Dipole Mode Analysis of Arrays of Dielectric and Plasmonic Particle Metamaterials. Seyedeh Shabnam Ghadarghadr Jahromi

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2 Dipole Mode Analysis of Arrays of Dielectric and Plasmonic Particle Metamaterials by Seyedeh Shabnam Ghadarghadr Jahromi Submitted to the Department of Electrical and Computer Engineering in partial fulfillment of the requirements for the degree of Doctorate of Philosophy in Electrical and Computer Engineering at the Northeastern University April 2010 c Northeastern University All rights reserved. Author Department of Electrical and Computer Engineering April 15, 2010 Certified by Hossein Mosallaei Assistant Professor Thesis Supervisor Certified by Anthony Devaney Professor Thesis Committee Certified by Carey Rappaport Professor Thesis Committee Accepted by Ali Abur Chairman, Department of Electrical and Computer Engineering

3 2 To my beloved family

4 Dipole Mode Analysis of Arrays of Dielectric and Plasmonic Particle Metamaterials by Seyedeh Shabnam Ghadarghadr Jahromi Submitted to the Department of Electrical and Computer Engineering on April 15, 2010, in partial fulfillment of the requirements for the degree of Doctorate of Philosophy in Electrical and Computer Engineering Abstract This dissertation presents a theoretical study and numerical evaluations of design and applications of metamaterials. We begin with a brief review of the history of metamaterials and then investigate the design and applications of metamaterials in electromagnetics and optics. The thesis is divided into two main parts part I (chapters 2, 3, 4, and 5) and part II (chapters 6 and 7). The first part is devoted to the design and development of metamaterials. The second half is dedicated to the applications of metamaterials in microwave and optical frequencies. To achieve a metamaterial with desired figure of merit, one needs to create appropriate electric and magnetic dipole moments (in small scale) and then tailor them to the application of interest. Arraying these dipole moments can lead to the required material properties. Since dielectric spheres have the potential to offer both electric and magnetic dipole modes, I consider an array of dielectric spheres. To investigate the behavior, I present a comprehensive method to solve the problem of multiple scattering of a plane wave incident on an arbitrary configuration of N spheres. By using multipole expansion method, I show that, if the spheres are small enough, such that the magnitude of higher order modes are negligible compared to the dipole modes, it is enough to consider only the first electric and magnetic modes. Namely, we can model each sphere with electric and magnetic dipolar modes. I then present an analytical solution for the problem of plane-wave scattering by 3D arrays of small spheres. We theoretically investigate the characteristics of electromagnetic (EM) waves supported by three dimensional (3D) periodic arrays of dielectric and magneto-dielectric spheres. It is assumed that either the spheres are sufficiently small or the frequency is such that only the dipole scattered modes are excited. Imposing the boundary conditions, will determine the required equations for obtaining kd βd dispersion characteristics. To ease the computer calculations, we transform slowly convergent summations to rapidly ones. A metamaterial constructed from unit-cells of two different spheres is created, where one set of spheres develops electric modes, and the other set establishes magnetic modes. It is demonstrated that such a composite of high dielectric spheres provides double negative (DNG) metama- 3

5 terial in a narrow frequency band spectrum where kd, βd < 1. The developments of DNG and backward wave metamaterials utilizing 3D array of magneto-dielectric spheres located inside free space, and dielectric spheres embedded in a negative µ host, are also addressed. The use of magnetic materials allows accomplishing wider dispersion characteristic bandwidths and tunable feasibility. Further, I investigate the creation of backward-wave metamaterial in optical frequencies. A three dimensional (3D) array of dielectric particles is designed, where the spheres operate in their magnetic modes and their couplings offer electric modes. We obtained electric and magnetic scattering coefficient resonances around the same frequency band. It is highlighted how a meta-patterned structure constructed from dielectric nanosphere unit-cells can provide electric and magnetic modes resulting in backward wave phenomenon. A comprehensive circuit model based on the RLC (resistor, inductor, and capacitor) realization is presented to successfully analyze the scattering performance of a dielectric nanosphere. The engineered dispersion diagram for a 3D array of identical highly coupled nanospheres is scrutinized, verifying that the high couplings between spheres can offer the backward wave characteristics. Metamaterials show potential for several novel applications in electromagnetics and optics. In this thesis, I present novel applications of metamaterials in building antenna devices. This works reviews the performance of small antennas enabled by metamaterials and directive nano-antennas. I first start with a hemispherical negative permittivity (ENG) resonator fed by a coaxial probe and a hemispherical negative permeability (MNG) resonator antenna using aperture coupling for excitation. For both cases exact and approximated Green s functions for the evaluation of the input impedances are obtained and expressed in a convenient form for numerical computation. It is illustrated how a resonator composed of negative permeability/permittivity medium can successfully establish a small antenna element. To achieve an antenna with higher impedance bandwidths or lower Q, a novel design is proposed. I show that if I embed the dipole antenna inside a core-shell structure, with magnetic shell and dielectric core, a quality factor as low as 1.08 times the Chu limit can be achieved. The Q of the antenna is attained with the use of Green s function analysis and input impedance as a function of frequency. The obtained observations may provide road maps for the future design of metamaterial-based subwavelength antennas. In addition to the applications of metamaterials in microwave frequencies, I also investigate the design and modeling of nano-antennas using plasmonic particles. To build antennas in optical frequency it has been suggested to use plasmonic particles (negative permittivity). Here, I theoretically characterize the performance of array of plasmonic core-shell nano-radiators located over layered substrates. Engineered substrates are investigated to manipulate the radiation performance of nanoantennas. I developed a rigorous analytical approach for the problem at hand by applying Green s function analysis of dipoles located above layered materials. I highlighted the effect of the dielectric substrate layers on the radiation performance. It is obtained that a dielectric substrate can drastically change the radiation pattern of a plasmonic nanoantenna from what is obtained for that when is located in free space. Integrating a silver layer can reduce the antenna interaction with the dielectric sub- 4

6 strate and suppress the back radiation. A composite substrate for an optimized 2D array of four-plasmonic nano core-shell radiators is engineered to tailor the dipolar modes of the nanoantennas and accomplish a pencil beam radiation. I established that by novel arraying of nano-particles and tailoring their multilayer substrates, one can successfully engineer the radiation patterns and beam angles. Thesis Supervisor: Hossein Mosallaei Title: Assistant Professor Thesis Committee: Anthony Devaney Title: Professor Thesis Committee: Carey Rappaport Title: Professor 5

7 Acknowledgments This work can not be accomplished without the support of many people. First and foremost, I must thank my family for their support and encouragement. My mother, Jila, deserves more credit than I can ever adequately acknowledge. Her support, critiques and care packages certainly sustained my effort throughout. I am also extremely fortunate to have two sisters capable of recasting my overwhelming trepidations into surmountable challenges. Especially I would like to thank my husband, Ali for his encouragement and understanding during the course of my studies. I wish to thank Prof. Mosallaei as he taught me gain the knowledge of science. My thanks also go to my colleagues and team mates, for their invaluable discussions and helps. I would like to thank all the wonderful staff at Electrical Engineering Department. In particular my special thanks to Ms. Faith Crisley and Ms. Linda Bonda, for their assistance as I found my way through the administrative maze of graduate school. Without all these contributions, indeed this work could not have been completed in the present status. 6

8 Contents 1 Introduction Metamaterials: Design and Development Dielectric and Magneto-Dielectric Based Metamaterials: Design and Performance Optical Metamaterials Various Applications of Metamaterials Antennas at Microwave Frequencies Novel Antenna Design at Optical Frequency Single-Sphere Electric and Magnetic Dipole Modes Electric and Magnetic Dipole Creation RLC Modeling Simulation Results Multiple Scattering by an Arbitrary Configuration of Spheres Modeling and Analysis Numerical Results Dipole Modes: Array of Spheres Conclusion Dispersion Diagram Characteristics of Periodic Array of Dielectric and Magnetic Materials Based Spheres Theoretical Model

9 4.2 Performance Analysis of Dielectric and Magneto-Dielectric Spheres Two-Sets and One-Set of Dielectric Spheres Magnetic Material Based Array of Spheres Summary D Array of Dielectric Spheres Manipulating Optical Metamaterials Characteristics Potential Applications of Electric and Magnetic Modes Optical Performances of Array of Coupled Spheres Dispersion Diagram Conclusion Metamaterial-Based Electrically Small Antennas Introduction Electrically Small Antenna: Monopole Inside Hemisphere Green s Function Analysis Small Antennas Enabled by Negative Permittivity Metamaterials Results and Physical Insight Aperture Coupled Hemispherical Electrically Small Antennas Computed Results Wide-Band Electrically Small Antennas Dipole Antenna Inside a Spherical Core-Shell General Definition of Bandwidths and Q Antenna Designs Conclusion Plasmonic Nanoparticles on Layered Substrates: Modeling and Radiation Characteristics Background Theory and Modeling Antenna Applications

10 7.3 Conclusion Conclusion and Possible Future Work 160 A Lowest Limits for Q 164 A.1 Minimizing Q A.2 Q Derivation : T M 01 Mode Bibliography 171 9

11 List of Figures 2-1 Geometry of a single sphere in the presence of plane wave Vector spherical wave functions field lines, (a) M 1 field lines, (b) N 1 field lines Comparison between FDTD and theory for a single sphere of ka = 1, ɛ = 4.431, (a) Scattering cross section in φ = 0 plane, (b)scattering cross section in φ = π/2 plane. In both figures, the blue solid line represent the theory while the red dashed line shows the FDTD values RLC schematics for Mie scattering coefficients, (a) Electric scattering coefficient, (b) Magnetic scattering coefficient Electric scattering coefficients for a single GaP sphere of radius a = 85 nm and ɛ r = Comparison between RLC ladder-type model versus exact Mie solution, (a) Magnitude, (b) phase Magnetic scattering coefficients for a single GaP sphere of radius a = 85 nm and ɛ r = Comparison between RLC ladder-type model versus exact Mie solution, (a) Magnitude, (b) phase Configuration of an aggregate of spheres Configuration of an aggregate of spheres Comparison of theoretical calculations with laboratory scattering measurements from [1] for angular distributions of polarization components of scattered intensity by two identical spheres with inter-sphere distance of Each sphere has a size parameter of ka = and a refractive index of n = i, (a) log(i 11 ), (b) log(i 22 )

12 3-4 Comparison of theoretical calculations with laboratory scattering measurements from [1] for angular distributions of polarization components of scattered intensity by two identical spheres with inter-sphere distance of Each sphere has a size parameter of ka = and a refractive index of n = i, (a) log(i 11 ), (b) log(i 22 ) Comparison of theoretical calculations with laboratory scattering measurements from [2] for angular distributions of polarization components of scattered intensity by two contacting identical BK7 glass spheres. Each BK7 sphere has a size parameter of ka = 7.86 and a refractive index of n = i, (a) log(i 11 ), (b) log(i 22 ) The configuration of an array of dielectric nanospheres Electric scattering coefficients for a Gap nanosphere with ɛ r = and a = 85 nm. (a) Normalized Mie scattering coefficient for the first three terms. The performance of electric scattering coefficient for a sphere inside a finite array with : L x = 4, L y = 4, L z = 2 and unitcell size of d/a = 2.23 and h x = h y = 2.1a, (b) Magnitude and, (c) Phase Magnetic scattering coefficients for a GaP nanosphere with ɛ r = and a = 85 nm. (a) Normalized Mie scattering coefficient for the first three terms. The performance of magnetic scattering coefficient for a sphere inside a finite array with : L x = 4, L y = 4, L z = 2 and unit-cell size of d/a = 2.23 and h x = h y = 2.1a, (b) Magnitude and, (c) Phase Geometry of two-sets of dielectric spheres Electric and magnetic Mie scattering coefficients of a single sphere for the first three modes with, (a) a = 0.5 cm and ɛ r = 40, (b) a = 0.5 cm and ɛ r = 21, (c) Comparison between FDTD results and theoretical approach for two-sets of dielectric spheres with a 1 = a 2 = 0.5 cm, d/a = 3, h x /a = h y /a = 5, h = h x /2, ɛ r1 =40 and ɛ r2 =

13 4-3 Electric and magnetic Mie scattering coefficients of a single sphere with, (a) a = 0.5 cm and ɛ r = 20.5, (b) a = cm and ɛ r = Dispersion diagrams for two-sets of dielectric spheres having the same size (a 1 = a 2 = 0.5 cm) and different dielectric material, (a) The performance of the first set described with ɛ r1 = 40 and d = 1.1 cm, h x = h y = 2.5 cm, (b) The performance of the second set described with ɛ r2 = 20.5, d = 1.1 cm and h x = h y = 2.5 cm, (c) The performance of the combinations of the two-sets with h = h x / (a) Dispersion diagrams for two-sets of dielectric spheres having the same size and different dielectric material ɛ r1 = 95, ɛ r2 = 46, a 1 = a 2 = 0.5 cm, d/a = 2.1, h x = h y = 2.5 cm and h = h x /2, (b) DNG backward wave spectrum shown in a larger scale Dispersion diagrams for two-sets of dielectric spheres having the same dielectric material (ɛ r1 = ɛ r2 = 40) and different radii, (a) The performance of the second set described with d = 1.1 cm, a = cm and h x = h y = 2.5 cm (The performance of the first set has been already demonstrated in Fig. 4-4(a)), (b) The performance of the combination of the two-sets when h = h x / Dispersion diagrams for two-sets of dielectric spheres having the same dielectric materials (ɛ r1 = ɛ r2 = 40) and different radii, (a) The performance of the first set described with d = 1.1 cm, a = 0.5 cm and h x = h y = 1.8 cm, (b) The performance of the second set described with ɛ r = 40, d = 1.1 cm, a = cm and h x = h y = 1.8 cm, (c) The performance of the combination of the two-sets with h = h x / Dispersion diagram characteristic for the second branch of array of highly coupled cubical unit-cell spheres having ɛ r = 20, a = 0.5 cm and h x = h y = d

14 4-9 (a) Mie scattering coefficients for a single magneto dielectric sphere with ɛ r = 18, µ r = 5, a = 0.85 cm., (b) Dispersion diagram characteristic for one-set of magneto-dielectric spheres with : ɛ r = 18, µ r = 5, a = 0.85 cm, d/a = 2.1 and h x = h y = d Effective permittivity and permeability for 3D array of magneto-dielectric spheres with cubical unit cell and a = 0.85 cm, ɛ r = 18, µ r = 5, d/a = 2.1. The bulk effective materials are plotted in the region where both kd, βd < (a) Plot of the Lorentzian permeability function given by (4.16). Dispersion diagram characteristic for one-set of dielectric spheres embedded in a Lorentzian host medium with cubical unit-cell and, a = 0.85 cm, d = h x = h y, d/a = 2.1, (b) ɛ r = 20, (c) ɛ r = Effective parameters for a cubical 3D array of dielectric spheres described with, ɛ r = 90, a = 0.85 cm, d/a = 2.1, inside a negative permeability material with Lorentzian function given by (4.16). The bulk effective materials are derived by using the Lewin mixing formula [3]. They are plotted in the region where both kd, βd < RLC circuit symbolizing the interactions between spheres (l l z ). Equivalent circuits for Σ ll, Σ llz, and Σ ll z The performance of the interaction coefficients Σ llz. Comparison between the exact solutions and RLC circuit model. (a) R ll (kd), (b) X ll (kd), (c) X llz (kd, 1) (l l z ) The performance of the coupling coefficients Σ ll z. Comparison between the exact solutions and RLC circuit model (l l z ). (a) R ll z (kd, 1), (b) X ll z (kd, 1)

15 5-4 Performance of a periodic array of GaP nanospheres with three layers in the direction of propagation (L z = 2) and ɛ r = 12.25, a = 85 nm, having unit-cell size of d = 2.23a and h x = h y = 2.1a, (a) Magnitude of 1./ 1 and 1./ 2 defined in (5.6), (b) Electric scattering coefficients (magnitude) and, (c) Magnetic scattering coefficients (magnitude) Performance of a periodic array of GaP nanospheres with three layers in the direction of propagation (l l z ) and ɛ r = 12.25, a = 85 nm, having unit-cell size of d = 2.23a and h x = h y = 2.1a, (a) Phases of 1/ 1 and 1/ 2 defined in (5.6), (b) Electric scattering coefficients (phase) and, (c) Magnetic scattering coefficients (phase) Electric scattering coefficients behavior for a periodic array of GaP nanospheres with three layers in the direction of propagation and ɛ r = 12.25, a = 85 nm, d = 2.23a versus different unit-cell sizes in the transverse direction (only the middle layer s scattering coefficient is plotted), (a) Magnitude, (b) Phase Magnetic scattering coefficients behavior for a periodic array of GaP nanospheres with three layers in the direction of propagation and ɛ r = 12.25, a = 85 nm, d = 2.23a versus different unit-cell sizes in the transverse direction (only the middle layer s scattering coefficient is plotted), (a) Magnitude, (b) Phase Performance of electric scattering coefficients for a periodic array of GaP nanospheres with three layers in the direction of propagation and ɛ r = 12.25, a = 85 nm, h x = h y = 2.23a versus different unit-cell sizes in the propagation direction (only the middle layer s scattering coefficient is depicted), (a) Magnitude, (b) Phase Performance of magnetic scattering coefficients for a periodic array of GaP nanospheres with three layers in the direction of propagation and ɛ r = 12.25, a = 85 nm, h x = h y = 2.23a versus different unit-cell sizes in the propagation direction (only the middle layer s scattering coefficient is depicted), (a) Magnitude, (b) Phase

16 5-10 Dispersion diagrams characteristics for the 3D array of nanospheres (a) Comparison between our approach and the technique introduced in [4] for an array of nanospheres with ɛ r = 20, a = 0.5 cm and cubical unitcell with a/d = A very good match is achieved, (b) Engineered dispersion diagram for an array of GaP nanospheres with ɛ r = 12.25, a = 85 nm and unit-cell size with d/a = 2.23 and h x = h y = 2.1a. The second branch shows the backward wave behavior The antenna configuration, a probe fed hemispherical resonator antenna, and the equivalent geometry of the antenna configuration Input impedance and return loss vs. frequency using delta gap source model. (a) Input impedance. (b) Return loss: a = 12.5 mm, b = 6.4 mm, l = 6.5 mm, ɛ r = 9.8, r 1 = mm comparison between approximation and numerical results for the Input impedance (real part and imaginary part) of a hemispherical resonator filled with a metamaterial described by Drude dispersion relation: a=7.5mm, l=6mm, r 1 =0.4mm, f p = 4GHz, γ p = 0.001ω p Return loss for a hemispherical resonator filled with a metamaterial described by Drude dispersion relation: a=7.5mm, l=6mm, r 1 =0.4mm, f p = 4GHz, γ p = 0.001ω p The Geometry of the aperture-coupled hemispherical antenna Real of the permeability/permittivity material inside the sphere described by Laurentz/Drude dispersion relation Input Impedance (Real part and Imaginary part) for a hemispherical resonator filled with a metamaterial described by Lorentz dispersion relation: a=7.5mm, L=11mm, W=0.9mm, Ls=0.57, d=0.635mm, W f = 1.45mm, f h = 2GHz,γ h = 0.001ω h and κ =

17 6-8 Return loss for a hemispherical resonator filled with a metamaterial described by Lorentz dispersion relation: a = 7.5mm, L =12mm,W = 0.9mm, Ls = 0.57,d=0.635mm,W f = 1.45mm,f h = 2GHz, γ h = 0.001ω h, κ = The antenna configuration, a dipole antenna inside a spherical coreshell resonator The input resistance and reactance. The antenna composed of a dipole of length 2l = 9mm, located at the center of a spherical core-shell resonator. Where a 1 = 5.5 mm,a 2 = 7.5 mm, ɛ r1 = 10 and µ r2 = (a) Scattering term for the first three modes. (b) The input resistance and reactance. The antenna composed of a dipole with total length of 2l = 9mm, located at the center of a spherical resonator filled with high dielectric material of ɛ r1 = 100. The radius of the sphere is a 1 = 7.5 mm The return loss of the antenna in Fig The return loss presents a very narrow-band bandwidths which is consistent with the Q= (a) Scattering term for the first three modes. (b) The input resistance and reactance. The antenna composed of a dipole with total length of 2l = 9mm, located at the center of a spherical resonator filled with magneto-dielectric material of ɛ r1 = 12 and µ r1 = 8. The radius of the sphere is a 1 = 7.5 mm The return loss of the antenna in Fig The return loss presents a very narrow-band bandwidths which is consistent with the Q= (a) Scattering term for the first three modes. (b) The input resistance and reactance. The antenna composed of a dipole with total length of 2l = 9mm, located at the center of a spherical resonator filled with magnetic material of µ r1 = 60. The radius of the sphere is a 1 = 7.5 mm The return loss of the antenna in Fig The return loss presents a very narrow-band bandwidths which is consistent with the Q=

18 6-17 (a) Scattering term for the first three modes. (b) The input resistance and reactance. The antenna composed of a dipole of length 2l = 9mm, located at the center of a spherical core-shell resonator. Where a 1 = 6.5 mm,a 2 = 7.5 mm, ɛ r1 = 4 and µ r2 = The return loss of the antenna in Fig The return loss presents a very narrow-band bandwidths which is consistent with the Q= The scattering term vs. different dielectric material of the core. The core-shell is described as follows: a 1 = 6.5 mm, a 2 = 7.5 mm, µ r2 = 90. A dipole of length 2l = 9 mm is located at the center of the core-shells (a) Scattering term for the first three modes. (b) The input resistance and reactance. The antenna composed of a dipole of length 2l = 9mm, located at the center of a spherical core-shell resonator. Where a 1 = 6.5 mm,a 2 = 7.5 mm, ɛ r1 = 60 and µ r2 = The return loss of the antenna in Fig The return loss presents a relatively large-bandwidths which is consistent with the Q = = 1.08Q chu The normalized near field behavior (in db) of antenna defined in Fig in y-z plane. The vertical axis is z (mm) while the horizontal axis is y (mm) Array of plasmonic nanoantennas located above a layered substrate Magnitude and phase of normalized polarizability of a plasmonic coreshell sphere versus r 1 /r 2 ratio for different frequencies for a structure with r 2 = 0.1λ, ɛ rc = i and silver coating. (a) Polarization magnitude, and (b) Polarization phase, (c) Silver permittivity behavior Maximum percentage of error in obtaining the integrals in (7.7) and (7.8) using our theoretical model compared to their exact values for different observation points. ρ and z 0 denote the location of the observation point

19 7-4 (a) A Yagi-Uda type antenna constructed from a dipole source exciting two nano core-shells, operating at frequency f = 650 T Hz. The reflector s radii ratio is r 1r /r 2r = 0.75 and the radii ratio of the director is r 1d /r 2d = Both core-shells have core made of SiO 2 (ɛ rc = i) and shell made of silver (ɛ rs = i). The normalized radiated power obtained by theory and FDTD, (b) in xy plane, and (c) in yz plane. A good comparison is observed. Note that FDTD characterizes the actual plasmonic core-shell structure whereas in theoretical model we approximate them with dipole modes Magnitude of E z (db) for the nanoantenna in Fig. 7-4(a) in xy (z = 0) plane and yz (x = 0) plane (a) Nanoantenna configuration in Fig. 7-4(a) located above a slab of InGaAs with the thickness of 0.35λ 0. (b) The 3D radiation pattern of the radiated power. The substrate considerably degrades the antenna radiation pattern Near field distribution for the antenna in Fig. 7-6(a) obtained by using the theoretical approach of this paper, (a) The magnitude of E y (db) in xy plane, (b) The magnitude of E z (db) in yz plane. Field penetration and propagation through the substrate is illustrated (a) A nanoantenna array of a horizontal dipole, p x, and two nano core-shells located above a InGaAs slab of 0.35λ 0 thickness. (b) The 3D radiation pattern. (c) The magnitude of E x (db) in yz plane (a) Configuration of the nanoantenna in Fig. 7-4(a) where the nano core-shells are located above a InGaAs slab of 0.35λ 0 which its top surface is coated with 0.2λ 0 of silver. (b) The 3D radiation pattern. The silver layer suppresses the back radiation Near field distribution for the antenna in Fig. 7-9(a). (a) The magnitude of E y (db) in xy plane, (b) The magnitude of E z (db) in yz plane

20 7-11 (a) A vertical dipole located at the top of a planar layered material. The layered substrate includes a slab of InGaAs with thickness of 0.35λ 0 where its top surface is coated with silver of thickness 0.2λ 0. A third layer of dielectric ɛ r2 is deposited at top of the silver. (b) The angle of maximum radiation in the yz plane vs. the thickness of the dielectric layer. The jump in angle is because the maximum radiation occurs in another direction (a) Nanoantenna configuration in Fig. 7-4(a) located above a 3 layered substrate including InGaAs slab of 0.35λ 0, silver with thickness of 0.2λ 0 and a 0.1λ 0 layer of SiO 2. (b) The 3D radiation pattern of the radiated power. Adding a third layer controls better the radiation pattern Near field distribution for the antenna in Fig. 7-12(a), (a) The magnitude of E y (db) in xy plane, (b) The magnitude of E z (db) in yz plane (a) 4-particle nanoantenna array with an engineered substrate demonstrating a broadside radiation characteristic. The radii ratio for the particles are r 1r /r 2r = 0.72 and r 1d /r 2d = 0.69, and d ref = 0.25λ 0 while d dir = 0.65λ 0. The layered material includes 0.5λ 0 of InGaAs, 0.2λ 0 of silver coating and a dielectric layer (SiO 2 ) with the thickness of 0.1λ 0. (b) The 3D radiation pattern Radiation pattern sensitivity to the change of r 1r for the configuration shown in Fig. 7-12(a). The radiation performance in yz plane, where r 2r = r 2d = 46 nm and r 1d = 30 nm

21 Chapter 1 Introduction The main establishments in the classic electromagnetic science is relatively old and may be traced back to the times of J. C. Maxwell [5] and his contemporaries. Maxwell s most important achievement was classical electromagnetic theory, synthesizing all previously unrelated observations, experiments and equations of electricity, magnetism and even optics into a consistent theory. Recently important advancements have been made in theoretical and numerical aspects of applied electromagnetics. These advancements significantly affected the applications related to this field, such as telecommunications and antennas. In particular, the necessity of going beyond the limitations that standard materials present in nature when applied to radiating systems has become an important issue, due to the increasing demands that requires improving performance of electromagnetic devices. For instance photonic and electromagnetic band-gap (EBG) materials [6]-[9] are known to represent a valid alternative to classic materials for the anomalous ways they interact with electromagnetic waves. Nowadays antennas loaded with EBG materials are one of the most investigated novel technologies in the specialized literature [9]-[12]. By inserting defect in a periodic EBG one can provide an effective way of tailoring the overall material response, which is useful in multiple applications. The main property that distinguishes EBG materials from standard materials, is based on the Bloch wave propagation, related to the periodicity of these artificial materials. Namely the stop-band characteristics of these materials are available at 20

