The Pennsylvania State University The Graduate School UNIAXIAL METAMATERIALS FOR MICROWAVE FAR-FIELD COLLIMATING LENSES

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1 The Pennsylvania State University The Graduate School UNIAXIAL METAMATERIALS FOR MICROWAVE FAR-FIELD COLLIMATING LENSES A Thesis in Electrical Engineering by Jeremiah Paul Turpin c 211 Jeremiah Paul Turpin Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 211

2 The thesis of Jeremiah Paul Turpin was reviewed and approved by the following: Douglas H. Werner Professor of Electrical Engineering Thesis Advisor Pingjuan Werner Professor of Electrical Engineering W. K. Jenkins Professor of Electrical Engineering Department Chair Signatures are on file in the Graduate School.

3 Abstract The ability to control and manipulate nature and natural phenomena for the solution of practical problems and invention of new tools and technologies has long been a goal of science and engineering. The development of metamaterials exemplifies this pursuit; metamaterial devices, such as materials with a negative index of refraction (NIM) where light travels backwards or low-index media where the effective phase velocity exceeds the speed of light, allow interactions with electromagnetic fields in ways unavailable to devices constructed from conventional materials. These properties are used to improve the operation of existing systems and also allow the creation of entirely new optical and electromagnetic devices. Thin slabs composed of negative-index materials, for example, can be used as near-field imaging lenses that can resolve details in the structure being imaged that are smaller than a wavelength in the medium. Metamaterials research is being performed for all regions of the electromagnetic spectrum, from applications in the LF to Microwave, IR, and optical devices. Metamaterials are useful for the improvement of antenna and radiation characteristics, with the capability to exert precise control over the behavior of the electromagnetic field within a region. This thesis describes the development of a microwave metamaterial for the implementation of a far-field collimating flat lens for use with a crossed-dipole antenna. The lens must be constructed from a material with an index of refraction that is nearly zero, a property that does not exist in natural materials. Thus, a metamaterial is required. The design and selection of appropriate unit cell structures are described, along with descriptions of unit cells that were tested and found to be inadequate. Like most microwave metamaterials, the lens uses printed circuit board (PCB) technology for straightforward fabrication and implementation. The final metamaterial design was simulated as a finite lens to confirm correct operation; the metamaterial lens improves the gain of the original crossed-dipole antenna by 6dB, without excessive return loss or absorption loss in the lens. iii

4 Table of Contents List of Figures List of Tables Acknowledgments vii xi xii Chapter 1 An Overview of Metamaterials What is a Metamaterial? Early Examples of Microwave Metamaterials The Advent of Magnetic Metamaterials Electric Metamaterial Elements Electric LC Resonator Complementary Split-Ring Resonator Babinet s Principle End-loaded dipoles Isotropy and Magneto-electric Coupling Symmetry Groups and coupling Chapter 2 Metamaterial Requirements Lens Layout, Material, and Dimensions Periodic Unit Cell Analysis Material Parameter Extraction Inversion Theory Inversion Issues Anisotropic Material Parameter Extraction iv

5 Chapter 3 Metamaterial Particle Development Magnetic Particle Split Ring Resonators SRR Equivalent Circuit Geometric Design and Parameters SRR Simulation Results for 5% arms and.3 mm gaps SRR Simulation Results for 25% arms and.3 mm gaps SRR Simulation Results for 25% arms and.6 mm gaps Summary Multilayer Split Ring Resonator SRR with Lumped Elements Multi-Split-Ring Resonator Dual-split SRR Four-split SRR Selected Design Electric Particle Electric LC Resonators Single-Capacitor Dual-Inductor (1C 2L) ELC Dual-Capacitor Single-Inductor (2C 1L) ELC ELC Summary Complementary Split-Ring Resonators Falcone CSRR CSRR Summary Volumetric End-loaded Dipoles VELD simulations Selected Design Combined Magneto-electric Particle Chapter 4 Metamaterial Lens Simulations Crossed-Dipole Simulation Results Metamaterial Lens Simulation Results Chapter 5 Conclusions and Summary 6 v

6 Bibliography 62 vi

7 List of Figures 1.1 (a) The electromagnetic interactions of a real material can be expressed in terms of a relative ɛ r and µ r. (b) The structures that compose a metamaterial, although larger than the molecular and atomic-scale effects of a real material, can, in many circumstances, be used to compute an averaged electromagnetic response and the assignment of an effective ɛ r and µ r Various SRR geometry designs from the literature (non-exhaustive) (a) Original Pendry SRR [9, 6] (b) Schurig SRR [18] (c) Marques SRR [12] (d) Smith SRR [13, 1, 19] (e) Baena dual-loop dual-gap SRR [15, 16] (f) Baena quad-loop quad-gap SRR [16] (a) Single-capacitor ELC design [2] (b) Dual-capacitor ELC design [1] (a) Pendry SRR (b) Original Complementary SRR (CSRR) (a) Unit cell of a simple wire-mesh plasma medium (b) Unit cell of a 2D ELD metamaterial Metalens design dimensions and geometry Single layer of stacked wine-crate metamaterial concept Three unit cell orientations for anisotropic equivalent material parameter extraction. (a) Normal Z-incidence (Actual metamaterial orientation) (b) Parallel X-incidence (c) Parallel Y-incidence The SRR may be analyzed as a Series LC resonant circuit (a) Single- SRR geometry (b) Equivalent LC circuit diagram (c) Equivalent RLC circuit including source Planar spiral inductor geometry and parameters vii

