρ1hn(1) ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ + k1µ1 ρ1hn(1) ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ . = − k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ 2hn(1) ( ρ1 )[ ρ1hn(1) ( ρ1 )]′]′ ; = − (1) hn ( ρ1 )[ ρ1 j n ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ (5.58a) when n is odd: 60 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere C N22 = k1µ1 ρ1hn(1) ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ + k1µ1 ρ1hn(1) ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′ k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ h(1) ( ρ )[ ρ h(1) ( ρ )]′ + hn(1) ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′ = n (1) 2 1 n 1 , hn ( ρ1 )[ ρ1 j n ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ 61 (5.58b) where [ ρ 2 hn(1) ( ρ 2 )]′ and hn(1) ( ρ 2 ) are expressed in equations (5.53c) and (5.53a). It is obvious that CM22 = CN22 , (5.59) wherever n is even or odd. 5.3.4 CALCULATION OF THE ELECTRICAL FIELD Rectangular coordinates can be transformed to spherical coordinates as: xˆ = sin θ cos φ rˆ + cos θ cos φθˆ − sin φφˆ . (5.60) For the infinitesimal horizontal electric dipole introduced in the beginning of 5.3, because θ ′ = 0, φ ′ = 0 , we have: xˆ = θˆ . (5.61) Because  n(n + 1) , m =1 mpnm (cos θ )  , = 2 lim θ →0 sin θ  0, otherwise (5.62a)  n(n + 1) , m =1 dpnm (cos θ )  . = 2 θ →0 dθ  0, otherwise (5.62b) lim Applying m = 1 to equation (5.39), we have: 61 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere Cmn = 2 × 2n + 1 1 × . n(n + 1) n(n + 1) 62 (5.63) From three equations above, the following equation will be obtained: mpnn (cos θ ) dp m (cosθ ) 2n + 1 . = Cmn × lim n = θ →0 θ → 0 sin θ dθ n(n + 1) Cmn × lim (5.64) Additionally, we have: dV ′ = R′2 dR′ sin θ ′dθ ′dφ ′. (5.65) Now the electrical fields will be calculated in three cases. For R > a : E1 (R ) = iωµ1 ∫∫∫ G e1 (R, R ′)iJ (R ′)dV ′ = iωµ1cG e1 (R , R ′)i xˆ = iωµ1cG e1 (R , R ′)iθˆ, (5.66) then derived from equations (3.10a), (3.10b) and (5.40a), the electric field can be obtained as: E1 (R ) = −  [ k2 rjn (k2 r )]′ A12 N (1) (k )  . k2ωµ1c ∞ 2n + 1  12 (1) j (0) A ( k ) lim i M +  ∑ 1 1  n M o1n N e1n r →0 4π n =1 n(n + 1)  k2 r   (5.67) Because we still have:  1, n = 0 jn (0) =  , 0, otherwise  2 , n =1 [k2 rjn (k2 r )]′  lim , = 3 r →0 k2 r 0, otherwise (5.68a) (5.68b) the electric field can be then represented as follows: E1 (R ) = − k2ωµ1c 12 (1)  AN N e11 (k1 )  . 4π  (5.69) In the metamaterial case, the above expression can be written as: 62 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere E1 (R ) = k1ωµ1c 12 (1)  AN N e11 (k1 )  . 4π  63 (5.70) We use the expressions of M e mn ( k ) and N e mn ( k ) , i.e., equations (3.10a) and (3.10b), o o together with the properties of the Hankel functions as given in equations (5.52a), (5.52c), (5.53a) and (5.53c), the following can be then derived: when n is even, M (1) ( k ) = M (1)e mn ( −k ) , e mn (5.71a) N (1) ( k ) = −N (1)e mn ( −k ) , e mn (5.71b) o o o o when n is odd, equations (5.53a) and (5.53c) should be applied. For R < a : E 2 (R ) = iωµ 2 ∫∫∫ G e 2 (R, R ′)iJ (R ′)dV ′ = iωµ 2 cG e 2 (R , R′)i xˆ = iωµ 2 cG e 2 (R , R′)iθˆ, (5.72) then derived from equations (3.10a), (3.10b) and (5.40b), the electric field can be obtained as: E2 (R ) = − k2ωµ 2 c ∞ 2n + 1 22  i jn (0) M (1) ∑ o1n ( k 2 ) + CM M o1n ( k 2 )  + 4π n =1 n(n + 1) { [ k rj (k r )]′  N (1) (k lim 2 n 2 r →0 k2 r  e1n    + C k N ) ( ) e1n 2 2  .  (5.73) 22 N Applying the equation (5.68a) and (5.68b) to (5.73), the following result can be given: E2 (R ) = − k2ωµ 2 c (1)  N e11 (k2 ) + CN22 N e11 (k2 )  . 4π (5.74) In the metamaterial case, the above expression can be written as: E 2 (R ) = k1ωµ 2 c (1)  N e11 (− k1 ) + CN22 N e11 (− k1 )  . 4π  (5.75) 63 Characterization of Spherical Metamaterials Chapter 5 Metamaterial Sphere 64 We use the expressions of N e mn ( k ) , i.e., equations (3.10b), together with the properties o of the Bessel functions given in equations (5.47), (5.49a) and (5.49b), the following can then be derived: when n is even, N e mn ( k ) = − N e mn ( −k ) , (5.76a) when n is odd, N e mn ( k ) = N e mn ( −k ) . (5.76b) o o o o 64 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell 65 6 METAMATERIAL SPHERICAL SHELL 6.1 INTRODUCTION The geometry of the spherically two-layered medium is considered as a single sphere with a coating layer superimposed by an unbounded homogeneous medium. In this chapter, the inner sphere is air. The radius of the outer sphere is a1 , the radius of the inner sphere is a2 . The two-layered sphere (spherical shell) under two different circumstances will be studied. One is for the current distribution located outside the spherical shell, the other is for the current distribution located in the center of the spherical shell. In each instance, the general case will be first analyzed, and then the special case: metamaterial sphere shell. 6.2 CURRENT DISTRIBUTION LOCATED OUTSIDE THE SPHERICAL SHELL 6.2.1 INTRODUCTION The spherical shell is centered at (0, 0, 0). An infinitesimal horizontal electric dipole with the current moment c pointes in the x -direction and locates at R ′ = b,θ ′ = 0, φ ′ = 0, which is at the top of the spherical shell. Thus we have: J (R′) = c δ ( R′ − b)δ (θ ′ − 0)δ (φ ′ − 0) xˆ. b 2 sin θ ′ (6.1) 65 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell 66 Scattering dyadic Green’s functions can be expressed generally as in equation (3.12). Applying this equation to a special case where N = 3 and s = 1, we can obtain the scattering DGFs of Region 1 (outside the spherical shell), Region 2 (between the outer sphere and the inner sphere) and Region 3 (inside the inner sphere): ′ G11 es ( R , R ) = G es21 (R, R′) = ik1 4π ∑C ik1 4π ∑ C { B mn m,n (1) ′(1) (k1 )  ,  BM11M (1) (k1 )M′(1) (k1 ) + B11 N N ( k1 ) N M (1) (k2 )M′(1) (k1 ) + BN21N (1) (k2 )N′(1) (k1 )  21 M mn m,n } (6.2a) (6.2b) +  D M (k2 )M′ (k1 ) + D N (k2 )N′ (k1 )  , 21 M ′ G 31 es ( R , R ) = (1) ik1 4π 21 N ∑C m,n mn (1) (1)  DM31M (1) (k3 )M′(1) (k1 ) + DN31N (1) (k3 )N′(1) (k1 )  (6.2c) where Cmn = (2 − δ 0 ) 2n + 1 (n − m)! , n(n + 1) (n + m)! (6.3) 21 21 21 21 31 31 and the coefficients BM11 , B11 N , BM , BN , DM , DN , DM , DN need to be solved for. 6.2.2 THE CALCULATION OF THE COEFFICIENTS The electric dyadic Green functions of Region 1-3 are presented as follows respectively: ′ G e1 (R, R′) = G e 0 (R, R ′) + G11 es ( R , R ), (6.4a) Ge2 (R, R′) = Ges21(R, R′), (6.4b) ′ Ge3(R, R′) = G31 es (R, R ), (6.4c) where the electric free space DGF is: 66 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell 67 when R > b G e 0 (R, R′) = − 1 ˆˆ ik RRδ (R − R ′) + 1 2 k1 4π ∑C mn M (1) (k1 )M′(k1 ) + N (1) (k1 )N′(k1 ) , (6.5a) 1 ˆˆ ik RRδ (R − R ′) + 1 2 k1 4π ∑C mn M (k1 )M′(1) (k1 ) + N(k1 )N′(1) (k1 ) , (6.5b) m,n when R < b G e 0 (R, R′) = − m,n where M e mn and N e mn are given in equations (3.10a) and (3.10b). o o The equations introduced in Chapter 3 will be used to obtain the coefficients of scattering dyadic Green’s functions. Applying N = 3 and s = 1 to equations (3.16i) and (3.17), we have: 31 11 BMN = DMN = 0,  1  f +1)1 (  BMN   TFfHV  ( f +1)1  =  HV  DMN   RPf  TPfHV RFfHV   f1 TFfHV   BMN  ⋅   Df1 +δ1 ,  f  1   MN HV TPf  (6.6a) (6.6b) where TFfHV , TPfHV , RFfHV , RPfHV are determined from equations (3.16a) to (3.16h), f will be either 1 or 2. For f = 2, equation (6.6b) can be rewritten as follows: 31 BMN = (6.7a) RPHV2 21 1 21 BMN + HV ( DMN + 0) , HV TP 2 TP 2 (6.7b) RFHV1 11 + 1) , ( DMN TFHV 1 (6.7c) TFHV 2 31 DMN = for f = 1, RFHV2 ( DMN21 + 0 ) , TFHV 2 1 21 BMN = 1 TFHV 1 21 BMN + 11 BMN + 67 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell 21 DMN = RPHV 1 11 11 1 BMN + HV ( DMN + 1) . HV TP1 TP1 68 (6.7d) From the above four equations together with equation (6.6a), the following equations will be derived: RFH2,V TFH1 ,V + RFH1,V TPH1 ,V , TPH1 ,V + TFH1 ,V RPH1,V RFH2,V BM11, N = − BM21, N = DM31, N = RFH2,V RPH1,V RFH1,V − RFH2,V , TPH1 ,V + TFH1 ,V RPH1,V RFH2,V 1 H ,V P2 T DM21, N =  RPH2,V BM21, N + DM21, N  , 1 H ,V P1 T  RPH1,V BM11, N + 1 , (6.8a) (6.8b) (6.8c) (6.8d) where RFH2 = µ2 k3∂ℑ32 ℑ22 − µ3k2∂ℑ22 ℑ32 , µ2 k3∂ℑ32 22 − µ3k2 ℑ32 ∂ 22 (6.9a) RFV 2 = µ2 k3ℑ32 ∂ℑ22 − µ3k2 ℑ22 ∂ℑ32 , µ2 k3 ℑ32 ∂ 22 − µ3k2∂ℑ32 22 (6.9b) TFH1 = µ1k2 (∂ℑ21 21 − ℑ21∂ 21 ) , µ1k2 ∂ℑ21 11 − µ2 k1ℑ21∂ 11 (6.9c) TFV1 = µ1k2 (ℑ21∂ 21 − ∂ℑ21 21 ) , µ1k2 ℑ21∂ 11 − µ2 k1∂ℑ21 11 (6.9d) RFH1 = µ1k2∂ℑ21ℑ11 − µ2 k1∂ℑ11ℑ21 , µ1k2 ∂ℑ21 11 − µ2 k1ℑ21∂ 11 (6.9e) RFV1 = µ1k2 ℑ21∂ℑ11 − µ2 k1ℑ11∂ℑ21 , µ1k2 ℑ21∂ 11 − µ2 k1∂ℑ21 11 (6.9f) 68 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell 69 TPH1 = µ1k2 (ℑ21∂ 21 − ∂ℑ21 21 ) , µ1k2 ℑ11∂ 21 − µ2 k1∂ℑ11 21 (6.9g) TPV1 = µ1k2 (∂ℑ21 21 − ℑ21∂ 21 ) , µ1k2 ∂ℑ11 21 − µ2 k1ℑ11∂ 21 (6.9h) RPH1 = µ1k2 ∂ 21 µ1k2 ℑ11∂ , (6.9i) , (6.9j) RPV1 = µ1k2 21∂ µ1k2 ∂ℑ11 − µ2 k1∂ 11 21 − µ 2 k1∂ℑ11 11 − µ2 k1 11∂ 21 − µ 2 k1ℑ11∂ 11 21 21 21 21 TPH2 = µ2 k3 (ℑ32 ∂ 32 − ∂ℑ32 32 ) , µ2 k3 ℑ22 ∂ 32 − µ3k2 ∂ℑ22 32 (6.9k) TPV2 = µ2 k3 (∂ℑ32 32 − ℑ32 ∂ 32 ) , µ2 k3∂ℑ22 32 − µ3k2 ℑ22 ∂ 32 (6.9l) RPH2 = µ2 k3∂ 32 µ2 k3ℑ22 ∂ RPV 2 = µ2 k3 32 ∂ µ2 k3∂ℑ22 − µ3k2∂ 22 32 − µ3 k 2 ∂ℑ22 22 − µ3k2 22∂ 32 − µ3 k 2 ℑ22 ∂ 22 32 , (6.9m) , (6.9n) 32 32 32 where ℑ32 = jn (k3 a2 ), ℑ22 = jn (k2 a2 ), ℑ21 = jn (k2 a1 ), ℑ11 = jn (k1a1 ), 32 = hn(1) (k3a2 ), ∂ℑ32 = 22 = hn(1) (k2 a2 ), 1 d [ ρ jn ( ρ ) ] , ρ dρ ρ =k a 21 = hn(1) (k2 a1 ), ∂ℑ22 = 3 2 ∂ℑ21 = 1 d [ ρ jn ( ρ ) ] , ρ dρ ρ =k a 2 1 11 = hn(1) (k1a1 ), (6.10a) (6.10b) 1 d [ ρ jn ( ρ ) ] , ρ dρ ρ =k a (6.10c) 1 d [ ρ jn ( ρ ) ] , ρ dρ ρ =k a (6.10d) 2 2 ∂ℑ11 = 1 1 69 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell ∂ 32 = (1) 1 d  ρ hn ( ρ )  ρ dρ (1) 1 d  ρ hn ( ρ )  ∂ 21 = ρ dρ , ∂ 22 = ρ = k3 a2 , ρ = k2 a1 70 (1) 1 d  ρ hn ( ρ )  ρ dρ (1) 1 d  ρ hn ( ρ )  ∂ 11 = ρ dρ , (6.10e) . (6.10f) ρ = k2 a2 ρ = k1a1 Because the structure is a spherical shell, we have additionally: ε 3 = ε1 , µ3 = µ1 , k3 = k1. (6.11) The same results can be obtained from equation (3.19a): CMf 1, N = TMf −, N1 ⋅⋅⋅ TM1 , N ( C Mf 1, N + A 22 ) . 