22 frequencies for which the wavelength is comparable with the period of the defects composing the EBG crystal. However, EBG technology, loses its unique properties at wavelengths much smaller than its intrinsic period. This implies that EBG materials are difficultly characterized as compact electromagnetic parameters having physical meaning (like permittivity and permeability) and that they are very sensitive to the periodicity, and therefore technologically challenging. A new class of artificial materials that go beyond these limitations have been the subject of extensive research. Metamaterials are a new class of artificially engineered structures exhibiting superior properties which are not attainable with naturally occurring materials. The prefix meta is originally a Greek work which means beyond. This signify that how such artificial materials are built with the purpose of going beyond the usual electromagnetic behavior of classic materials. Increasing the range of naturally accessible materials and filling the enormous voids that exist in the currently available designs have been the motivating factors for many governmental and industrial agencies to fund several research projects on this topic. Metamaterials have become a new subdiscipline within physics and electromagnetism (especially optics and photonics). They show promise for optical and microwave applications such as new types of beam steerers, modulators, band-pass filters, lenses, microwave couplers, and antenna radomes. Potential applications of metamaterials are diverse and include remote aerospace applications, sensor detection and infrastructure monitoring, smart solar power management, public safety, radomes, high-frequency battlefield communication and lenses for high-gain antennas, improving ultrasonic sensors, and even shielding structures from earthquakes [13]-[15]. To accomplish a new material a mixture of dielectric, magnetic, and metallic elements is used as the building block unit-cell of medium. The new composites with state of the art figures of merit, i.e., ±ɛ and/or ±µ, have significant applications in many exciting areas. The main difference between conventional materials and metamaterials is the resonance property of small inclusions that often compose the material at the frequency of interest. Note that, proper care should be used in in designing the inclusions composing the artificial crystals, such that, at the desired 21

23 wavelength, (which is much larger than the inclusion size) the bulk metamaterial should be close to an overall resonance. These resonance offers interesting applications in many fields [13]-[16]. However, various challenges are still present in metamaterial science, both from a theoretical perspective, and a technological point of view. In the next section, I present novel approaches for design and modeling of metamaterials. 1.1 Metamaterials: Design and Development Significant amounts of research have been performed to realize metamaterials for novel applications [13]-[15]. Among these interesting features are negative index and negative refraction behaviors, usually known as left handed materials (LHMs) [17]. One of the greatest potential of metamaterials is probably the possibility to create a structure with a negative refractive index, since this property is not found in any non-synthetic material. In a LHM, the phase and group velocities are in the opposite directions. Negative refractive index materials appear to permit the creation of superlenses and near field imaging which can have a spatial resolution below that of the wavelength, and a form of invisibility has been demonstrated at least over a narrow wave band [18, 19], and superlensing [17]. To achieve a metamaterial with desired figure of merit, one needs to first create appropriate electric and magnetic dipole moments (in small scale) and then tailor them to the application of interest. Primarily, the electric and magnetic dipole moments are the basic foundations for making metamaterials. Novel arrangements of these dipole moments can lead to the required material properties [20]. So far, most of the metamaterial designs are metallic based [21]. However, there are some major drawbacks due to using metallic inclusions; such as, conduction loss, polarization dependence, and metal frequency dispersive behavior in optics. In addition to these, fabrication of metallic metamaterials are challenging especially in the optical and infrared regimes. Recently, a new paradigm for metamaterial development was introduced by Holloway et al. [22], that a double negative material (DNG) can be designed by 22

24 embedding an array of magneto-dielectric spherical particles in a background matrix. The work of Holloway et al. is based on the mixing formulas, obtained by Lewin [3]. Later on, Venidk et al. proposed a more practical approach utilizing only dielectric particles, where they used two-sets of spheres, having the same dielectric material, but different radii in a building block unit-cell [23, 24]. Ahmadi et al. presented a detailed physical insight of this concept in reference [20]. The electromagnetic fields inside such a composition behave as the superposition of electric and magnetic dipole modes (provided that higher order multipoles are negligible). That is to say, one set offers the electric dipole moments, while the other provides magnetic dipole modes. In the next part, I go over the design and characteristics of periodic arrays of dielectric and magnetic materials based spheres [25] Dielectric and Magneto-Dielectric Based Metamaterials: Design and Performance In this thesis, I present an advanced theoretical approach to characterize the performance of metamaterials created by using dielectric and magneto-dielectric particles. The advantages of dielectric based metamaterials compared to metallic based structures are being free of conduction loss, promise for isotropic pattern, and better potential for fabrication. Furthermore, the dielectric spheres can offer wider bandwidths owing to the larger volume density that they can occupy. I also investigate the use of magnetic materials for metamaterial creation. Using magnetic materials generates unique opportunities to achieve metamaterials with enhanced performance, including wider backward wave bandwidth and tunable characteristic. The goal is to theoretically determine the dispersion diagrams for three dimensional (3D) array(s) of dielectric and magneto-dielectric sphere metamaterials. The intention is to derive the kd βd equation, which studies the propagation of electromagnetic waves supported by the array. In order to achieve the kd βd relation, the problem of multiple scattering of an EM plane wave by an array of spheres is solved utilizing the multipole expansion method [26]. This enables a very efficient 23

25 theoretical model for characterizing the periodic configurations. It is worth mentioning that, one can also apply powerful numerical techniques, such as, finite difference time domain (FDTD) [27]-[29], finite element method (FEM) [30]-[31], and method of moments (MoM) to fully characterize the dispersion characteristics of periodic structures. These methods can be extremely accurate and comprehensive for arbitrary complex geometries. Nevertheless, in compare to these numerical techniques, our theoretical approach is simple enough to successfully predict the physical insights of the composite structure. I generalize the reciprocity method to characterize the case of two different spheres as a unit-cell building block [25, 32, 33]. Implicit transcendental equations for the propagation constant (β) that can exist in the direction of the array axis, are obtained. It is illustrated, how a metamaterial constructed from unit-cells of two different spheres can provide a medium with DNG parameters in a narrow bandwidth frequency spectrum. Note that, the two-sphere lattices treated here, are indeed, pseudo-3d arrays in the sense that all planes of the spheres normal to the array axis are identical. By choosing the same two spheres in one unit-cell, the performance of 3D array of identical spheres is developed. For a 3D array of dielectric spheres, it is exhibited that by bringing the dielectric spheres close to each other, the couplings between them are increased such that, the hybrid modes (the combination of both electric and magnetic dipole moments) and the backward wave performance (for kd > 1) are established. Also, waves with low group velocity are shown to exist in a narrow waveband by arrays of highly coupled spheres. Since the equations are inclusive, based on availability of natural materials, other designs can also be established. For instance, Holloway et al. [22] obtained that one can make a DNG medium from only one-set of spheres having appropriate permittivity and permeability values. We also investigate a structure consisting of magneto-dielectric spheres which can provide the backward wave behavior in the region where kd,βd <1. It is possible to define effective material parameters in such spectrum. Another interesting design is to embed an array of dielectric spheres in a host with natural ferrite material operating above its resonant frequency (where 24

26 the µ is negative). By using this technique, an isotropic metamaterial design with DNG performance is established for kd, βd <1. The obtained structure shows more robustness than double-spheres lattice design in terms of fabrication challenges. Computer calculations are performed to illustrate the theory of wave propagation in the arrays of dielectric and magneto-dielectric spheres, and derive kd βd dispersion characteristics [25, 32]. The series in the dispersion relations are very slowly convergent series, therefore an excessive number of terms is required to obtain β in terms of kd. To facilitate the computer calculations, I use the Floquet mode method to transform slowly convergent summations to rapidly ones [34]. The accuracy of our theoretical model is validated very well with the use of an in-house developed FDTD method [35]. It is demonstrated that the dipolar spherical waves can provide accurate description of the traveling waves supported by the array. In all particular examples studied here, the magnitude of dipole modes scattering coefficients are dominant (compared to the higher order terms); and thus only the electric and magnetic dipole moments are considered for accurate characterization of the spheres. Note that the operating frequencies in above designs are in the Giga hertz range. In the next section the characterizations of arrays of dielectric spheres in optical frequencies will be discussed briefly Optical Metamaterials During the last few years there have been considerable efforts to develop metamaterials in RF/microwave frequencies as we shown in the firs part; however, this progress has not been as rapid as expected in the optical domain. The goal here is to theoretically demonstrate the optical performance of arrays of dielectric nanoparticles enabling coupled electric and magnetic mode features. As mentioned earlier, to achieve an optical material with the functionality of interest one needs to create appropriate electric and magnetic dipole modes in a building block unit-cell [20, 25]. However, the low dielectric material of the nanospheres in optical regimes generates some difficulties in tuning the THz DNG medium, if two sets of nanospheres are used. In the current work, we use one set of dielectric nanoparticles to accomplish the backward wave be- 25

27 havior in THz spectrum. We illustrate that, by bringing the dielectric spheres close to each other, the couplings between them are increased such that both electric and magnetic resonances can be achieved around the same frequency region. Moreover, I obtain accurate closed-form solutions for the scattering coefficients by utilizing RLC (resistor, inductor, and capacitor) circuit modeling. The developed model intuitively describes the physics of the scattering coefficients and their dependence on the material properties and the array configuration. The proposed circuit model, is based on the theoretical derivations and is confirmed by comparing it with the exact solutions. So far, I talked about the design and performance of metamaterial structures, in the next section some of the possible applications of metamaterials will be presented. 1.2 Various Applications of Metamaterials Several applications of metamaterials have been proposed in the literature, where some of which have also been realized experimentally [13]-[15]. It was first suggested by John Pendry et. al, [17] that a superlens could be enabled by negative refractive index, due to: 1. A wave propagating in a negative-refractive medium exhibits a phase advance instead of a phase delay in conventional materials. 2. Evanescent waves in a negative-refractive medium increase in amplitude as they move away from their origin. The first superlens with a negative refractive index provided resolution three times better than the diffraction limit and was demonstrated at microwave frequencies. Subsequently, the first optical superlens was created and demonstrated,[36, 37] but the lens did not rely on negative refraction. Instead, a thin silver film was used to enhance the evanescent modes through surface plasmon coupling. This is one of the applications of metamaterials, below are two other potential applications of metamaterials in antennas and guiding. Antennas are one of the most common electronic devices used in the modern society. An antenna is defined as that part of a transmitting or receiving system 26

28 that is designed to radiate or to receive electromagnetic waves [38]. In other words, an antenna is a device designed to efficiently distribute the energy into the open space so that long distance communication through electromagnetic waves is possible. Since the first real application of antennas, the antenna theories and techniques has experienced dramatic developments. Nowadays, antennas are one of the indispensable devices in many aspects of the modern society Antennas at Microwave Frequencies An interesting aspect of metamaterials is to build antennas with smaller physical size, wider bandwidth and higher efficiency. In recent years, considerable efforts have been devoted towards antenna miniaturization. Fundamentally the ability of any antenna to radiate effectively depends on k 0 a(where k 0 is the wave number in free space and a is the radius of the smallest sphere enclosing the antenna). According to Chu limit [39], the Q of any antenna is inversely proportional to the radius of the smallest sphere which completely surrounds it, so the smaller the radius, the higher the Q and the narrower the bandwidth. Thus, the challenges are to make the physical size of the antenna as small as possible along with a wider bandwidth. Recently there have been some works to embed the antenna inside a negative permittivity material, it can be shown that in this way Chu limit can be approached but not exceeded [40]. Stuart et. al. [40] used the quasi-static analysis to show how ENG material can be a building block for constructing effective small antenna. In this thesis by finding the Green s function for electric fields we study the exact behavior of a dipole antenna embedded inside an ENG sphere. In the same manner the use of negative permeability would also provide us with the same performance however in this case the structure should be dual of the structure used before. Namely instead of electric current we have magnetic current and so on which will be discussed completely. The use of double negative (DNG) materials has been considered previously as a means of improving power radiated by electrically small antenna [41]. Our work is distinguished from earlier studies in that we are specifying using negative permeability materials in addition to use ENG materials. Moreover, I find a direct formula for obtaining 27

29 resonance frequencies for such small antennas. Another novel approach to reduce the quality factor of the antenna and hence increase the bandwidth, is to implement the antenna inside a concentric sphere. Since we have different layers inside the sphere we have more degrees of freedom. If we embed the antenna inside a core-shell sphere with dielectric core and magnetic shell, the bandwidth performance can enhance significantly. To investigate the performance, I used the Green s function analysis along with method of moments. It is worth mentioning that, by using a magnetic shell, we can achieve an electrically small antenna with relatively wide bandwidths in microwave frequencies. Although, the use of ENG material is proven to be effective, but a negative permittivity medium is accessible only at optical frequencies, whereas a magnetic material can be attained in microwave frequencies Novel Antenna Design at Optical Frequency At microwave frequencies, in order to achieve a high gain antenna performance, one needs to use the concept of array antennas where different radiators are tailored in unique arrangements to control the fields amplitudes and phases in the far-field spectrum enabling directive emission. This concept can be realized in optics with the use of plasmonic particles acting like small dipole radiators. Although several results came out recently, it is still an evolving question how we can design devices that radiate and receive electromagnetic signals coherently in the infrared and optical domains. The first difficulty could be the fabrication challenge. As a resonating device, an antenna is often designed to have a size comparable to the wavelength (which is from tens of micrometers-infrared to hundreds of nanometers-visible range) in the surrounding medium. Li et al. [42] used the concept of array antenna for providing a working THz Yagi-Uda antenna in free-space. Similar performance is observed in [43]. From practical point of view, however, one needs to deposit the nano radiators on a substrate. In this case, the radiation performance changes drastically as for instance illustrated for an array of plasmonic rods located above a half-space medium through a numerical analysis [44]. 28

30 It can be of great benefit if one establishes a theoretical formulation for performance analysis of nanoantennas composed of an array of core-shell spherical particles located above layered substrates. This will allow successful study of the complex structure via understanding the physics and obtaining the optimal radiation behavior and beam scanning for the application of interest. To establish the theory of nanoantennas, array of plasmonic core-shells over a layered material is approximated with electric dipoles where their induced dipoles are calculated through the Green s function analysis of dipoles over the layered material [45]. I apply our theory-modeling to obtain the radiation performance of a dipole exciting an array of plasmonic nanoantennas located above a planar layered material. The induced dipole on each nano core-shell is the key parameter determining the antenna radiation characteristics. Optimal patterning of plasmonic particles along with engineering their layered substrate can successfully tailor the optical parameters for achieving a directive emission. It is illustrated that a focused-beam can be obtained for a 2D array of plasmonic particles-nanoantenna deposited on a layered materials and arranged in a Yagi-Uda type configuration. I show that by engineering the layered substrate one can control the radiation beam for focusing it in the desired direction, or suppressing the back radiation [46]-[48]. 29

31 Chapter 2 Single-Sphere Electric and Magnetic Dipole Modes As mentioned in the introduction, to achieve a metamaterial with functionality of interest one needs to create appropriate electric and magnetic dipole moments inside the unit-cell of array configuration. It has been demonstrated that dielectric spheres have the potential to offer both electric and magnetic dipole modes [49, 20]. Arraying dielectric spheres and tailoring the appropriate dipole modes can lead to the required material properties. So it is essential to investigate the behavior of a single sphere. 2.1 Electric and Magnetic Dipole Creation In this chapter, we highlight the concept of electric and magnetic dipoles creation by using dielectric and magneto-dielectric spheres. To achieve a functional optical metamaterial with desired parameters, it is required to first create the appropriate electric and magnetic dipole moments and then tailor them to the applications of interest. Hence, the electric and magnetic dipoles are envisioned as the alphabet for making new structures [25, 50]. To provide a physical insight of dipole modes creation by using dielectric and magneto-dielectric particles, let us begin with the problem of a plane wave and a dielectric and/or magneto-dielectric sphere (refer to Fig. 2-1) centered at (r p, θ p, φ p ). We assume that a sphere with radius a, and 30

32 dielectric constant ɛ rp and µ rp is embedded in free space (in general it can be any other background matrix). A plane wave excitation with the propagation vector k which lies in the xz plane with harmonic time dependence exp (jωt) is assumed. The electric field is considered to be y polarized. The incoming incident field on the sphere can be expanded in terms of spherical wave vectors of the first kind M (1) mn and N (1) mn [26]: n E i p = ηh i p = j n=1 m= n n=1 m= n [A ip mnn (1) n mn(r p, θ p, φ p ) + BmnM ip (1) mn(r p, θ p, φ p )], [A ip mnm (1) mn(r p, θ p, φ p ) + BmnN ip (1) mn(r p, θ p, φ p )], (2.1a) (2.1b) where η is the free space intrinsic impedance and the incident-field expansion coeffi- Figure 2-1: Geometry of a single sphere in the presence of plane wave cients are derived in [51]. For the especial case of end-fire incidence we have [49]: A ip = B ip = 1 j n 2n + 1 n(n + 1) δ m,±1e jk.rp. (2.2) 31

33 Following the same routine, the scattered field can be represented as a weighted sum of spherical wave vector of the third kind M (3) mn and N (3) mn. n E s p = ηh i p = j [A sp mnn (3) n=1 m= n n [A sp mnm (3) n=1 m= n mn(r p, θ p, φ p ) + B sp mnm (3) mn(r p, θ p, φ p ) + B sp mn(r p, θ p, φ p )], mnn (3) mn(r p, θ p, φ p )], (2.3a) (2.3b) A sp mn and B sp mn are the unknown scattered coefficients to be found by matching the boundary conditions at the surface of the nanosphere. Note that, M (1) mn(m (3) mn) and N (1) mn(n (3) mn) are the spherical vector wave functions representing the incoming (outgoing) waves associated with spherical Bessel (Hankel) functions. It must be recalled that, spherical vector wave function M has the circular field lines, that is to say, the radial component of every function M is zero. Hence, if only the A sp mn coefficients are excited, the field has a radial component of E (E N), where the magnetic field is always perpendicular to the radial vector (H M). In other words, the distribution of electromagnetic (EM) or optical fields is such as it would be produced by electric charges on the surface of the sphere. Therefore, A sp mn can be envisioned as electric scattered coefficients [49]. If on the other hand, only B sp mn are excited, the field is such as would be produced by oscillating magnetic charges on the surface of the sphere and the field is said to be of magnetic type. Hence, B sp mn are the magnetic scattered coefficients [49]. The field lines for M 11 and N 11 are shown in Fig Manifestly, the near-field pattern of M 11 is very similar to the electric field pattern of a magnetic dipole while the near-field pattern of N 11 resembles to the electric field pattern of an electric dipole. Hence, if only A sp 11 is excited, the nanosphere is equivalent to an electric dipole moment, where if only B sp 11 exists, the nanosphere reveals the behavior of a magnetic dipole. Note that, the quantities in (2.1) are defined with respect to a local spherical coordinate system. Forcing the boundary conditions at the surface of the nanosphere 32

34 (a) (b) Figure 2-2: Vector spherical wave functions field lines, (a) M 1 field lines, (b) N 1 field lines. directs us to: A sp mn = ς n (a)a ip mn, B sp mn = ξ n (a)b ip mn, (2.4a) (2.4b) where ς n (a) and ξ n (a) are the normalized Mie electric and magnetic scattering coefficients for a single dielectric sphere [49],[52]. ς n (a) = µ rpj n (ka)(k p aj n (k p a)) µ rp ɛ rp j n (k p a)(kaj n (ka)) µ rp h (2) n (ka)(k p aj n (k p a)) µ rp ɛ rp j n (k p a)(kah (2) n (ka)), (2.5a) ξ n (a) = µ rpj n (k p a)(kaj n (ka)) µ r j n (ka)(k p aj n (k p a)) µ rp j n (k p a)(kah (2) n (ka)) µ r h (2) n (ka)(k p aj n (k p a)). (2.5b) Once the scattering coefficients are obtained, we can obtain the scattering cross section. For a single sphere of size ka = 1 I obtain the scattering cross section and compare the theoretical solution for the scattering cross section in two planes (φ = 0, π/2) with the one obtained with FDTD in Fig A very good comparison is achieved. The scattering coefficients are very important in any application that involves the creation and use of electric and magnetic moments. Their resonances are the principal parameters for manipulating the performance of a nanosphere. The formal mathematical solutions for these scattering coefficients are known for years, however 33

35 the physics of these parameters may not be grasped efficiently. It is then, important to develop a model that can easily explain the physical features of these coefficients [53]. In the next section, we present a circuit model to successfully tailor the scattering coefficients. This idea was initially proposed by Alam et al.[54] where they used RLC circuit realization for modeling the scattering coefficients of a plasmonic nanosphere. 2.2 RLC Modeling Over the past few years, there has been much improvement in the fabrication, design, and modeling of nanostructures [55]. Most of the current modeling techniques of nanostructures are based on the discrete dipole approximation (DDA), finite-difference time domain (FDTD), finite element method (FEM), and T-matrix [56]-[59]. Numerical methods can be extremely accurate and comprehensive, but they are somehow time consuming. Theoretical approaches have the great advantage of providing fast solutions, where at the same time they can illustrate the physics and concept of nanostructures. As an example of efficient techniques, Alam et al. [60, 54] investigated an accurate analytical method based on the circuit model analogy to calculate the resonance frequency and scattering parameters of a single plasmonic particle. Hanson et al. [61] suggested another efficient approach for obtaining the electromagnetic interactions between a carbon nanotube and an electrically-small plasmonic sphere (where the sphere is characterized by its dipole moment). The characteristic equations and electric and magnetic resonant phenomena for the nanostructures are derived. An intuitive circuit analogy is also exploited to obtain a better understanding of the resonant behavior of the array of spheres. Accurate closed-form solutions for the scattering coefficients are obtained by utilizing RLC (resistor, inductor, and capacitor) circuit modeling. The developed model intuitively describes the physics of the scattering coefficients and their dependence on the material properties and the array configuration. The proposed circuit model, is based on the theoretical derivations and is confirmed by comparing it with the exact solutions. 34

36 Due to the nature of spherical Bessel functions and their derivatives, Mie s solutions in its present form may not be suitable to characterize the physics of the problem. Expressing the scattering coefficients in terms of polynomial continued fractions helps to effectively elucidate the optical scattering and resonance properties of a nanostructure. Here, we present an accurate circuit model based on the RLC realization for the scattering coefficients of a single dielectric sphere in the optical spectrum. Basically, we reduce the scattering problem to a circuit type problem. Notice that, the scattering coefficients are indeed the input impedance of the circuit model. Let us first expressed the scattering coefficients as: 1 ς n (a) = 1 jx E (n, ka, ɛ r ), (2.6a) 1 ξ n (a) = 1 jx H (n, ka, ɛ r ), (2.6b) where X E and X H are: X E (n, ka, ɛ r ) = y n(ka)(k p aj n (k p a)) ɛ r j n (k p a)(kay n (ka)) j n (ka)(k p aj n (k p a)) ɛ r j n (k p a)(kaj n (ka)), X H (n, ka, ɛ r ) = j n(k p a)(kay n (ka)) y n (ka)(k p aj n (k p a)) j n (k p a)(kaj n (ka)) j n (ka)(k p aj n (k p a)), (2.7a) (2.7b) j n and y n are the spherical Bessel and Neumann functions respectively. Without loss of generality, in this work we only investigate the circuit models for the first electric and magnetic scattering coefficients. Spherical Bessel and Neumann functions of the first order are: j 1 x = sin x x 2 y 1 x = cos x x 2 cos x x, (2.8a) sin x x (2.8b) In order to derive an accurate simple model for the coefficients, the spherical Bessel and Neumann functions are evaluated by using the rational function expansions [62]. Since higher order terms contribute negligible values to the scattering coefficients, we only keep the terms with order of 8 or less (for ka < 1). If we define x = ka and 35

37 m = ɛ r, we end up with: where: X E (1, x, m) 65 m 2 (5m ) 144x 3 5m m 2 13 x 8 + a 1 x 6 + a 2 x 4 + a 3 x 2 + a 4 x (m 4 1) 756m 2 (5m 4 +13m 2 13) x (m + 2 1) 756m 2 (5m 4 +13m 2 13), (2.9a) X H (1, x, m) 21(m4 6m 2 + 5) 4m 4 x 5 x4 + e 1 x 2 + e 2 x 2 14(m2 1) m 4, (2.9b) a 1 = 4m2 (35m m ), 13m 4 (5m ) (2.10a) a 2 = 216(m6 + 20m m 2 70), 13m 4 (5m ) (2.10b) a 3 = 6048(m4 + 9m 2 10), 13m 4 (5m ) (2.10c) a 4 = 60480(m2 + 2) 13m 4 (5m ), (2.10d) e 1 = 20(m2 1) (m 4 6m 2 + 5), (2.10e) 120 e 2 = (m 4 6m 2 + 5). (2.10f) For simplicity, we refer to the denominator coefficients of (2.9a) and (2.9b) as b 1 -b 2 and f (m 4 1) b 1 = 756m 2 (5m m 2 13), b 2 = (m 2 1) 756m 2 (5m m 2 13), (2.11a) (2.11b) f 1 = 14(m2 1) m 4. (2.11c) To derive explicit RLC models, the rational functions given by (2.9) are expanded as continued fraction forms of polynomial terms. Hence the scattering coefficients are 36

38 achieved as the input impedances of RLC-ladder types circuits (see also Fig. 2-4): 1 ς 1 (a) = jx 1e + jx 2e ξ 1 (a) = 1 + jx 1h + 1, jx 2h jx 3e, (2.12a) (2.12b) where and: X 1e = 65 m 2 (5m ) ( ) x 4 + (a 144x 3 5m m 2 1 b 1 )x 2 + a 2 + b 2 1 b 2 a 1 b 1, (2.13a) 13 ( ) x 2 X 2e = x 3 m 2 (5m ) 5m m 2 13 c 1 + b 1c 1 c 2 X 3e = x 3 m 2 (5m ) 5m m 2 13 (c 1x 2 + c 2 ) c 2 1, (2.13b) ), (2.13c) ( c1 b 1 c b 2 + c2 1 c 2 2 X 1h = 21(m4 6m 2 + 5) 4m 4 x 5 (x 2 + (e 1 f 1 )), (2.13d) X 2h = 21(m4 6m 2 + 5) 4m 4 x 5 x 2 + f 1 f e 2 e 1 f 1, (2.13e) c 1 =a 1 (b 2 1 b 2 ) a 2 b 1 + a 3 b 1 (b 2 1 2b 2 ), c 2 =a 1 b 1 b 2 a 2 b 2 + a 4 (b 2 1 b 2 )b 2. (2.14a) (2.14b) It is worth mentioning that X 1e X 3e and X 1h X 2h denote the reactance parts of the scattering coefficients. Depending on the material property (m) and frequency they can be either inductive or capacitive. Notice that, if mka 1 and ka > 1 these approximations might not be valid Simulation Results To evaluate the accuracy of our approach we compare the results of RLC technique with the exact Mie solutions. For a nano GaP sphere with ɛ r = and radius of 85 nm, the normalized Mie scattering coefficients are depicted in Figs. 2-5 and 2-6. As it is illustrated, there is a negligible difference between the RLC modeling and 37

39 (a) (b) Figure 2-3: Comparison between FDTD and theory for a single sphere of ka = 1, ɛ = 4.431, (a) Scattering cross section in φ = 0 plane, (b)scattering cross section in φ = π/2 plane. In both figures, the blue solid line represent the theory while the red dashed line shows the FDTD values. the exact Mie solutions, indicating that our circuit model can properly predict the behavior of the scattering coefficients for a dielectric sphere. Note that, in this scenario ka is between zero and one and m is as high as It is worth highlighting again that these circuit models are obtained only for the first electric and magnetic scattering coefficients. The capability of representing the scattering coefficients with RLC circuit models, allows one to apply the well-known concepts in circuit theory to effectively design required elements for providing desired electric and magnetic resonances. (a) (b) Figure 2-4: RLC schematics for Mie scattering coefficients, (a) Electric scattering coefficient, (b) Magnetic scattering coefficient. 38

40 Electric scattering coefficients (magnitude) ka Exact Fraction Form Electric scattering coefficient (phase) ka Exact Fraction form Frequency (THz) (a) Frequency (THz) (b) Figure 2-5: Electric scattering coefficients for a single GaP sphere of radius a = 85 nm and ɛ r = Comparison between RLC ladder-type model versus exact Mie solution, (a) Magnitude, (b) phase. It is interesting to mention that, Figs. 2-5 and 2-6 show that at the magnetic resonance (f = 485 T Hz), A sp 11 B sp 11 or ς 1 (a) ξ 1 (a), namely at the resonance of the magnetic scattering coefficient the nanosphere behaves as a magnetic dipole (E M). The same argument can be done for the electric resonance at f = 643 T Hz (however here the plots are demonstrated only in the frequency spectrum of interest). In summary, the dielectric spheres can provide electric and magnetic dipole moments at the electric and magnetic resonances of the scattering coefficients. is worth mentioning that, the same performance for magneto-dielectric spheres can also be obtained. Novel combinations of these modes can offer desired metamaterial features. It 39