8 3.3 (a) The Pendry SRR has significant bi-anisotropy. (b) Using stacked, concentric resonators as suggested by [12] removes some of the asymmetry from the SRR and decreases the magneto-electric coupling that leads to bi-anisotropy. (c) With the addition of arms to the capacitive gaps, the Schurig SRR has a wider tuning range of possible behaviors. (d) The actual unit cell for simulations places the SRR elements on the inside of a hollow cube of dielectric slabs. (e) adjacent unit cells are combined to model the full-thickness substrates SRR dimensions and variable parameters Extracted material parameters for 1-SRR with 5% arm length (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (c) Tangential permeability Extracted material parameters for 1-SRR with 25% arm length (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (c) Tangential permeability Extracted material parameters for 1-SRR with 25% arm length and.6 mmgaps. The ZIM frequency has shifted higher relative to the smaller gap and longer arms. (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability Multilayer SRR structures using vias for lower resonant frequencies without lumped elements. (a) one loop (b) two loops (c) three loops (a) Single gap SRR (b) Dual-gap SRR (c) Four-gap SRR Extracted material parameters for Dual-split SRR with 25% arms (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability Extracted material parameters for Dual-split SRR with higher resonant frequency due to arm length reduction (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability Extracted material parameters for 4-SRR with 25% arm length.(a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability Extracted material parameters for 4-SRR without arms (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability (a) Single-Capacitor ELC structure and parameters. (b) Dual- Capacitor ELC structure and parameters viii

9 3.15 Extracted material parameters for Single 1C 2L ELC (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability Extracted material parameters for Four 1C 2L ELC (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability Extracted material parameters for Four 1C 2L ELC with smaller capacitive gap (g =.1) for reduced ZIM frequency (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability Extracted material parameters for Single 2C 1L ELC (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability Extracted material parameters for Four 2C 1L ELC (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability Extracted material parameters for Four 1C 2L ELC, g =.1 (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability (a) Modified dual-gap symmetric SRR (b) Modified Complementary SRR (CSRR) Extracted material parameters for CSRR (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability (a) TE-mode and (b) TM-mode scattering parameters at normal incidence from a single layer of Modified CSRR resonators Extracted material parameters for Falcone CSRR (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability (a) TE-mode and (b) TM-mode scattering parameters at normal incidence from a single layer of Falcone CSRR resonators (a) Unit cell of a simple wire-mesh plasma medium (b) Unit cell of a 2D ELD metamaterial (a) Unit cell of a uniaxial volumetric end-loaded dipole (VELD) metamaterial (b) Unit cell of an isotropic VELD metamaterial (a) Large coupling magnetic field coupling region (red) for a dual- SRR particle. (b) Small electric field coupling region (blue) of an ELC device. (c) Larger electric field coupling regions (blue) of an ELD structure ix

10 3.29 Extracted material parameters for 3-arm VELD (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability Extracted material parameters for 6-arm VELD (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability (a) Dual-SRR structure and parameters (b) End-loaded dipole structure and parameters (c) Four-element VELD (d) Combined cubic magneto-electric ZIM/LIM metamaterial unit cell Extracted material parameters for matched uniaxial Magneto-electric ZIM/LIM metamaterial (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability Z-oriented permittivity and permeability of matched uniaxial Magnetoelectric ZIM/LIM metamaterial Crossed-dipole simulation setup and dimensions design dimensions and geometry (a) Return loss from a 15 mm crossed-dipole antenna. (b) Broadside co-pol gain and peak cross-pol (off-normal) gain for the crosseddipole antenna Simulated 2D radiation co- and cross-polarized gain pattern slices of a 15 mm crossed-dipole antenna over a finite ground plane at (a) 7.75 GHz (b) 8. GHz (c) 8.25 GHz (d) 8.5 GHz (e) 8.75 GHz (f) 9. GHz Metalens design dimensions and geometry (a) Return loss from a 15 mm dipole antenna feeding the metalens. (b) Broadside co-polarized gain and peak cross-polarized (offnormal) gain for the meta-lens Simulated 2D radiation co- and cross-polarized gain pattern slices of a 15 mm dipole antenna feeding a 6x6x5 mm meta-lens over a finite ground plane at (a) 7.75 GHz (b) 8. GHz (c) 8.25 GHz (d) 8.5 GHz (e) 8.75 GHz (f) 9. GHz Simulated 3D radiation co-polarized gain patterns of a 15 mm crossed-dipole antenna feeding a 6x6x5mm meta-lens over a finite ground plane at (a) 7.75 GHz (b) 8. GHz (c) 8.25 GHz (d) 8.5 GHz (e) 8.75 GHz (f) 9. GHz x