6.2.3 COEFFICIENTS IN THE SPECIAL CASE OF METAMATERIAL Now let us consider the metamaterial case, that is, k2 = − k1 = − k3 , µ2 = − µ1 = − µ3 . (6.12) The equations (6.9a) to (6.9n) can be written as: RFH2 = RFV 2 = − µ1k1∂ℑ32 ℑ22 + µ1k1∂ℑ22 ℑ32 ∂ℑ32 ℑ22 − ∂ℑ22 ℑ32 = , − µ1k1∂ℑ32 22 + µ1k1ℑ32 ∂ 22 ∂ℑ32 22 − ℑ32 ∂ 22 − µ1k1ℑ32 ∂ℑ22 + µ1k1ℑ22 ∂ℑ32 ℑ32 ∂ℑ22 − ℑ22 ∂ℑ32 = , − µ1k1ℑ32 ∂ 22 + µ1k1∂ℑ32 22 ℑ32 ∂ 22 − ∂ℑ32 22 TFH1 = TFV1 = − µ1k1 (∂ℑ21 21 − ℑ21∂ 21 ) ∂ℑ21 = − µ1k1∂ℑ21 11 + µ1k1ℑ21∂ 11 ∂ℑ21 − µ1k1 (ℑ21∂ 21 − ∂ℑ21 21 ) ℑ21∂ = − µ1k1ℑ21∂ 11 + µ1k1∂ℑ21 11 ℑ21∂ − ℑ21∂ 11 − ℑ21∂ 21 − ∂ℑ21 11 − ∂ℑ21 21 21 (6.13a) (6.13b) , (6.13c) , (6.13d) 11 21 11 70 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell 71 RFH1 = − µ1k1∂ℑ21ℑ11 + µ1k1∂ℑ11ℑ21 ∂ℑ21ℑ11 − ∂ℑ11ℑ21 , = − µ1k1∂ℑ21 11 + µ1k1ℑ21∂ 11 ∂ℑ21 11 − ℑ21∂ 11 (6.13e) RFV1 = − µ1k1ℑ21∂ℑ11 + µ1k1ℑ11∂ℑ21 ℑ21∂ℑ11 − ℑ11∂ℑ21 = , − µ1k1ℑ21∂ 11 + µ1k1∂ℑ21 11 ℑ21∂ 11 − ∂ℑ21 11 (6.13f) TPH1 = − µ1k1 (ℑ21∂ 21 − ∂ℑ21 21 ) ℑ ∂ = 21 − µ1k1ℑ11∂ 21 + µ1k1∂ℑ11 21 ℑ11∂ TPV1 = RPH1 = RPV1 = TPH2 = TPV2 = RPH2 = RPV 2 = − ∂ℑ21 21 − ∂ℑ11 21 − µ1k1 (∂ℑ21 21 − ℑ21∂ 21 ) ∂ℑ21 = − µ1k1∂ℑ11 21 + µ1k1ℑ11∂ 21 ∂ℑ11 − ℑ21∂ 21 − ℑ11∂ − µ1k1∂ 21 − µ1k1ℑ11∂ ∂ 21 ℑ11∂ − ∂ 11 21 − ∂ℑ11 ∂ − 11∂ 21 − ℑ11∂ − µ1k1 21∂ − µ1k1∂ℑ11 + µ1k1∂ 11 21 + µ1k1∂ℑ11 11 21 = 21 + µ1k1 11∂ 21 + µ1k1ℑ11∂ 11 21 21 = 21 21 ∂ℑ11 11 11 − µ1k1 (ℑ32 ∂ 32 − ∂ℑ32 32 ) ℑ ∂ = 32 − µ1k1ℑ22 ∂ 32 + µ1k1∂ℑ22 32 ℑ22 ∂ − ∂ℑ32 32 − ∂ℑ22 − µ1k1 (∂ℑ32 32 − ℑ32 ∂ 32 ) ∂ℑ = 32 − µ1k1∂ℑ22 32 + µ1k1ℑ22 ∂ 32 ∂ℑ22 − ℑ32 ∂ 32 − ℑ22 ∂ − µ1k1∂ 32 − µ1k1ℑ22 ∂ − µ1k1 32 ∂ − µ1k1∂ℑ22 + µ1k1∂ 22 32 + µ1k1∂ℑ22 22 + µ1k1 22 ∂ 32 + µ1k1ℑ22 ∂ 22 32 = 32 32 32 = ∂ 32 ℑ22∂ 32 ∂ ∂ℑ22 , (6.13g) , (6.13h) , (6.13i) , (6.13j) 21 21 21 21 21 21 21 21 32 32 − ∂ 22 32 − ∂ℑ22 22 − 22∂ 32 − ℑ22 ∂ 22 32 , (6.13k) , (6.13l) , (6.13m) , (6.13n) 32 32 32 32 32 32 32 where ℑ32 = jn (k1a2 ), ℑ22 = jn (−k1a2 ), ℑ21 = jn (−k1a1 ), ℑ11 = jn (k1a1 ), 32 = hn(1) (k1a2 ), 22 = hn(1) (−k1a2 ), ∂ℑ32 = [ ρ jn ( ρ ) ]′ , ρ = k1a2 21 = hn(1) (− k1a1 ), 11 ∂ℑ22 = [ ρ jn ( ρ )]′ (6.14a) = hn(1) (k1a1 ), (6.14b) , (6.14c) ρ =− k1a2 71 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell ∂ℑ21 = [ ρ jn ( ρ ) ]′ ∂ ∂ 32 21 =  ρ hn(1) ( ρ ) ′ =  ρ hn(1) ( ρ ) ′ 72 ∂ℑ11 = [ ρ jn ( ρ ) ]′ , ρ =− k1a1 ∂ , 22 ρ = k1a2 ∂ , ρ =− k1a1 =  ρ hn(1) ( ρ ) ′ 11 (6.14d) , ρ = k1a1 , (6.14e) . (6.14f) ρ =− k1a2 =  ρ hn(1) ( ρ ) ′ ρ = k1a1 We still have the following results.  j (− z ), When n is even, jn ( z ) =  n  − jn (− z ), When n is odd. (6.15a) When n is even: [ ρ 2 jn ( ρ 2 )]′ = [ ρ1 jn ( ρ1 )]′ ρ =− ρ , 1 (6.15b) 2 when n is odd: [ ρ 2 jn ( ρ 2 )]′ = − [ ρ1 jn ( ρ1 )]′ ρ =− ρ . 1 (6.15c) 2 When n is even: hn(1) ( ρ 2 ) = hn(1) ( ρ1 ) ρ1 =− ρ2 [ ρ 2 hn(1) ( ρ 2 )]′ = [ ρ1hn(1) ( ρ1 )]′ , ρ1 =− ρ 2 (6.15d) (6.15e) ; when n is odd: hn(1) ( ρ 2 ) = (1) [ ρ 2 hn ( ρ 2 )]′ = j− n ( ρ1 ) + jn ( ρ1 ) e − iπ n i sin ( nπ ) j− n ( ρ1 ) + jn ( ρ1 ) e− iπ n i sin ( nπ ) (6.15f) , ρ1 =− ρ 2  j− n ( ρ1 ) + jn ( ρ1 ) e− iπ n ′ + ρ1   i sin ( nπ )   . (6.15h) ρ1 =− ρ 2 72 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell 73 It is obvious that we have the following results in this special case:  RFH2 = RFV 2 , TFH1 = TFV1 , RFH1 = RFV1 , TPH1 = TPV1 ,  RPH1 = RPV1 , TPH2 = TPV2 , RPH2 = RPV 2 ,  (6.16) hence,  BM11 = B11 BM21 = BN21 , N ,  31 31 21 21  DM = DN , DM = DN . (6.17) 6.2.4 CALCULATION OF THE ELECTRICAL FIELD Rectangular coordinates can be transformed to spherical coordinates as: xˆ = sin θ cos φ rˆ + cos θ cos φθˆ − sin φφˆ, (6.18) For the infinitesimal horizontal electric dipole introduced in the beginning of 6.2, because θ ′ = 0, φ ′ = 0 , we have: xˆ = θˆ . (6.19) Because  n(n + 1) , m =1 mpnm (cos θ )  , = 2 lim θ →0 sin θ  0, otherwise (6.20a)  n(n + 1) , m =1 dpnm (cos θ )  = 2 , lim θ →0 dθ  0, otherwise (6.20b) Apply m = 1 to equation (6.3), we have: Cmn = 2 × 2n + 1 1 . × n(n + 1) n(n + 1) (6.21) From the above three equations, the following equation can be obtained: 73 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell mpnn (cos θ ) dp m (cos θ ) 2n + 1 = Cmn × lim n = . θ →0 θ → 0 sin θ dθ n(n + 1) Cmn × lim 74 (6.22) Additionally, we have: dV ′ = R′2 dR′ sin θ ′dθ ′dφ ′. (6.23) Now the electrical fields will be calculated of each region. When R > b : E1 (R ) = iωµ1 ∫∫∫ G e1 (R, R ′)iJ (R ′)dV ′ = iωµ1cG e1 (R, R ′)i xˆ = iωµ1cG e1 (R , R ′)iθˆ, (6.24) together with equation (6.5a), then we have: E1 (R ) = − k1ωµ1c ∞ 2n + 1 i  jn ( ρb ) + BM11 hn(1) ( ρb )  M (1) ∑ o1n ( k1 ) 4π n =1 n(n + 1) {   ′  (1) (1) + ( ρb jn ( ρb ) )′ + B11 N ( ρ b hn ( ρ b ) )  N e1n ( k1 ) ρ b  ,    (6.25) where ρb = k1b. When a1 < R < b : E 2 (R ) = iωµ1 ∫∫∫ G e1 (R, R ′)iJ (R ′)dV ′ = iωµ1cG e1 (R , R ′)i xˆ = iωµ1cG e1 (R , R ′)iθˆ, (6.26) together with equation (6.5b), then we have: E 2 (R ) = − k1ωµ1c ∞ 2n + 1  i hn(1) ( ρb ) M o1n (k1 ) + BM11M (1) ∑ o1n ( k1 )  4π n =1 n(n + 1) {  (1)  + ( ρ h ( ρb ) )′  N e1n (k1 ) + B11 N N e1n ( k1 )  ρ b  .  (6.27) (1) b n When a2 < R < a1 : E3 (R ) = iωµ 2 ∫∫∫ G e 2 (R, R ′)iJ (R ′)dV ′ = iωµ2 cG e 2 (R , R ′)i xˆ = iωµ 2 cG e 2 (R , R′)iθˆ, (6.28) applying the related equations to the above equation, we have: 74 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell E3 ( R ) = − k1ωµ2 c ∞ 2n + 1 i hn(1) ( ρb )  DM21M o1n (k2 ) + BM21M o(1)1n (k2 )  ∑ 4π n =1 n(n + 1) 75 {   + ( ρ h ( ρb ) )′  DN21N e1n (k2 ) + BN21N (1) e1n ( k 2 )  ρ b  .  (6.29) (1) b n In the metamaterial case, the above expression can be written as: E3 ( R ) = k1ωµ1c ∞ 2n + 1  i hn(1) ( ρb )  DM21M o1n (− k1 ) + BM21M (1) ∑ o1n ( − k1 )  4π n =1 n(n + 1) {   + ( ρ h ( ρb ) )′  DN21N e1n (−k1 ) + BN21N (1) e1n ( − k1 )  ρ b  .  (6.30) (1) b n We use the expressions of M e mn ( k ) and N e mn ( k ) , i.e., equations (3.10a) and (3.10b), o o together with the properties of the Bessel functions and the Hankel functions given in equations (6.15a)-(6.15h), the following can then be derived: when n is even, M e mn ( k ) = M e mn ( −k ) , (6.31a) N e mn ( k ) = − N e mn ( −k ) , (6.31b) M (1) ( k ) = M (1)e mn ( −k ) , e mn (6.31c) N (1) ( k ) = −N (1)e mn ( −k ) , e mn (6.31d) M e mn ( k ) = −M e mn ( −k ) , (6.31e) N e mn ( k ) = N e mn ( −k ) . (6.31f) o o o o o o when n is odd, o o o o o o Equations (6.15f) and (6.15h) should also be applied in the odd case. When R < a2 : E 4 (R ) = iωµ3 ∫∫∫ G e 3 (R, R ′)iJ (R′)dV ′ = iωµ3cG e3 (R , R′)i xˆ = iωµ3cG e 3 (R , R ′)iθˆ, (6.32) 75 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell 76 applying the related equations to the above equation, we have: E 4 (R ) = − k1ωµ3c ∞ 2n + 1  (1)  ihn ( ρb ) DM31M o1n (k3 ) + ( ρb hn(1) ( ρb ) )′ DN31N e1n (k3 ) ρb  . (6.33) ∑ 4π n =1 n(n + 1)   Because k1 = k3 , the above expression can be written as: E 4 (R ) = − 6.3 k1ωµ1c ∞ 2n + 1  (1)  i hn ( ρb ) DM31M o1n (k1 ) + ( ρb hn(1) ( ρb ) )′ DN31N e1n (k1 ) ρb  . (6.34) ∑ 4π n =1 n(n + 1)   CURRENT DISTRIBUTION LOCATED INSIDE THE SPHERICAL SHELL 6.3.1 INTRODUCTION The spherical shell is centered at (0, 0, 0). An infinitesimal horizontal electric dipole with a current moment c pointing in the x -direction and located at R′ = 0,θ ′ = 0, φ ′ = 0 (the center of the spherical shell), let: J (R′) = c δ ( R′ − 0)δ (θ ′ − 0)δ (φ ′ − 0) xˆ. R′2 sin θ ′ (6.35) Scattering dyadic Green’s functions can be expressed generally in equation (3.12). Applying this equation to special case where N = 3 and s = 3, we can obtain the scattering DGFs of Region 1 (outside the spherical shell), Region 2 (between the outer sphere and the inner sphere) and Region 3 (inside the inner sphere): ′ G13 es ( R , R ) = G es23 (R, R′) = ik3 4π ik3 4π ∑C mn m ,n  AM13M (1) (k1 )M′(k3 ) + AN13 N (1) (k1 )N′(k3 )  , ∑ C { A 23 M mn m ,n M (1) (k2 )M′(k3 ) + AN23 N (1) (k2 )N′(k3 )  } (6.36a) (6.36b) + C M (k2 )M′(k3 ) + C N(k2 )N′(k3 )  , 23 M 23 N 76 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell ′ G 33 es ( R , R ) = ik3 4π ∑C mn m,n CM33M (k3 )M′(k3 ) + C N33 N(k3 )N′(k3 )  , 77 (6.36c) where Cmn = (2 − δ 0 ) 2n + 1 (n − m)! , n(n + 1) (n + m)! (6.37) and the coefficients AM13, N , AM23, N , CM23, N , CM33, N need to be solved for. 6.3.2 THE CALCULATION OF THE COEFFICIENTS The electric dyadic Green functions of Regions 1-3 are presented as follows respectively: ′ G e1 (R, R′) = G13 es ( R , R ), (6.38a) Ge2 (R, R′) = Ges23(R, R′), (6.38b) ′ Ge3(R, R′) = Ge0 (R, R′) +G33 es (R, R ), (6.38c) where the electric free space DGF is: because R > 0 G e 0 (R, R′) = − ik 1 ˆˆ RRδ (R − R ′) + 3 2 4π k3 ∑C m ,n mn M (1) (k3 )M′(k3 ) + N (1) (k3 )N′(k3 ) , (6.39) where M e mn and N e mn are given in