41 Magnetic scattering coefficient (magnitude) ka Exact Fraction form Frequency (THz) (a) Electric scattering coefficient (phase) ka Exact fraction form Frequency (THz) (b) Figure 2-6: Magnetic scattering coefficients for a single GaP sphere of radius a = 85 nm and ɛ r = Comparison between RLC ladder-type model versus exact Mie solution, (a) Magnitude, (b) phase. 40

42 Chapter 3 Multiple Scattering by an Arbitrary Configuration of Spheres In the previous chapter, the creation of electric and magnetic dipole modes by using dielectric/magneto-dielectric spheres was discussed. By novel arrangement of these dipole modes, a metamaterial with desired features is achieved. In this chapter I present a comprehensive method to solve the problem of multiple scattering of a plane wave incident on an arbitrary configuration of N spheres. The multipole expansion method is applied, and the boundary conditions is imposed using the translational addition theorem for vector spherical wave functions. The Cruzan s original theorem on the vector addition theorems is used [63]. I compare our theoretical results with the experimental results obtained by Y. L. Xu and B. A. S. Gustafon [64, 65, 1], a good comparison is obtained. Note that, although many believes that Cruzan s theorem contains an overall sign error, the comparison with experimental results reveals that the original Cruzan s theorem is correct. 3.1 Modeling and Analysis The geometry of the an aggregate of spheres is depicted in Fig Each sphere (for instance the p th one) is represented with dielectric constant of ɛ rp and radius a p. The goal is to determine the scattering coefficients for every sphere by forcing 41

43 (a) Figure 3-1: Configuration of an aggregate of spheres the boundary conditions at the surface of that particle. To implement the boundary conditions at the surface of each sphere (e.g. p th sphere) the outgoing scattered fields from all other spheres are expressed in the local coordinate of the reference sphere (p th sphere). The continuity of the tangential electric and magnetic fields is applied by using the spherical vector translational addition theorem [63]. The solution for the unknown scattering coefficients is then obtained as [51]: ( A sp mn = ς n (a) A ip mn + q q p ( Bmn sp = ξ n (a) Bmn ip + q q p ν ν=1 µ= ν ν ν=1 µ= ν [ Amnµν (d pq, θ pq, φ pq )A sq µν + B mnµν (d pq, θ pq, φ pq )B sq µν] ), (3.1a) [ Amnµν (d pq, θ pq, φ pq )B sq µν + B mnµν (d pq, θ pq, φ pq )A sq µν] ). (3.1b) Where the summation over q is the summation on all nanoparticles except the reference sphere. d pq is the center distance between p th sphere and q th one [51]. A µν mn and B µν mn are the translational addition theorem coefficients [63]. Equations (3.1) are a coupled set of complex linear algebraic equations, and should be solved simultaneously to yield the unknown scattering coefficients. In addition, the infinite series must be truncated to a finite number n = ν = N and m = µ = M. To obtain the scattering 42

44 coefficient first we have to determine the vector translation additional coefficients. Base on the Cruzan s theorem, the translation addition coefficients are [63]: A mnµν (d, θ, φ) = ( 1) m e i(µ m)φ a( m, n, µ, ν, p)a(n, v, p)h (1) p (kd)p p µ m (cos θ) (3.2a) p B mnµν (d, θ, φ) = ( 1) m+1 e i(µ m)φ a( m, n, µ, ν, p, p 1)b(n, v, p)h (1) p (kd)p p µ m (cos θ) (3.2b) p In the above equations h (1) p is the associated Legendre function of the first kind. a( m, n, µ, ν, p, p 1) are the Gaunt coefficients [66]. closely related to Wigner 3jm symbol and defined by: represents the Hankel function of the first kind, P m n The a( m, n, µ, ν, p) and The Gaunt coefficient are [ (n m)!(v + µ)!(p + m µ)! a( m, n, µ, ν, p) =( 1) (µ m) (2p + 1) (n + m)!(v µ)!(p m + µ)! n ν p n ν p m µ m µ [ (n m)!(v + µ)!(p + m µ)! a( m, n, µ, ν, p, p 1) =( 1) (µ m) (2p + 1) (n + m)!(v µ)!(p m + µ)! n ν p n ν p m µ m µ ] 1/2 (3.3a) ] 1/2 (3.3b) 43

45 The other coefficients in (3.2) are defined as: a(n, ν, p) = ( j)(n ν+p) 2n(n + 1) [ 2n(n + 1)(2n + 1) + (n + 1)(ν + n p)(ν + p n + 1) ] n(n + ν + p + 2)(n + p ν + 1) (3.4a) b(n, ν, p) = ( j)(n ν+p) (2n + 1) 2n(n + 1) [ (n + ν + p + 1)(n + p ν)(ν + p n)(n + ν p + 1)] 1/2 (3.4b) Note that, the integer p in the summation of (3.2a) takes the values of n ν, n ν +2,...,n+ν. While this integer (p) in (3.2b) varies from n ν +1, n ν +3,...,n+ν. The symbol j1 j2 j3 m1 m2 m3 is the Wigner 3jm symbol and is defined as: j1 j2 j3 = ( 1) j1+j2+m3 m1 m2 m3 [ (j1 m1)!(j1 + m1)!(j2 m2)!(j2 + m2)!(j3 m3)!(j3 + m3)! (j1 + j2 j3)!(j1 j2 + j3)!( j1 + j2 + j3)!(j1 + j2 + j3 + 1)! k max j1 + j2 j3 j1 j2 + j3 j1 + j2 + j3 ( 1) k k=k min k j1 m1 k j2 + m2 k ] 1/2 (3.5a) k min = max(0, j2 j3 m1, j1 j3 + m2) k max = min(j1 + j2 j3, j1 m1, j2 + m2) (3.5b) (3.5c) 44

46 Two numerical examples of Wigner 3jm symbol are shown: = (3.6) = (3.7) According to (3.5) the Wigner 3jm symbol consists of factorials. For large values of j1, j2, j3 or m1, m2, m3 the accurate factorials might lead to incorrect value. The wigner 3jm for j1 = 98, j2 = 115, m1 = 69, m2 = 100, m3 = 169 is calculated for different j3. 98 W = 115 j3 (3.8) j3 = 213 j3 = 212 j3 = 211 j3 = 207 W = W = W = W = j3 = 169 j3 = 180 j3 = 175 j3 = 172 j3 = 193 j3 = 197 W = W = W = W = W = W = In the above, W stands for the Wigner 3jm calculated using (3.5). Evidently, the values for Wigner 3jm symbol is not correct owing to the over flow problem caused by calculation of factorials with large numbers. To overcome this problem, logarithms 45

47 of factorials might be used in decimal calculations. As in the definition of Wigner 3jm the factorials are often divided by another factorial, we end up with the subtraction of logarithms which leads to small numbers. There is a very neat approximation derived by Lanczos specifically to the gamma function [67]. z! = 2π(z + γ + 1/2) z+1/2 e (z+γ+1/2) A γ (z) (3.9a) A γ (z) = ρ ρ z 1 z ρ z(z 1) 2... (3.9b) (z + 1)(z + 2) k ρ k = C2αF 2k (α) (3.9c) F (α) = α=0 2 π (α 1/2)!(α + γ + 1/2) α 1/2 e α+γ+1/2 (3.9d) and C 2k 2α are the coefficients of the Chebyshev polynomial. The logarithm of a factorial can be calculated from the equation: ln(z!) = ln2π 2 (z + γ + 1/2) + (z + 1/2)ln(z + γ + 1/2) + ln(c 0 + c 1 z c ɛ) (3.10) z + 2 where ɛ is the truncation error. There is a published subroutine using Lanczos approximation method in the book Numerical Recipes [68], which uses γ = 5 and N = 6 that results in ɛ < 2 e 10. To apply Lanczos approximation in doubleprecision calculations we use γ = 10 and N = 11, and consequently ɛ <

48 for n up to [65]. The corresponding values of c 0, c 1,... are given as: c 0 = 1 c 1 = c 2 = c 3 = c 4 = c 5 = c 6 = c 7 = c 8 = c 9 = c 10 = c 11 = (3.11) By using the Lanczos approximation, the large factorials can be calculated accurately and hence the Wigner 3jm symbol is computed for large values without the overflow problem. The computed values for Wigner 3jm in (3.8) using the approximation for factorial are listed as W lz. We compare these values to the one obtained in [65] and a very good agreement is observed. j3 = 213 W lz = j3 = 212 W lz = j3 = 211 W lz = j3 = 207 W lz = j3 = 169 W lz =

49 j3 = 180 W lz = j3 = 175 W lz = j3 = 172 W lz = j3 = 193 W lz = j3 = 197 W lz = (3.12) Now that we have all the required parameters we can calculate the addition translation coefficients. Below shows some practical examples of the Gaunt coefficients using the Wigner 3jm formulations. We compare the values with the one obtained in [65] and the results reveals a very good comparison. m n µ ν p a(m, n, µ, ν, p) (3.13) Once we calculated all the addition theorems, the scattering coefficients can be calculated using (3.2). Now that we have the scattering coefficients, the total scattering field of the entire cluster can be obtained. In the far-field region the relation between the electric vector components of the incident field and those of the scattered filed is given by: E s = e jkr jkr (S 2E i + S3E i ) E s = e jkr jkr (S 4E i + S1E i ) (3.14a) (3.14b) where k is the wave number and r is the usual spherical coordinate for a point in space. The electric vector components (E i, E i ) and (E s, E s ) of the incident 48

50 and scattered fields are parallel and perpendicular respectively to the scattering plane defined by the direction of the incident wave vector and the scattering direction. To check the accuracy of the scattering coefficient, we compare the theoretical results of components of polarized scattered intensity i 11 and i 22 with the experimental results in [1, 2]. i 11 and i 22 are functions of the scattering angle θ alone,when referred to the scattering plane only (φ = 0) and can be calculated from the amplitude scattering matrix through the equations i 11 = S y 1(θ) 2 i 22 = S x 2 (θ) 2 (3.15a) (3.15b) where the superscript x or y indicates that the incident plane wave is x or y polarized (note that the incident plane wave propagates along the z axis). For each sphere, S y 1 and S x 1 are defined as follows [1]: S y 1(θ) = S x 2 (θ) = N n e jk.d p A sp mn p=1 n=1 m= n N n e jk.dp Bmn sp p=1 n=1 m= n m sin θ P n m (cos θ) + B sp mn m sin θ P n m (cos θ) + A sp mn d dθ P n m (cos θ) d dθ P n m (cos θ) (3.16a) (3.16b) In the next section we show some results comparing our theoretical model based on the Cruzan s original theorem for additional translation coefficients with the experimental results obtained by Xu in [1, 2] Numerical Results The formulation described in this report has been implemented in a computer code. Some of our numerical calculations are presented for illustration. The theoretical predictions are compared with laboratory scattering measurement done in [1, 2]. The examples shown here are the angular distributions (phase functions i 11 and i 22 at a fixed orientation for different sets of sphere chains, each consisting of two identical spheres. For all the cases, the axis of symmetry of each sphere chain is perpendicular 49

51 to either the scattering plane xz plane or the incident wave vector 1along the z axis. In other words, it is always parallel to the y axis (refer to Fig. 3-2). These dimensionless quantities are independent of the measurement or computational units. (a) Figure 3-2: Configuration of an aggregate of spheres 6 4 Theory Experiment (a) (b) Figure 3-3: Comparison of theoretical calculations with laboratory scattering measurements from [1] for angular distributions of polarization components of scattered intensity by two identical spheres with inter-sphere distance of Each sphere has a size parameter of ka = and a refractive index of n = i, (a) log(i 11 ), (b) log(i 22 ) As shown in Figs. 3-3 to Figs. 3-5, our theoretical calculations agree favorably with the experimental measurements. We have shown that by comparing with available experimental data that the original Cruzan s theory for vector translational 50

52 (a) (b) Figure 3-4: Comparison of theoretical calculations with laboratory scattering measurements from [1] for angular distributions of polarization components of scattered intensity by two identical spheres with inter-sphere distance of Each sphere has a size parameter of ka = and a refractive index of n = i, (a) log(i 11 ), (b) log(i 22 ) addition coefficients provides correct numerical values, though it has been considered to have a wrong overall sign for more than two decades. In the next section, by obtaining the scattering coefficient for an array of spheres, I illustrate that if the spheres are small enough, electric and magnetic modes are sufficient to model the spheres. 3.2 Dipole Modes: Array of Spheres In this section, I obtain the scattering coefficients for an array of spheres The geometry of the 3D array of spheres is depicted in Fig Each sphere is represented with dielectric constant of ɛ r and radius a. The spheres are centered at x = l x h x, y = l y h y and z = l z d where l x = ±1, ±2, ±3,... ± L x, l y = ±1, ±2, ±3,... ± L y and l z = 0, 1, 2,..., L z. The goal is to determine the scattering coefficients for every sphere by forcing the boundary conditions at the surface of that particle. The solution for the unknown scattering coefficients is obtained in (3.1). If the spheres are small enough, or the frequency is such that the magnitude of higher order modes are negligible compared to the magnitude of the first electric and 51

53 (a) (b) Figure 3-5: Comparison of theoretical calculations with laboratory scattering measurements from [2] for angular distributions of polarization components of scattered intensity by two contacting identical BK7 glass spheres. Each BK7 sphere has a size parameter of ka = 7.86 and a refractive index of n = i, (a) log(i 11 ), (b) log(i 22 ). magnetic modes, only the dipole scattered fields are excited. And thus the scattered coefficients can be well approximated by only considering n = 1. To verify this idea we analyze the performance of the scattered coefficients for an array of dielectric nanospheres with radius of a = 85 nm, ɛ r = 12.25, L x = L y = 4 and L z = 2. A plane wave excitation having a y-polarized electric field which propagates in the z direction is assumed. The results are described in Figs. 3-7 and 3-8. As it is shown, in the frequency bands where ς n (a), ξ n (a) n>1 ς 1 (a), ξ 1 (a) one can assume that only the first electric and magnetic dipole modes are excited. It is worth noting that, Figs. 3-7 and 3-8 depict the electric and magnetic scattered coefficients for a sphere located in the middle of the array (l x = l y = 0, l z = 1). In the following chapters, we assume that either the spheres are sufficiently small or the frequency is such that, the spheres scattering can be treated with only dipole vector spherical waves. 52

54 Figure 3-6: The configuration of an array of dielectric nanospheres. 3.3 Conclusion In this chapter, theoretical investigation of arbitrary configuration of spheres is addressed. A full wave spherical modal analysis is applied to express the optical fields in terms of the electric and magnetic dipole modes and the higher order terms. Imposing the boundary conditions at the surface of each nanosphere, using the translational addition theorem for vector spherical wave functions, required equations to determine the scattered coefficients are obtained. We show that, if the spheres are small enough or the frequency is such that the magnitude of higher order modes are negligible compared to dipole modes, it is enough to model each particle with only dipole spherical waves. 53

55 Magnitude of Mie electric scattering coefficients ς 1 ς 2 ς Frequency(THz) (a) Electric scattering coefficients (magnitude) with highr order modes dipole modes only Frequency (THz) (b) 180 Electric scattering coefficient (phase) with highr order modes dipole modes only Frequency (THz) (c) Figure 3-7: Electric scattering coefficients for a Gap nanosphere with ɛ r = and a = 85 nm. (a) Normalized Mie scattering coefficient for the first three terms. The performance of electric scattering coefficient for a sphere inside a finite array with : L x = 4, L y = 4, L z = 2 and unit-cell size of d/a = 2.23 and h x = h y = 2.1a, (b) Magnitude and, (c) Phase. 54

56 Magnitude of Mie magnetic scattering coefficients Frequency(THz) (a) ξ 1 ξ 2 ξ 3 Magnetic scattering coefficients (magnitude) with highr order modes dipole modes only Frequency (THz) (b) Magnetic scattering coefficient (phase) with highr order modes dipole modes only Frequency (THz) (c) Figure 3-8: Magnetic scattering coefficients for a GaP nanosphere with ɛ r = and a = 85 nm. (a) Normalized Mie scattering coefficient for the first three terms. The performance of magnetic scattering coefficient for a sphere inside a finite array with : L x = 4, L y = 4, L z = 2 and unit-cell size of d/a = 2.23 and h x = h y = 2.1a, (b) Magnitude and, (c) Phase. 55

57 Chapter 4 Dispersion Diagram Characteristics of Periodic Array of Dielectric and Magnetic Materials Based Spheres In this chapter, the characteristics of electromagnetic (EM) waves supported by three dimensional (3D) periodic arrays of dielectric and magneto-dielectric spheres are theoretically investigated. The sphere particles have the potential to offer electric and magnetic dipole modes, where their novel arrangements engineer desired metamaterial performances. A full wave spherical modal analysis is applied to express the electromagnetic fields in terms of the electric and magnetic dipole modes and the higher order terms. Imposing the boundary conditions, will determine the required equations for obtaining kd βd dispersion characteristics. A metamaterial constructed from unit-cells of two different spheres is created, where one set of spheres develops electric modes, and the other set establishes magnetic modes. It is demonstrated that such a composite of high dielectric spheres provides double negative (DNG) metamaterial in a narrow frequency band spectrum where kd, βd < 1. We also investigate the dispersion diagram for a 3D array of one-set of highly coupled dielectric spheres. 56

58 Here, the coupling between the electric and magnetic dipoles of the spheres generates hybrid modes, resulting in a backward wave medium for kd > 1. The developments of DNG and backward wave metamaterials utilizing 3D array of magneto-dielectric spheres located inside free space, and dielectric spheres embedded in a negative µ host, are also addressed. The use of magnetic materials allows accomplishing wider dispersion characteristic bandwidths and tunable feasibility. 4.1 Theoretical Model To achieve a metamaterial with functionality of interest one needs to create appropriate electric and magnetic dipole moments inside the unit-cell of array configuration. It has been demonstrated that dielectric spheres have the potential to offer both electric and magnetic dipole modes [49, 20]. A unit-cell composed of two different spheres with the same radii and different materials or the same dielectric materials and different radii can provide the required electric and magnetic dipole modes [23, 24]. Tailoring the appropriate dipole modes, allows controlling the performance of the structure. In this section, we develop the required formulations for characterizing the performance of periodic arrays of spheres. The intention is to accomplish the dispersion relations. The obtained formulations is general and one can model both dielectric and magneto-dielectric spheres inside a background media. We use the reciprocity approach, [4]-[34], to model pseudo-3d metamaterial configuration constructed from two different spheres in one unit-cell (all planes of the spheres normal to the array axis are identical). Since we are exploring the traveling waves supported by the periodic array of spheres, the EM fields are identical except for a phase shift in the direction of propagation. Using these, the desired system of equations for kd βd diagram is achieved. The method of this work is comprehensive and can be applied to other periodic metamaterial structures with more or less particles in a building block unit-cell. The geometry of two-sets of spheres is depicted in Fig The array is con- 57

59 Figure 4-1: Geometry of two-sets of dielectric spheres. structed from unit-cells of two spheres with relative permittivity and permeability of ɛ r1, µ r1 and ɛ r2, µ r2. The spheres radii are a 1 and a 2. The spheres with constitutive parameters ɛ r1 and µ r1 are centered at x = l x h x, y = l y h y and z = l z d where l x, l y, l z = 0, ±1, ±2, ±3,... The second set of spheres are centered at x = l x h x + h, y = l y h y and z = l z d. The z axis is taken to be the array axis. Plane wave excitation with x polarized electric field and y polarized magnetic field is considered. The electric and magnetic dipole components of each sphere are oriented in the x and y direction respectively. It is assumed that the array can support an EM-plane wave with the propagation vector β which lies in the direction of array axis (z). As mentioned earlier, in this work a pseudo-3d configuration is analyzed (doubtlessly, the analysis procedure can be extended to isotropic patterns of two-sets of spheres). Using the multipole expansion, the incoming EM field on each sphere (e.g. p th sphere) can be expanded in terms of spherical wave vectors of the first kind M (1) mn and N (1) mn [51] where η is the free space intrinsic impedance and the incident-field expansion coefficients (A ip mn and B ip mn) are derived in [51]. The scattered field from each sphere however, can be expanded as a weighted sum of spherical wave vector of the third kind (2.3) M (3) mn and N (3) mn [51]. A sp mn and B sp mn 58

60 are the unknown scattering coefficients for each mode, to be determined by matching the boundary conditions. Note that, M (1) mn(m (3) mn) and N (1) mn(n (3) mn) are the spherical vector wave functions representing the incoming (outgoing) waves associated with spherical Bessel (Hankel) functions. The incident field on each sphere arise a scattered field which itself, is an incident wave on all other spheres. Also, the scattered field from every sphere is related to the incident field upon that sphere, through the Mie scattering coefficients. In order to derive the kd βd relation, we apply the reciprocity approach and equate the EM fields on the reference spheres with the sum of scattered field coming from all other spheres. This is the similar approach that was proposed by Shore et al. for modeling the performance of one-set of identical spheres [69, 4]. To apply the reciprocity technique, we first consider one reference sphere for each set of spheres. For simplicity the first set s reference sphere is assumed to be centered at (0, 0, 0) of Cartesian coordinate, while the other set s reference sphere is centered at (h, 0, 0). It is assumed that either the spheres are sufficiently small or the frequency is such that only the dipole scattered modes are excited. Hence from (2.2) and (2.3) the dipole modes approximation for the electromagnetic field of a small sphere can be achieved by considering only M (3) 1,1 and N (3) 1,1. In order to verify the accuracy of our assumption, we compare the dispersion diagram performance of a periodic configuration obtained using our approach with an in-house developed full wave FDTD method [35] (see Fig. 4-2). Let E ι 0 and H ι 0 (ι is either 1 or 2) be the incident electric and magnetic fields illuminating the reference spheres of each set resulting from all other spheres in the array (ι, τ {1, 2} and ι τ): E ι 0 = H ι 0 = l z = lx= l y = (l x,n,l) (0,0,0) l z = lx= l y = (l x,n,l) (0,0,0) E ιι l x l y l z + H ιι l xl yl z + 59 l z = l x = l y = l z = l x = l y = E τι l x l y l z, H τι l xl yl z, (4.1a) (4.1b)

61 where for ι, τ {1, 2} we have: E ιτ l xl yl z = b ι l z N (3) 1,1(r ιτ l xl yl z, θ ιτ l xl yl z, φ ιτ l xl yl z ) + b ι +l z M (3) 1,1(r ιτ l xl yl z, θ ιτ l xl yl z, φ ιτ l xl yl z ), ηh ιτ l xl yl z = jb ι +l z N (3) 1,1(r ιτ l xl yl z, θ ιτ l xl yl z, φ ιτ l xl yl z ) jb ι l z M (3) 1,1(r ιτ l xl yl z, θ ιτ l xl yl z, φ ιτ l xl yl z ). (4.2a) (4.2b) The quantities in (4.2) are defined with respect to a local spherical coordinate system with origin at r ιτ l xl yl z, and: r 11 l x l y l z = (0, 0, 0) (l x h x, l y h y, l z d) = l x h xˆx l y h y ŷ l z dẑ, r 21 l xl yl z = (0, 0, 0) (h + l x h x, l y h y, l z d) = (h + l x h x )ˆx l y h y ŷ l z dẑ, r 12 l xl yl z = (h, 0, 0) (l x h x, l y h y, l z d) = (h l x h x )ˆx l y h y ŷ l z dẑ, r 22 l xl yl z = (h, 0, 0) (h + l x h x, l y h y, l z d) = l x h xˆx l y h y ŷ l z dẑ. (4.3a) (4.3b) (4.3c) (4.3d) From [34] the field distribution defined in (4.2) can be further expressed as: E ιτ l xl yl z jηh ιτ l xl yl z = b ι e jkrιτ lxlylz l z krl ιτ x l y l z + ( 1 j kr ιτ l x l y l z + b ι +l z e jkrιτ l xlylz kr ιτ l x l y l z = b ι +l z e jkrιτ lxlylz + ( kr ιτ l xl yl z 1 j kr ιτ l xl yl z + b ι l z e jkrιτ l xlylz kr ιτ l xl yl z [ 2j kr ιτ l x l y l z ( [ ( 1 (kr ιτ l x l y l z ) 2 1 j kr ιτ l x l y l z 2j kr ιτ l xl yl z ( ( 1 (kr ιτ l xl yl z ) 2 1 j kr ιτ l xl yl z 1 j ) krl ιτ x l y l z ) e jφιτ lxlylz sin θ ιτ lxl yl z ˆr ιτ l xl yl z ] (e jφιτ lxlylz cos θ ιτ ˆθιτ lxlyl z lxlyl z + je jφιτ lxlylz ) ) ( ) e jφιτ lxlylz ˆθιτ lxlyl z + je jφιτ lxlylz cos θ ιτ ˆφιτ lxlyl z lxlyl z, 1 j ) krl ιτ xl yl z ) e jφιτ lxlylz sin θ ιτ lxl yl z ˆr ιτ l xl yl z ] (e jφιτ lxlylz cos θ ιτ ˆθιτ lxlyl z lxlyl z + je jφιτ lxlylz ) (4.4a) ) ( ) e jφιτ lxlylz ˆθιτ lxlyl z + je jφιτ lxlylz cos θ ιτ ˆφιτ lxlyl z lxlyl z. (4.4b) Basically E ιτ l xl yl z and H ιτ l xl yl z (ι, τ {1, 2}) are the scattered electric and magnetic field from the (l x l y l z ) th element of the ι th set at the center of τ th set s reference 60

62 sphere. Substituting (4.4) in (4.1) along with the proper transformation to Cartesian coordinate, we obtain: (Notice that in the following, we only keep the even function with respect to l x or l y. Since the odd functions vanish when we take the summation over l x and l y from to ) E ι 0ˆx = ˆx l z = lx= l y = (l x,l y,l z ) (0,0,0) [ 2j krl ιι x l y l z + [ ( 1 + j krl ιι x l y l z l z = l x = l y = 2j kr τι l x l y l z ( 1 + j kr τι l x l y l z ) b ι l z e jkrιι lxlylz kr ιι l xl yl z (l x h x ) 2 (r ιι l x l y l z ) 2 + b τ l z e jkrτι lxlylz ) krl τι x l y l z ( (l x h x + h) 2 + (rl τι x l y l z ) j kr ιι l x l y l z ( 1 + j kr τι l x l y l z ) 1 (krl ιι x l y l z ) 2 ) 1 (krl τι x l y l z ) 2 ] (l y h y ) 2 + (l z d) 2 (r ιι l x l y l z ) 2 ] (l y h y ) 2 + (l z d) 2 (r τι l x l y l z ) 2 ηh ι 0ŷ = ŷ lz= n 0 b ι +l z l x = l y = b τ +l z l z = l x = l y = l z = lx= l y = (l x,l y,l z ) (0,0,0) [ ( 2j krl ιι x l y l z 1 + j krl ιι x l y l z l z = l x = l y = [ 2j krl τι x l y l z lz= n 0 ( ) 1 + j krl τι x l y l z e jkrιι lxlylz kr ιι l xl yl z e jkrτι lxlylz kr τι l x l y l z ( ( b ι +l z e jkrιι lxlylz kr ιι l x l y l z (l y h y ) 2 (r ιι l x l y l z ) 2 + b τ +l z e jkrτι lxlylz ) b ι l z l x= l y= b τ l z l z= l x= l y= krl τι x l y l z (l y h y ) 2 (rl τι x l y l z ) + 2 e jkrιι lxlylz kr ιι l x l y l z e jkrτι lxlylz kr τι l x l y l z 61 ( 1 + j kr ιι l xl yl z ( ( ( 1 + j kr τι l x l y l z ) 1 + j kr ιι l x l y l z 1 + j kr τι l x l y l z 1 + j kr ιι l x l y l z 1 + j kr τι l x l y l z ) ) ) l z d r ιι l xl yl z l z d r τι l x l y l z, ) 1 (krl ιι x l y l z ) 2 ) 1 (krl τι x l y l z ) 2 l z d r ιι l x l y l z l z d r τι l x l y l z, (4.5a) ] (l x h x ) 2 + (l z d) 2 (r ιι l x l y l z ) 2 ] (l x h x + h) 2 + (l z d) 2 (r τι l x l y l z ) 2 (4.5b)

63 where E0 ι (H0) ι is the x (y) directed electric (magnetic) field illuminating the reference sphere of ι th set. The constants b ι l z, b ι +l z (for ι {1, 2}) are related to El ι z ˆx and ηhl ι z ŷ with the normalized electric and magnetic dipole Mie scattering coefficients (S and ι S+) ι for each sphere of the first and second set [4, 34]. Normalize means that b ι l z (b ι +l z ) is the coefficient of exp( jkr)/(kr) in the outgoing electric (magnetic) dipole field. Since all the spheres in the array are excited identical except for a phase shift, we have (ι {1, 2}): b ι l z = b ι 0e jl zβd, (4.6a) b ι +l z = b +0 e jl zβd. (4.6b) Using the above, after some manipulations, the desired system of equations for extracting the dispersion relation is determined as: b ι 0 S ι b ι +0 S ι + =b ι 0Σ eeιι + b τ 0Σ eeτι b ι +0Σ ehιι b τ +0Σ ehτι, (4.7a) = b ι 0Σ heιι b τ 0Σ heτι + b ι +0Σ hhιι + b τ +0Σ hhτι, (4.7b) S ι and S ι + for ι {1, 2} are the normalized Mie electric and magnetic scattering coefficients, S = 3j 2 ς 1(a), (4.8a) S + = 3j 2 ξ 1(a), (4.8b) ς 1 (a) and ξ 1 (a) are the first Mie electric and magnetic scattering coefficient [49]. ς 1 (a) = µ rpj 1 (ka)(k p aj 1 (k p a)) µ rp ɛ rp j 1 (k p a)(kaj 1 (ka)) µ rp h (2) 1 (ka)(k p aj 1 (k p a)) µ rp ɛ rp j 1 (k p a)(kah (2) 1 (ka)), (4.9a) ξ 1 (a) = µ rpj 1 (k p a)(kaj 1 (ka)) µ r j 1 (ka)(k p aj 1 (k p a)) µ rp j 1 (k p a)(kah (2) 1 (ka)) µ r h (2) 1 (ka)(k p aj 1 (k p a)). (4.9b) Here k p is the wave-number inside the p th sphere while j 1 and h (2) 1 are the spherical 62

64 Bessel and spherical Hankel function of the second kind. The summations defined in (4.7) are given by: Σ eeιι = l z = [ ( 2j krl ιι x l y l z e jlzβd 1 + j kr ιι l x l y l z lx= l y = (l x,l y,l z ) (0,0,0) ) (l x h x ) 2 (r ιι l x l y l z ) 2 + ( e jkrιι lxlylz kr ιι l xl yl z 1 + j kr ιι l x l y l z ) ] 1 (l y h y ) 2 + (l z d) 2, (krl ιι x l y l z ) 2 (rl ιι x l y l z ) 2 (4.10a) Σ eeτι = l z= [ ( 2j krl τι xl yl z Σ hhιι = e jl zβd 1 + j kr τι l xl yl z l z= [ ( 2j krl ιι xl yl z Σ hhτι = e jl zβd 1 + j kr ιι l xl yl z l z= [ ( 2j krl τι xl yl z e jl zβd 1 + j kr τι l xl yl z l x= l y= ) e jkrτι lxlylz kr τι l xl yl z (l x h x + h) 2 + (rl τι xl yl z ) 2 ( lx= l y= (l x,l y,l z) (0,0,0) ) (l y h y ) 2 (r ιι l xl yl z ) 2 + l x= l y= ) (l y h y ) 2 (r τι l xl yl z ) 2 + ( 1 + j kr τι l xl yl z e jkrιι lxlylz kr ιι l xl yl z 1 + j kr ιι l xl yl z e jkrτι lxlylz kr τι l xl yl z ( 1 + j kr τι l xl yl z ) ] 1 (l y h y ) 2 + (l z d) 2, (krl τι xl yl z ) 2 (rl τι xl yl z ) 2 (4.10b) ) ] 1 (l x h x ) 2 + (l z d) 2, (krl ιι xl yl z ) 2 (rl ιι xl yl z ) 2 (4.10c) ) ] 1 (l x h x + h) 2 + (l z d) 2. (krl τι xl yl z ) 2 (rl τι xl yl z ) 2 (4.10d) Σ ehιι = Σ heιι = Σ ehιτ = Σ heιτ = lz= n 0 l z = e jl zβd e jl zβd l x = l y = l x = l y = e jkrιι lxlylz kr ιι l x l y l z e jkrτι lxlylz kr τι l x l y l z ( ( 1 + j kr ιι l x l y l z 1 + j kr τι l x l y l z ) ) l z d r ιι l x l y l z, l z d r τι l x l y l z. (4.11a) (4.11b) 63

65 It is worth noticing that, the summations in equations (4.10) show the interactions between the scattered electric (magnetic) fields from other spheres and the electric (magnetic) fields of reference spheres while the terms in equations (4.11) are the couplings between the scattered electric (magnetic) fields from other spheres and the magnetic (electric) fields of reference spheres. Also, one can divide each of the series in (4.10), and (4.11) into two parts, one the contributions from the the spheres of the self-plane (l z = 0) and the other, the contribution form all others. For instance, Σ ee11 can be expressed as: Σ ee11 = Σ s ee11 + Σ o ee11, (4.12) where Σ s ee11 is the summation over the scattered fields from the spheres in the selfplane (the plane of the reference sphere) and Σ o ee11 is the summation of the scattered field over all other planes except l z = 0. These expressions help us better understand which part provides more contribution to the calculation of kd βd diagram. Since in (4.7), ι takes the values of 1 and 2, we have a set of four homogenous equations to be solved for deriving the desired kd βd relation. The dispersion relation is then obtained, equating the determinant of the coefficients to zero (note that the variables are b 1 0, b 2 0, b 1 +0 and b 2 +0). It is worth mentioning that, the series in (4.10, 4.11) are slowly convergent series and hence excessive number of terms are required for the desired preciseness. Acceleration techniques, similar to the models derived in [70],[34], are applied to effectively solve the above transcendental equations for the propagation constant β. Thereafter, the series in (4.10, 4.11) are expressed as: 64

66 Σ ee11 = Σ ee22 = Σ hh11 = Σ hh22 = 2π sin(kd) (kh x )(kh y ) cos(βd) cos(kd) 2π kh x 4π (kh x ) 3 8 (kh x ) 3 cos(nβd) n=1 l=1 m=1 m= Y 0 (lkh y ) l=1 [(2mπ) 2 (kh x ) 2 ] l= [(2mπ) 2 (kh x ) 2 ]K 0 (lh y /h x (2mπ)2 (kh x ) 2 ) + 4 (kh x ) 2 CL 2(kh x ) + 4 (kh x ) 3 CL 3(kh x ) j 2 3 Σ ee21 = Σ ee12 = Σ hh21 = Σ hh12 = 2π sin(kd) (kh x )(kh y ) cos(βd) cos(kd) 2jπ(kh x) 2 4π (kh x ) 3 8 (kh x ) 3 + m=1 cos(nβd) n=1 l=1 m=1 e jkhx(m 0.5) m 0.5 m= l=1 [(2mπ) 2 (kh x ) 2 ] 4π 2 (l 2 +mh y/h x) 2 (kh y) 2 e nd/hy 4π2 (l 2 + mh y /h x ) 2 (kh y ) 2 H (2) 0 (lkh y ) l= (4.13a) 4π 2 (l 2 +mh y/h x) 2 (kh y) 2 e nd/hy 4π2 (l 2 + mh y /h x ) 2 (kh y ) 2 e jmπ [(2mπ) 2 (kh x ) 2 ]K 0 (lh y /h x (2mπ)2 (kh x ) 2 ) j m 0.5 (kh x Σ eh11 = Σ eh22 = Σ he11 = Σ he22 = 2π sin(βd) (kh x )(kh y ) cos(βd) cos(kd) 4π sin(nβd) (kh x )(kh y ) n=1 m= Σ eh21 = Σ eh12 = Σ he21 = Σ he12 = 2π sin(βd) (kh x )(kh y ) cos(βd) cos(kd) 4π sin(nβd) (kh x )(kh y ) n=1 m= j m 0.5 ) m= m= (4.13b) e nd (2mπ/h x) 2 +(2lπ/h y) 2 k 2 (4.13c) e jmπ e nd (2mπ/h x) 2 +(2lπ/h y) 2 k 2 (4.13d) 65

67 Σ ee21 = Σ ee12 = 2π sin(kd) (kh x )(kh y ) cos(βd) cos(kd) 2π kh x 4π (kh x ) 3 8 (kh x ) m=1 cos(nβd) n=1 l=1 m=1 m= Y 0 (lkh y ) l=1 [(2mπ) 2 (kh x ) 2 ] l= 4π 2 (l 2 +mh y/h x) 2 (kh y) 2 e nd/hy 4π2 (l 2 + mh y /h x ) 2 (kh y ) 2 e jmπ [(2mπ) 2 (kh x ) 2 ]K 0 (lh y /h x (2mπ)2 (kh x ) 2 ) e jkhx(m 0.5) m 0.5 [ (kh x ) 2 jkh ] x m (m 0.5) 2 (4.13e) (4.13f) Where CL 2 and CL 3 are the Clausen functions and are given by: CL 2 (a) = CL 3 (a) = n=1 n=1 sin(na) n 2 cos(na) n 2 (4.14a) (4.14b) A fast convergent series for the slowly convergent Schlomilch series Σ l (Y 0 (lx)) is also given in [34]. Note that, all of the exponential terms in the above definitions involve negative exponentials which decays very fast, so that only a few terms of these series rises to the desired accuracy. It must be recalled that, if we choose the two spheres in a unit-cell the same, then the case of two-sets of dielectric spheres simplifies to 3D array of identical spheres with the following dispersion relation [4]: 1 S 1 (Σ ee11 + Σ ee21 ) S 1 (Σ eh11 + Σ eh21 ) = S1 + (Σ he11 + Σ he21 ) 1 S 1 + (Σ hh11 + Σ hh21 ). (4.15) The formulations derived in this section enable us to comprehensively characterize the performance of metamaterial spheres in the next section. 66

68 4.2 Performance Analysis of Dielectric and Magneto- Dielectric Spheres This section is devoted to the performance analysis of periodic arrays of dielectric and magneto-dielectric spheres, and dielectric spheres embedded in a negative permeability ferrite host. The presented results are obtained by solving the transcendental equations derived in (4.7) and (4.15) for propagation constant β. Since our model restricts to the electric and magnetic dipole wave functions only, the accuracy of our technique is limited to the frequency range before the quadrupole resonances can occur Two-Sets and One-Set of Dielectric Spheres The focus of this section is to explore the development of dielectric metamaterials. As mentioned earlier, to obtain the backward wave and DNG behaviors, appropriate electric and magnetic dipole moments should be created in building-block unit-cells of array configurations. In order to design such a structure, one can utilize 3D array of dielectric spheres with two different spheres as a unit-cell. In this scenario, one set of spheres offers the required magnetic dipole resonances while the other provides the electric dipole resonances. The configuration which we study in this section is a pseudo-3d arrays of two different spheres as a unit-cell; where the spheres have either the same dielectric material and different sizes or the same sizes and different materials. The challenge is to establish both the electric and magnetic dipole resonances around the same frequency band. Although the dipole resonances of an array is different from the resonances associated with Mie s dipole coefficients (2.5), still ς 1 (a) and ξ 1 (a) can successfully predict the dipole resonances of the array. In this sense, the parameters of the two sets are chosen such that, the Mie electric dipole resonances of one set occur around the Mie magnetic dipole resonances of the other set. Based on the resonances of S + and S (4.8) we optimally choose the dielectric constant for a given sphere size, or obtain the sphere size for a given dielectric constant. 67

69 Magnitude of Mie scattering coefficients ξ 1 ξ 2 ξ 3 ς 1 ς 2 ς 3 Magnitude of Mie scattering coefficients ξ 1 ξ 2 ξ 3 ς 1 ς 2 ς Frequency (GHz) Frequency (GHz) (a) (b) kd Theoretical approach FDTD results βd (c) Figure 4-2: Electric and magnetic Mie scattering coefficients of a single sphere for the first three modes with, (a) a = 0.5 cm and ɛ r = 40, (b) a = 0.5 cm and ɛ r = 21, (c) Comparison between FDTD results and theoretical approach for two-sets of dielectric spheres with a 1 = a 2 = 0.5 cm, d/a = 3, h x /a = h y /a = 5, h = h x /2, ɛ r1 =40 and ɛ r2 = 21. Magnitude of Mie scattering coefficients ξ 1 ς 1 Magnitude of Mie scattering coefficients ξ 1 ς Frequency (GHz) Frequency (GHz) (a) (b) Figure 4-3: Electric and magnetic Mie scattering coefficients of a single sphere with, (a) a = 0.5 cm and ɛ r = 20.5, (b) a = cm and ɛ r =

70 kd 1 kd βd βd (a) (b) 1.5 kd βd (c) Figure 4-4: Dispersion diagrams for two-sets of dielectric spheres having the same size (a 1 = a 2 = 0.5 cm) and different dielectric material, (a) The performance of the first set described with ɛ r1 = 40 and d = 1.1 cm, h x = h y = 2.5 cm, (b) The performance of the second set described with ɛ r2 = 20.5, d = 1.1 cm and h x = h y = 2.5 cm, (c) The performance of the combinations of the two-sets with h = h x /2. 69

71 Let us first check the accuracy of our approach in modeling the dispersion characteristic of periodic array of spheres with the full wave numerical analysis based on FDTD. Figs. 4-2(a) and (b) show the magnitude of electric and magnetic Mie scattering coefficients for the first three terms (n = 1, 2, 3) of dielectric spheres with radius of a = 0.5 cm and relative permittivity of ɛ r1 = 40 and ɛ r2 = 21. It is evident that the magnitude of the scattering coefficients for higher order modes are negligible in compare to the magnitude of ξ 1 (a) and ς 1 (a). One can claim that in this region the behavior of each sphere can be well approximated by only the electric and magnetic dipoles modes. To verify this idea, the dispersion diagrams obtained utilizing the theoretical technique of our work and FDTD approach for a periodic array with unit-cells constructed from two different dielectric spheres with the same radius of a = 0.5 cm and different dielectrics of ɛ r1 = 40, and ɛ r2 = 21, h x = h y = 5a, d/a = 3, and h = h x /2 are depicted in Fig. 4-2(c). A very good agreement is illustrated. We now demonstrate the evolvement of backward wave characteristics using two-sets of dielectric spheres. Fig. 4-1 depicts the geometry of the structure. We consider dielectric spheres with a = 0.5 cm, d = 1.1 cm, h x = h y = 2.5 cm, and h = h x /2. The dielectric constant of one-set of spheres is ɛ r1 = 40 (providing electric resonance) and the other array has dielectric material of ɛ r2 = 20.5 (offering magnetic resonance). Mie scattering coefficients for the single dielectric spheres with a = 0.5 cm, ɛ r1 = 40 and ɛ r2 = 20.5 are shown in Figs. 2(a) and 3(a), respectively. As highlighted above, the materials of the spheres are optimized such that the magnetic and electric resonances occur around the same frequency (f = 6.55 GHz). Note that the first resonance of a positive-permittivity sphere is associated with the magnetic dipole and the second resonance represents the electric dipole mode [20]. To explain the performance of array of two-spheres unit-cell, let us start with the dispersion diagram analysis of each set separately. The kd βd dispersion diagrams for each set with unit-cell size h x = h y = 2.5 cm, d = 1.1 cm are presented in Figs. 4-4(a) and (b). In Fig. 4-4(a) two stop bands (gaps) are observed. The first stop band is associated with the magnetic resonance where the second band is associated with the electric resonance. Notice that, the first gap has a wider bandwidth in compare 70

72 to the second one. For the other set, only the first band gap (magnetic resonance) is shown in the frequency spectrum of interest. As anticipated, the electric gap of the first set and the magnetic gap of the other set are around the same frequency region. Fig. 4-4(c) demonstrates the kd βd dispersion diagram for two-sets of pseudo-3d dielectric spheres. By combining the two-sets, or basically by uniting the electric and magnetic dipole modes, a negative slope backward wave behavior is achieved. Note that, one may not be able to define effective material parameters for this case, as the kd has a relatively large value (1.48 kd 1.53). kd kd β d (a) β d (b) Figure 4-5: (a) Dispersion diagrams for two-sets of dielectric spheres having the same size and different dielectric material ɛ r1 = 95, ɛ r2 = 46, a 1 = a 2 = 0.5 cm, d/a = 2.1, h x = h y = 2.5 cm and h = h x /2, (b) DNG backward wave spectrum shown in a larger scale. If the objective is to obtain a bulk medium with DNG effective materials, the backward wave should be created in the frequency spectrum where kd < 1. It was explained that for the two-spheres lattice, the backward wave characteristic occurs as a result of the electric and magnetic resonances. Hence to restrict the values of kd to be less than one, one way is to utilize spheres with higher dielectrics. Fig. 4-5(a) shows the dispersion diagram for a structure with a 1 = a 2 = 0.5 cm, h x = h y = 2.5 cm, h = h x /2, d/a = 2.1 and ɛ r1 = 95, ɛ r2 = 46. As obtained, a narrow-bandwidth backward wave (0.945 kd 0.974) in the frequency spectrum where kd <1 is developed (see Fig. 4-5(b)). In this region the electric and magnetic modes of the 71

73 kd 1 kd βd βd (a) (b) Figure 4-6: Dispersion diagrams for two-sets of dielectric spheres having the same dielectric material (ɛ r1 = ɛ r2 = 40) and different radii, (a) The performance of the second set described with d = 1.1 cm, a = cm and h x = h y = 2.5 cm (The performance of the first set has been already demonstrated in Fig. 4-4(a)), (b) The performance of the combination of the two-sets when h = h x /2. array of spheres can in fact, provide effective permittivity and permeability materials parameters. Basically, above the resonances of the electric and magnetic modes negative effective permittivity and negative effective permeability are anticipated. Thus, the development of backward wave can be a result of the DNG medium behavior [20]. One also should notice that the higher the dielectric the narrower the bandwidths of the backward wave. It is also possible to design a two-sphere lattice metamaterial using the spheres with the same dielectric material and different sizes. The geometry is shown in Fig. 4-1, where the spheres have ɛ r1 = ɛ r2 = 40, a 1 = 0.5 cm, a 2 = cm, d = 1.1 cm, h x = h y = 2.5 cm and h = 1.25 cm. The radius of the second set s spheres is optimized at a 2 = cm such that, it can provide the magnetic resonant frequency around the electric resonance of the first set. Mie scattering coefficients for the single spheres are shown in Figs. 4-2(a) and 4-3(b), where the electric and magnetic resonances are around f = 6.55 GHz. The dispersion diagram for each set is presented in Figs. 4-4(a) and 4-6(a), illustrating that, the electric gap of the first set is at the magnetic gap of the second set. The kd βd engineered dispersion diagram of the combination 72

74 kd 1 kd βd βd (a) (b) 1.5 kd βd (c) Figure 4-7: Dispersion diagrams for two-sets of dielectric spheres having the same dielectric materials (ɛ r1 = ɛ r2 = 40) and different radii, (a) The performance of the first set described with d = 1.1 cm, a = 0.5 cm and h x = h y = 1.8 cm, (b) The performance of the second set described with ɛ r = 40, d = 1.1 cm, a = cm and h x = h y = 1.8 cm, (c) The performance of the combination of the two-sets with h = h x /2. 73

75 kd Frequency (GHz) d/a=2.22 d/a=2.35 d/a=2.5 d/a=2.58 d/a= β d (a) βd (b) Figure 4-8: Dispersion diagram characteristic for the second branch of array of highly coupled cubical unit-cell spheres having ɛ r = 20, a = 0.5 cm and h x = h y = d. of spheres is depicted in Fig. 4-6(b). Backward wave performance is accomplished in the region where first set offers the electric resonance and the second set provides the magnetic resonance (1.494 kd 1.531). The effect of spheres spacing on the kd βd dispersion diagram performance is also investigated. Basically, by bringing the spheres closer to each other, one can expect more couplings between them (and more radiation through the array) resulting in lowering the Q of the system. Thus a wider bandwidth can be anticipated [20]. We now investigate the case where ɛ r1 = ɛ r2 = 40, a 1 = 0.5 cm, a 2 = cm, d = 1.1 cm and h x = h y = 1.8 cm, h = h x /2. The dispersion diagrams for each set separately and the combined array are shown in Figs As can be observed from Fig. 4-7(c), the backward wave has a wider bandwidth (1.489 kd 1.537) in compare to that in Fig. 4-6(b) where the spheres are located in a farther distance from each other (h x = h y = 2.5 cm). The coupling between the spheres can be integrated in an interesting way to achieve a backward wave metamaterial utilizing one-set of spheres. The dielectric spheres can provide magnetic and electric resonances occurring around different resonant frequencies. If the dielectric material of spheres is not very large, and if the coupling between the spheres is increased, one may expect to couple the electric and magnetic dipole modes almost around the same frequency. In other words, the 74

76 spheres can operate in their magnetic dipole moments where their couplings offer required electric dipole modes. Note that, when the dielectric material of a single sphere is decreased, the value of electric Mie scattering coefficient around the magnetic resonant frequency is increased (this can be obtained from equations (2.5) - see also Fig. 4-2(a)). Here, we will investigate the performance of an array of dielectric spheres with ɛ r = 20 and radius a = 0.5 cm having cubical unit-cell with h x = h y = d. Bringing the spheres close to each other (making d/a smaller) will increase the dipolar electric coupling between the spheres where it can combine with the magnetic dipole mode performance of spheres above their magnetic resonances, offering a hybrid mode with backward wave phenomenon. To verify this idea, the dispersion diagram performance of the second branch (the mode right after the magnetic resonance) is explored in Fig. 4-8(b). The more the coupling the wider the bandwidth. As obtained, for d/a = 2.66 forward wave with positive slope is developed. Making the unit-cell smaller changes the positive slope to a negative slope. At d/a = 2.22 negative slope backward wave is established. The complete dispersion diagram for this case is depicted in Fig. 4-8(a). It is worth mentioning that, the backward wave is supported in the frequency band where kd>1 (1.65 kd 1.76) Magnetic Material Based Array of Spheres Magnetic materials have great microwave features enabling novel characteristics in electromagnetics [71]. The major challenge is how to naturally achieve a highperformance magnetic material in gigahertz (GHz) spectrum. Currently, there is a significant work in progress to achieve GHz self-biased hexaferrites with magnetic resonant frequency of above 2GHz, establishing ɛ and µ of around 18 and 5 in GHz, respectively [72]. Of course, applying a DC magnetic field allows achieving tunable larger permeability values at any desired GHz frequency. Using the expressions derived by Lewin [3], Holloway et al. [22] obtained that a 3D array of magneto-dielectric spheres with proper values of ɛ r and µ r can provide the frequency bands (near the diploe resonances of these spheres) in which the bulk 75

77 permittivity and permeability are both negative. Shore et al. [34] also investigated the dispersion diagrams for an array of magneto-dielectric spheres, where they showed in the spectrum band in which kd, βd <1 one can define an isotropic homogeneous metamaterial performance with negative ɛ eff and µ eff. In this work, we apply our dispersion diagram modeling tool to characterize the performance of array of one-set of magneto-dielectric spheres, and dielectric spheres embedded in a host negative permeability material. It is demonstrated that the magnetic materials will enhance the metamaterials features in achieving backward wave wide-bandwidth and isotropic behavior. The loss of magnetic material is ignored in the analysis (estimated loss-tangent is around 0.05 [72]). This will add an attenuation content to the obtained propagation constant. Fig. 4-9(b) exploits the dispersion behavior of an array of magneto-dielectric spheres with ɛ r = 18, µ r = 5. The backward wave performance is accomplished around the region where kd One can attain the negative effective bulk material constants to this periodic configuration using either Lewin mixing formula [3] or Shore-Yaghjian method [34]. The latter technique determines the effective material based on the available parameters by solving the kd βd equation. Fig exhibits the effective relative parameters for the bulk medium, in the backward wave spectrum using Shore-Yaghjian formula. Evidently, in the region where both kd, βd are less than one, DNG material is achieved. We also scrutinize the possibility of realizing DNG metamaterials by embedding dielectric spheres inside a ferrite material operating above its material resonance. Here the structure requires only electric resonances, hence it shows more robustness to fabrication difficulties. In addition to this, the larger dispersion diagram bandwidth and further potentially less propagation loss are expected. Essentially, this design is the dual of implanting dielectric spheres inside a negative permittivity host medium, which was initially proposed by Seo et al. [73] to accomplish DNG metamaterial in optics. It should be mentioned that, the dispersion diagrams in this scenario are merely analyzed utilizing (4.15) where k = ω µ host ɛ host. And, since k is imaginary for a negative permeability material, acceleration techniques are no longer necessary. 76

78 We investigate the feasibility of providing backward wave and DNG characteristics by embedding dielectric spheres inside a material with Lorentzian behavior. Lorentzian permeability function is given by (also, plotted in Fig. 4-11(a)): µ = µ 0 (1 The ) ω 2. (4.16) ω 2 (2π ) Two cases are considered, one is a 3D array of dielectric spheres with ɛ r = 20, and the second case is the dielectric spheres with ɛ r = 90. The dispersion diagrams are obtained in Figs. 4-11(b) and (c). The spheres operate in their electric resonances (above the resonance, negative permittivity can be obtained), and the host material offers negative permeability. Combination of these two features develops negative slope backward wave. For the first case the kd is around ( ), and one may not be able to relate the backward wave performance to the effective material parameters (although in [73] a full wave analysis reveals that still the effective material parameters can be defined). For the second case, the backward wave is achieved in the spectrum where kd and βd can be smaller than one and effective material parameters can successfully be obtained for the periodic configuration. In the backward-wave region based on the Lewin s work [3], the effective µ is between and -1.03, and the effective permittivity can be estimated around -0.6 (the effective relative bulk materials are depicted in Fig. 4-12). This structure has a cubical unit-cell and offers an isotropic microwave DNG metamaterial performance. One interesting aspect of operating above the material resonance of a ferrite host matrix is that a lower loss can be anticipated. In other words, a large value for µ is not required to achieve a DNG performance and one can get some distance from the resonance. In this spectrum the µ is also less sensitive to the frequency dispersion. Further, if one operates in the frequency spectrum where the µ rhost is relatively small, the effective wavelength of the host matrix is increased helping to successfully define the effective material parameters even for smaller-dielectric spheres ingredients. This can be of great practical interest. 77