equations (3.10a) and (3.10b), and Cmn is shown in o o equation (6.37). The equations introduced in Chapter 3 will be used to obtain the coefficients of scattering dyadic Green’s function. Applying N = 3 and s = 3 to equation (3.16i) and equation (3.17), we have: 33 13 AMN = CMN = 0, (6.40a) 77 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell  1  f +1)3 ( 3  AMN + δ f +1   TFfHV   =  HV ( f +1)3  CMN   RPf  TPfHV RFfHV   f3 TFfHV   AMN   ⋅ C f 3  , 1   MN  TPfHV  78 (6.40b) where TFfHV , TPfHV , RFfHV , RPfHV are determined from equation (3.16a) to equation (3.16h), and f will be either 1 or 2. For f = 2, equation (6.40b) can be rewritten as follows: AM33, N + 1 = CM33, N = for f = 1, AM23, N = CM23, N = 1 RFH2,V 23 CM , N , TFH2,V AM23, N + TFH2,V (6.41a) RPH2,V 23 1 AM , N + H ,V CM23, N , H ,V TP 2 TP 2 (6.41b) RFH1,V 13 CM , N , TFH1 ,V (6.41c) RPH1,V 13 1 AM , N + H ,V CM13, N . H ,V TP1 TP1 (6.41d) 1 TFH1 ,V AM13, N + From the above four equations together with equation (6.40a), the following coefficients will be derived: 13 M ,N A TFH1 ,V TFH2,V TPH1 ,V = H ,V , TP1 + TFH1 ,V RPH1,V RFH2,V 23 M ,N A = CM23, N = CM33, N = AM13, N TFH1 ,V , RPH1,V AM13, N TPH1 ,V (6.42a) (6.42b) , 1  RPH2,V AM23, N + CM23, N  , TPH2 ,V  (6.42c) (6.42d) where 78 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell 79 RFH2 = µ2 k3∂ℑ32 ℑ22 − µ3k2∂ℑ22 ℑ32 , µ2 k3∂ℑ32 22 − µ3k2 ℑ32 ∂ 22 (6.43a) RFV 2 = µ2 k3ℑ32 ∂ℑ22 − µ3k2 ℑ22 ∂ℑ32 , µ2 k3 ℑ32 ∂ 22 − µ3k2∂ℑ32 22 (6.43b) TFH1 = µ1k2 (∂ℑ21 21 − ℑ21∂ 21 ) , µ1k2 ∂ℑ21 11 − µ2 k1ℑ21∂ 11 (6.43c) TFV1 = µ1k2 (ℑ21∂ 21 − ∂ℑ21 21 ) , µ1k2 ℑ21∂ 11 − µ2 k1∂ℑ21 11 (6.43d) TFH2 = µ2 k3 (∂ℑ32 32 − ℑ32 ∂ 32 ) , µ2 k3∂ℑ32 22 − µ3k2 ℑ32 ∂ 22 (6.43e) TFV2 = µ2 k3 (ℑ32∂ 32 − ∂ℑ32 32 ) , µ2 k3ℑ32 ∂ 22 − µ3k2∂ℑ32 22 (6.43f) TPH1 = µ1k2 (ℑ21∂ 21 − ∂ℑ21 21 ) , µ1k2 ℑ11∂ 21 − µ2 k1∂ℑ11 21 (6.43g) TPV1 = µ1k2 (∂ℑ21 21 − ℑ21∂ 21 ) , µ1k2∂ℑ11 21 − µ2 k1ℑ11∂ 21 (6.43h) RPH1 = µ1k2 ∂ 21 µ1k2 ℑ11∂ , (6.43i) , (6.43j) TPH2 = µ2 k3 (ℑ32 ∂ 32 − ∂ℑ32 32 ) , µ2 k3ℑ22 ∂ 32 − µ3 k2 ∂ℑ22 32 (6.43k) TPV2 = µ2 k3 (∂ℑ32 32 − ℑ32 ∂ 32 ) , µ2 k3∂ℑ22 32 − µ3k2 ℑ22 ∂ 32 (6.43l) RPV1 = RPH2 = µ1k2 21∂ µ1k2 ∂ℑ11 µ2 k3∂ 32 µ2 k3ℑ22 ∂ − µ2 k1∂ 11 21 − µ 2 k1∂ℑ11 11 − µ2 k1 11∂ 21 − µ 2 k1ℑ11∂ 11 − µ3 k2∂ 22 32 − µ3 k 2 ∂ℑ22 22 21 21 21 21 32 , (6.43m) 32 79 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell RPV 2 = µ2 k3 32 ∂ µ2 k3∂ℑ22 − µ3 k2 22∂ 32 − µ3 k 2 ℑ22 ∂ 22 80 32 , (6.43n) 32 where ℑ32 = jn (k3 a2 ), ℑ22 = jn (k2 a2 ), ℑ21 = jn (k2 a1 ), ℑ11 = jn (k1a1 ), 32 = hn(1) (k3 a2 ), ∂ℑ32 = 22 = hn(1) (k2 a2 ), 1 d [ ρ jn ( ρ ) ] , ρ dρ ρ =k a 21 = hn(1) (k2 a1 ), ∂ℑ22 = 3 2 ∂ℑ21 = 1 d [ ρ jn ( ρ ) ] , ρ dρ ρ =k a ∂ 32 (1) 1 d  ρ hn ( ρ )  ∂ 21 = ρ dρ = hn(1) (k1a1 ), (6.44b) 1 d [ ρ jn ( ρ ) ] , ρ dρ ρ =k a (6.44c) 1 d [ ρ jn ( ρ ) ] , ρ dρ ρ =k a (6.44d) 2 2 ∂ℑ11 = 2 1 (1) 1 d  ρ hn ( ρ )  = ρ dρ 11 (6.44a) 1 1 , ∂ ρ = k3 a2 , ρ = k2 a1 22 (1) 1 d  ρ hn ( ρ )  = ρ dρ (1) 1 d  ρ hn ( ρ )  ∂ 11 = ρ dρ , (6.44e) , (6.44f) ρ = k2 a2 ρ = k1a1 Additionally, because the structure is a spherical shell, we have: ε 3 = ε1 , µ3 = µ1 , k3 = k1. (6.45) The same results can be obtained from equation (3.19c): CMfN, N = TMf −, N1 ⋅⋅⋅ TM1 , N C1MN, N − H ( f − N ) A11. 6.3.3 COEFFICIENTS IN THE SPECIAL CASE OF METAMATERIAL Now let us consider the metamaterial case, that is, k2 = − k1 = − k3 , µ2 = − µ1 = − µ3 . (6.46) The equations (6.43a) to (6.43n) can be written as: 80 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell RFH2 = RFV 2 = 81 − µ1k1∂ℑ32 ℑ22 + µ1k1∂ℑ22 ℑ32 ∂ℑ32 ℑ22 − ∂ℑ22 ℑ32 = , − µ1k1∂ℑ32 22 + µ1k1ℑ32 ∂ 22 ∂ℑ32 22 − ℑ32 ∂ 22 − µ1k1ℑ32 ∂ℑ22 + µ1k1ℑ22 ∂ℑ32 ℑ32 ∂ℑ22 − ℑ22 ∂ℑ32 = , − µ1k1ℑ32 ∂ 22 + µ1k1∂ℑ32 22 ℑ32 ∂ 22 − ∂ℑ32 22 TFH1 = TFV1 = TFH2 = TFV2 = TPH1 = TPV1 = RPH1 = RPV1 = TPH2 = TPV2 = RPH2 = − µ1k1 (∂ℑ21 21 − ℑ21∂ 21 ) ∂ℑ21 = − µ1k1∂ℑ21 11 + µ1k1ℑ21∂ 11 ∂ℑ21 − ℑ21∂ 11 − ℑ21∂ 21 − µ1k1 (ℑ21∂ 21 − ∂ℑ21 21 ) ℑ ∂ = 21 − µ1k1ℑ21∂ 11 + µ1k1∂ℑ21 11 ℑ21∂ − ∂ℑ21 11 − ∂ℑ21 − ℑ32 ∂ 22 − ℑ32 ∂ − µ1k1 (ℑ32 ∂ 32 − ∂ℑ32 32 ) ℑ ∂ = 32 − µ1k1ℑ32 ∂ 22 + µ1k1∂ℑ32 22 ℑ32 ∂ − ∂ℑ32 22 − ∂ℑ32 − µ1k1∂ 21 − µ1k1ℑ11∂ ∂ 21 ℑ11∂ − ∂ 11 21 − ∂ℑ11 ∂ − 11∂ 21 − ℑ11∂ − µ1k1 21∂ − µ1k1∂ℑ11 + µ1k1 11∂ 21 + µ1k1ℑ11∂ 11 21 = 21 21 21 = 21 ∂ℑ11 − µ1k1 (ℑ32 ∂ 32 − ∂ℑ32 32 ) ℑ ∂ = 32 − µ1k1ℑ22 ∂ 32 + µ1k1∂ℑ22 32 ℑ22 ∂ − µ1k1 (∂ℑ32 32 − ℑ32 ∂ 32 ) ∂ℑ = 32 − µ1k1∂ℑ22 32 + µ1k1ℑ22 ∂ 32 ∂ℑ22 − µ1k1∂ 32 − µ1k1ℑ22 ∂ + µ1k1∂ 22 32 + µ1k1∂ℑ22 22 32 32 = ∂ 32 ℑ22∂ 11 11 (6.47d) 32 , (6.47e) , (6.47f) 22 21 , (6.47g) , (6.47h) , (6.47i) , (6.47j) 21 21 21 21 21 21 − ∂ℑ32 32 − ∂ℑ22 32 − ℑ32 ∂ 32 − ℑ22 ∂ 32 − ∂ 22 32 − ∂ℑ22 22 , 22 − ∂ℑ21 21 − ∂ℑ11 − ℑ21∂ 21 − ℑ11∂ (6.47c) 32 21 21 , 11 32 − µ1k1 (∂ℑ21 21 − ℑ21∂ 21 ) ∂ℑ21 = − µ1k1∂ℑ11 21 + µ1k1ℑ11∂ 21 ∂ℑ11 + µ1k1∂ 11 21 + µ1k1∂ℑ11 21 32 − µ1k1 (ℑ21∂ 21 − ∂ℑ21 21 ) ℑ ∂ = 21 − µ1k1ℑ11∂ 21 + µ1k1∂ℑ11 21 ℑ11∂ (6.47b) 11 21 − µ1k1 (∂ℑ32 32 − ℑ32∂ 32 ) ∂ℑ = 32 − µ1k1∂ℑ32 22 + µ1k1ℑ32 ∂ 22 ∂ℑ32 11 21 (6.47a) 21 32 , (6.47k) , (6.47l) , (6.47m) 32 32 32 32 32 81 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell RPV 2 = − µ1k1 32 ∂ − µ1k1∂ℑ22 + µ1k1 22 ∂ 32 + µ1k1ℑ22 ∂ 22 32 32 = 32 ∂ ∂ℑ22 82 − 22∂ 32 − ℑ22 ∂ 22 32 , (6.47n) 32 where ℑ32 = jn (k1a2 ), ℑ22 = jn (−k1a2 ), ℑ21 = jn (−k1a1 ), ℑ11 = jn (k1a1 ), 32 = hn(1) (k1a2 ), 22 = hn(1) (−k1a2 ), ∂ℑ32 = [ ρ jn ( ρ ) ]′ ∂ℑ21 = [ ρ jn ( ρ ) ]′ ∂ ∂ 32 21 21 = hn(1) (− k1a1 ), ∂ℑ22 = [ ρ jn ( ρ )]′ , ρ = k1a2 =  ρ hn(1) ( ρ ) ′ , ∂ 22 ρ = k1a2 , ρ =− k1a1 = hn(1) (k1a1 ), (6.48b) , (6.48c) ρ =− k1a2 ∂ℑ11 = [ ρ jn ( ρ ) ]′ , ρ =− k1a1 =  ρ hn(1) ( ρ ) ′ 11 ∂ =  ρ hn(1) ( ρ ) ′ 11 (6.48a) , (6.48d) ρ = k1a1 , (6.48e) . (6.48f) ρ =− k1a2 =  ρ hn(1) ( ρ ) ′ ρ = k1a1 We still have the results represented in equation (6.15a)-(6.15h). It is obviously that we can get the following results in this special case:  RFH2 = RFV 2 , TFH1 = TFV1 , TFH2 = TFV2 , TPH1 = TPV1 ,  RPH1 = RPV1 , TPH2 = TPV2 , RPH2 = RPV 2 ,  (6.49) Therefore, we have  AM13 = AN13 , AM23 = AN23 ,  33 33 23 23 CM = CN , CM = C N . (6.50) 82 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell 83 6.3.4 CALCULATION OF THE ELECTRICAL FIELD Before we calculate the electric field, we should know some equations first. Rectangular coordinates can be transformed to spherical coordinates as: xˆ = sin θ cos φ rˆ + cos θ cos φθˆ − sin φφˆ . (6.51) Because θ ′ = 0, φ ′ = 0 , we have: xˆ = θˆ . (6.52) Because  n(n + 1) , m =1 mpnm (cos θ )  , lim = 2 θ →0 sin θ  0, otherwise (6.53a)  n(n + 1) , m =1 dpnm (cos θ )  , = 2 θ →0 dθ  0, otherwise (6.53b) lim Applying m = 1 to equation (6.37), we have: Cmn = 2 × 2n + 1 1 . × n(n + 1) n(n + 1) (6.54) From three equations as above, the following equation will be obtained: mpnn (cos θ ) dp m (cosθ ) 2n + 1 = Cmn × lim n = . θ →0 θ →0 sin θ dθ n(n + 1) Cmn × lim (6.55) Moreover, we have: dV ′ = R′2 dR′ sin θ ′dθ ′dφ ′. (6.56) Now the electrical fields will be calculated in three cases. For R > a1 : 83 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell E1 (R ) = iωµ1 ∫∫∫ G e1 (R, R ′)iJ (R ′)dV ′ = iωµ1cG e1 (R , R′)i xˆ = iωµ1cG e1 (R , R′)iθˆ, 84 (6.57) then derived from equations (3.10a), (3.10b) and (6.38a), the result can be obtained as: E1 (R ) = −  [ k3rjn (k3r )]′ A13N (1) (k )  . k3ωµ1c ∞ 2n + 1  13 (1) (0) ( ) lim + i M j A k  n ∑ M o1n 1 N e1n 1  r →0 4π n =1 n(n + 1)  k3 r   (6.58) Because we still have:  1, n = 0 jn (0) =  , 0, otherwise (6.59a)  2 , n =1 [k3rjn (k3r )]′  = 3 lim , r →0 k3 r 0, otherwise (6.59b) 2n + 1 3 = , n(n + 1) n =1 2 (6.59c) the electric field can be then represented as follows: E1 (R ) = − k3ωµ1c 13 (1)  AN N e11 (k1 )  . 4π  (6.60) Because k1 = k3 , the above expression can be written as: E1 (R ) = − k1ωµ1c 13 (1)  AN N e11 (k1 )  . 4π  (6.61) For a2 < R < a1 : E 2 (R ) = iωµ 2 ∫∫∫ G e 2 (R, R′)iJ (R′)dV ′ = iωµ 2 cG e 2 (R, R′)i xˆ = iωµ 2 cG e 2 (R, R′)iθˆ, (6.62) then derived from equations (3.10a), (3.10b) and (6.38b), the result can be obtained as: 84 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell E 2 (R ) = − k3ωµ2 c ∞ 2n + 1  i jn (0) CM23M o1n (k2 ) + AM23M (1) ∑ o1n ( k 2 )  + 4π n =1 n(n + 1) { k3 rjn (k3 r ) ]′ 23 [ C N lim r →0 85  k3 r N    (k2 )   . e1n ( k 2 ) + A N  23 N (6.63) (1) e1n Applying the equations (6.59a), (6.59b) and (6.59c) to (6.63), the following result can be given: E 2 (R ) = − k3ωµ2 c 23 CN N e11 (k2 ) + AN23 N (1)  e11 ( k 2 )  . 4π  (6.64) In the metamaterial case, the above expression can be written as: E 2 (R ) = k1ωµ1c 23 CN N e11 (−k1 ) + AN23N (1)  e11 ( − k1 )  . 4π (6.65) We use the expressions of N e mn ( k ) , i.e., equations (3.10b), together with the properties o of the Bessel function and the Hankel function given in equations (6.15a)-(6.15h), follows can then be derived: when n is even, N e mn ( k ) = −N e mn ( −k ) , o (6.66a) N (1) ( k ) = −N (1)e mn ( −k ) , e mn (6.66b) N e mn ( k ) = N e mn ( −k ) . (6.66c) o when n is odd, o o o o equations (6.15f) and (6.15h) should also be applied in the odd case. When R < a2 : E3 (R ) = iωµ3 ∫∫∫ G e 3 (R, R′)iJ (R′)dV ′ = iωµ3cG e3 (R, R′)i xˆ = iωµ3cG e 3 (R, R′)iθˆ, (6.67) Then derived from equations (3.10a), (3.10b) and (6.38c), we have: 85 Characterization of Spherical Metamaterials Chapter 6 Metamaterial Spherical Shell E3 ( R ) = − k3ωµ3c ∞ 2n + 1 i jn (0) M o(1)1n (k3 ) + CM33M o1n (k3 )  + ∑ 4π n =1 n(n + 1) { k3 rjn (k3 r ) ]′ (1) [  N (k ) + C 33N lim r →0 86 k3 r  e1n 3 N    e1n ( k3 )   .  (6.68) Applying the equations (6.59a), (6.59b) and (6.59c) to (6.68), we have: E3 ( R ) = − k3ωµ3c (1)  N e11 (k3 ) + CN33N e11 (k3 )  . 4π (6.69) Because k1 = k3 , the above expression can be written as: E3 ( R ) = − k1ωµ1c (1)  N e11 (k1 ) + CN33N e11 (k1 )  . 4π  (6.70) 86 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials 7 7.1 87 MULTI-SPHERICAL LAYERS OF METAMATERIALS INTRODUCTION In this chapter, the general case will be considered. The structure was represented in Figure 3.1. One should notes that there is air in the layers n, n − 2, n − 4, n − 6...... , therefore, the structure is a periodical structure or a spherical photonic bandgap structure. The multi-spherical layers under two different circumstances will be studied. One is for the current distribution located outside the spherical multilayers, the other is for the current distribution located in the center of the spherical multilayers. In each instance, the normal case will be first analyzed, and then the special case: spherical layers of metamaterials. 7.2 CURRENT DISTRIBUTION LOCATED OUTSIDE THE SPHERICAL MULTILAYERS 7.2.1 INTRODUCTION The spherical multilayers has a center at (0, 0, 0). An infinitesimal horizontal electric dipole with current moment c pointing in the x -direction and located at R ′ = b,θ ′ = 0, φ ′ = 0, which is at the top of the spherical multilayers. Thus we have: J (R′) = c δ ( R′ − b)δ (θ ′ − 0)δ (φ ′ − 0) xˆ. b 2 sin θ ′ (7.1) 87 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials 88 Scattering dyadic Green’s functions can be expressed generally in equation (3.12). Applying this equation to the case where s = 1, we can get the scattering DGFs of each region. For f = 1, ′ G11 es ( R , R ) = ∞ n ik1 4π ∑∑ C ik1 4π ∞ n =0 m=0 mn (1)  BM11M (1)  (7.2a) ′(1) (k1 )M′e(1)mn (k1 ) + B11 e N N e mn ( k1 ) N e mn ( k1 ) . o mn o o o   For f ≠ 1, N , G (esf 1) (R, R′) = n ∑∑ C n =0 m =0 mn { B M (1) (k f )M′e(1)mn (k1 ) + BNf 1N (1) (k f )N′e(1)mn (k1 )  e e o mn o o mn o  f1 M } (7.2b) +  D M e mn (k f )M′ (k1 ) + D N e mn (k f )N′ (k1 )  . o o   f1 M (1) e o mn f1 N (1) e o mn For f = N , G es( N 1) (R, R′) = ik1 4π ∞ n ∑∑ C n =0 m=0 mn  DMN 1M e (k N )M′e(1) (k1 ) + DNN 1N e (k N )N′e(1) (k1 )  o mn o mn o mn o mn   (7.2c) where Cmn = (2 − δ 0 ) 2n + 1 (n − m)! , n(n + 1) (n + m)! (7.3) f1 f1 f1 f1 N1 N1 the coefficients BM11 , B11 N , BM , BN , DM , DN , DM , DN need to be determined, and the first superscript of DMN 1 , DNN 1 and the subscript of k N all indicate the layer number N . 7.2.2 THE CALCULATION OF THE COEFFICIENTS The electric dyadic Green functions of each region are presented as : f = 1, ′ G e1 (R, R′) = G e 0 (R, R′) + G11 es ( R , R ), (7.4a) 88 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials Gef (R, R′) = Gesf 1(R, R′), f ≠ 1, N , GeN (R, R′) = GesN1(R, R′), f = N, 89 (7.4b) (7.4c) where the electric free space DGF is: when R > b G e 0 (R, R′) = − 1 垐 ik RRδ (R − R′) + 1 2 4π k1 ∑C mn M (1) (k1 )M′(k1 ) + N (1) (k1 )N′(k1 ) , (7.5a) ik 1 ˆˆ RRδ (R − R′) + 1 2 k1 4π ∑C mn M (k1 )M′(1) (k1 ) + N(k1 )N′(1) (k1 ) , (7.5b) m,n when R < b G e 0 (R, R′) = − m,n where M e mn and N e mn are given in equations (3.10a) and (3.10b). o o The equations introduced in Chapter 3 will be used to obtain the coefficients of scattering dyadic Green’s functions. The equations (3.18a), (3.18b), (3.18c) and (3.19a) should be applied:  A fs CMfs , N =  Mfs, N CM , N f M ,N T  1  TFfH ,V =  H ,V RPf  TPfH ,V  1 0  A11 =   , and 0 0 BMfs , N  , DMfs, N    , 1 H ,V  TPf  H ,V RFf TFfH ,V 0 0 A 22 =  . 0 1  CMf 1, N = TMf −, N1 ⋅⋅⋅ TM1 , N ( C11 M , N + A 22 ) . 89 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials 90 We can first assume that the field point is located in the last layer: f = N . Because the coefficients satisfy the equation (3.16i):  AMNs, N   CM1s , N   Ns  =  1s  = 0,  BM , N   DM , N  the coefficients in the first and the last layers can be found. After obtaining the coefficients in the first and last layers, the rest of the coefficients can be derived from the equations (3.18a), (3.18b), (3.18c) and (3.19a). Now we apply the method mentioned above to the s = 1 case. First, assume f = N , the following can be obtained:   0 BM11, N  0 0   0 BMN 1, N  N −1 1 = ⋅⋅⋅ + T T .  M ,N M ,N   N1   0 DM11, N  0 1   0 DM , N      (7.6) From equation (3.16i): N1 11 BMN = DMN = 0. (7.7) TN = TMN ,−N1 TMN ,−N2 ⋅⋅⋅ TM1 , N , (7.8) Let then equation (7.6) can be written as: 0   TN11 TN12   0 BM11, N  0 0 D N 1  =  .  T T N N 0 1 , M N 21 22       (7.9) From above equation, we have: TN11 BM11, N + TN12 = 0, (7.10a) DMN 1, N = TN 21 BM11, N + TN 22 , (7.10b) 90 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials 91 then: BM11, N = − DMN 1, N = − TN12 , TN11 TN 21TN12 + TN 22 . TN11 (7.11a) (7.11b) The coefficients of the first and the last layers are given. We will study with the rest of the coefficients. For f ≠ 1, N , from equations (3.19a) and (7.11a):  0 BM11, N  0 0   0 BMf 1, N  f −1 1 = TM , N ⋅⋅⋅ TM , N    +  . f1   0 0  0 1   0 DM , N   (7.12) Let Tf = TMf −, N1 TMf −, N2 ⋅⋅⋅ TM1 , N , (7.13) together equation (7.11a), then equation (7.12) can be written as: TN12   0 BMf 1, N   Tf11 Tf12  0 − TN11  . =   f1   T T f f 0 D  21 22  M ,N   1  0 (7.14) From above equation, we have: BMf 1, N = − Tf11TN12 + Tf12 , TN11 (7.15a) DMf 1, N = − Tf 21TN12 + Tf 22 . TN11 (7.15b) TMf , N is represented as equation (3.18b). TFfH ,V , RFfH ,V , TPfH ,V , RPfH ,V were given as equations from (3.16a) to (3.16h): 91 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials µ f k f +1∂ ( f +1) f ff − µ f +1k f ∂ ff µ f k f +1ℑ ff ∂ ( f +1) f − µ f +1k f ∂ℑ ff RPfH = ( f +1) f , ( f +1) f RFfH = µ f k f +1∂ℑ( f +1) f ℑ ff − µ f +1k f ∂ℑ ff ℑ( f +1) f , µ f k f +1∂ℑ( f +1) f ff − µ f +1k f ℑ( f +1) f ∂ ff RPfV = µ f k f +1 ( f +1) f ∂ ff − µ f +1k f ff ∂ µ f k f +1∂ℑ ff ( f +1) f − µ f +1k f ℑ ff ∂ TPfH = ( f +1) f , ( f +1) f µ f k f +1ℑ( f +1) f ∂ℑ ff − µ f +1k f ℑ ff ∂ℑ( f +1) f , µ f k f +1ℑ( f +1) f ∂ ff − µ f +1k f ∂ℑ( f +1) f ff RFfV = µ f k f +1 (ℑ( f +1) f ∂ ( f +1) f − ∂ℑ( f +1) f µ f k f +1ℑ ff ∂ ( f +1) f − µ f +1k f ∂ℑ ff µ f k f +1 (∂ℑ( f +1) f µ f k f +1∂ℑ( f +1) f TFfH = ( f +1) f ff − ℑ( f +1) f ∂ TFfV = µ f k f +1 (ℑ( f +1) f ∂ µ f k f +1ℑ( f +1) f ∂ ℑil = jn ( ki al ) , il ( f +1) f ff − ∂ℑ( f +1) f ) ( f +1) f , ( f +1) f ) ( f +1) f − µ f +1k f ℑ( f +1) f ∂ µ f k f +1 (∂ℑ( f +1) f ( f +1) f − ℑ( f +1) f ∂ µ f k f +1∂ℑ ff ( f +1) f − µ f +1k f ℑ ff ∂ TPfV = where 92 , ff ( f +1) f ) , ( f +1) f ( f +1) f − µ f +1k f ∂ℑ( f +1) f ) , ff = hn(1) ( ki al ) , ∂ℑil = 1 d [ ρ jn ( ρ ) ] , dρ ρ ρ =k a i l ∂ il (1) 1 d  ρ hn ( ρ )  = ρ dρ because there , as given in equations from (3.14a) to (3.14d). Additionally, ρ = ki al is air in layers N , N − 2, N − 4, N − 6...... , when f = N , N − 2, N − 4, N − 6...... : ε f = ε1 , µ f = µ1 , k f = k1. (7.16) 92 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials 93 7.2.3 COEFFICIENTS IN THE SPECIAL CASE OF METAMATERIAL Now let us consider the metamaterial case, that is, when f = N − 1, N − 3, N − 5......2 : µ f = − µ1 . k f = −k1 , (7.17) Whatever f is, we have µ f k f +1 = µ f +1k f = − µ1k1. (7.18) The equations from (3.16a) to (3.16h) can then be written as: RPfH = RFfH = RPfV = RFfV = TPfH = TFfH = TPfV = TFfV = − µ1k1∂ ( f +1) f − µ1k1ℑ ff ∂ ff ( f +1) f + µ1k1∂ ff ( f +1) f + µ1k1∂ℑ ff ( f +1) f − µ1k1∂ℑ( f +1) f ℑ ff + µ1k1∂ℑ ff ℑ( f +1) f − µ1k1∂ℑ( f +1) f ff + µ1k1ℑ( f +1) f ∂ − µ1k1 ff + µ1k1 ( f +1) f − µ1k1∂ℑ ff ∂ ( f +1) f ( f +1) f + µ1k1ℑ ff ∂ ( f +1) f − µ1k1ℑ( f +1) f ∂ℑ ff + µ1k1ℑ ff ∂ℑ( f +1) f − µ1k1ℑ( f +1) f ∂ − µ1k1 (ℑ( f +1) f ∂ − µ1k1ℑ ff ∂ ( f +1) f − µ1k1 (∂ℑ( f +1) f − µ1k1∂ℑ( f +1) f − µ1k1 (∂ℑ( f +1) f − µ1k1∂ℑ ff ( f +1) f − µ1k1 (ℑ( f +1) f ∂ − µ1k1ℑ( f +1) f ∂ ( f +1) f ff ) = ( f +1) f − ℑ( f +1) f ∂ − ℑ( f +1) f ∂ + µ1k1ℑ ff ∂ ( f +1) f ff − ∂ℑ( f +1) f ( f +1) f + µ1k1ℑ( f +1) f ∂ ( f +1) f ( f +1) f + µ1k1∂ℑ( f +1) f + µ1k1∂ℑ ff ( f +1) f ff ff ) = ff ( f +1) f ) = ( f +1) f − ∂ℑ( f +1) f + µ1k1∂ℑ( f +1) f ( f +1) f ff ) = ff ∂ ff = = = = ∂ ( f +1) f ℑ ff ∂ ff ( f +1) f −∂ ff ( f +1) f − ∂ℑ ff ( f +1) f , (7.19a) , (7.19b) ∂ℑ( f +1) f ℑ ff − ∂ℑ ff ℑ( f +1) f ∂ℑ( f +1) f ( f +1) f ∂ℑ ff ∂ ff ff ( f +1) f − ℑ( f +1) f ∂ − ff ∂ ( f +1) f − ℑ ff ∂ ( f +1) f ff ℑ( f +1) f ∂ℑ ff − ℑ ff ∂ℑ( f +1) f ℑ( f +1) f ∂ ℑ( f +1) f ∂ ff − ∂ℑ( f +1) f , (7.19c) , (7.19d) ff ( f +1) f − ∂ℑ( f +1) f ( f +1) f − ∂ℑ ff ( f +1) f − ℑ( f +1) f ∂ ( f +1) f ff − ℑ( f +1) f ∂ ff ∂ℑ( f +1) f ( f +1) f − ℑ( f +1) f ∂ ∂ℑ ff ( f +1) f − ℑ ff ∂ ( f +1) f − ∂ℑ( f +1) f ( f +1) f − ∂ℑ( f +1) f ff ℑ ff ∂ ∂ℑ( f +1) f ∂ℑ( f +1) f ℑ( f +1) f ∂ ℑ( f +1) f ∂ ff ( f +1) f , (7.19e) ( f +1) f ( f +1) f , (7.19f) , (7.19g) ( f +1) f . (7.19h) When f = N − 1, N − 3, N − 5......2 : 93 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials 94  ′  ℑ ff = jn (− k1a f ), ∂ℑ ff = [ ρ jn ( ρ ) ] ρ =− k a , 1 f    ff = hn(1) (− k1a f ), ∂ ff =  ρ hn(1) ( ρ ) ′ .    k a ρ =− 1 f  (7.20a) When f = N , N − 2, N − 4, N − 6...... :  ′  ℑ( f +1) f = jn (−k1a f ), ∂ℑ( f +1) f = [ ρ jn ( ρ )] ρ =− k a , 1 f    ( f +1) f = hn(1) (−k1a f ), ∂ ( f +1) f =  ρ hn(1) ( ρ ) ′ .    =− k a ρ 1 f  (7.20b) We still have the following results.  