79 Mie scattering coefficients Magnetic scattering Electric scattering Frequency (GHz) (a) kd β d (b) Figure 4-9: (a) Mie scattering coefficients for a single magneto dielectric sphere with ɛ r = 18, µ r = 5, a = 0.85 cm., (b) Dispersion diagram characteristic for one-set of magneto-dielectric spheres with : ɛ r = 18, µ r = 5, a = 0.85 cm, d/a = 2.1 and h x = h y = d. 0 1 ε reff Effective permittivity 2 3 µ reff Effective permeability kd Figure 4-10: Effective permittivity and permeability for 3D array of magnetodielectric spheres with cubical unit cell and a = 0.85 cm, ɛ r = 18, µ r = 5, d/a = 2.1. The bulk effective materials are plotted in the region where both kd, βd < 1. 78

80 Relative permeability Frequency(GHz) (a) kd β d (b) kd β d (c) Figure 4-11: (a) Plot of the Lorentzian permeability function given by (4.16). Dispersion diagram characteristic for one-set of dielectric spheres embedded in a Lorentzian host medium with cubical unit-cell and, a = 0.85 cm, d = h x = h y, d/a = 2.1, (b) ɛ r = 20, (c) ɛ r = Effective material ε eff µ eff kd Figure 4-12: Effective parameters for a cubical 3D array of dielectric spheres described with, ɛ r = 90, a = 0.85 cm, d/a = 2.1, inside a negative permeability material with Lorentzian function given by (4.16). The bulk effective materials are derived by using the Lewin mixing formula [3]. They are plotted in the region where both kd, βd < 1. 79

81 4.3 Summary In this chapter, electromagnetic characteristics of periodic arrays of 3D dielectric and magnetic materials based spheres are theoretically investigated. The focus is to characterize the interactions of EM waves with the 3D arrays of spheres, and compute the kd βd dispersion diagrams. A full wave spherical modal analysis is applied to express the electromagnetic fields in terms of the electric and magnetic dipole modes and the higher order terms. By enforcing the boundary conditions, transcendental equations for obtaining β in terms of k are obtained. It is assumed that, the spheres are small enough, or the frequency is such that the spheres scattering coefficients can be treated by considering only the electric and magnetic vector spherical waves. We apply the reciprocity technique to the modeling of periodic arrays composed of two different spheres as unit-cells building-blocks. The dispersion diagrams for pseudo-3d array configurations of unit-cells with two-spheres, either having the same size but different materials, or the same material but different sizes are fully characterized. The structure is optimized to establish the spheres electric and magnetic resonances around the same frequency band. A DNG medium in the spectrum band with kd and βd <1 is developed (although the bandwidth is relatively small). The concept of electric and magnetic dipole modes creation for metamaterial development using array of one-set of dielectric spheres is also explored. It is illustrated that an array of highly coupled dielectric spheres can combine the electric and magnetic modes (hybrid modes) and create negative slope dispersion diagram. Although, this backward wave is in the frequency spectrum where kd has relatively large value. Metamaterial creation utilizing array of one-set of magneto-dielectric spheres located inside a free space background, and dielectric spheres embedded in a negative permeability host, is also explored. Magneto-dielectric spheres can be optimized to create both electric and magnetic modes around the same frequency band realizing backward wave phenomenon. Array of dielectric spheres inside a ferrite is another novel design where the spheres offer electric modes resonant which are combined with 80

82 the negative permeability of the ferrite host above its resonance. The acquired magnetic materials based structures provide isotropic backward wave and DNG metamaterials characteristics. Wider dispersion bandwidths and tuning capability are interesting features of these designs. 81

83 Chapter 5 3D Array of Dielectric Spheres Manipulating Optical Metamaterials Characteristics In this chapter I investigate the performance of 3D array of dielectric spheres in optical frequencies. During the last few years there have been considerable efforts to develop metamaterials in RF/microwave frequencies as we shown in the pervious chapter; however, this progress has not been as rapid as expected in the optical domain. The goal here is to theoretically demonstrate the optical performance of arrays of dielectric nanoparticles enabling coupled electric and magnetic mode features. As mentioned earlier, to achieve a metamaterial including optical metamaterials with the functionality of interest one needs to create appropriate electric and magnetic dipole modes in a building block unit-cell [20, 25]. In Microwave frequencies, one way is to utilize two sets of dielectric particles, where one set provides the electric modes as the other produces the magnetic modes. However, the low dielectric material of the nanospheres in optical regimes generates some difficulties in tuning the THz DNG medium, if two sets of nanospheres are used. In the current chapter, I use one set of dielectric nanoparticles to accomplish the desired performance (backward wave behavior) in THz spectrum. Here, the 3D array of gallium phosphide (GaP) particles is tailored to establish 82

84 backward wave phenomena [50]. The objective is to create electric and magnetic dipole modes by utilizing 3D array of nanospheres. Hence the problem of multiple scattering of a plane wave by an array (either finite or infinite) of dielectric spheres is solved, with the use of multipole expansion method and the translational addition theorem for the vector spherical wave functions [26]-[63]. I have already shown that if the particles are small enough or the frequency is such that the magnitudes of higher order modes are negligible compared to dipole modes, dipole modes give accurate descriptions of the traveling waves in the array. We illustrate that, by bringing the dielectric spheres close to each other, the couplings between them are increased such that both electric and magnetic resonances can be achieved around the same frequency region. 5.1 Potential Applications of Electric and Magnetic Modes A 3D array of spheres has the potential to manipulate the coefficients of M and N in a novel fashion that the metamaterial with functionality of interest can be achieved. In this section, we develop the required formulations for characterizing the performance of a 3D array of dielectric spheres. Full wave electric and magnetic multipole couplings are considered for the performance modeling. The problem of multiple scattering is solved by matching the boundary conditions at the surfaces of all spheres. By utilizing these, the desired system of linear equations for the scattering coefficients is achieved. We also obtain the engineered dispersion relation for an infinite 3D array of dielectric spheres by generalizing this concept. Moreover, I demonstrate RLC circuit theory to model the interactions and couplings between spheres. Modern nano-material performances are also highlighted. We now, address the potential applications of electric and magnetic dipoles for developing novel metamaterials by tailoring their scattering coefficients. As illustrated earlier, based on the Mie series, dielectric spheres can offer electric and 83

85 magnetic dipole moments and higher order modes. To obtain a backward wave behavior, one should create the electric and magnetic dipole resonances around the same frequency regime. Here, we investigate the feasibility of making an array of identical dielectric spheres for nano-metamaterial development. Spheres operate in their magnetic resonant modes and by increasing their couplings electric modes are created. Hybrid modes (combination of electric and magnetic dipole moments) will manipulate a meta-patterned structure with the desired figures-of-merit. The scattered coefficients are the key parameters which dictate the performance of the dielectric resonator. Therefore, to generate the backward wave metamaterial behavior using one set of dielectric spheres, one should tune the resonances of the electric scattering coefficients to occur at the resonances of the magnetic scattering coefficients. For a sphere inside an array the characteristic frequencies are derived solving (3.1) thus, by changing the array configuration one can achieve the electric and magnetic resonances around the same frequency region. To show how the array formation can control the performance of the electric and magnetic resonances, we consider a periodic array of dielectric nanospheres. The array is taken to be infinite in the x and y directions (L x = L y = ) and finite in the z direction. The plane wave incident on the array is set to be y polarized with the z directed propagation vector k. Since we are exploring the traveling waves supported by the periodic array of spheres, the spheres scattering coefficients in a layer are identical. Hence, we have a linear system of 2(L z + 1) equations to be solved for the electric and magnetic scattered coefficients. For l {0, 1,..., L z } we obtain (the frequency is considered such that only the first dipole modes can be excited): [ ] [ ] A sl 11 ς1 (a)a [Λ] [ ] = il 11 [ ]. (5.1) B sl 11 ξ1 ab11 il 84

86 where: [Λ] = I [ ] ς 1 (a)σ llz [ ] ξ 1 (a)σ ll z Σ llz = Σ ll z = ly= l x = (l x,l y,l z ) (0,0,l) ly= l x = (l x,l y,l z ) (0,0,l) [ ] ς 1 (a)σ ll z I [ ], ξ 1 (a)σ llz (5.2a) h (2) 0 (kr) 1 2 h(2) 2 (kr)p 2 (cos θ), (5.2b) 3j 2 h(2) 1 (kr)p 1 (cos θ), (5.2c) r = l x h xˆx + l y h y ŷ + (l z l)dẑ, (5.2d) where A sl 11 and B sl 11 represent the electric and magnetic scattered coefficients for the layer denoted by l. I is the identity matrix. Solving the above matrix equation, the unknown scattering coefficients are achieved. Primarily, Σ llz can be envisioned as the interactions between the electric (magnetic) fields of the reference nanosphere (the sphere centered at (0, 0, l)) with the electric (magnetic) fields of all other spheres. For l = l z, Σ ll (self interaction) denotes the interactions between the electric (magnetic) fields of the reference nanosphere with electric (magnetic) fields of all other spheres located at the same plane as the reference sphere (l). In the same manner, Σ ll z is the coupling between the magnetic (electric) fields of reference sphere with the electric (magnetic) fields of all other nanospheres. It is worth mentioning that Σ ll (self coupling) is zero because P 1 (cos θ) l=lz = 0. Note that if Σ llz and Σ ll z are zero then [Λ] is simply the identity matrix, meaning that if there is no interaction or coupling between the spheres, the scattering coefficients are determined by Mie equations. To better understand the physics of the coupling coefficients and their impact on the metamaterial performance, we will investigate the RLC circuit model for these parameters. By implementing the circuit analogy and acceleration techniques [70], [34] the summations defined in (5.2) can be realized as the input impedances of RLC type circuits. For a cubic unit-cell with d = h x = h y where (l l z ) we end up with 85

87 (the schematics of these circuits are depicted in Fig. 5-1.), Σ ll = R ll (kd) + jx ll (kd), (5.3a) Σ llz = jx llz (kd, l l z ), (5.3b) Σ ll = 0, (5.3c) Σ ll z = R ll z (kd, l l z ) + jx ll z (kd, l l z ), (5.3d) where for 0 kd π: R ll (kd) 3π (kd) 2 1, R ll z (kd, l l z ) 3π (kd) 2 cos( l l z kd), (5.4a) (5.4b) X ll (kd) 3 [ 4kd ln ( kd 4π ) ] (2π ) 2 3 ln ( 2 sin(kd/2) ) + CL 2(kd), kd 2kd (kd) 2 (5.4c) X llz (kd, l l z ) 12π 2(kd) 3 e l lz 2π 2 e l lz 2(kd) 2 8π, (5.4d) X ll z (kd, l l z ) 3π (kd) 2 sin( l l z kd). (5.4e) And CL 2 (kd) is the Clausen function of the second kind. Equation (5.4d) exhibits that X llz is always negative. This reveals that the interaction factor (X llz ) between electric (magnetic) fields of the reference sphere and electric (magnetic) fields of the spheres in other planes (rather than the self plane) are of capacitive type. Also, as can be seen from (5.4b), (5.4e), Σ ll z has the kind of sinusoidal variation with kd. For the verification purposes we compare the exact solutions (5.2) and the RLC circuit models in Figs. 5-2 and 5-3. A very good agreement is observed. Circuit models presented here allow one to intuitively explain the optical properties of a periodic array of nanospheres. As an instance, for a cubical array of spheres with only one layer in the direction of propagation (L z = 0) the scattering coefficients for 86

88 Figure 5-1: RLC circuit symbolizing the interactions between spheres (l l z ). Equivalent circuits for Σ ll, Σ llz, and Σ ll z. all spheres are identical (an end-fire incidence is considered): A s1 11 = B s1 11 = A i ς 1 (a) Σ 00 B i ξ 1 (a) Σ 00 = = A i1 11 [ 1 + R 00 (kd) ] + j [ X E (1, ka, ɛ r ) X 00 (kd) ], B i1 11 [ 1 + R 00 (kd) ] + j [ X H (1, ka, ɛ r ) X 00 (kd) ]. (5.5a) (5.5b) If the aim is to accomplish both electric and magnetic resonances around the same region, then X E = X 00 and X H = X 00. In other words, for a specified structure X E (ka) should be equal to X H (ka), which is intrinsically impossible for a dielectric sphere (2.9). This observation (5.5) also reveals that to obtain both resonances around the same frequency band, the couplings between electric (magnetic) fields of reference sphere with the magnetic (electric) fields of the other spheres (Σ ll z ) are also necessitated in addition to the interactions between electric (magnetic) fields of the reference spheres and electric (magnetic) fields of the other spheres. To better understand the effect of couplings between nanospheres on the scattered coefficients, a 3-layer (L z = 2) array of dielectric spheres is investigated. Notice that, the eigenfrequencies of the electric and magnetic scattered coefficients are achieved equating the determinant of the [Λ] to zero. Referring to (5.4d) it can be shown that Σ 01 and Σ 02 have small contributions to the determinant of [Λ] compared to the other elements. Hence the characteristic equations (the determinant of [Λ]) 87

89 Exact RLC model Exact RLC model R ll X ll kd (a) kd (b) Exact RLC model X llz kd (c) Figure 5-2: The performance of the interaction coefficients Σ llz. Comparison between the exact solutions and RLC circuit model. (a) R ll (kd), (b) X ll (kd), (c) X llz (kd, 1) (l l z ). can be well approximated by: 1 =(1 ς 1 (a)σ 00 )(1 ξ 1 (a)σ 00 ) + 2ς 1 (a)ξ 1 (a)σ ς 1 (a)ξ 1 (a)σ 2 02, (5.6a) 2 =(1 ς 1 (a)σ 00 )(1 ξ 1 (a)σ 00 ). (5.6b) It is worth mentioning that, the resonant frequencies of the nano-metamaterial can be predicted from the characteristic equations when 1 = 0 or 2 = 0. Equations (5.6) are in fact the closed-form solutions to characterize the resonance phenomena in nanostructures composed of three layers (L x = L y = and L z = 2) of dielectric spheres. These closed-form models help to intuitively explain the scattering behavior of dielectric spheres inside a 3D array of spheres. In addition to that, the derived closed-form solutions can highlight the dependence of the resonance behavior on dielectric material, spheres sizes, and the array configuration. 88

90 If the inter-sphere spacings are large enough, then the interactions between spheres (Σ 00, Σ 01 and Σ 02) are so small that Λ simplified to the identity matrix, therefore the eigenfrequencies are accurately described with electric and magnetic resonance frequencies of a single nanosphere. 5.2 Optical Performances of Array of Coupled Spheres Couplings between the spheres can offer interesting behaviors. Basically by making the spheres closer to each other the couplings between them are increased, and can be optimally combined with the magnetic resonances of the spheres causing the electric and magnetic resonances to occur around the same frequency band. To investigate the impact of couplings between spheres, the performance of scattered coefficients for a 3D array of highly coupled nanospheres is scrutinized. The scattering coefficients for a 3- layer array of GaP (ɛ r = 12.5) nanospheres with radius of a = 85 nm, d/a = 2.23, and the periodicity of h x /a = h y /a = 2.1 are depicted in Figs. 5-4 and 5-5. As expected, the array configuration manipulate the electric and magnetic scattering coefficients for all layers to have the resonance behavior around the same frequency region. Figs. 5-4(b) and (c), establish that both electric and magnetic scattering coefficients for all layers resonate around f = 520 T Hz (λ = nm). It is worth mentioning that, at this frequency a/λ = 0.14, d/λ = 0.33 and the periodicity in the transverse direction is h x /λ = h y /λ = The magnitude of 1/ 1 and 1/ 2 (characteristic equations defined in (5.6)) are represented in Fig. 5-4(a), where it can be seen that, at f = 520 T Hz where electric and magnetic scattering coefficients resonate, 1 tends to zero (1/ 1 goes to resonance). From Fig. 5-4(a), one can observe that, (5.6a) precisely determines the location of the resonance frequency, which indicates that for a highly coupled array of dielectric spheres, all the interactions and couplings (Σ 00, Σ 01 and Σ 02) between one layer and others (including itself) are inevitable. On the other hand, if (5.6b) anticipates the locations of the resonance frequencies, only interactions between electric (magnetic) fields of each layer with its electric (magnetic) field (self interaction), would be taken into the account. 89

91 Figs. 5-5 represent the phase behaviors of the characteristic equations and the scattering coefficients. Figs. 5-5 (b) and (c) exhibit negative slopes for the phases of scattering coefficients after the resonance, which can be interpreted as a result of backward wave performance. Our simulation results demonstrate that using an array of highly coupled nanospheres one can achieve backward wave characteristics. R llz Exact RLC model kd (a) X llz Exact RLC model kd (b) Figure 5-3: The performance of the coupling coefficients Σ ll z. Comparison between the exact solutions and RLC circuit model (l l z ). (a) R ll z (kd, 1), (b) X ll z (kd, 1). It can be deduced that, the backward phenomena is created because of the high coupling between the nanospheres. To verify this further, the performance of the scattered coefficients decrease the coupling (enlarging the unit-cell sizes) is also analyzed. Figs. 5-6 and 5-7 present the electric and magnetic scattering coefficients for a 3-layer array of GaP nanospheres with a = 85 nm, d/a = 2.23 and h x = h y. As observed, For h x /a = h y /a 4, the electric scattering coefficients do not provide the resonant behavior at the resonance of magnetic scattering coefficients. Also, Figs. 5-6(b) and 5-7(b) exhibit that by increasing the transverse distance, the phases of scattering coefficients do not show the negative slope performances after the resonance, even for h x /a = h y /a = 3. These signify that, to provide the backward wave phenomena the coupling in the transverse direction should be high enough. Figs. 5-8 and 5-9 illustrate the dependence of scattering coefficients on d, given h x and h y. The magnitude of electric and magnetic scattering coefficients show the resonance performance around the same frequency spectrum for all the cases presented in Figs. 5-8(a) and 5-9(a), however the negative slope of the phase switches to positive slope for 90

92 Characteristic equations (magnitude) / 1 1/ Frequency (THz) (a) Electric scattering coefficient (magnitude) first layer second layer third layer Frequency (THz) (b) Magnetic scattering coefficients (magnitude) first layer second layer third layer Frequency (THz) (c) Figure 5-4: Performance of a periodic array of GaP nanospheres with three layers in the direction of propagation (L z = 2) and ɛ r = 12.25, a = 85 nm, having unit-cell size of d = 2.23a and h x = h y = 2.1a, (a) Magnitude of 1./ 1 and 1./ 2 defined in (5.6), (b) Electric scattering coefficients (magnitude) and, (c) Magnetic scattering coefficients (magnitude). d/a = 4. It is interesting to note that by decreasing the periodicity in the transverse direction (h x and h y ), increasing the coupling, the electric resonance is moved back over the frequency (and appears in the spectrum of interest) where the magnetic resonance is slightly moved forward, where for an optimal design both modes occur around the same frequency band. Making the spheres closer in the transverse plane, one can envision increasing the capacitances between the dielectric spheres. This has a positive effect for electric mode creation while slightly perturbs the magnetic mode development. By increasing the periodicity in the longitudinal direction (d) both electric and magnetic resonances are moved in the same direction, towards the lower frequency (for large d the brag modes play important roles in the performance). 91

93 Characteristic equations (phase) Frequency (THz) (a) 1/ 1 1/ 2 Electric scattering coefficients (phase) first layer second layer third layer Frequency (THz) (b) 180 Magnetic scattering coefficient (angle) first layer second layer third layer Frequency (THz) (c) Figure 5-5: Performance of a periodic array of GaP nanospheres with three layers in the direction of propagation (l l z ) and ɛ r = 12.25, a = 85 nm, having unit-cell size of d = 2.23a and h x = h y = 2.1a, (a) Phases of 1/ 1 and 1/ 2 defined in (5.6), (b) Electric scattering coefficients (phase) and, (c) Magnetic scattering coefficients (phase). To better demonstrate the progression of backward wave behavior, we also explore the dispersion diagram for an infinite array of spheres in the following section. 5.3 Dispersion Diagram Dispersion diagram of a periodic configuration can provide significant physical features. In this section, the dispersion performance of the 3D array of dielectric nanospheres is characterized. The array is considered to be infinite in all directions (L x = L y =, l z = 0, ±1,..., ± ). We take the z axis to be the array axis. It is assumed that the array can support a plane wave with the propagation vector β 92

94 Electric scattering coefficients (magnitude) h x =h y =5a h x =h y =4a h x =h y =3a Frequency (THz) (a) Electric scattering coefficients (phase) h =h =5a x y h x =h y =4a 90 h x =h y =3a Frequency (THz) (b) Figure 5-6: Electric scattering coefficients behavior for a periodic array of GaP nanospheres with three layers in the direction of propagation and ɛ r = 12.25, a = 85 nm, d = 2.23a versus different unit-cell sizes in the transverse direction (only the middle layer s scattering coefficient is plotted), (a) Magnitude, (b) Phase. Magnetic scattering coefficients (magnitude) h x =h y =5a h x =h y =4a h x =h y =3a Frequency (THz) (a) Magnetic scattering coefficients (phase) 180 h =h =5a 135 x y h x =h y =4a 90 h x =h y =3a Frequency (THz) (b) Figure 5-7: Magnetic scattering coefficients behavior for a periodic array of GaP nanospheres with three layers in the direction of propagation and ɛ r = 12.25, a = 85 nm, d = 2.23a versus different unit-cell sizes in the transverse direction (only the middle layer s scattering coefficient is plotted), (a) Magnitude, (b) Phase. 93

95 Electric scattering coefficients (magnitude) d=4a d=3a d=2.5a Frequency (THz) (a) Electric scattering coefficients (phase) d=4a 135 d=3a d=2.5a Frequency (THz) (b) Figure 5-8: Performance of electric scattering coefficients for a periodic array of GaP nanospheres with three layers in the direction of propagation and ɛ r = 12.25, a = 85 nm, h x = h y = 2.23a versus different unit-cell sizes in the propagation direction (only the middle layer s scattering coefficient is depicted), (a) Magnitude, (b) Phase. where: β = β xˆx + β y ŷ + βẑ (5.7) To obtain the dispersion diagram the eigenvalue problem is solved, namely we force the incident expansion coefficients to be zero (A ip mn = Bmn ip = 0). Inasmuch we have a uniform array of infinite dielectric nanospheres, the electric and magnetic scattering coefficients due to a plane wave with the propagation vector β are identical except for a phase shift in the direction of propagation: (A/B) sp mn = (A/B) sq mne jβ.dqp (5.8) Substituting (5.8) in (3.1) we end up with: ( A s0 mn = ς n (a) q q 0 ( Bmn s0 = ξ n (a) q q 0 ν=1 µ= ν ν=1 µ= ν ν e [ ] ) jβ.d 0q A µν mn(d pq, θ pq, φ pq )A s0 µν + B mn(d µν pq, θ pq, φ pq )Bµν s0 (5.9a) ν e [ ] ) jβ.d 0q A µν mn(d pq, θ pq, φ pq )Bµν s0 + B mn(d µν pq, θ pq, φ pq )A s0 µν (5.9b) 94

96 The desired dispersion equation is then achieved resolving the above system of equations for A s0 mn and B s0 mn. Notice that (5.9) is exact and no approximation is taken into the account. Moreover the derived equations are general and the dispersion performance for both normal and oblique incidence waves can be accomplished. Now, if the nanospheres are sufficiently small or the frequency is such that pairs of crossed electric and magnetic dipoles are enough to model the sphere, the above linear system can be simplified further. For the special case of end-fire incidence: (A/B) s0 11 = (ς 1 (a)/ξ 1 (a)) e [ jβẑ.d 0q (A/B) s0 11A 11 11(d 0q, θ 0q, φ 0q ) + (B/A) s0 11B11(d 0q, θ 0q, φ 0q ) ] (5.10) q q 0 One can numerically solve (5.3) for βd, given values of k using for instance a secant search procedure. It is worth highlighting that, the series in (5.3) are slowly convergent and an excessive number of terms is required to acquire the desired accuracy. Hence to ease the computer programming, rapidly convergent solutions are applied [70], [4]. To evaluate the accuracy of our approach we compare the dispersion diagram obtained using the approach of our work and the technique discussed in [4]. For a 3D array of high dielectric spheres with ɛ r = 20, a = 85 nm and cubical unit-cell of a/d = 0.45, the diagrams in Fig. 5-10(a) reveal a very good agreement Fig. 5-10(b) displays the dispersion diagram for the array of dielectric GaP nanospheres with ɛ r = and radius of a = 85 nm having the unit-cell size of h x = h y = 2.1a and d/a = As it is shown a negative slope backward wave with relatively large bandwidth is established. This occurs in the spectrum where the scattering coefficients show the resonance behavior (Figs. 5-4 and 5-5). Basically, bringing the spheres closer to each other will increase the electric coupling between the spheres where it can be combined with the magnetic mode performance of spheres above their magnetic resonances, offering a hybrid mode with backward wave phenomenon. 95

97 Magnetic scattering coefficients (magnitude) d=4a d=3a d=2.5a Frequency (THz) (a) Magnetic scattering coefficients (phase) d=4a d=3a d=2.5a Frequency (THz) (b) Figure 5-9: Performance of magnetic scattering coefficients for a periodic array of GaP nanospheres with three layers in the direction of propagation and ɛ r = 12.25, a = 85 nm, h x = h y = 2.23a versus different unit-cell sizes in the propagation direction (only the middle layer s scattering coefficient is depicted), (a) Magnitude, (b) Phase. 5.4 Conclusion In this chapter, theoretical investigation of all dielectric nano-metamaterials is addressed. A full wave spherical modal analysis is applied to express the optical fields in terms of the electric and magnetic dipole modes and the higher order terms. Imposing the boundary conditions at the surface of each nanosphere, using the translational addition theorem for vector spherical wave functions, required equations to determine the scattered coefficients are obtained. We show that, if the nanospheres are small enough or the frequency is such that the magnitude of higher order modes are negligible compared to dipole modes, it is enough to model each particle with only dipole spherical waves. The concept of electric and magnetic dipole mode generation for meta-patterned structure development is presented. We demonstrate that by bringing the dielectric spheres close to each other the electric coupling between them are increased such that both electric and magnetic resonances can be achieved around the same frequency region. After the resonances, the phases of the scattering coefficients show the negative slope behavior. Dispersion diagram characteristic for an array of highly coupled spheres is investigated and the backward wave phenomenon is achieved. 96

98 kd Shore Yaghjian method The paper s approach β d (a) Frequency (THz) βd (b) Figure 5-10: Dispersion diagrams characteristics for the 3D array of nanospheres (a) Comparison between our approach and the technique introduced in [4] for an array of nanospheres with ɛ r = 20, a = 0.5 cm and cubical unit-cell with a/d = A very good match is achieved, (b) Engineered dispersion diagram for an array of GaP nanospheres with ɛ r = 12.25, a = 85 nm and unit-cell size with d/a = 2.23 and h x = h y = 2.1a. The second branch shows the backward wave behavior. A capable modeling technique by utilizing RLC circuit theory for characterizing the scattering coefficients is presented. Following the same analogy, the couplings between spheres are also described in a type of circuit realization. Computational results verify the accuracy of circuit models in predicting the performance of nanostructures. The success of this work may open new paradigms for high-performance electric and magnetic dipole mode creations in optics, using non-magnetic particles. 97

99 Chapter 6 Metamaterial-Based Electrically Small Antennas So far, the design and development of metamaterials in microwave and optical frequencies were discussed. In this chapter, I investigate the use of metamaterials for enhancing the performance of electrically small antennas. Smaller physical size, wider bandwidth and higher efficiency are three desired characteristics of antennas for mobile systems. In recent years, considerable efforts have been devoted towards antenna miniaturization. Fundamentally the ability of any antenna to radiate effectively depends on k 0 a (where k 0 is the wave number in free space and a is the radius of the smallest sphere enclosing the antenna). According to Chu limit [39], the Q of any antenna is inversely proportional to the radius of the smallest sphere which completely surrounds it, so the smaller the radius, the higher the Q and the narrower the bandwidth. Thus, the challenges are to make the physical size of the antenna as small as possible along with a wider bandwidth. Recently, there have been some efforts in this regard to present wide-band small antennas by exciting negative constitutive parameters metamaterial resonators. For instance, Stuart et al in [40] excited an ENG sphere with a dipole feed to produce the appropriate polarization required for the resonance performance. In fact, they demonstrated that one can achieve a very small antenna (as much as interested) by operating the sphere in ɛ r = 2 region. Spherical geometry of the structure 98

100 offers wide-band performance approaching the Chu limit. Ziolkowski et al. have also demonstrated other novel small antennas designs utilizing combinations of ENG and MNG or Double Negative (DNG) metamaterials [41]. In this work by finding the Green s function for electric fields we study the exact behavior of such an antenna and obtain some properties like resonance frequencies. I also explore the possibility of miniaturizing the antenna with the use of magneto-dielectric and negative permittivity materials for constructing the small antenna. Moreover, I show that magnetic-dipoles inside negative permeability would provide the same miniaturization. In this scenario, the structure is the dual of the pervious structure, namely instead of electric current we have magnetic current and so on, which will be discussed completely. In this chapter, I also investigate the behavior of a dipole antenna inside a concentric sphere. Embedding the antenna inside a core-shell shows potential for improving the bandwidths performance of the antenna. The idea is to combine both electric and magnetic effects by using a coreshell. To implement this theory, a spherical core-shell is considered, where shell is made of magnetic materials and core is dielectric. I show that by proper choosing of materials, the quality factor as low as Chu limit can be achieved. 6.1 Introduction The purpose of this chapter is to theoretically investigate the behavior of an electrically small antenna enclosed in a metamaterial sphere. I consider the use of magnetic, magneto-dielectric and negative permittivity materials for constructing the small antennas. The Greens functions for the evaluation of the input impedance are derived and the method of moment with Galerkins procedure is used to determine the probe current from which the input impedance of the resonator is calculated. A physical insight is provided, and the effect of metamaterials for bandwidth enhancement is addressed. The use of double negative (DNG) materials has been considered previously as a means of improving power radiated by electrically small antennas [41]. My work 99

101 is distinguished from earlier studies in that we are specifying using negative permeability materials (MNG) in addition to use negative permittivity (ENG), magnetic and magneto-dielectric materials. Also I directly find a formula for obtaining resonance frequencies for such a small antenna. From practical points of view, obviously, one should also consider the availability of metamaterials. For instance, a negative permittivity medium is naturally accessible only at optical frequencies and at microwave frequencies, negative permittivity material can be realized using straight wires. However negative permeability material can be achieved naturally in microwave frequencies less than 4GHz. This chapter is divided to three main parts, first I investigate the behavior of a monopole antenna inside a hemispherical resonator. To accomplish the desired miniaturization, the sphere is filled with and ENG materials. The Greens function for electric field is decomposed into the incident and scattered terms. This will allow effectively formulating the problem in order to achieve a rapid convergence in the numerical analysis. The results are simplified to study the resonant condition and radiation performance of the metamaterial antenna. In the second section, I illustrate the behavior of an aperture coupled hemispherical electrically small antenna using negative permeability material. The use of MNG material will allow achieving the similar functionality (as a dipole inside an ENG resonator), if the dual of ENG feeding system is implemented. This means that one must use magnetic current excitation instead of the electric dipole feed. In the last part, I explore the quality factor behavior of small antennas embedded in multi-layer sphere. The Green s functions for electric fields are attained and by using method of moment the input impedance is derived. The quality factor is then obtained from the input impedance. I illustrate that by optimizing the structure and the materials quality factor as low as Chu limit can be attained. 100

102 6.2 Electrically Small Antenna: Monopole Inside Hemisphere It is known that a properly constructed self-resonant sphere gives good small antenna bandwidth. The use of ENG materials clearly provides one possible approach for synthesizing a self-resonant spherical object. Rather, it is the geometry of a spherical resonator which allows us to approach the Chu limit [40]. Now in order to form a spherical resonator we consider a configuration like what is shown in Fig The hemisphere resonator of radius a and permittivity of ɛ and permeability of µ is fed with a transmission line with a small monopole stub of length l which is located at a distance b from the center of sphere. Using the equivalence principle and image theory this structure can be considered as a dipole inside a sphere but with one-half input impedance. Obviously, the input impedance of the original problem is equal to one-half of the new structure Green s Function Analysis To obtain the electric field in the z-direction due to the z-directed current element, we first decompose the current vector into the r- and θ-components. Note that r- directed current can only generate T M r modes, where the θ-directed current can generate both T M r and T E r modes. For the θ-directed current, we represent three Green s functions as incident field (dipole only), scattering field inside the sphere (the field inside the sphere caused by the sphere discontinuity), and the scattering field outside the sphere; as given below 101

103 [32, 33, 74, 75]. G A i,f i J θ = G As i,fs i J θ = G A so.f so J θ = n n=0 m= n n n=0 m= n n n=0 m= n C A,F mn P m n (cos θ) e jmφ { Ĵn (kr) r < r Ĥ (2) n (kr) r > r (6.1a) D A,F mn P m n (cos θ) e jmφ.ĵn (kr) r a (6.1b) E A,F mn P m n jmφ (cos θ) e.ĥ(2) n (k 0 r) r a (6.1c) Where, Ĵn (x), Ĥ(2) n (x) are spherical Bessel and Hankel functions respectively. P m n (x) is the associated Legendre function. It is worth noticing that, A and F stands for the electric and magnetic vector potentials. The unknown coefficients of incident and scattered fields are obtained by applying the boundary conditions. The unknown incident expansion coefficients in (6.1)(a) are obtained by employing the continuity of E θ, E φ and H θ at r = r. H φ is discontinuous by θ-directed surface current distributed on r = r (source point). The unknown scattering coefficients in (6.1)(b,c) are achieved by applying the continuity of the tangential electric and magnetic fields at r = a. It is worth mentioning that (r, θ, φ ) refers to source point while (r, θ, φ) refers to field point. Note that, in this part the Green s function due to general a θ-directed current located at r is obtained where r can be at any location including b. It will be shown that the Green s function is related to (r, θ, φ ) hence the location of the dipole antenna (for instance the distance b from the center) is defined within r, θ and φ. Solving the boundary conditions the unknown scattering coefficients and hence the Greens function associated with the θ-directed current are accomplished. Now that we have the Green s functions associated with θ-directed current, by obtaining the r-directed Green s functions, we can derive the input impedance of the antenna. The solution for the r-directed Green s function due to the incident dipole can be obtained by solving the following differential equation for electric vector potentials 102

104 Figure 6-1: The antenna configuration, a probe fed hemispherical resonator antenna, and the equivalent geometry of the antenna configuration. [76], [pp ]: ( 2 + k 2) GA i J r r = 1 δ ( ) ( ) ( ) r r δ θ θ δ φ φ r r 2 sin θ (6.2) By matching the boundary conditions at r=a the scattering solutions for inside and outside of the sphere can be derived. By having the Green s functions for the vector potentials (G A J r and G A/F J θ ) the Green s functions for the electric field can be obtained [76]. The next step in obtaining the antenna input impedance is to derive the following integral equation for the z-component of electric field: E z = S 0 [( ) ( ) cos θ G Er J r sin θ G E θ J r cos θ cos θ G Er J θ sin θ G E θ J θ sin θ ] J zds (6.3) Where S 0 is the surface on which the current is flowing and J z is the z-directed current. Also G E r J r and G E θ J r are the Greens functions for r and θ directed electric fields associated with the r directed current. In the same manner G E r J θ and G E θ J θ the Green s functions representing the r and θ directed electric fields associated with the θ directed current. It is found that the incident solution (dipole field) is a slowly convergent series of Hankel functions and hence an excessive number of terms are required [77]. To avoid this difficulty, we use the well known Green s function of a cylindrical dipole with the reduced kernel [76, 77]. Combining the incident field and 103 are

105 the scattering field, the z-directed Green s function due to the z-directed current can be written as (note that for a thin dipole the φ, φ variation can be ignored) G Es inside J z = 1 θ cos θ.cos 4πωɛk r 2 r sin θ cos θ 4πωɛ r 2 r n (n + 1) h n L(n, θ, θ, φ, φ )Ĵn (kr ) Ĵn (kr) n=1 n=1 + 1 cos θ sin θ 4πωɛ rr 2 k 1 b 4πωɛ rr n φ n=1 k sin θ sin θ 4πωɛ rr n=1 h n θ L(n, θ, θ, φ, φ )Ĵ n (kr ) Ĵn (kr) h n θ L(n, θ, θ, φ, φ )Ĵn (kr ) Ĵ n (kr) φ L(n, θ, θ, φ, φ )Ĵn (kr ) Ĵn (kr) h n n (n + 1) θ n=1 θ L(n, θ, θ, φ, φ )Ĵ n (kr ) Ĵ n (kr) (6.4a) G E i J z = j ( ) 2 e jk ( z z ) 2 +r ωɛ z k2 4π (z z ) 2 + r1 2 (6.4b) where ( L(n, θ, θ, φ, φ ) = P n (cos θ cos θ + sin θ sin θ cos φ φ )) (6.5) G E i J z represents the Green s function of the incident source radiating in an unbounded dielectric medium, while G E s inside J z stands for the Green s function of the scattered electric field inside the sphere associated with the z-directed current. h n = b n = k (2n + 1) 0 n (n + 1) k (2n + 1) 0 n (n + 1) ɛ 0 Ĥ n (2) (ka) Ĥ(2) n k 0 ɛ 0 Ĵ n (ka) Ĥ(2) n µ 0 Ĥ n (2) k 0 µ 0 Ĥ n (2) (ka) Ĥ(2) n (k 0 a) k ɛ Ĥ(2) n (k 0 a) Ĥ(2) n (ka) (k 0 a) + k ɛ Ĥ(2) n (k 0 a) Ĵ n (ka) (k 0 a) k µĥ(2) n (k 0 a) Ĥ(2) n (ka) (k 0 a) Ĵn (ka) + k µĥ(2) n (k 0 a) Ĵ n (ka) (6.6) (6.7) 104

106 The characteristic equations for TM and TE modes are: T M n = k 0 ɛ 0 T E n = k 0 Ĵ n (ka) Ĥ(2) n Ĥ (2) n µ 0 (k 0 a) + k ɛ Ĥ(2) n (k 0 a) Ĵ n (ka) (6.8) (k 0 a) Ĵn (ka) + k µĥ(2) n (k 0 a) Ĵ n (ka) (6.9) It should be pointed out again that for an antenna located at the center of sphere only T M r modes exist. Finding the electric field Greens functions, the radiator input impedance can be then determined by applying the method of moment with Galerkins procedure, and piecewise sinusoidal (PWS) basis functions, and delta gap source model for the feed. The unknown coefficients of dipole current (I n ) are obtained from the following matrix equation [77, 75, 74, 78, 79]. [ ] Z i mn + Zmn s [In ] = [V m ] (6.10) where Z i mn = Z s mn = z z f m (z) z ( f m (z) z ( ) G E i J z f n (z ) dz dz (6.11) ) G E s inside J z f n (z ) dz dz (6.12) A computer program is developed based on the above equations to successfully characterize the performance of an antenna embedded inside the metamaterial sphere. Before we study the behavior of antenna inside an ENG material, let us first validate the accuracy of numerical method by computing the input impedance of a dipole antenna embedded inside a dielectric hemisphere, as discussed in [77]. Fig. 6-2 shows the results and reveals a good comparison. Two resonant frequencies are observed. The first resonance is at 3.62 GHz and is related to the resonant frequency of the dipole solely inside the dielectric material, and the second resonance is at 5.58 GHz occurs as a result of spherical discontinuity. Moreover the second resonant frequency has a narrower bandwidths (about 7.46%) compared to the resonant frequency caused 105

107 by the dipole (about 14.92%). In the next section, approximated Green s function for a small antenna inside a negative permittivity sphere is discussed. (a) (b) Figure 6-2: Input impedance and return loss vs. frequency using delta gap source model. (a) Input impedance. (b) Return loss: a = 12.5 mm, b = 6.4 mm, l = 6.5 mm, ɛ r = 9.8, r 1 = mm Small Antennas Enabled by Negative Permittivity Metamaterials In this section, I illustrate some approximation for the achieved Green s functions. The dipole is located at the center of the sphere, where only T M r modes are excited. Using proper approximations, I can model the resonant behavior. When ka, k 0 a < 0.5 we can approximate the exponential and the spherical Bessel and Hankel functions using the first terms of the Taylor series. We have (for x << 1) e x 1 + x 2 /2 + x 3 /6 Ĵ n (x) xn+1 (n + 0.5)! (6.13a) (6.13b) 106

108 By Utilizing the above approximations, after some manipulation we end up with: ( ) G E i 1 2 J z = âz jωɛ z + 2 k2 1 jk (z z ) 2 + r1 2 k2 2 4π (z z ) 2 + r1 2 G Es inside 1 J = âz n (n + 1) z 4ωɛ k h 1 n (n + 0.5)! 2 n=1 ( (z z ) 2 ) + r1 2 + j k 6 3 ( (z z ) 2 + r 2 1 ) 1.5 (6.14a) ( ) 2n+2 k (zz ) n 1 (6.14b) 2 where ( (k0 ) ) 2n+1 (n + 1) ɛ r + n h n = (2n + 1) k nɛ r + n j 4n ( ) n 4 ɛ r 1 (n + 0.5)!2 aπ k k 2 a 2 nɛ r + n + 1 (6.15) h n is an approximation of the scattering coefficient inside the sphere due to the r- directed current. From the above equations (as along the dipolez, z < l) one can see that the scattering solution is proportional to (l 2 /a 2 ). Therefore, to consider only the first term, for a successful analysis, the antenna length should be much smaller than the size of sphere. For comparable values of antenna length with respect to the sphere radius, higher order terms must be considered. The characteristic equation which determines the resonance performance is estimated by: T M n = j (k/k0 ) n k (2n + 1) ɛ (nɛ r + n + 1) (6.16) In the region where ɛ r = (n + 1) /n one can expect a singularity in the scattering field that can be associated with a resonant frequency. As an example, the first dominant mode (n = 1) offers a resonant frequency for ɛ r = 2. Higher order modes can be resonant in other negative permittivity values, as will be demonstrated later. With the Green s function for electric field the input impedance of the structure can be attained by using method of moments with Galerkin s procedure, pulse basis functions, and delta gap source model. I demonstrate that by using 9 elements in the moment 107

109 method is sufficient to present a very good approximation for the input impedance. For the case of small antennas located at the center of the sphere, the dipole incident field depends only on r r. Also the Z i matrix is a symmetric Toeplitz matrix and hence only one row is enough to describe the whole matrix. Now to theoretically investigate the behavior of the antenna two cases are considered: 1- l/a is so small that only the first term is enough to describe the scattering field. 2- l/a is not that small but the first three terms are sufficient to obtain a well approximation for the scattering solution. Small-Size Dipole For a small dipole (compared to the radius of the sphere) one may consider only the first dominant term which shows that the scattering solution has a singularity at ɛ r = 2 and hence a potential resonant frequency in this region is expected, as addressed in [77]. The scattering matrix Z s can be found using the first dominant term: Z s mn = (2 l/9)2 h1 k 3 / (32ωɛ (1.5)! 2 ) (6.17) Finding the Z s mn, Z i mnelements, the hemisphere antenna input impedance can be now successfully obtained in a closed form as Z in = 1 j4πωɛl/9 ( ) j (kl/9) h ) j (kl/9) 3 ( 0.67 h (6.18) The above equation shows that if ɛ r > 1 then input impedance of the structure behaves like a capacitor while for the region when ɛ r 2 the resonator acts like an inductor, it is of value to mention that for other region of ɛ r the behavior of input impedance also depends on l/a. The scattering electric field inside the sphere can be obtained using the probe current: E sinside = [ k 3 h1 (l/9) 16ωɛ (1.5)! 2 9 k=1 108 I 1 k ] (cos θ â r sin θ â θ ) (6.19)

110 where I 1 k are the coefficients of the probe current with these approximation. This shows that the near field inside the sphere caused by scattering only depends on cos θ and sin θ. The radiated field of the antenna is, E soutside ( 0.16j ωµ 9 0 ɛ r + 2 (2l/9) k=1 I 1 k ) sin θ e jk 0r r â θ (6.20) A Larger-Size Dipole For a longer size dipole one needs to consider a more number of terms. Let s investigate the situation considering 3-terms for the summation in (6.1). Thus, the Z s matrix is achieved: Z s mn = (l/9)2 k 3 h 1 8ωɛ(1.5)! 2 + (5 m) (5 n) 3 (l/9)4 k 5 h 2 8ωɛ(2.5)! m 3 9 2m n 3 9 2n 3 (l/9) 6 k 7 h (6.21) 3 768ωɛ(3.5)! 2 Notice that Z s is a 9 9 matrix. The input impedance of the hemisphere structure is thus: Z in = where H 1 = E S inside r where: 1 j4πωɛ(l/9) 28.3 (kl/9) 10 H1 h3 + j2.85 (kl/9) 3 H1 + j3.09 (kl/9) 7 h (kl/9) 10 H1 h3 + j1.495 (kl/9) 3 H1 + j3.28 (kl/9) 7 h (6.22) ( ) 0.45 h The scattering field inside the sphere is described by: = â r [( k 3 h1 (l/9) 16ωɛ (1.5)! 2 9 k=1 I3 k +â θ [( k 3 h1 (l/9) 16ωɛ(1.5!) 2 9 k=1 I3 k ) ( ) ] cos θ + l 3 (k/2) 7 h 3 C (3.5!) 2 ωɛ I r 2 (5 cos 3θ + 3 cos θ) ) ( ) ] sin θ + (k/2) 7 l 3 h 3 r 2 (5 sin 3θ + sin θ) (3.5!) 2 ωɛ C I (6.23) C I =I 3 1 ( 1 (7/9) 3 ) + I 3 2 ( (7/9) 3 (5/9) 3) + I 3 3 ( (5/9) 3 (3/9) 3) + I 3 4 ( (3/9) 3 (1/9) 3) + I 3 5/9 3 (6.24) 109

111 Equation (6.23) shows that the electric field inside the sphere depends only on the first and third harmonic of θ, however as we consider three terms one may expect to see some dependence on the second harmonics as well, but because of the symmetry of the structure even terms are simply removed from the electric field and that is why we do not see a resonance frequency at ɛ r = 2/3 in the input impedance. The far field radiation pattern in this case is: E soutside = e jk 0r [( r â θ 0.15j (k 0/2) 2 l 3 C I ωɛ 0 (3ɛ r + 4) ) (5 sin 3θ + sin θ) + ( 0.16j ωµ 9 0 ɛ r + 2 (2l/9) k=1 I 3 k ) sin θ ] (6.25) Results and Physical Insight To verify the validity of our work, we compare the approximated results for input impedance of small antenna embedded in material which its permittivity represented by Drude dispersion relation with the results obtained from computing the input impedance based on the complete formulation for Green s function (no approximation). The Drude dispersion relation is specified by (refer to Fig. 6-6): ɛ (ω) = ɛ 0 (1 ωp 2 ) ω (ω + jγ p ) (6.26) Fig.6-3 shows the result of this comparison for a 1.5mm dipole enclosed in a 7.5mm Drude metamaterial sphere. A good comparison is observed. Two resonant frequencies around 2.31 GHz and 2.65 GHz are associated with ɛ r = 2 (dominant mode), and ɛ r = 4/3 (third mode). Also, we investigate the return loss for an antenna with a=7.5mm, l=6mm,l/r 1 = 15 (Fig. 6-4). The antenna performance is obtained using the complete form Green s function. The resonant frequencies are again around 2.3 GHz and 2.65 GHz, validating our theoretical investigation. Therefore, for an electrically small antenna the resonant frequencies can be predicted around the regions where ɛ r = (n + 1)/n. It is also found that in the absence of material loss (which is 110

112 (a) (b) Figure 6-3: comparison between approximation and numerical results for the Input impedance (real part and imaginary part) of a hemispherical resonator filled with a metamaterial described by Drude dispersion relation: a=7.5mm, l=6mm, r 1 =0.4mm, f p = 4GHz, γ p = 0.001ω p Figure 6-4: Return loss for a hemispherical resonator filled with a metamaterial described by Drude dispersion relation: a=7.5mm, l=6mm, r 1 =0.4mm, f p = 4GHz, γ p = 0.001ω p not a practical design) these approximations may not provide a good accuracy for the value of input impedance; however they predict a very good approximation for the resonance frequencies. In the next section I investigate the behavior of an aperture coupled hemispherical small antenna. 111

113 6.3 Aperture Coupled Hemispherical Electrically Small Antennas As discussed before negative permeability material is more accessible at microwave frequencies than negative permittivity materials, so we consider the use of MNG materials to build effective small antennas. Another thing is that coaxial probes usually introduce ohmic loss and large probe self-reactance in high frequencies, the use of an aperture coupled source does not have these problems and also the drilling hole for the probe penetration is no longer necessary. The geometry of the antenna Figure 6-5: The Geometry of the aperture-coupled hemispherical antenna is shown is shown in Fig.6-5, where a slot of length L and width W couples to the hemispherical resonator of radius a, permittivity of ɛ a and permeability of µ a at its center. The feeding system is a microstripline of width W f. The grounded dielectric slab of height d is represented with dielectric constant of ɛ rs.by using the reciprocity procedure introduce by Pozar [80], along with the assumption of infinite length for microstrip feed line we end up with [81] [V n ] = { [Y a mn + Y s mn] [ v m][ v n ] t } 1 [ v m ] (6.27) Where Y a mn is the admittance caused by the discontinuity of the sphere and Y s mnis the substrate admittance which are defined in [81]. The evaluation of Y s mn, v m were well 112

114 studied [80, 82] and can be easily performed in spectral domain: Ymn s = 1 4π 2 F 2 u (k x ) F 2 p (k y ) G HM yy (k x, k y ) cos (k y (y m y n )) dk x dk y (6.28) G HM yy v m = = j ωµ 0 1 G HJ yx ( β f, k y ) F u (k y ) F p (k x ) cos (k y y m ) dk y (6.29) 2π Z c [ j (k1 cos k 1 d + jk 2 ɛ rs sin k 1 d) ( ɛ rs k0 2 ky) 2 jk2 yk 1 (ɛ rs 1) k 1 T m T e T m ] + ( k 2 k 2 y) ωµ a k 3 (6.30) G HJ yx = k 1 T e + jk2 x (ɛ rs 1) sin k 1 d T e T m (6.31) Where: k 2 = ω 2 µ a ɛ a k 2 3 = k 2 β 2, Im k 3 < 0 (6.32a) (6.32b) And β f is the propagation constant of an infinite micro-strip line which can be found in [81] and Z c is the characteristic impedance of the strip line. The expressions for other quantities are given in the appendix of [80, 81]. The scattering Green s function for the y-directed magnetic current located at the center of the sphere is obtained using the same procedure as the monopole inside the hemisphere [81, 80]: ( G H s inside M y = â y 1 sin φ cos φ 4πωµ rr 2 Where: 1 sin φ sin φ. 4πωµ k r 2 r 2 n=1 n (n + 1) b ( ( )) np n cos φ φ Ĵn (kr ) Ĵn (kr) ) n=1 b n φ P n ( cos ( φ φ )) Ĵn (kr ) Ĵ n (kr) (6.33) b n = (2n + 1) k 0 µ 0 Ĥ n (2) (ka) Ĥ(2) n (k 0 a) k µ a Ĥ n (2) k 0 µ 0 Ĵ n (ka) Ĥ(2) n (k 0 a) + k µ a Ĥ n (2) (k 0 a) Ĥ(2) n (ka) (k 0 a) Ĵ n (ka) (6.34) 113

115 It s worth noting that for the slot located at the center of sphere only T E r (dual of the dipole) modes with the following characteristic equation exist: T n E = k 0 Ĵ n (ka) µ Ĥ(2) n (k 0 a) + k a Ĥ n (2) (k 0 a) 0 µ Ĵ n (ka) (6.35) a Like the previous section, appropriate approximations are applied: G H s inside M y 1 = ây 4ωµ k n=1 n (n + 1) b n 1 (n + 0.5)! 2 ( ) 2n+2 k (yy ) n 1 (6.36) 2 ( (k0 ) ) 2n+1 (n + 1) µ ra + n bn = (2n + 1) k nµ ra + n j 4n ( ) n 4 µ ra 1 (n + 0.5)!2 aπ k k 2 a 2 nµ ra + n + 1 (6.37) The resonance frequencies can be predicted as the zeros of the characteristic equation T E n = 0: T E n = j (k/k0 ) n k (2n + 1) µ (nµ ra + n + 1) (6.38) Finally applying the method of moment with Galerkin s procedure we end up with: Where S 0 is the area of the slot and: Ymn a = 2 S 0 f m (x, y) G H (x, y ) ds ds S 0 (6.39) f m (x, y) = 1 sin(k e (h y y m )) x < W/2, y y m < h (6.40a) W sin(k e h) y m = L + mh (6.40b) 2 µa ɛ a + µ 0 ɛ s k e = ω (6.40c) 2 114

116 After Y a mn and hence [v n ] are obtained, one can calculate the series impedance Z e of the slot [81, 80], Z e = Z c [ v n ] t [V n ] [ v n] t [V n ] (6.41) Once Z e is found, the input impedance of the antenna configuration with the slot terminated by an open circuit stub of length L s is evaluated [80]: Z in = Z e j cot (β f L s ) (6.42) Just like the case of the dipole (electric current) the magnetic field is also an even function of y,y. This shows that resonance frequencies happen only when n is an odd number Computed Results physical interpretation of the antenna as a resonator can suggest a general methodology for designing electrically small antennas. To validate the accuracy of our work, we compare the final formula for input impedance of small antenna embedded in metamaterial with the results obtained from computing the input impedance based on the complete formulation for Green s function (no approximation) Numerical results for a hemisphere of radius 7.5 mm fed by a slot coupled to a microstripline are also presented validating the theory discussed (Figs. 6-7 and 6-8). The permeability of the material inside it represented by (cf Fig. 6-6): ( ) µ a = µ 0 1 κ 2 ω 2 ω 2 ωh 2 j2δ hω (6.43) As predicted by the Theory, the resonant frequencies are around the region where µ r = 2, 4/3. However in this case due to the effect of slot and the open circuit stub the calculated resonance frequency is a little different from predicted resonance frequencies. It should mention that in this case we match the antenna at the first resonance frequency by finding the proper length for the open circuit stub. In these designs, we showed that higher bandwidths can be achieved, however the Chu limit 115

117 (a) Figure 6-6: Real of the permeability/permittivity material inside the sphere described by Laurentz/Drude dispersion relation was not achieved. In the next section, I study the performance of an antenna surrounded by a core-shell sphere. This antenna has the potential to reach the Chu limit and hence exhibits a higher bandwidths. (a) (b) Figure 6-7: Input Impedance (Real part and Imaginary part) for a hemispherical resonator filled with a metamaterial described by Lorentz dispersion relation: a=7.5mm, L=11mm, W=0.9mm, Ls=0.57, d=0.635mm, W f = 1.45mm, f h = 2GHz,γ h = 0.001ω h and κ = Wide-Band Electrically Small Antennas To achieve a high bandwidth usually we can use several antennas. However, due to limited space, integration of several antennas to a platform is challenging. Hence there 116