j (− z ), When n is even, jn ( z ) =  n − jn (− z ), When n is odd. (7.21a) When n is even: [ ρ 2 jn ( ρ 2 )]′ = [ ρ1 jn ( ρ1 )]′ ρ =− ρ , 1 (7.21b) 2 when n is odd: [ ρ 2 jn ( ρ 2 )]′ = − [ ρ1 jn ( ρ1 )]′ ρ =− ρ . 1 (7.21c) 2 When n is even: hn(1) ( ρ 2 ) = hn(1) ( ρ1 ) ρ1 =− ρ2 [ ρ 2 hn(1) ( ρ 2 )]′ = [ ρ1hn(1) ( ρ1 )]′ , ρ1 =− ρ 2 (7.21d) ; (7.21e) when n is odd: 94 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials j− n ( ρ1 ) + jn ( ρ1 ) e − iπ n hn ( ρ 2 ) = i sin ( nπ ) (1) 95 , (7.21f) ρ1 =− ρ2  j− n ( ρ1 ) + jn ( ρ1 ) e− iπ n ′ j− n ( ρ1 ) + jn ( ρ1 ) e− iπ n [ ρ 2 hn ( ρ 2 )]′ = + ρ1   i sin ( nπ ) i sin ( nπ )   (1) . (7.21g) ρ1 =− ρ 2 It is obvious that we can get the following results in this special case:  RFfH = RFfV , TFfH = TFfV ,  H V H V  RPf = RPf , TPf = TPf , (7.22)  BM11 = B11 BMf 1 = BNf 1 , N ,  f1 f1 N1 N1  DM = DN , DM = DN . (7.23) hence 7.2.4 CALCULATION OF THE ELECTRICAL FIELD Rectangular coordinate can be transformed to spherical coordinate as: xˆ = sin θ cos φ rˆ + cos θ cos φθˆ − sin φφˆ, (7.24) For the infinitesimal horizontal electric dipole introduced in the beginning of 7.2, because θ ′ = 0, φ ′ = 0 , we have: xˆ = θˆ . (7.25) Because  n(n + 1) , m =1 mpnm (cos θ )  = 2 , θ →0 sin θ  0, otherwise lim (7.26a) 95 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials  n(n + 1) , m =1 dpnm (cos θ )  , = 2 lim θ →0 dθ  0, otherwise 96 (7.26b) Substituting m = 1 into equation (7.3), we have: Cmn = 2 × 2n + 1 1 × . n(n + 1) n(n + 1) (7.26c) From the three equations above, the following equation will be obtained: mpnn (cos θ ) dp m (cosθ ) 2n + 1 = Cmn × lim n = . θ →0 θ → 0 sin θ dθ n(n + 1) Cmn × lim (7.27) Additionally, we have: dV ′ = R′2 dR′ sin θ ′dθ ′dφ ′. (7.28) The electrical fields will be calculated of each region. When R > b : E1 (R ) = iωµ1 ∫∫∫ G e1 (R, R′)iJ (R′)dV ′ = iωµ1cG e1 (R, R′)i xˆ = iωµ1cG e1 (R, R′)iθˆ. (7.29) Applying the expression of G e1 (R, R′) to the above equation, we have: E1 (R ) = − k1ωµ1c ∞ 2n + 1 i  jn ( ρb ) + BM11hn(1) ( ρb )  M (1) ∑ o1n ( k1 ) 4π n =1 n(n + 1)  {    + ( ρb jn ( ρb ) )′ + BN11 ( ρb hn(1) ( ρ b ) )′  N e(1)1n (k1 ) ρ b  ,    (7.30) where ρb = k1b. When a1 < R < b : E 2 (R ) = iωµ1 ∫∫∫ G e1 (R, R′)iJ (R′)dV ′ = iωµ1cG e1 (R, R′)i xˆ = iωµ1cG e1 (R, R′)iθˆ, (7.31) Together with the equation (7.5b), we have: 96 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials E 2 (R ) = − k1ωµ1c ∞ 2n + 1 i hn(1) ( ρb ) M o1n (k1 ) + BM11M o(1)1n (k1 )  ∑ 4π n =1 n(n + 1) 97 {  (1)  + ( ρ h ( ρb ) )′  N e1n (k1 ) + B11 N N e1n ( k1 )  ρ b  .  (7.32) (1) b n For a f < R < a f −1 , where f ranges from 2 to N − 1 : E f +1 (R ) = iωµ f ∫∫∫ G ef (R, R′)iJ (R′)dV ′ = iωµ f cG ef (R, R′)i xˆ = iωµ f cG ef (R, R′)iθˆ, (7.33) apply the related equations to the above equation, we have: E f +1 (R ) = − k1ωµ f c 4π ∞ 2n + 1 ∑ n(n + 1) i{h (1) n n =1  ( ρb )  DMf 1M o1n (k f ) + BMf 1M (1) o1n ( k f )    + ( ρ h ( ρb ) )′  DNf 1N e1n (k f ) + BNf 1N (1) e1n ( k f )  ρ b  .  (7.34a) (1) b n when f = N − 2, N − 4, N − 6...... , the following expression will be obtained: E f +1 (R ) = − k1ωµ1c ∞ 2n + 1  i hn(1) ( ρb )  DMf 1M o1n (k1 ) + BMf 1M (1) ∑ o1n ( k1 )  4π n =1 n(n + 1) {   + ( ρ h ( ρb ) )′  DNf 1N e1n (k1 ) + BNf 1N (1) e1n ( k1 )  ρ b  .  (7.34b) (1) b n In the metamaterial case, when f = N − 1, N − 3, N − 5......2, equation (7.34a) can be rewritten as: E f +1 (R ) = k1ωµ1c ∞ 2n + 1  i hn(1) ( ρb )  DMf 1M o1n (− k1 ) + BMf 1M (1) ∑ o1n ( − k1 )  4π n =1 n(n + 1) {   + ( ρ h ( ρb ) )′  DNf 1N e1n (− k1 ) + BNf 1N (1) e1n ( − k1 )  ρ b  .  (7.34c) (1) b n From the expressions of M e mn ( k ) and N e mn ( k ) , i.e., equations (3.10a) and (3.10b), o o together with the properties of the Bessel function and the Hankel function shown in equations (7.21a)-(7.21g), the following can then be obtained: 97 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials when n is even, M e mn ( k ) = M e mn ( −k ) , (7.35a) N e mn ( k ) = − N e mn ( −k ) , (7.35b) M (1) ( k ) = M (1)e mn ( −k ) , e mn (7.35c) N (1) ( k ) = −N (1)e mn ( −k ) , e mn (7.35d) M e mn ( k ) = −M e mn ( −k ) , (7.35e) N e mn ( k ) = N e mn ( −k ) . (7.35f) o o o o o o o when n is odd, 98 o o o o o When R < aN −1 : E N +1 (R ) = iωµ N ∫∫∫ G eN (R, R′)iJ (R′)dV ′ = iωµ N cG eN (R, R′)i xˆ = iωµ N cG eN (R, R′)iθˆ. (7.36) Applying the related equations to the above equation, we have: E N +1 (R ) = − k1ωµ N c ∞ 2n + 1 i{hn(1) ( ρb ) DMN 1M o1n (k N ) ∑ 4π n =1 n(n + 1)  + ( ρ h ( ρb ) )′ DNN 1N e1n (k N ) ρb  .  (7.37a) (1) b n Because k1 = k N , (7.37a) can be rewritten as: E N +1 (R ) = − k1ωµ1c ∞ 2n + 1 i{hn(1) ( ρb ) DMN 1M o1n (k1 ) ∑ 4π n =1 n(n + 1)  + ( ρ h ( ρb ) )′ DNN 1N e1n (k1 ) ρb  .  (7.37b) (1) b n 98 Characterization of Spherical Metamaterials Chapter 7 7.3 Muti-spherical Layers of Metamaterials CURRENT DISTRIBUTION LOCATED INSIDE 99 THE SPHERICAL MULTILAYERS 7.3.1 INTRODUCTION The spherical multilayers has a center at (0, 0, 0). An infinitesimal horizontal electric dipole with current moment c pointing in the x -direction and located at R′ = 0,θ ′ = 0, φ ′ = 0 (the center of the spherical multilayers), let: J (R′) = c δ ( R′ − 0)δ (θ ′ − 0)δ (φ ′ − 0) xˆ. R′2 sin θ ′ (7.38) Scattering dyadic Green’s functions can be expressed generally in equation (3.12). Applying this equation to the special case that s = N , we can get the scattering DGFs of each region. For f = 1, G1esN (R, R′) = ik N 4π ∑C mn m,n  AM1N M (1) (k1 )M′(k N ) + AN1N N (1) (k1 )N′(k N )  . (7.39a) For f ≠ 1, N , G esfN (R, R′) = ik N 4π ∑ C { A fN M mn m,n M (1) (k f )M′(k N ) + ANfN N (1) (k f )N′(k N )  } (7.39b) + C M (k f )M′(k N ) + C N(k f )N′(k N )  . fN M fN N For f = N , G esNN (R, R′) = ik N 4π ∑C m,n mn CMNN M (k N )M′(k N ) + CNNN N(k N )N′(k N )  , (7.39c) where Cmn = (2 − δ 0 ) 2n + 1 (n − m)! , n(n + 1) (n + m)! (7.40) 99 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials 100 the coefficients AMN 1 , ANN 1 , AMfN , ANfN , CMfN , CNfN , CMNN , CNNN need to be determined, and the superscript of coefficients and the subscript of k N all indicate the layer number N . 7.3.2 THE CALCULATION OF THE COEFFICIENTS The electric dyadic Green functions of each region are presented as follows respectively: f = 1, f ≠ 1, N , G e1 (R, R′) = G1esN (R, R′), (7.41a) Gef (R, R′) = GesfN (R, R′), (7.41b) GeN (R, R′) = Ge0 (R, R′) +GesNN (R, R′), f = N, (7.41c) where the electric free space DGF is: because R > 0 G e 0 (R, R′) = − ik 1 ˆˆ RRδ (R − R′) + N 2 4π kN ∑C m,n mn M (1) (k N )M′(k N ) + N (1) (k N )N′(k N ) , (7.42) where M e mn and N e mn are given in equation (3.10a) and (3.10b), and Cmn is shown as o o equation (7.40). The equations introduced in Chapter 3 will be used to obtain the coefficients of scattering dyadic Green’s functions. Because s = N , equations from (3.18a), (3.18b), (3.18c) and (3.19c) should be applied:  A fs CMfs , N =  Mfs, N CM , N BMfs , N  , DMfs, N  100 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials  1  TFfH ,V =  H ,V RPf  TPfH ,V  f M ,N T 1 0  A11 =   , and 0 0 101   , 1 TPfH ,V   H ,V RFf TFfH ,V 0 0 A 22 =  . 0 1  CMfN, N = TMf −, N1 ⋅⋅⋅ TM1 , N C1MN, N − H ( f − N ) A11. We can first assume that the field point is located in the last layer: f = N . Because the coefficients satisfy equation (3.16i):  AMNs, N   CM1s , N   Ns  =  1s  = 0,  BM , N   DM , N  the coefficients in the first and the last layers can be found. After obtaining the coefficients in the first and last layers, the rest of the coefficients can be derived from the equations from (3.18a), (3.18b), (3.18c) and (3.19c). Firstly, by assuming f = N , we have:  AMNN, N  NN CM , N 0 N −1 1  = TM , N ⋅⋅⋅ TM , N 0  AM1N, N  1N CM , N 0  1 0  . − 0  0 0  (7.43) From equation (3.16i): NN 1N AMN = CMN = 0. (7.44) TN = TMN ,−N1 TMN ,−N2 ⋅⋅⋅ TM1 , N , (7.45) Let then the equation (7.43) can be written as: 101 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials  0 C NN  M ,N 0   TN11 TN12   AM1N, N = 0  TN 21 TN 22   0 102 0  1 0  − . 0 0 0 (7.46) Deriving the above equation, we have: TN11 AM1N, N − 1 = 0, (7.47a) CMNN, N = TN 21 AM1N, N , (7.47b) then: AM1N, N = 1 , TN11 (7.48a) CMNN, N = TN 21 . TN11 (7.48b) The coefficients of the first and the last layers now are given. We will study the rest of the coefficients. For f ≠ 1, N , from equations (3.19c) and (7.48a):  AMfN, N  fN CM , N 0  AM1N, N f −1 1 = ⋅⋅⋅ T T  M ,N M ,N  0  0 0 . 