118 Figure 6-8: Return loss for a hemispherical resonator filled with a metamaterial described by Lorentz dispersion relation: a = 7.5mm, L =12mm,W = 0.9mm, Ls = 0.57,d=0.635mm,W f = 1.45mm,f h = 2GHz, γ h = 0.001ω h, κ = is a significant interest in ultra-wide band electrically small antennas. According to Chu limit [39], electrically small antennas present high-q impedance characteristics. In this section, I investigate the behavior of dipole antennas inside core-shell spheres. The use of core-shell gives us the opportunity to achieve the quality factor as low as the Chu limit. The idea is to combine both electric and magnetic materials to improve the bandwidths. I also investigate the behavior of an electrically small antenna enclosed in a metamaterial sphere. I consider the use of magneto-dielectric and high dielectric permittivity materials for constructing the small antenna. A physical insight is provided, and the effect of metamaterials for bandwidth enhancement is addressed Dipole Antenna Inside a Spherical Core-Shell Here, I represent a rigorous analysis of electromagnetic waves radiation and scattering in the presence of a core-shell sphere. The required dyadic Greens function has been expressed in the form of an infinite series of spherical eigen-modes. This series is convergent and hence can be truncated using a finite number of terms. The geometry of a dipole antenna inside a spherical core-shell is shown in Fig The inner and outer radii of the core-shell are a 1 and a 2 respectively. The material of the core has the permittivity of ɛ 1 and permeability of µ 1, where the shell is described with ɛ 2 and µ 2. The dipole of length 2l is located at the center of the core-shell. 117

119 To obtain the z-directed electric field due to the z-directed current element, I first Figure 6-9: The antenna configuration, a dipole antenna inside a spherical core-shell resonator. express the z-directed current as r-directed current. Note that r-directed current can only generate T M r modes. Similar to previous section, the total Green s function is equal to the summation of the Green s function due to the dipole and the Green s function associated with the core-shell. The green s functions in different layers are defined as follows (r > r ). Note that prime coordinate refers to the source where non-prim coordinate is describing the field points. G A 1 J r = G A 2 J r = G A 3 J r = n 2n + 1 [Ĥ(2) ] n(n + 1) P n(cos ζ)ĵn(k 1 r ) n (k 1 r) + R 12 Ĵ n (k 1 r) n C 1 (n) 2n + 1 [Ĥ(2) ] n(n + 1) P n(cos ζ)ĵn(k 2 r ) n (k 2 r) + R 23 Ĵ n (k 2 r) n C 2 (n) 2n + 1 [Ĥ(2) ] n(n + 1) P n(cos ζ)ĵn(k 3 r ) n (k 3 r) n=0 n=0 n=0 (6.44a) (6.44b) (6.44c) where G A i J r for i {1, 2, 3} are the Green s functions for the electric vector potential in region i. k i for i {1, 2, 3} refers to the wave number in each region (region i). Equation (6.44a) refers to the Green s function in the first layer (core) for r > r where Ĥ(2) n represent the field of the dipole for r > r and Ĵn shows the behavior of the scattered field associated with the discontinuity of the core-shell inside the core. It is worth mentioning that the Green s function associated with the dipole in (6.44a) 118

120 is the same as the one in (6.2a) for r > r. Also for a dipole located at the center of the core-shell we have: cos ζ = cos θ cos θ. (6.45) As shown in (6.44) the Green s functions in each layer is related to the summation of Ĥ (2) n (kr) and Ĵn(kr). For the first layer Ĥ(2) n (k 1 r) is associated with the dipole and Ĵ n (k 1 r) is associated with the effect of the core-shell. In the third layer, we only have radiation and hence G A 3 J r is proportional to Ĥ(2) (k 3 r) only. By matching the boundary conditions at r = a 1 and r = a 2 the unknown coefficients are derived, we have: R 23 = R 12 = k 2 ɛ 2 Ĥ (2) k 3 ɛ 3 Ĥ n (2) (k 2 a 2 ) Ĥ2 n (k 3 a 2 ) k 2 ɛ Ĥ n (2) n (k 3 a 2 ) Ĥ(2) n (k 2 a 2 ) k 3 ɛ 3 Ĵ n (k 2 a 2 ) Ĥ(2) n (k 3 a) + k 2 ɛ 2 Ĥ n (2) (k 3 a 2 ) Ĵ n (k 2 a 2 ) n (k 1 a 1 ) k 2 ɛ 2 Ĵ n (k 1 a 1 ) [Ĥ (2) ] n (k 2 a 1 ) + R 23 Ĵ n(k 2 a 1 ) [Ĥ (2) ] n (k 2 a 1 ) + R 23 Ĵ n(k 2 a 1 ) k 1 ɛ 1 Ĥ (2) [Ĥ (2) n (k 1 a 1 ) k 1 ɛ 1 Ĵ n (k 1 a 1 ) (6.46a) ] n (k 2 a 1 ) + R 23 Ĵ n(k 2 a 1 ) [Ĥ (2) ] n (k 2 a 1 ) + R 23 Ĵ n(k 2 a 1 ) (6.46b) R 12 and R 23 are the generalized reflection coefficients in a multi-layer sphere. It is worth highlighting that R 23 is basically the TM scattering coefficient of a simple sphere filled with ɛ 2 and µ 2 inside a homogeneous medium of ɛ 3 and µ 3. Once we obtained the generalized reflection coefficients other coefficients can be achieved: C 1 = Ĥ(2) n (k 1 a 1 ) + R 12 Ĵ n (k 1 a 1 ) Ĥ n (2) (k 2 a 1 ) + R 23 Ĵ n (k 2 a 1 ) C 2 = C 1 Ĥ (2) n (k 2 a 2 ) + R 23 Ĵ n (k 2 a 2 ) Ĥ (2) n (k 3 a 2 ) (6.47a) (6.47b) 119

121 Combining the incident and scattering Green s functions, the z-directed Green s function is attained: G E i J = ẑ 1 ( 2 z jωɛ 1 z + 2 k2 1 G E s inside J z ) e jk 1 (z z ) 2 +r 2 1 4π (6.48a) (z z ) 2 + r1 2 1 cos θ cos θ = ẑ n (n + 1) R 4πωɛ 1 k 1 r 2 r 2 12 P n (cos θ cos θ )Ĵn (k 1 r ) Ĵn (k 1 r) n=1 (6.48b) Having the z-directed Green s function, one can compute the input impedance due to a delta gap input voltage by applying the moment method. A computer program is developed to obtain the current distribution and the input impedance. Once the input impedance is obtained, the quality factor can be calculated from (6.58). In the next section, the relation between quality factor and input impedance is discussed General Definition of Bandwidths and Q In this part, the general definitions for bandwidths and quality factor are illustrated. Consider a general transmitting antenna composed of electromagnetically linear materials and fed by a waveguide or transmission line that carries just one propagating mode at the time harmonic frequency. The propagating mode in the feed line can be characterized at a reference plane by a complex voltage V (ω), complex current I(ω) and complex input impedance Z(ω) defined as: Z(ω) = R(ω) + jx(ω) = V (ω)/i(ω) (6.49) Assume the antenna is tuned at a frequency ω 0 with a series reactance X s (ω). The total reactance is zero at ω 0. X 0 (ω 0 ) = X(ω 0 ) + X s (ω 0 ) = 0 Z 0 (ω) = R 0 (ω) + jx 0 (ω) (6.50a) (6.50b) 120

122 Because the tuning inductor or capacitor is assumed lossless and in series with the antenna, at the tuning resonance the reflection is zero. Also, note that we assume the feed line is pure resistive and is matched to the antenna, namely Z ch = R 0 (ω). The bandwidth of an antenna tuned to zero, is often defined by the matched VSWR bandwidth. The matched VSWR bandwidth, unlike the conductance bandwidth is well-defined for all frequencies ω 0 at which the antenna is tuned to be lossless. We have Γ = Z 0(ω) Z ch Z 0 (ω) + Z ch (6.51a) Γ 2 = Z 0(ω) R 0 (ω 0 ) 2 Z 0 (ω) + R 0 (ω 0 ) 2 = R 0(ω) R 0 (ω 0 ) 2 + X 0 (ω) X 0 (ω 0 ) 2 R 0 (ω) + R 0 (ω 0 ) 2 + X 0 (ω) X 0 (ω 0 ) 2 (6.51b) Γ 2 = R 0(ω) R 0 (ω 0 ) 2 + X 0 (ω) 2 R 0 (ω) + R 0 (ω 0 ) 2 + X 0 (ω) 2 (6.51c) The bandwidth is defined as the difference between the two frequencies on other side of ω 0 at which the VSWR equals to a constant (α 1/2). Hence α = R 0(ω ± ) R 0 (ω 0 ) 2 + X 0 (ω ± ) 2 R 0 (ω ± ) + R 0 (ω 0 ) 2 + X 0 (ω ± ) 2 (6.52a) X0(ω 2 ± ) + [R 0 (ω ± ) R 0 (ω 0 )] 2 = 4 α 1 α R 0(ω 0 )R 0 (ω ± ) (6.52b) Now if ω/ω 0 1, the solution for bandwidth is obtained as: α ω = 2 1 α R 0 (ω 0 ) Z (ω 0 ) (6.53) The quality factor for an antenna tuned to have zero reactance at the frequency ω 0 121

123 can be defined as: Q(ω 0 ) = ω 0 W (ω 0 ) P A (ω 0 ) (6.54a) W (ω 0 ) = W m (ω 0 ) + W e (ω 0 ) X 0 (ω) = (6.54b) 4ω I 0 (ω) 2 [W m(ω) W e (ω)] 0 W m (ω) W e (ω) (6.54c) X 0(ω) = 4 I 0 2 [W (ω 0) + W L (ω 0 ) + W R (ω 0 )] (6.54d) Using the above relations, the quality factor of the antenna is derived as: Q(ω 0 ) = ω 0 2ω 0 2R 0 (ω 0 ) X 0(ω 0 ) I 0 2 R 0 (ω 0 ) [W L(ω 0 ) + W R (ω 0 )] (6.55) Note that, we can estimate the total dispersion energy W L (ω 0 ) + W R (ω 0 ) to get an approximation of Q which can be immediately related to bandwidths. Using the RLC circuit realization it can be shown that [83]: X 0(ω 0 ) 4 I 0 [W L(ω 2 0 ) + W R (ω 0 )] X 0 2 (ω 0 ) + R 0 2 (ω 0 ) (6.56) X 0(ω 0 ) 4 I 0 [W L(ω 2 0 ) + W R (ω 0 )] Z 0(ω 0 ) (6.57) Inserting this in (6.58) yields: Q(ω 0 ) ω 0 2R 0 (ω 0 ) Z 0(ω 0 ) (6.58) According to the above equation, the quality factor is directly related to the input impedance. Hence by having the input impedance we can easily obtain the Q. In the next section I investigate different cases of dipole embedded inside a metamaterial sphere. I illustrate that by using magnetic shell and dielectric core, the quality factor of the antenna approaches the Chu limit. 122

124 6.4.3 Antenna Designs The simplest way to achieve antenna miniaturization is to use a high permittivity material. However, because of a strong concentration of electromagnetic fields inside the material, the performance of the antenna is significantly degraded leading to narrow-band characteristics. Recently there have been some works to properly overcome this problem by either printing the antenna on a magneto-dielectric substrate [71], or embedding the antenna inside a magnetic material [40]. Here, we apply the developed formulations to present a better physical insight of these approaches. Moreover, I illustrate a novel design for lowering the Q by embedding the antenna inside a spherical core-shell where core is made of high dielectric and shell is made of magnetic materials. As highlighted in [71], by using a magneto-dielectric material one can deduce the capacitive behavior of the high dielectric resonator. Basically, by choosing moderate values for ɛ r and µ r the same miniaturization factor as a high permittivity nonmagnetic material can be achieved, while the capacitive property of the resonator is now decreased by addition of some inductance behavior caused by µ r. This will allow us to improve the antenna impedance bandwidth. It has to be mentioned that in this case a higher input resistance is realized since the impedance is related to the ratio of µ r /ɛ r. In another approach, instead of using a magneto-dielectric material, I integrate magnetic materials (µ) inside the resonator. In this scenario, the same miniaturization is achieved even with smaller permeability by introducing artificial inductance. Furthermore, the input impedance is even higher than using a magneto-dielectric material. Also, since the effect of inductance is stronger than the artificial inductance caused by magneto-dielectric materials the bandwidths is higher and hence the quality factor is lower. Another novel design is to combine the effect of magnetic material and dielectric in a core-shell structure. Thus a spherical core-shell with dielectric core and magnetic shell is designed to increase the bandwidths and reduce Q. By proper choose of the dielectric and magnetic materials Q as low as Chu limit can be achieved. 123

125 First let us validate the accuracy of our computer code. The geometry of the structure is shown in Fig. 6-9, where a 1 = 5.5mm and a 2 = 7.5mm and l = 4.5mm. The material inside the core is a dielectric described with ɛ r1 = 10 where the shell is magnetic (only permeability) with µ r2 = 2. The input impedance attained by MoM (Method of Moments) and HFSS is plotted in Fig A very good comparison is observed validating our program. Figure 6-10: The input resistance and reactance. The antenna composed of a dipole of length 2l = 9mm, located at the center of a spherical core-shell resonator. Where a 1 = 5.5 mm,a 2 = 7.5 mm, ɛ r1 = 10 and µ r2 = 2. The electric field inside the core is proportional to P n (cos θ cos θ )R 12 Ĵ n (k 1 r )Ĵn(k 1 r). For computation of the input impedance r and r varies over the dipole, therefore cos θ cos θ = 1 and r, r < l. Thus, each term is basically proportional to R 12 Ĵ n (k 1 r )Ĵn(k 1 r) where r, r < l. So let us introduce a new term called scattering 124

126 term for more clarity. The scattering term is defined as: ST = R 12 Ĵ 2 n(k 1 l) (6.59) So investigating a scattering term such as (6.59) seems appropriate to provide a physical insight of the electric field. As discussed earlier to achieve the desired miniaturization, the simplest way is to use high-dielectric material. The objective it to have a wide-band antenna working around 2GHz. We embed a dipole antenna at the center of a dielectric sphere filled with a high dielectric material. The spherical radiator has a radius of a 1 = 7.5 mm and is filled with dielectric material with ɛ r1 = 100. It is worth mentioning that, a spherical resonator is basically a core-shell where the material of the shell is free space. The scattering terms (for the first three modes) and input impedance are shown in Fig As is shown, the first scattering term behavior exhibits a resonance around 2.8GHz, so we expect to have a resonance in the input impedance around this region. The input impedance shows the resonance performance around 2.3GHz. The return loss for the tuned antenna is plotted in Fig. 6-12, revealing a narrow-band resonance at f = 2.32 GHz. For this resonance, the quality factor of the tuned antenna calculated by using (6.58) is Q = 293.1, whereas the Chu limit is Q chu = 1 ka ka = The obtained Q from the antenna is more than 12 times the Chu limit which was expected from the return loss behavior. The input reactance in Fig. 6-11(b) demonstrates a very high capacitive behavior. To increase the bandwidths and hence reduce the quality factor, we can introduce an artificial inductance. The artificial inductor caused by permeability reduces the capacitance characteristics of the input impedance and hence a wider bandwidths can be achieved. To verify this theory, instead of embedding the antenna inside a highdielectric material, one can utilize magneto-dielectric materials. Fig demonstrates the scattering terms and the input impedance for a dipole antenna. Notice that, the scattering term shows a resonance performance with a higher bandwidths than the one illustrated in Fig The return loss is shown in Fig exhibiting 125

127 300 Scattering Terms n=1 n=2 n=3 Input Impedance Resistance Reactance Frequency(GHz) (a) Frequency(GHz) (b) Figure 6-11: (a) Scattering term for the first three modes. (b) The input resistance and reactance. The antenna composed of a dipole with total length of 2l = 9mm, located at the center of a spherical resonator filled with high dielectric material of ɛ r1 = 100. The radius of the sphere is a 1 = 7.5 mm. 0 2 Return Loss Frequency(GHz) Figure 6-12: The return loss of the antenna in Fig The return loss presents a very narrow-band bandwidths which is consistent with the Q=

128 a resonant around f = 2.19 GHz. Clearly the bandwidth is increased, meaning the quality factor should be reduced. From (6.58), quality factor is equal to Q = which is 3.64 times the Q chu = This result verifies that by adding magnetic material Q has decreased while bandwidths has increased. Scattering Terms n=1 n=2 n=3 Input Impedance x Resistance Reactance Frequency(GHz) (a) Frequency(GHz) (b) Figure 6-13: (a) Scattering term for the first three modes. (b) The input resistance and reactance. The antenna composed of a dipole with total length of 2l = 9mm, located at the center of a spherical resonator filled with magneto-dielectric material of ɛ r1 = 12 and µ r1 = 8. The radius of the sphere is a 1 = 7.5 mm. 0 2 Return Loss 4 6 8, Frequency(GHz) Figure 6-14: The return loss of the antenna in Fig The return loss presents a very narrow-band bandwidths which is consistent with the Q= As shown, by adding magnetic material the Q reduces and hence antenna provides a better performance. So, we embed the antenna inside a magnetic material to achieve the same miniaturization and higher bandwidths. Fig demonstrates 127

129 the scattering terms and the input impedance. The return loss is shown in Fig exhibiting a resonant around f = 2.32 GHz. Clearly the bandwidths is increased, meaning the quality factor should be reduced. From (6.58), quality factor is equal to Q = which is 3.4 times the Q chu = This result verifies that by adding magnetic material Q has decreased while bandwidths has increased. Scattering Terms n=1 n=2 n=3 Input Impedance x Resistance Reactance Frequency(GHz) (a) Frequency(GHz) (b) Figure 6-15: (a) Scattering term for the first three modes. (b) The input resistance and reactance. The antenna composed of a dipole with total length of 2l = 9mm, located at the center of a spherical resonator filled with magnetic material of µ r1 = 60. The radius of the sphere is a 1 = 7.5 mm. 0 2 Return Loss 4 6 8, Frequency(GHz) Figure 6-16: The return loss of the antenna in Fig The return loss presents a very narrow-band bandwidths which is consistent with the Q= Although, embedding the antenna inside magneto-dielectric or magnetic materials increases the bandwidths of the antenna, but Q Chu limit is still not accessible. It 128

130 has be demonstrated in [40] that by embedding the antenna inside an ENG material, a quality factor as low as 1.5 times of the Chu limit can be achieved. However, to the best of our knowledge there is no report of achieving the actual Chu limit. In this thesis, I present a novel design to approach the Chu limit by embedding the antenna inside a magnetic shell. Fig demonstrates the scattering terms and the input impedance for a dipole of length 2l = 9mm embedded inside a core-shell. The inner and outer radii of the core are a 1 = 6.5 mm and a 2 = 7.5 mm. The core is filled with a dielectric with permittivity of ɛ r1 = 4, while the shell is a magnetic material of µ r2 = 90. The scattering term shows a significant increase in the bandwidths of the resonance hence, we expect to have a higher bandwidths and lower Q for the antenna. The return loss is shown in Fig exhibiting a resonant around f = 2.36 GHz. Clearly the bandwidths is increased remarkably, meaning the quality factor should be reduced. From (6.58), quality factor is equal to Q = 34.97, which is times the Q chu = This is an important result, the antenna is surrounded by a spherical core-shell. Where the core is dielectric material and shell is magnetic material. The magnetic shell can reduce the capacitance effect of the dielectric core, while both dielectric and magnetic materials helps miniaturizing the antenna. This Scattering Coefficient n=1 n=2 n=3 Input Impedance Resistance Reactance Frequency(GHz) (a) Frequency(GHz) (b) Figure 6-17: (a) Scattering term for the first three modes. (b) The input resistance and reactance. The antenna composed of a dipole of length 2l = 9mm, located at the center of a spherical core-shell resonator. Where a 1 = 6.5 mm,a 2 = 7.5 mm, ɛ r1 = 4 and µ r2 = 90. design demonstrates a new approach to achieve the Chu limit. As discussed earlier, 129

131 0 2 Return Loss 4 6 8, Frequency(GHz) Figure 6-18: The return loss of the antenna in Fig The return loss presents a very narrow-band bandwidths which is consistent with the Q= the scattering term can provide the resonance behavior of the antenna. Thus let us first investigate the behavior of the scattering term with changing the material of the core. According to Figs. 6-19, if the dielectric material of the cores increases the, Scattering Coefficient ε r1 =5 ε r1 =10 ε r1 =20 ε r1 =40 ε r1 = Frequency(GHz) (a) Scattering Coefficient ε r1 =50 ε r1 =60 ε r1 =70 ε r1 =80 ε r1 = Frequency(GHz) (b) Figure 6-19: The scattering term vs. different dielectric material of the core. The core-shell is described as follows: a 1 = 6.5 mm, a 2 = 7.5 mm, µ r2 = 90. A dipole of length 2l = 9 mm is located at the center of the core-shells. bandwidths of the scattering term increases. From Fig. 6-19(b) it can be seen that the scattering terms do not vary much by increasing the dielectric material of the core. I optimize the design to achieve the lowest quality factor for having a resonance around 2 GHz. Fig demonstrates the scattering terms and the input impedance for a dipole of length 2l = 9mm embedded inside a core-shell. The inner and outer radii 130

132 of the core are a 1 = 6.5 mm and a 2 = 7.5 mm. The core is filled with a dielectric with permittivity of ɛ r1 = 60, while the shell is a magnetic material of µ r2 = 90. The scattering term shows a significant increase in the bandwidths of the resonance. Hence, we expect a higher bandwidths and lower Q for the antenna. The return loss is shown in Fig exhibiting a resonant around f = 2.16 GHz where the bandwidths has increased remarkably. The quality factor obtained (6.58), is Q = 31.07, whereas the Q Chu = Note that Q/Q Chu = Chu [39] obtained his limit by Scattering Coefficient n=1 n=2 n=3 Input Impedance Resistance Reactance Frequency(GHz) (a) Frequency(GHz) (b) Figure 6-20: (a) Scattering term for the first three modes. (b) The input resistance and reactance. The antenna composed of a dipole of length 2l = 9mm, located at the center of a spherical core-shell resonator. Where a 1 = 6.5 mm,a 2 = 7.5 mm, ɛ r1 = 60 and µ r2 = 90. Return Loss , Frequency(GHz) Figure 6-21: The return loss of the antenna in Fig The return loss presents a relatively large-bandwidths which is consistent with the Q = = 1.08Q chu. ignoring stored energy inside the sphere surrounding the antenna. Hence, if one can 131

133 reduce the absorption and loss inside the sphere surrounding the antenna the quality factor is closer to the Chu limit. For instance, if we embed a dipole antenna inside a resonator, to achieve a Q near the Chu limit the magnitude of scattering field inside the resonator compared to the dipole field should be negligible. To better understand the performance, the near field is also plotted in Fig Fig reveals that the magnitude of electric field inside the core is negligible compared to the field inside the shell and the dipole. This means we can ignore the stored energy inside the core. Also since we have a thin shell, the stored energy inside shell is small and a quality factor close to Chu can be obtained. It is worth mentioning that in this case, we ignored the material loss. From practical points of view, obviously,, Figure 6-22: The normalized near field behavior (in db) of antenna defined in Fig in y-z plane. The vertical axis is z (mm) while the horizontal axis is y (mm). one should also consider the availability of metamaterials. For instance, the naturally available magnetic materials have always a narrow-band resonant behavior. Also, a negative permittivity medium is accessible only at optical frequencies. 6.5 Conclusion In this chapter, I investigate the behaviors of small resonators composed of negative permittivity/permeability materials by obtaining the exact and approximated Greens functions for the electric/magnetic field, respectively. Also, we present a novel 132

134 approach for achieving the desired miniaturization while maintains the low Q performance, by embedding the dipole antenna inside a core-shell. The scattering resonant properties of sub-wavelength resonators are well described by a set of analytical formulas predicting the locations of resonances. We also demonstrate for small antennas consisting of ENG/MNG material the locations of resonant frequencies are almost independent of the structure size, and mainly derived from the material property. We illustrate through the method of moment with Galerkins procedure and considering the exact form for Greens function analysis how well our prediction responds. I also present a novel design for reducing the quality factor. The idea is to embed the antenna inside a core-shell, where core is dielectric and shell is magnetic. I illustrate that by proper choosing of the materials, quality factor as low as 1.08 times the Chu limit can be achieved. 133

135 Chapter 7 Plasmonic Nanoparticles on Layered Substrates: Modeling and Radiation Characteristics In this chapter, the applications of plasmonics materials in building antenna devices are investigated. I theoretically characterize the performance of array of plasmonic core-shell nano-radiators located over layered substrates. Engineered substrates are investigated to manipulate the radiation performance of nanoantennas. A rigorous analytical approach for the problem in hand is developed by applying Green s function analysis of dipoles located above layered materials. It is illustrated that around the electric scattering resonances of the subwavelength spherical particles, each particle can be viewed as an induced electric dipole which is related to the total electric field upon that particle by a polarizability factor. Utilizing this, we can effectively study the physical performance of such structures. 7.1 Background Recently, the field of optical nano-metals and plasmonics has experienced an extensive growth in research activities, owing to the fascinating possibility of confining electromagnetic radiation to subwavelength spatial domains via surface plasmon 134

136 polaritons (SPPs) [84]-[85]. These surface modes are established by the couplings of electromagnetic fields to conduction electrons at the boundary between a noble metal and a dielectric. This phenomenon offers unique opportunity for nanooptics physics enabling, areas such as plasmonic waveguides, nano-interconnects and plasmonic light sources [84]. Other newly proposed applications include plasmonic THz nano-antennas [42]-[43] and solar-cells energy-efficient guiding [86]-[88]. Patterning plasmonic particles can offer significant advantage for achieving desired optical properties. One of the interesting applications of the plasmonic particles is to build antenna devices in optical frequencies featuring photonic communication. However, design and fabrication of nanoantennas is the most challenging part mainly due to the optical properties of metal. For instance, at optical frequencies metal is well described by a frequency dispersive negative permittivity material rather than a highconductivity structure as what is observed in microwave spectrum. As a consequent, one can offer resonant nanoparticles using core-shell dielectric-plasmonic elements enabling nanoantenna radiation [43]. The goal of this part is to theoretically exploit the performance of arrays of plasmonic nanoparticles featuring enhanced radiation characteristics when placed over layered substrates. At microwave frequencies, in order to achieve a high gain antenna performance, one needs to use the concept of array antennas where different radiators are tailored in unique arrangements to control the fields amplitudes and phases in the far-field spectrum enabling directive emission. This concept can be realized in optics with the use of plasmonic particles acting like small dipole radiators. Li et al. [42] used this concept for providing a working THz Yagi-Uda antenna in free-space. Similar performance is observed in [43]. From practical point of view, however, one needs to deposit the nano radiators on a substrate. In this case, the radiation performance changes drastically as for instance illustrated for an array of plasmonic rods located above a half-space medium through a numerical analysis [44]. It can be of great benefit if one establishes a theoretical formulation for performance analysis of nanoantennas composed of an array of core-shell spherical particles 135

137 located above layered substrates. This will allow successful study of the complex structure via understanding the physics and obtaining the optimal radiation behavior and beam scanning for the application of interest. To establish the theory of nanoantennas, array of plasmonic core-shells over a layered material is approximated with electric dipoles where their induced dipoles are calculated through the Green s function analysis of dipoles over the layered material [45]. The obtained Sommerfeld integrals are efficiently calculated by defining a robust integral path [45]. I apply our theory-modeling to obtain the radiation performance of a dipole exciting an array of plasmonic nanoantennas located above a planar layered material. The induced dipole on each nano core-shell is the key parameter determining the antenna radiation characteristics. We validate the results by comparing the analytic pattern with the ones obtained by using full-wave finite difference time domain (FDTD) technique [46]. Optimal patterning of plasmonic particles along with engineering their layered substrate can successfully tailor the optical parameters for achieving a directive emission. 7.2 Theory and Modeling The configuration of a layered substrate plasmonic particle-antenna is shown in Fig The geometry consists of subwavelength nano core-shells implanted above an infinitely long (in x and y directions) semiconductor substrate where layer(s) of plasmonic or dielectric materials are coating the semiconductor surfaces. Here, the plasmonic particles are spherical core-shells with dielectric cores (k c, ɛ rc ) and plasmonic coatings (k s, ɛ rs ). The use of plasmonic material as the coating allows greater control on scattering characteristics of the nano core-shells [42]. At the plasmonic resonance the scattered field from a small spherical core-shell under the influence of an incident field is dominated by dipole terms. Thus, a subwavelength particle at the scattering resonance can be viewed as an induced dipole (p), which is related to the 136

138 Figure 7-1: Array of plasmonic nanoantennas located above a layered substrate. total electric field upon that particle by a polarizability factor (α). p = α. E total = j 6πɛ 0 Γ (e) k E total, (7.1) where k 0 is the wave number of the surrounding medium and, Γ (e) 1 is the first electric scattering coefficient of the plasmonic core-shell [42, 49]: Γ (e) n = U (e) n U (e) n jv (e) n, (7.2) U (e) n and V (e) n are given by: U (e) n = V (e) n = j n (k c r 1 ) j n (k s r 1 ) y n (k s r 1 ) 0 j n (k c r 1 )/ɛ c j n (k s r 1 )/ɛ s ỹ n (k s r 1 )/ɛ s 0, (7.3a) 0 j n (k s r 2 ) y n (k s r 2 ) j n (k 0 r 2 ) 0 j n (k s r 2 ) ỹ n (k s r 2 )/ɛ s j n (k 0 r 2 )/ɛ 0 j n (k c r 1 ) j n (k s r 1 ) y n (k s r 1 ) 0 j n (k c r 1 )/ɛ c j n (k s r 1 )/ɛ s ỹ n (k s r 1 )/ɛ p 0. (7.3b) 0 j n (k s r 2 ) y n (k s r 2 ) y n (k 0 r 2 ) 0 j n (k s r 2 ) ỹ n (k s r 2 )/ɛ s ỹ n (k 0 r 2 )/ɛ 0 137

139 freq(thz)=450 freq(thz)=550 freq(thz)= α /(ε 0 λ 0 3 ) Phase(α)/π freq(thz)=450 freq(thz)=550 freq(thz)= r 1 /r 2 (a) r 1 /r 2 (b) 4 Permittivity of Silver real imaginary Freq(THz) (c) Figure 7-2: Magnitude and phase of normalized polarizability of a plasmonic coreshell sphere versus r 1 /r 2 ratio for different frequencies for a structure with r 2 = 0.1λ, ɛ rc = i and silver coating. (a) Polarization magnitude, and (b) Polarization phase, (c) Silver permittivity behavior. where r 1 and r 2 are the inner and outer radii respectively. It is well known that, the scattering resonance of a plasmonic concentric core-shell structure can be obtained at desired frequencies by tuning the (r 1 /r 2 ) radii ratio [42]. Figures 7-2(a) and (b) present the magnitudes and phases of normalized polarizability versus the r 1 /r 2 ratio for a concentric core-shell with silver coating and dielectric core made of SiO 2, wherein the total radius r 2 = 0.1λ 0 is taken to be constant and the core s radius (r 1 ) varies. As it is shown, by tuning the r 1 /r 2 ratio, we can tailor the resonance frequency of the plasmonic nano-particle, for instance, the nano-sphere goes to resonance at f = 650 T Hz (λ 0 = 461 nm) if r 1 /r 2 = Another interesting feature in these figures, is that at the resonance the phase of polarizability is close to π/2 (phase of 138

140 i), which means that at the resonance the induced dipole is almost in phase with the incident field. Notice that, to tune the resonant frequency for a simple metallic nanoparticle, one should change the total size of the particle, while in many applications it is required to maintain a desired size. Nevertheless, with the use of core-shell structures, the total size can be remained the same, whereas at the same time the resonant frequency can be tuned by changing the radii ratio. In order to analyze the frequency dependency of the scattering resonance, it is of value to consider the frequency dispersion of the plasmonic material. The permittivity of the silver in optical regime can be described with a Drude model, in which the numerical parameters are achieved by fitting the available experimental data of the Silver s permittivity to the Drude model (see also Fig. 7-2(c)). ɛ rs = ω(ω j ). (7.4) In addition, note that silver is the material of choice, due to its small loss in optical frequencies. Now, let us investigate the case of Fig If a non-periodic array percentage of error z 0 =0.1λ z 0 =0.5λ z 0 =λ ρ Figure 7-3: Maximum percentage of error in obtaining the integrals in (7.7) and (7.8) using our theoretical model compared to their exact values for different observation points. ρ and z 0 denote the location of the observation point. of plasmonic core-shells are placed over a layered material, with the presence of an 139

141 incident field, each nano-particle can be viewed as an induced dipole around the scattering resonance of that nano core-shell. From (7.1) it is clear that, the induced dipole on each nano core-shell is proportional to the total field upon that particle. In this scenario, the total field upon each nano core-shell can be expressed as the summation of three terms. The first part is associated with the total incident field in the absence of the nano core-shells (E total inc ), the second part is the electric field due to the couplings between the nano core-shells in the absence of the layered material (Ḡ l dipole (r l, r q )p q ). Since this term represents the couplings between the nano coreshells, for the l th nano core-shell we consider the fields of all other nano core-shell except itself (excluding the field of the l th particle). The last term, is associated with the reflected fields from the layered substrate (Ḡ l reflected (r l, r q )p q ). Note that for the computing the last two terms we approximate each nano core-shell with an electric dipole. Hence for the second term we calculate the dipolar couplings and the Green s function analysis of dipoles over layered material is applied for evaluation of the final part [45]. Also, it is worth mentioning that the fields associated with every nano coreshells (both couplings and reflected field) is directly proportional to the induce dipole moment, thus the induced dipole moment for each particle is derived by solving the following linear system of equations. For l, q {1, 2,..., N} with N being the total number of particles, we obtain: p l = α l ( E total inc (r l ) + Ḡ l dipole(r l, r q )p q + q,q l q Ḡ l reflected(r l, r q )p q ), (7.5) where E total inc denotes the sum of the incident field and its reflection from the layered material in the absence of the nano-spheres. Ḡ l dipole is the dyadic Green s function of the q th nano core-shell evaluated at the position of the l th particle. Ḡ l reflected is the reflected Greens function of the q th nano-sphere (from the layered material ) computed at the location of the l th one, i.e., 140

142 (k Ḡ l dipole(r l, r q ) = 2 x y 2 x z ) 2 2 x x y Ge lx rx(r l r q ) Ḡ l reflected(r l, r q ) = Ge ly rx(r l r q ) Ge lz rx(r l r q ) 2 x z (k y ) 2 y z 2 y z Ge lx ry(r l r q ) Ge ly ry(r l r q ) Ge lz ry(r l r q ) (k z ) e jk 1 r l r q 4πɛ 1 r l r q, Ge lx rz(r l r q ) Ge ly rz(r l r q ). Ge lz rz(r l r q ) (7.6a) (7.6b) By defining r = r l r q, the elements of the matrix in equation (7.6b) are derived as [45]: Ge lz rx = 1 cos φ dk ρ k 8πɛ ρh 2 (2) 1 (k ρ ρ) R 1,2 T M e jk 1z(z+2d 1 ), (7.7a) 1 Ge lz ry = 1 Ge lz rz = sin φ 8πɛ 1 i 8πɛ 1 dk ρ k 2 ρh (2) 1 (k ρ ρ) R T M 1,2 e jk 1z(z+2d 1 ), (7.7b) dk ρ k 3 ρ k 1z H (2) 0 (k ρ ρ) R T M 1,2 e jk 1z(z+2d 1 ), (7.7c) where for example, Ge lz rx is Green s function for the z-directed electric field associated with a x-directed dipole. Let G be the kernel of the integral, for w {x, y, z} and s {x, y} we have [45]: Ge ls rw(k ρ ) = 1 k 2 ρ [ z ] ls Gerw(k ρ ) lz + jωµẑ s Gh z rw (k ρ ), (7.8) where Gh lz rw is the Green s function for the z directed magnetic field due to a w directed dipole located above the layered material [45]-[46]. RT M/T E 1,2 is the generalized reflection coefficient (that incorporates subsurface reflections) at the interface between region 1 and region 2 [45]. The generalized reflection coefficient at the inter- 141

143 face of region i and i + 1 is defined as [45]: R T M/T E i,i+1 = RT M/T E i,i+1 R T M/T E i,j R T M/T E i,i+1 R T M/T E i+1,i+2 e 2jk i+1,z(d i+1 d i ) R T M/T E i+1,i+2 e 2jk i+1,z(d i+1 d i ), (7.9a) = (ɛ j/µ j )k iz (ɛ i /µ i )k jz (ɛ j /µ j )k iz + (ɛ i /µ i )k jz, (7.9b) In the above equations, k i is the wave number in the layer i and: k iz = k 2 i k2 ρ. (7.10) The integrals in (7.7) and (7.8) are calculated by defining a robust numerical path whereby the discretization of the integration steps is chosen to adapt to the integrand s oscillation [45]. For the verification purposes, the integrals in (7.7) and (7.8) are computed in a homogenous medium (ɛ = ɛ 0,µ = µ 0 ) where their exact values are given by the response of a dipole in the same homogenous regime. Figure 7-3 shows the maximum percentage of error between all the computed integrals (nine integrals) in (7.7) and (7.8) for each observation point marked as (ρ,z 0 ). As it is exhibited, the maximum error is less than 0.1% for different observation points, validating our method of integration. It is worth noticing that, if instead of the array only one nano core-shell is implanted above a layered material then the induced dipole is achieved utilizing the following equation: p =(Ī αḡ reflected ) 1 αe total, (7.11) where Ḡ reflected in this particular case simplifies to a diagonal matrix, meaning that there is no coupling between the components of the induced dipole. It is worth highlighting again, that in this case the induced dipole is only related to the reflected field of that nano-particle from the layered material and the incident field. As mentioned earlier, regardless of the structure, we assume that each nano core-shell can be modeled as an electrically small dipole with the induced dipole p. 142

144 Yet, the induced dipole for each nano core-shell is derived by using (7.5), in which the parameters of the Ḡ reflected are determined by the geometry and materials of the layered structure. Moreover, the far-field radiation pattern is also affected by the presence of the substrate. Hence, to understand the behavior of the antenna located above a layered material, let us first analyze the radiation pattern of such antennas. Since each nano core-shell behaves like an electric dipole, the far-field radiation pattern of such a structure depicted in Fig. 7-1, consists of summation of the radiation patterns of electric dipoles located above a layered material. The far zone electric field for a dipole deposited over a layered-substrate can be evaluated using the conventional steepest decent contour (SDC) technique with the transformation k ρ = k sin θ, where the θ is the spherical angle from the z axis. So for each dipole (p x, p y, p z ) located at (x 0, y 0, z 0 ), the upper half-space and lower half-space far-field radiation pattern can be represented as [90]: E = E θ E φ = k2 J e jkj r 4πɛ J r ( px cos φ + p y sin φ ) cos θ Φ 2 J p z sin θφ 1 J ( p x sin φ p y sin φ ) Φ 3 J. (7.12) Where the index J [1, N] is to distinguish between the upper-half (ɛ 1, µ 1 ) and lower-half (ɛ N, µ N ). The potential parameters are defined as [90]: Φ 1 1 =e jk 1r [1 + R T M 1,2 (θ)e jk 1z 0 ], (7.13a) Φ 2 1 =e jk 1r [1 R T M 1,2 (θ)e jk 1z 0 ], (7.13b) Φ 3 1 =e jk 1r [1 + R T E 1,2 (θ)e jk 1z 0 ], (7.13c) Φ 1 N = Φ 2 N = n N cos θ T T M (θ)e ik N [z 0 s z +(d N d 1 ) cos θ], n 1 s z (7.13d) Φ 3 N = cos θ T T E (θ)e jk N [z 0 s z +(d N d 1 ) cos θ], s z (7.13e) with n 1 and n N being the refractive indices for the upper and lower half-spaces respectively. The T T M/T E is the generalized T M/T E transmission coefficient for a layered 143

145 material [45]. And k J r and k J z are specified as: k J r =k J [ x0 cos φ sin θ + y 0 sin φ sin θ + z 0 cos θ ], k J z =2k J (z 0 + d 1 ) cos θ, (7.14a) (7.14b) also: s z = k 1z = ( n 1 ) k N n 2 sin 2 θ. (7.15) N Notice that, from (7.13) a remarkable result is achieved. The first term in the potentials (Φ 1 J Φ3 J ) can be interpreted as the far zone radiation pattern of a dipole, where the second term can be identified as the radiation from a dipole located at the image plane (at the distance z 0 beneath the d 1 ) weighted by generalized reflection coefficients. It is also worth mentioning that, at θ = π/2, k 1z = 0 and hence R T M/T E 1,2 T M/T E is zero, which leads to R 1,2 = 1 and consequently, the electric field is zero at θ = π/2 direction. So clearly the radiation pattern of a nanoantenna located above a planar layered material, is much different from that in the absence of layeredsubstrate. By utilizing the dipole approximation for the nano-particles, in the next section we discuss the required steps for designing a narrow beamwidth antenna Antenna Applications In practice, it is often required to design an antenna system that will yield the prescribed far-field radiation pattern. For example, a very common request is to design an antenna whose pattern has a narrow beamwidth and small side lobes. At radio frequencies, an approach to achieve this, is based on the concept of Yagi-Uda antennas [91]. The high directivity is achieved by placing a reflector and several directors arranged in proper locations around a resonant feed element. The directors are designed to be more capacitive while the reflector is more inductive, resulting in a narrow radiation beam toward the direction of directors and a minimum toward the reflector. To take advantage of this principle in optical regime, it has been suggested 144

146 (a) (b) (c) Figure 7-4: (a) A Yagi-Uda type antenna constructed from a dipole source exciting two nano core-shells, operating at frequency f = 650 T Hz. The reflector s radii ratio is r 1r /r 2r = 0.75 and the radii ratio of the director is r 1d /r 2d = Both core-shells have core made of SiO 2 (ɛ rc = i) and shell made of silver (ɛ rs = i). The normalized radiated power obtained by theory and FDTD, (b) in xy plane, and (c) in yz plane. A good comparison is observed. Note that FDTD characterizes the actual plasmonic core-shell structure whereas in theoretical model we approximate them with dipole modes. 145

147 Figure 7-5: Magnitude of E z (db) for the nanoantenna in Fig. 7-4(a) in xy (z = 0) plane and yz (x = 0) plane. to use plasmonic nano-particles as the antenna element and to put an optical dipole source as the feed element [42]-[43]. In reality the antenna must be placed on a substrate where surface wave couplings-interactions with the plasmonic particles antenna elements can drastically change the radiation performance. This phenomenon is addressed carefully in our work by applying the developed formulations. A composite of indium gallium arsenide (InGaAs)-silver (Ag)-silicon dioxide (SiO 2 ) substrate is engineered to first suppress the surface waves propagation and second tailor the radiation beam in a desired direction enabling an on-chip nanoantenna. Figure 7-4(a) illustrates a Yagi-Uda type optical nanoantenna constructed from a reflector core-shell sphere located at r ref = 0.2λ 0 y and a director located at r dir = 0.3λ 0 y. Both particles have the core made of SiO 2 (ɛ rc = j) and the shell made of silver (ɛ rs = i). For the working frequency of f = 650 T Hz, the outer radius of each particle is 0.1λ 0 whereas the radii ratios for reflector and director are r 1r /r 2r = 0.75 and r 1d /r 2d = 0.65 respectively. The ratios of radii r 1 /r 2 are chosen to set the nano spheres operate around their resonances. Further, under this design, the phase of the scattering coefficient for the reflector compared to the incident field is greater than π/2 (inductive), and that for the director is less than π/2 (capacitive). These conditions ensure successful Yagi-Uda antenna realization. Note 146

148 that, unless otherwise mentioned, the optical dipole source throughout this work is perpendicular to the top surface of the multilayer substrate (p = z). The radiation patterns for the 2-element Yagi-Uda depicted in Fig. 7-4(a), are obtained using our theoretical model and are demonstrated in Figs. 7-4(b) and 7-4(c) in xy and yz planes, respectively. In this scenario, since the optical dipole is polarized in the z direction, the induced dipoles on the nano core-shells are also polarized in the z direction. The polarizabilities for the reflector and the director associated with the radii ratio are α r = 0.059e i0.77π ɛ o λ 3 0 and α d = 0.052e i0.21π ɛ o λ 3 0 respectively. From the polarizabilities one can notice that the phases of the reflector and the director are shifted from π/2 indicating the inductive and capacitive behavior for the reflector and the director respectively. It is worth noticing that, all the radiation patterns plotted here are the radiated power normalized to the maximum value of the radiated power of the optical dipole source in the absence of the nano core-shells (and the layered substrate). (a) (b) Figure 7-6: (a) Nanoantenna configuration in Fig. 7-4(a) located above a slab of InGaAs with the thickness of 0.35λ 0. (b) The 3D radiation pattern of the radiated power. The substrate considerably degrades the antenna radiation pattern. 147

149 To validate the accuracy of our dipole-mode model, a full wave numerical analysis based on an in-house developed FDTD method [35] is applied for characterizing the plasmonic core-shell structures and determining the radiation patterns (as shown in Figs. 7-4(b) and (c) with the red dashed lines). The near-fields based on theory are also shown in Figs Good comparisons are observed. It is worth highlighting that the near field characteristics of the antenna obtained by applying our theoretical approach show the field behavior of dipoles. This implies that, the electric field has local maxima at the location of the dipoles (centers of the core-shells). Nevertheless, considering the small size of particles and the fact that we are interested in the field performance outside the particles, this is a very good approximation as observed from the comparisons. The next step is to investigate the performance of the array antenna when it is located above an InGaAs substrate. This offers potential advantage for making on-chip nanoantennas. The 2-particle nanoantenna of the previous example is now located above an InGaAs slab of 0.35λ 0, see Fig. 7-6(a) (in our theoretical modeling a refractive index of about n 3.4 is assumed). The radiation pattern and the electric fields in xy and yz planes are shown in Figs. 7-6(b) and 7-7. As highlighted earlier, due to the surface wave propagation along the substrate and its strong interactions with the plasmonic antennas the radiation beams are degraded and the radiated power is split in different directions. Similar observation has been reported in [44, 90]. Far field behaviors reveals the antenna beam has a maximum in the xz plane. Although the source is polarized along the z, but because of the plasmonic particles locations along the y and their slab interactions, there are some y-polarized induced dipoles for the plasmonic particles, cf. Fig. 7-7(a). This can be seen from (7.8) where Ge ly rz is not zero, and hence the induced dipole have both y and z components. In addition, the near field in the xz plane in Fig. 7-7(b) shows that the electric field above and below the substrate are comparable verifying the field propagation through the substrate. For the sake of completion, we also investigate the behavior of a nanoantenna, when the source feed is in parallel direction with the slab surface, cf. Fig. 7-8(a). The radiation pattern and the near-field (normalized to the maximum value of E x ) 148

150 in yz plane are demonstrated in Figs. 7-8 (b) and (c), respectively. In this scenario, the feed dipole is polarized along x (p = x) and hence the induced dipoles on the nano core-shells are also x-polarized. Figure 7-8(b) exhibits that the surface wave generation has a dominant effect on the radiation performance. The back radiation can be suppressed by coating the silicon surface with a layer of silver material with ɛ = i and thickness of 0.2λ 0. The two plasmonic radiators are now implanted at the top of the silver, refer to Fig. 7-9(a). Figures 7-9(b) and 7-10 represent the far field radiation and the near field behavior of the antenna. As it is demonstrated, the back-radiation is stopped and the nanoantenna array radiates towards the up where the side lobe level gets smaller. Also the the near field distribution of the antenna proves that the magnitude of the field below the substrate is considerably smaller (around 50dB) than that in the forward region (upper half-space). That is to say, by adding a plasmonic layer we prevent the wave to transmit on the other side of the substrate. From the above, and considering the strong effect of the layered materials on the plasmonic nanoantenna performance, one may engineer the substrate of an array antenna properly to achieve a desired radiation pattern characteristic. Basically, the substrate and the reflected wave from that can control the dipolar polarization of each plasmonic-particle array elements, shaping the radiation beam. To provide some physical understanding, it is valuable to study the performance of a single optical dipole located at the top surface of a planar layered material, refer to Fig. 7-11(a). Figure 7-11(b) shows the angle of the maximum beam (in the yz plane) for an optical dipole located above an InGaAs slab (of 0.35λ 0 ) which its top surface is coated with 0.2λ 0 of silver. The dashed line exhibits the angle of maximum radiation for the antenna, versus the distance from the layered material (ɛ r2 = 1). As observed, with changing the distance between the dipole and the multi layer substrate the angle of radiation changes from 30 to around 75. Note that, as we increase the distance between the dipole and the layered material the number of side lobes increases (see also [90]) so the beam of the main lobe changes and a jump in the maximum radiation is exhibited. Adding another layer to this configuration will provide better opportunity for controlling the beam radiation angle. We consider 149

151 a case where a third layer of SiO 2 medium is grown over the silver and the dielectric slab (InGaAs). The solid line in Fig. 7-11(b) demonstrates that the beam angle (for maximum radiation) varies between 30 to around 90. Evidently, by depositing the third dielectric layer, we have a better control on the angle of radiation. A three-layers substrate can be engineered to manipulate the plasmonic particles dipolar modes and tilt the beam of a two-particles free-space Yagi-Uda antenna from endfire-type radiation to close broadside performance. The layered-substrate includes an InGaAs slab with thickness of 0.35λ 0, silver with thickness of 0.2λ 0 and a coating dielectric with ɛ r = i and thickness of 0.1λ 0 (refer to Fig. 7-12(a)). The radiation performance is obtained in Fig. 7-12(b). It shows the main lobe around 16 with suppressed back radian (due to negative permittivity of the silver layer). This is an interesting result, because we started with a z-directed optical dipole source which has a maximum radiation around θ = 90 and we end up with a maximum radiation around 16, by tailoring the plasmonic particles array and the substrate. The near-field performance is plotted in Figs. 7-13(a) and (b). As observed from Fig. 7-12, the two nano core-shells parasitic radiators array and its engineered substrate provides a narrow beam only in the yz plane. To control the 3D radiation performance in both planes (xz and yz) and achieve a pencil-beam broadside radiation one may change the distances of the nano core-shells and/or place four plasmonic nano core-shells around the dipole excitation. Figure 7-14(a) shows the configuration for an optimized design where four plasmonic nano core-shells are placed such that the antenna has a maximum radiation around θ = 0. In this case by using the data from Fig. 7-2(a), the radii ratios for the nano particles are chosen to be closer to the 0.71 (which is the radii ratio of the resonance for f = 650 T Hz). Hence the polarizability of the particles are larger and the antenna can have higher power transmission. Figure 7-14(b) demonstrates the 3D radiation pattern of the radiated power. As expected a pencil-beam around the z-axis is successfully achieved. The four plasmonic nanoantennas manipulate the induced dipole polarizations, and hence control the radiation characteristic to the application of interest. From practical points of view, it is challenging to precisely fabricate the core- 150

152 shell antenna. Specially one might be concerned about the exact thickness of the plasmonic shell that goes beyond one nanometer. Therefore, it is important to investigate the effect of changing the radii ratio. Figure 7-15 shows the change in the far field behavior, if the thickness of the shell varies. As it is evident the direction of the maximum beam is not so sensitive to the plasmonic shell thickness. Nevertheless the power transmission of the antenna changes. We need to highlight again that the designed structures presented in this paper are achieved for infinite-size substrates (along the x and y directions), where a finite substrate size antenna can have different performance due to the type of surface wave generation, the fields propagation in and along the substrate, and the edges diffractions. Since the infinite case is the limit of the finite one, when the size of the structure approaches to infinity the field at the back can be rather small leading to a strong directional beam above, as the one we obtain from the theoretical computation. Although, in this case the edges diffractions can have some undesirable effects on the radiation pattern increasing the side-lobes. The optimum design for a plasmonic array nanoantenna located above a finite substrate is currently under investigation. 7.3 Conclusion In this chapter, the radiation performance of an array of plasmonic core-shell nanoantennas located above engineered substrates are explored. We develop a theoreticalmodeling framework base on the Green s function analysis of dipole excitation above layered structures, to successfully characterize the antenna performance. The plasmonic core-shells are approximated with electric dipoles where their induced dipoles are calculated through the Greens function modeling of dipoles over the layered materials. The theoretical approach and developed formulations are verified through applying a full wave numerical analysis utilizing FDTD method. Both the radiation patterns and near-field characteristics are determined. The effect of the dielectric substrate layers on the radiation performance is highlighted. It is obtained that a dielectric substrate can drastically change the 151

153 radiation pattern of a plasmonic nanoantenna from what is obtained for that when is located in free space. Integrating a silver layer can reduce the antenna interaction with the dielectric substrate and suppress the back radiation. A composite substrate for an optimized 2D array of four-plasmonic nano core-shell radiators is engineered to tailor the dipolar modes of the nanoantennas and accomplish a pencil beam radiation. The results of this chapter can be of great value for creating on-chip nanoantennas with desired high-performance radiation characteristics. 152

154 (a) (b) Figure 7-7: Near field distribution for the antenna in Fig. 7-6(a) obtained by using the theoretical approach of this paper, (a) The magnitude of E y (db) in xy plane, (b) The magnitude of E z (db) in yz plane. Field penetration and propagation through the substrate is illustrated. 153

155 (a) (b) (c) Figure 7-8: (a) A nanoantenna array of a horizontal dipole, p x, and two nano coreshells located above a InGaAs slab of 0.35λ 0 thickness. (b) The 3D radiation pattern. (c) The magnitude of E x (db) in yz plane. 154

156 (a) (b) Figure 7-9: (a) Configuration of the nanoantenna in Fig. 7-4(a) where the nano coreshells are located above a InGaAs slab of 0.35λ 0 which its top surface is coated with 0.2λ 0 of silver. (b) The 3D radiation pattern. The silver layer suppresses the back radiation. (a) (b) Figure 7-10: Near field distribution for the antenna in Fig. 7-9(a). (a) The magnitude of E y (db) in xy plane, (b) The magnitude of E z (db) in yz plane. 155

157 (a) θ ε r2 =1 ε r2 = i d/λ 0 (b) Figure 7-11: (a) A vertical dipole located at the top of a planar layered material. The layered substrate includes a slab of InGaAs with thickness of 0.35λ 0 where its top surface is coated with silver of thickness 0.2λ 0. A third layer of dielectric ɛ r2 is deposited at top of the silver. (b) The angle of maximum radiation in the yz plane vs. the thickness of the dielectric layer. The jump in angle is because the maximum radiation occurs in another direction. 156

158 (a) (b) Figure 7-12: (a) Nanoantenna configuration in Fig. 7-4(a) located above a 3 layered substrate including InGaAs slab of 0.35λ 0, silver with thickness of 0.2λ 0 and a 0.1λ 0 layer of SiO 2. (b) The 3D radiation pattern of the radiated power. Adding a third layer controls better the radiation pattern. (a) (b) Figure 7-13: Near field distribution for the antenna in Fig. 7-12(a), (a) The magnitude of E y (db) in xy plane, (b) The magnitude of E z (db) in yz plane. 157

159 (a) (b) Figure 7-14: (a) 4-particle nanoantenna array with an engineered substrate demonstrating a broadside radiation characteristic. The radii ratio for the particles are r 1r /r 2r = 0.72 and r 1d /r 2d = 0.69, and d ref = 0.25λ 0 while d dir = 0.65λ 0. The layered material includes 0.5λ 0 of InGaAs, 0.2λ 0 of silver coating and a dielectric layer (SiO 2 ) with the thickness of 0.1λ 0. (b) The 3D radiation pattern. 158