0 (7.49) Let Tf = TMf −, N1 TMf −, N2 ⋅⋅⋅ TM1 , N , (7.50) together with equation (7.48a), then equation (7.49) can be rewritten as:  AMfN, N  fN CM , N  1  , 0 0   Tf11 Tf12   TN11 . =  0  Tf 21 Tf 22   0   0 (7.51) Deriving the above equation, we have: 102 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials 103 AMfN, N = Tf11 , TN11 (7.52a) CMfN, N = Tf 21 . TN11 (7.52b) TMf , N is represented in equation (3.18b). TFfH ,V , RFfH ,V , TPfH ,V , RPfH ,V were given in equations from (3.16a) to (3.16h): RPfH = µ f k f +1∂ ( f +1) f ff − µ f +1k f ∂ ff µ f k f +1ℑ ff ∂ ( f +1) f − µ f +1k f ∂ℑ ff ( f +1) f , ( f +1) f RFfH = µ f k f +1∂ℑ( f +1) f ℑ ff − µ f +1k f ∂ℑ ff ℑ( f +1) f , µ f k f +1∂ℑ( f +1) f ff − µ f +1k f ℑ( f +1) f ∂ ff RPfV = µ f k f +1 ( f +1) f ∂ ff − µ f +1k f ff ∂ µ f k f +1∂ℑ ff ( f +1) f − µ f +1k f ℑ ff ∂ RFfV = TPfH = TPfV = TFfV = , ( f +1) f µ f k f +1ℑ( f +1) f ∂ℑ ff − µ f +1k f ℑ ff ∂ℑ( f +1) f , µ f k f +1ℑ( f +1) f ∂ ff − µ f +1k f ∂ℑ( f +1) f ff µ f k f +1 (ℑ( f +1) f ∂ ( f +1) f − ∂ℑ( f +1) f µ f k f +1ℑ ff ∂ ( f +1) f − µ f +1k f ∂ℑ ff TFfH = ( f +1) f µ f k f +1 (∂ℑ( f +1) f µ f k f +1∂ℑ( f +1) f ( f +1) f ff − ℑ( f +1) f ∂ µ f k f +1 (ℑ( f +1) f ∂ µ f k f +1ℑ( f +1) f ∂ ( f +1) f ff − ∂ℑ( f +1) f , ( f +1) f ) ( f +1) f − µ f +1k f ℑ( f +1) f ∂ µ f k f +1 (∂ℑ( f +1) f ( f +1) f − ℑ( f +1) f ∂ µ f k f +1∂ℑ ff ( f +1) f − µ f +1k f ℑ ff ∂ ) ( f +1) f , ff ( f +1) f ) , ( f +1) f ( f +1) f − µ f +1k f ∂ℑ( f +1) f ) , ff 103 Characterization of Spherical Metamaterials Chapter 7 where Muti-spherical Layers of Metamaterials ℑil = jn ( ki al ) , il 104 = hn(1) ( ki al ) , ∂ℑil = 1 d [ ρ jn ( ρ ) ] , dρ ρ ρ =k a i l ∂ il (1) 1 d  ρ hn ( ρ )  = ρ dρ because there , as shown in equations from (3.14a) to (3.14d). Additionally, ρ = ki al is air in layers N , N − 2, N − 4, N − 6...... of , when f = N , N − 2, N − 4, N − 6...... : ε f = ε1 , µ f = µ1 , k f = k1. (7.53) 7.3.3 COEFFICIENTS IN THE SPECIAL CASE OF METAMATERIAL Let us consider the metamaterial case, that is, for f = N − 1, N − 3, N − 5......2 : µ f = − µ1 . k f = −k1 , (7.54) It is obvious that whatever f is, µ f k f +1 = µ f +1k f = − µ1k1. (7.55) The equations from (3.16a) to (3.16h) can then be written as: RPfH = RFfH = RPfV = − µ1k1∂ ( f +1) f − µ1k1ℑ ff ∂ ff ( f +1) f + µ1k1∂ ff ( f +1) f + µ1k1∂ℑ ff ( f +1) f − µ1k1∂ℑ( f +1) f ℑ ff + µ1k1∂ℑ ff ℑ( f +1) f − µ1k1∂ℑ( f +1) f ff + µ1k1ℑ( f +1) f ∂ − µ1k1 ff + µ1k1 ( f +1) f − µ1k1∂ℑ ff ∂ ( f +1) f = ff ∂ ( f +1) f + µ1k1ℑ ff ∂ ( f +1) f ff = = ∂ ( f +1) f ℑ ff ∂ ff ( f +1) f −∂ ff ( f +1) f − ∂ℑ ff ( f +1) f , (7.56a) , (7.56b) ∂ℑ( f +1) f ℑ ff − ∂ℑ ff ℑ( f +1) f ∂ℑ( f +1) f ( f +1) f ∂ℑ ff ∂ ff ff ( f +1) f − ℑ( f +1) f ∂ − ff ∂ ( f +1) f − ℑ ff ∂ ( f +1) f ff , (7.56c) 104 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials RFfV = TPfH = TFfH = TPfV = TFfV = − µ1k1ℑ( f +1) f ∂ℑ ff + µ1k1ℑ ff ∂ℑ( f +1) f − µ1k1ℑ( f +1) f ∂ − µ1k1 (ℑ( f +1) f ∂ − µ1k1ℑ ff ∂ ( f +1) f ( f +1) f − µ1k1 (∂ℑ( f +1) f − µ1k1 (∂ℑ( f +1) f − µ1k1∂ℑ ff ff − µ1k1 (ℑ( f +1) f ∂ ff ( f +1) f ff ) = ( f +1) f − ℑ( f +1) f ∂ − ℑ( f +1) f ∂ + µ1k1ℑ ff ∂ ( f +1) f ) = ff ( f +1) f ) = ( f +1) f − ∂ℑ( f +1) f ( f +1) f + µ1k1∂ℑ( f +1) f ff ( f +1) f − µ1k1ℑ( f +1) f ∂ − ∂ℑ( f +1) f + µ1k1ℑ( f +1) f ∂ ( f +1) f ( f +1) f + µ1k1∂ℑ( f +1) f + µ1k1∂ℑ ff ( f +1) f − µ1k1∂ℑ( f +1) f ff ) = = 105 ℑ( f +1) f ∂ℑ ff − ℑ ff ∂ℑ( f +1) f ℑ( f +1) f ∂ ff − ∂ℑ( f +1) f , ℑ( f +1) f ∂ ( f +1) f − ∂ℑ( f +1) f ℑ ff ∂ ( f +1) f − ∂ℑ ff ( f +1) f − ℑ( f +1) f ∂ ( f +1) f ff − ℑ( f +1) f ∂ ff ∂ℑ( f +1) f ( f +1) f − ℑ( f +1) f ∂ ∂ℑ ff ( f +1) f − ℑ ff ∂ ( f +1) f − ∂ℑ( f +1) f ( f +1) f − ∂ℑ( f +1) f ff ∂ℑ( f +1) f ∂ℑ( f +1) f ℑ( f +1) f ∂ ℑ( f +1) f ∂ ff (7.56d) ff ( f +1) f , (7.56e) ( f +1) f ( f +1) f , (7.56f) , (7.56g) ( f +1) f . (7.56h) When f = N − 1, N − 3, N − 5......2 :  ′  ℑ ff = jn (− k1a f ), ∂ℑ ff = [ ρ jn ( ρ ) ] ρ =− k a , 1 f    ff = hn(1) (− k1a f ), ∂ ff =  ρ hn(1) ( ρ ) ′ .    ρ =− k a 1 f  (7.57a) When f = N , N − 2, N − 4, N − 6...... :  ′  ℑ( f +1) f = jn (−k1a f ), ∂ℑ( f +1) f = [ ρ jn ( ρ )] ρ =− k a , 1 f    ( f +1) f = hn(1) (−k1a f ), ∂ ( f +1) f =  ρ hn(1) ( ρ ) ′ .    =− ρ k a 1 f  (7.57b) Equations (7.21a)- (7.21g) should be applied to (7.57). We can obtain easily the following results in this special case: H V H V  RFf = RFf , TFf = TFf ,  H V H V  RPf = RPf , TPf = TPf , (7.58) 105 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials 106 therefore,  AM1N = AN1N , AMfN = ANfN ,  fN fN NN NN CM = CN , CM = CN . (7.59) 7.3.4 CALCULATION OF THE ELECTRICAL FIELD Rectangular coordinate can be transformed to spherical coordinate as: xˆ = sin θ cos φ rˆ + cos θ cos φθˆ − sin φφˆ. (7.60) For the infinitesimal horizontal electric dipole introduced in the beginning of 7.3, for θ ′ = 0, φ ′ = 0 , we have: xˆ = θˆ . (7.61) Because  n(n + 1) , m =1 mpnm (cos θ )  , = 2 lim θ →0 sin θ  0, otherwise (7.62a)  n(n + 1) , m =1 dpnm (cos θ )  . = 2 θ →0 dθ  0, otherwise (7.62b) lim Applying m = 1 to equation (7.40), we have: Cmn = 2 × 2n + 1 1 × . n(n + 1) n(n + 1) (7.62c) From the three equations as above, the following equation will be obtained: mpnn (cos θ ) dp m (cosθ ) 2n + 1 = Cmn × lim n = . θ →0 θ →0 sin θ dθ n(n + 1) Cmn × lim (7.63) Additionally, we have: 106 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials dV ′ = R′2 dR′ sin θ ′dθ ′dφ ′. 107 (7.64) The electrical fields will be calculated in each region. When R > a1 : E1 (R ) = iωµ1 ∫∫∫ G e1 (R, R′)iJ (R′)dV ′ = iωµ1cG e1 (R, R′)i xˆ = iωµ1cG e1 (R, R′)iθˆ. (7.65) Applying the expression of G e1 (R, R′) to the above equation, we have:   k N rjn (k N r ) ]′ 1N (1) [ k N ωµ1c ∞ 2n + 1   1N (1) E1 (R ) = − i jn (0) AM M o1n (k1 ) + lim AN N e1n (k1 )  . (7.66) ∑ r 0 → 4π n =1 n(n + 1)  kN r   Because we still have:  1, n = 0 jn (0) =  , 0, otherwise (7.67a)  2 , n =1 [k3rjn (k3r )]′  = 3 lim , r →0 k3 r 0, otherwise (7.67b) 2n + 1 3 = , n(n + 1) n =1 2 (7.67c) the electric field can be then represented as follows: E1 (R ) = − k N ωµ1c 1N (1)  AN N e11 (k1 )  . 4π  (7.68) Because k1 = k N , (7.68) can be rewritten as: E1 (R ) = − k1ωµ1c 1N (1)  AN N e11 (k1 )  . 4π  (7.69) 107 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials a f < R < a f −1 , For f ranges from 108 2 to N −1 : E f (R ) = iωµ f ∫∫∫ G ef (R, R′)iJ (R′)dV ′ = iωµ f cG ef (R, R′)i xˆ = iωµ f cG ef (R, R′)iθˆ, (7.70) applying the related equations to the above equation, we have: E f (R ) = − k N ωµ f c 4π 2n + 1 ∞ ∑ n(n + 1) i{ j (0) C n n =1 [ k rj (k r )]′ C fN N lim N n N r →0 kN r  N fN M  M o1n (k f ) + AMfN M (1) o1n ( k f )  +    + N ( ) ( ) k A k e1n f f  .  fN N (7.71) (1) e1n Applying equation (7.67), the following result can be obtained: E f (R ) = − k N ωµ f c 4π CNfN N e11 (k f ) + ANfN N (1)  e11 ( k f )  . (7.72a) When f = N − 2, N − 4, N − 6...... , the following expression will be derived: E f (R ) = − k1ωµ1c C NfN N e11 (k1 ) + ANfN N (1)  e11 ( k1 )  . 4π  (7.72b) In the metamaterial case, when f = N − 1, N − 3, N − 5......2, equation (7.34a) can be written as: E f (R ) = k1ωµ1c CNfN N e11 (−k1 ) + ANfN N (1)  e11 ( − k1 )  . 4π  (7.72c) We use the expressions of N e mn ( k ) , i.e., equation (3.10b), together with the properties of o the Bessel function and the Hankel function given in equation (7.21), we have: when n is even, N e mn ( k ) = −N e mn ( −k ) , (7.73a) N (1) ( k ) = −N (1)e mn ( −k ) , e mn (7.73b) N e mn ( k ) = N e mn ( −k ) . (7.73c) o o o when n is odd, o o o 108 Characterization of Spherical Metamaterials Chapter 7 Muti-spherical Layers of Metamaterials 109 E N (R ) = iωµ N ∫∫∫ G eN (R, R′)iJ (R′)dV ′ = iωµ N cG eN (R, R′)i xˆ (7.74) When R < aN −1 : = iωµ N cG eN (R, R′)iθˆ. Applying the related equations to the above equation, we have: E N (R ) = − k N ωµ N c ∞ 2n + 1 NN  i jn (0) M (1) ∑ o1n ( k N ) + CM M o1n ( k N )  4π n =1 n(n + 1) { [ k rj (k r )]′  N (1) (k + lim N n N r →0 kN r  e1n N )+C NN N   N e1n (k N )   .  (7.75) From equation (7.67), we obtain: E N (R ) = − k N ωµ N c (1)  N e11 (k N ) + CNNN N e11 (k N )  . 4π  (7.76a) Because k1 = k N , the above expression can be derived as: E N (R ) = − k1ωµ1c (1)  N e11 (k1 ) + CNNN N e11 (k1 )  . 4π (7.76b) 109 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 8 110 SIMULATION RESULTS In this chapter, the simulation results are presented and discussed. Here we consider the case that every other layer is air. The chapter is divided into three parts. The first part is about the case of metamaterial sphere, i.e., m = 2, where m is the number of the layers. Metamaterial spherical shell ( m = 3 ) is studied in the second part. The last part is concentrated on the case m = 5 . 8.1 SIMULATION RESULTS AND DISCUSSION OF METAMATERIAL SPHERE a 1 2 k2 k1 Figure 8.1 Structure of the metamaterial sphere Regarding the truncation number n for the expansion expressions of the field components, it is dependent upon the electric radius of the sphere: the bigger the radius is; the larger is the number needed. We usually truncate the series based on experience and 110 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 111 numerical test. Here, n is from 1 to 20 when the radius of the sphere is 5λ or less. CPU frequency used in our simulation is 800MHz, the simulations ( n = 30 ) each took half an hour to finish. The bright circles in the figures from Fig. 8.2 to Fig. 8.6 exist because of the discontinuity of the free space dyadic Green’s function in R = b . When k2 = k1 , as shown in Figure 8.2, region 2 can also be considered as air. After coming out from the dipole, the radiated energy will spread very quickly while not focus. When k2 = 2k1 , as shown in Figure 8.3, the refractive angles are less than those in the k2 = k1 case, therefore, some of the refractive waves are focused near the surface of the sphere in the opposite side. The smaller the angles between the incident waves and z -axis are, the more likely the refractive waves will be focused. When k2 = 4k1 , as shown in Figure 8.4, the angles between the refractive waves and the normal are even less than those in the k2 = 2k1 case. Therefore, there are more refractive waves that will be focused. We then find that the density of the focused area when k2 = 4k1 is much more than that in the k2 = 2k1 case. Now let us consider metamaterial cases. When k2 = − k1 , as shown in Figure 8.5, because of the negative refraction, i.e., the incident waves and the refractive waves are in the same side of the normal, the refractive waves focus near the metamaterial spherical surface quickly. When k2 = −2k1 , as shown in Figure 8.6, the angles between the refractive waves and the normal are even smaller than that in the k2 = −k1 case, together 111 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 112 with the negative refraction, the focused area is much larger and the density is much less than that in the k2 = −k1 case. Figure 8.2 Near field of metamaterial sphere when k2 = k1 = k0 , a = 1λ , b = 2λ . Figure 8.3 Near field of metamaterial sphere when k2 = 2k1 , a = 1λ , b = 2λ . 112 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 113 Figure 8.4 Near field of metamaterial sphere when k2 = 4k1 , a = 1λ , b = 2λ . Figure 8.5 Near field of metamaterial sphere when k2 = − k1 , a = 1λ , b = 2λ . 113 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 114 Figure 8.6 Near field of metamaterial sphere when k2 = −2k1 , a = 1λ , b = 2λ . We can find that the density of the focused area when k2 = 4k1 even can be comparable with the case when k2 = − k1 . 8.2 SIMULATION RESULTS AND DISCUSSION OF METAMATERIAL SPHERICAL SHELL Regarding the truncation number n for the expansion expressions of the field components, it is dependent upon the electric radius of the sphere: the bigger the radius is; the larger is the number needed. We usually truncate the series based on experience and 114 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 115 numerical test. Here, n is from 1 to 30 when the radius of the sphere is 10λ or less. CPU used in our simulation is 2GHz, the simulations ( n = 30 ) each took six hours to finish. Current distribution is located outside the spherical shell. When k2 = k1 = k3 , as shown in Figure 8.8, the radiated energy spreads from the dipole. When k2 = 2k1 = 2k3 , as shown in Figure 8.9, some of the refractive waves are focused, but most of the refractive waves radiate. When k2 = −k1 = − k3 , because of the negative refraction, the waves radiate after they are focused near the interface between region 1 and region 2. The refractive waves focus again very near the interface between region 2 and region 3 when they arrive at the region 3. After the second focus, the waves spread again. a1 1 a2 3 2 k3 k2 k1 Figure 8.7 Structure of the metamaterial spherical shell 115 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 116 1 Figure 8.8 Near field of metamaterial spherical shell when k2 = k1 , a1 = 4λ , a2 = 2λ . Figure 8.9 Near field of metamaterial spherical shell when k2 = 2k1 . a1 = 4λ , a2 = 2λ . 116 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 117 Figure 8.10 Near field of metamaterial spherical shell when k2 = − k1 , a1 = 4λ , a2 = 2λ . Figure 8.11 Near field of metamaterial spherical shell when k2 = −2k1 , a1 = 4λ , a2 = 2λ . 117 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 118 Because the focus is very near to the interface, the radiate waves will spread around near the interface, as shown in Figure 8.10. When k2 = −2k1 = −2k3 , as shown in Figure 8.11, the refractive waves are focused on the area that is far away from the interface more than on the area in the k2 = −k1 = − k3 case. Then, the refractive waves will focus again when leaving region 3 and entering region 2. 8.3 SIMULATION RESULTS AND DISCUSSION OF METAMATERIAL SPHERICAL MULTILAYERS Here we consider the m = 5 case, i.e., region 1, 3 and 5 are air. a1 a2 a3 1 2 3 5 k5 4 a4 k4 k3 k2 k1 Figure 8.12 Structure of the spherical mutilayers ( m = 5 ) 118 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 119 Figure 8.13 Near field of 5-layer metamaterial spheres when k2 = k1 , a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ . Figure 8.14 Near field of 5-layer metamaterial spheres when k2 = 2k1 , a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ . 119 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 120 Figure 8.15 Near field of 5-layer metamaterial spheres when k2 = − k1 , a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 10 . Figure 8.16 Near field of 5-layer metamaterial spheres when k2 = − k1 , a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 20 . 120 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 121 Figure 8.17 Near field of 5-layer metamaterial spheres when k2 = − k1 , a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 30 . Figure 8.18 Near field of 5-layer metamaterial spheres when k2 = − k1 , a1 = 5λ , a2 = 4λ , a3 = 3λ , a4 = 2λ , b = 5.5λ , n = 30 . 121 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 122 Figure 8.19 Near field of 5-layer metamaterial spheres when k2 = −2k1 , a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 20 . Figure 8.20 Near field of 5-layer metamaterial spheres when k2 = −2k1 , a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 30 . 122 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 123 Figure 8.21 Near field of 5-layer metamaterial spheres when k2 = −2k1 , a1 = 5λ , a2 = 4λ , a3 = 3λ , a4 = 2λ , b = 5.5λ , n = 30 . The truncation number n for the expansion expressions of the field components is dependent upon the electric radius of the sphere: the bigger the radius is; the larger is the number needed. The series are usually truncated based on experience and numerical test. Here, n is from 1 to 30 when the radius of the sphere is 10λ or less. The computer used in our simulation has a CPU frequency of 2GHz, each of the simulations ( n = 30 ) took nine hours to finish. When k2 = k4 = k1 , as shown in Figure 8.13, the waves radiated from the dipole will not focus. When k2 = k4 = 2k1 , as shown in Figure 8.14, refractive waves focus in each region, with the density of the focused area becoming less. Most of the refractive waves radiate. When k2 = k4 = −k1 , as shown in Figures 8.17 and 8.18, because of the negative 123 Characterization of Spherical Metamaterials Chapter 8 Simulation Results 124 refraction, the refractive waves focus on the area near the interface in every region, and then they radiate. We can find from the figures that the waves focus very well in this case. When k2 = k4 = −2k1 , as shown in Figures 8.20 and 8.21, the angles between the refractive waves and the normal in this case are much smaller than the angles in the case k2 = k4 = −k1 . The refractive waves and the incident waves are in the same side of the normal because of the negative refraction. Therefore, the waves focus slower than the k2 = k4 = −k1 case, and much more energy will be radiated because they do not been focused. From the discussion above, we know that the spherical metamaterials are able to focus waves. The smaller the absolute value of k is, the stronger the focus capability is. We also see that the normal spheres can also focus waves to some extent, especially when k is big, which is contrary to the metamaterial cases. Because it causes singularity when the dipole is located in the center of the spheres, the results of these cases have already been studied, so we only pay attention to the cases when the dipole is located outside the spheres in this chapter. 124 Characterization of Spherical Metamaterials Chapter 9 Summary 125 9 SUMMARY In this degree thesis project, metamaterial spherical multilayers are studied by using the spherical dyadic Green’s function. Basic theories of metamaterials, the Green’s functions and scattering of spheres are described. The special and general structures of spherical multilayers are examined. Metamaterials are very different materials from the conventional materials. They behaves with very interesting characteristics. Metamaterials have negative refractive index. They can maketheoretically perfect lens. Electromagnetic waves travel in them with quite different routes from the normal materials. Spherical dyadic Green’s functions are described in Chapter 3. Eigenfunctions used for dyadic Green’s functions in unbounded media, i.e., free space DGF, are first given. Then, the scattering dyadic Green’s functions for spherical multilayered media are described. To obtain the coefficients of the DGF, the recurrence matrix equations for the coefficients of scattering dyadic Green’s functions are given. The scattering of normal material spheres is introduced in Chapter 4. Rayleigh scattering and Mie scattering are described firstly. These two kinds of scattering are the basis of understanding the scattering of spheres. In succession, scattering of two separate spheres as a special example is analyzed in particular. 125 Characterization of Spherical Metamaterials Chapter 9 Summary 126 Three cases of the spherical multilayers are considered respectively: the single sphere, the spherical shell and the spherical multilayers (general case). Both the conventional materials and the metamaterials are studied. Two circumstances where the dipole is located outside the structure, and the dipole locates inside the structure are examined, respectively. The analysis and discussion of the simulation results are given afterwards. This project can be extended to cylindrical metamaterials case, which is a very promising area. 126 Characterization of Spherical Metamaterials Reference 127 REFERENCES [1] Y. Zhang, T. M. Grzegorczyk, and J. A. Kong, “Propagation of electromagnetic waves in a slab with negative permittivity and negative permeability”, Progress in Electromagnetics Research, PIER 35, pp. 271-286, 2002. [2] J. A. Kong, “Electromagnetic wave interaction with stratified negative isotropic media”, Progress In Electromagnetic Research, PIER 35, pp. 1-52, 2002. [3] V.G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeability”, Soviet Physics Uspekhi 10, pp. 509-514, 1968. [4] C. D. Moss, T.M. Grzegorczyk, Y. Zhang and J.A. Kong, “Numerical studies of left handed metamaterials”, Progress In Electromagnetic Research, PIER 35, pp. 315334, 2002. [5] S. Kozaki, “A new expression for the scattering of a Gaussian beam by a conducting cylinder”, IEEE Transactions on Antennas and Propagation, Vol.AP-30, pp. 881887, September 1982. [6] J. A. Kong, B. L. Wu and Y. Zhang, “Lateral displacement of a Gaussian beam reflected from a grounded slab with negative permittivity and permeability”, Applied Physics Letters, Vol. 80, No.12, pp. 2084-2086, Mar. 2002. [7] D. R. Smith and N. Knoll, “Negative refractive index in left-handed materials”, Physical Review Letters, Vol. 85, No 14, pp. 2933-2936, Oct. 2000. [8] J. B. Pendry, “Negative refraction makes a perfect lens”, Physical Review Letters, Vol. 85, No 18, pp. 3966-3969, 30 Oct. 2000. [9] R. A. Shelby, D.R. Smith and S. Schultz, “Experimental verification of a negative index of refraction”, Science, Vol. 292, pp. 77-79, April 2001. [10] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser and S. Schultz, “A composite medium with simultaneously negative permeability and permittivity”, Physical Review Letters, Vol. 84, No 18, pp. 4184-4187, May 2000. [11] J. Yonekura, M. Ikeda, and T. Baba, “Analysis of finite 2-D photonic crystals of columns and lightwave devices using the scattering matrix method”, Journal of Lightwave Technology, Vol. 17, No.8, pp. 1500-1508, August 1999. 127 Characterization of Spherical Metamaterials Reference 128 [12] A. K, Iyer, and G.V. Eleftheriades, 2002 IEEE MTT-S Digest, pp.1067-1070, 2002. [13] R.W. Ziolkowski, D. P. Marsland, L. F. Libelo, JR. and G. E. Pisane, “Scattering from an open spherical shell having circular aperture and enclosing a concentric dielectric sphere”, IEEE Transaction on Antennas and Propagation, Vol.36, No.7, pp. 985-999, July 1988. [14] J. A. Kong, B. L. Wu and Y. Zhang, “A unique lateral displacement of a Gaussian beam transmitted through a slab with negative permittivity and permeability”, Microwave and Optical Technology Letters, Vol.33, No.2, pp. 136-139 Apr. 2002. [15] J. Pacheco, Jr., T. M. Grzegorczyk, B. I. Wu, Y. Zhang and J.A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media”, Physical Review Letters, Vol. 89, No.25, pp. 257401-1-4, December 2002. [16] C. Li and Z. Shen, “Electromagnetic scattering by a conducting cylinder coated with metamaterials”, Progress In Electromagnetics Research, PIER 42, pp. 91-105, 2003. [17] L. W. Li, D. You, M. S. Leong and T. S. Yeo, “Electromagnetic scattering by multilayered chiral-media structrures”, Progress In Electromagnetics Research, PIER 26, pp. 249-291, 2000. [18] L. W. Li, Y. Dan, M. S. Leong and T. S. Yeo, “Plane wave scattering by an achiral multilayered sphere in an infinitely extended chiral host medium”, Progress In Electromagnetics Research, PIER 33, pp. 261-298, 2001. [19] Y. L. Xu and Bo A.S. Gustafson, “An analytical solution to electromagnetic multisphere-scattering”, Department of Astronomy, P.O.Box 112055,University of Florida, Gainesville, FL 32611-2055. [20] C. Liang and Y. T. Lo, “Scattering by two spheres”, Radio Science, Vol.2 (New Series), No.12, pp. 1481-1495, December 1967. [21] L. W. Li, M. S. Leong, T. S. Yeo, and P. S. Kooi, “Electromagnetic Dyadic Green’s functions in spectral domain for multilayered cylinders”, J.Electromagn. Waves Applicat., Vol. 14, No.7, pp. 961-986, July 2000. [22]S. Kozaki, “High frequency scattering of a Gaussian beam by a conducting cylinder”, IEEE Transactions on Antennas and Propagation, Vol.AP-31, pp.795-799, Sep. 1983. 128 Characterization of Spherical Metamaterials Reference 129 [23] A. Z. Elsherbeni, G. Tian, and M. Hamid, “Scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders”, IEEE Southeastcon '92, Birmingham, AL, pp. 542-545, April 1992. [24] C. T. Tai, Dyadic Green Functions in Electromagnetic Theory, IEEE Antenna and Propagation Society and IEEE Microwave Theory and Techniques Society, Cosponsors, The Institute of Electrical and Electronics Engineers, Inc., New York. [25] W. C. Chew, Waves and Fields in Inhomogeneous Media, University of Illinois Urbana-Champaign, Van Nostrand Reinhold, New York. [26] L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “Electromagnetic Dyadic Green’s functions in spherically multilayered media”, IEEE Transactions on Microwave Theory and Techniques, Vol. 42, No. 12, pp. 2302-2310, Dec. 1994. 129 Characterization of Spherical Metamaterials [...]... (field) layer ( f = 1, 2, , N ) of the spherically N -layered system Figure 3.1 Geometry of a spherically multilayered medium 18 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 19 3.2 FUNDAMENTAL FORMULATION 3.2.1 EIGENFUNCTION EXPANSION OF DGF IN UNBOUNDED MEDIA Electromagnetic problems can always be solved by beginning with the analysis of Maxwell’s equations The... of the refractive index of a nearly transparent and passive medium is usually taken to have only positive values Through an analysis of a current source radiating into a metamaterial, it can be determined that the sign of the real part of the refractive index is actually negative The regime of negative index leads to unusual electromagnetic wave 5 Characterization of Spherical Metamaterials Chapter 2... xz plane for a 30 incidence of a Gaussian beam upon a grounded slab of thickness d = 6λ with ε1 = −ε 0 and µ = − µ0 This is the simulated result by using the formula developed in reference [6] 16 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 17 3 SPHERICAL DYADIC GREEN’S FUNCTIONS 3.1 INTRODUCTION Green’s function is named in honor of English mathematician and... geometry requiring a new formulation This is the main limitation of Green’s function technique 17 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 18 This chapter will present a detailed description of electromagnetic dyadic Green’s function in spherically multilayered media Figure 3.1 shows the spherically N -layered geometry The transmitter with an arbitrary... simulation results of the single sphere and spherical multi layers are given Chapter 9 gives a summary of the whole thesis 1.4 ORIGINAL CONTRIBUTION Conference paper: 3 Characterization of Spherical Metamaterials Chapter1 Introduction 4 Le-Wei Li, Ningyun Huang, Qun Wu and Zhong-Cheng Li, "Macroscopic Characteristics of Electromagnetic Waves Radiated by a Dipole in the Presence of Metamaterial Sphere... incidence of a Gaussian This is the simulated result by using the formula developed in reference [6] 15 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 16 2 0 x/λ -2 -4 -6 -8 -10 -2 -1 0 1 2 3 4 5 6 7 8 9 10 z/λ Figure 2.5 Time-averaged power density on the xz plane for a upon a grounded slab of thickness d = 6λ with ε1 = −ε 0 30° incidence of a Gaussian... [7]: ε (ω ) = 1 − ω 2p , ω2 (2.6) where the plasma frequency ω p is related to the geometry of the wire array 7 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 8 We can assume that the wire and the SRR arrays do not interact directly, the index of the refraction of this structure can be presented as: n(ω ) = ε (ω ) µ (ω ) Thus, it can be obtained that... Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 22 the multiple reflected and transmitted waves due to the presence of the spherical boundary of the dielectric layered media Using the contour integration method in the complex h -plane, the dyadic Green’s function in the unbounded medium can be represented in terms of the normalized spherical vector wave functions as a result of the... 1, 2, , N ) are the coefficients of the scattering DGF to be solved, and the superscript N is the number of the layers of the multi-layer medium Same as the expression of the unbounded dyadic Green’s function, the superscript (1) indicates that the third-type spherical Bessel function or the first-type spherical Hankel function should be chosen in the function of the spherical wave vector functions:... a Dipole in the Presence of Metamaterial Sphere ({Invited})", Proc of 5th Asia-Pacific Engineering Research Forum on Microwaves and Electromagnetic Theory, Kyushu University, Fukuoka, Japan, July 29-30, 2004 4 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 5 2 BACKGROUND INTRODUCTION OF METAMATERIAL After metamaterial was first introduced in 1968 by ... layer ( f = 1, 2, , N ) of the spherically N -layered system Figure 3.1 Geometry of a spherically multilayered medium 18 Characterization of Spherical Metamaterials Chapter Spherical Dyadic Green’s... contribution of 21 Characterization of Spherical Metamaterials Chapter Spherical Dyadic Green’s Function 22 the multiple reflected and transmitted waves due to the presence of the spherical boundary of. .. limitation of Green’s function technique 17 Characterization of Spherical Metamaterials Chapter Spherical Dyadic Green’s Function 18 This chapter will present a detailed description of electromagnetic