11 List of Tables 1.1 Illustrations of symmetry groups described by Schöenflies notation PCB fabrication constraints for microwave metamaterials Geometric parameters for near-zim metamaterial unit cell xi

12 Acknowledgments This research was partially funded by the Lockheed Martin University Research Initiative (URI) program. Dr. Erik Lier and Bonnie Martin of Lockheed Martin Commercial Space Systems were instrumental in specifying the requirements, performance, and manufacturing constraints of the metamaterial lens. The final design was manufactured and characterized by Erik Lier s team at LMC. The author thanks LMC for their assistance and the provision of the measured data for comparison with the design results. xii

13 Chapter 1 An Overview of Metamaterials Metamaterials are one of many examples of the engineering urge to improve and extend beyond the capabilities and restrictions imposed by nature. Skyscrapers, jet planes, and modern industry are made possible by the development of advanced materials - steel and fiberglass composites, ceramics and plastics. Computing technology has been carried forward on the back of the materials industry, where improvements in silicon refinement and other semiconducting materials can be translated into higher clock speeds or more compact circuits. By purifying and recombining natural substances in new ways, useful properties can be exploited for the advancement of almost every field in modern science and technology. Because of the myriad interactions of matter with electromagnetic waves, materials are vitally important to the field of electromagnetics; the unique properties of new dielectrics and material substrates have been used to create better lenses, antireflective coatings, radar-absorbing coatings, and more efficient antennas. Despite the rapid pace of recent developments, even the most advanced materials will be limited in their properties and applications. Engineering a material to meet specific constraints in its interaction with electromagnetic waves offers great advantages over selecting a material from a library of available materials, as a substance may be tailored for a particular application. An engineered material that may be characterized in its electromagnetic interactions as possessing some electric permittivity and magnetic permeability created through the use of structures arranged in a specific geometric pattern can be called a metamaterial. Metamaterials derive their properties from their subwavelength structure, not the material

14 2 composition alone. Electromagnetic metamaterials are quite useful tools for the development of new antennas, filters, waveguides, transmission lines, and lenses, allowing more compact, broadband, and potentially low-loss operation. Although the field is still in infancy, metamaterials show great promise for transforming all of electromagnetics. This thesis describes the design of a metamaterial for use in a low-index far-field collimating or aperture-expanding lens for a single antenna, as a step in the development of compact high-gain circularly-polarized horn antenna arrays. 1.1 What is a Metamaterial? A metamaterial is a structure composed of a collection of artificial engineered molecules designed to behave as a homogeneous material with electromagnetic wave interactions within some frequency band of interest that are uncommon or impossible to achieve with an ordinary real material. A structure defined in this way emulates the electromagnetic field interactions with actual materials, where the atomic, molecular, and inter-molecular resonances and interactions determine the response of a material to an electromagnetic impulse, as in Fig. 1.1(a). Most metamaterials use periodic arrays of identical molecules for simplicity of analysis and construction, where the molecules are macro-sized structures that are still much smaller than a wavelength at the desired operational frequency band. Since the periodic structures are on a macro scale, the electromagnetic field is clearly inhomogeneous inside the metamaterial. However, the fields may be homogenized for analysis and design by using small unit cells, thus allowing the wave interactions of the electromagnetic field and the metamaterial to be approximately specified by an effective electric permittivity and magnetic permeability as in Fig. 1.1(b). Although metamaterials are considered here only for electromagnetic waves, they may also be defined and constructed for other physical systems governed by wave equations, such as surface waves on a liquid [1] and longitudinal acoustic waves [2, 3, 4]. More specifically, this thesis will focus on metamaterials for microwave applications. Designs for the terahertz, infrared, and optical regimes require different design and fabrication techniques than those used here, and should be considered

15 3 separately. Although the equations that govern electromagnetism are identical throughout the EM spectrum, the natural materials that form the building blocks of metamaterials have distinct properties in the different frequency bands. The differences can be opportunities for improvement as well as roadblocks that must be overcome. D = ɛē B = µ H Inhomogeneous field interactions Homogenized model: ɛ eff, µ eff Ē Ē (a) (b) Fig (a) The electromagnetic interactions of a real material can be expressed in terms of a relative ɛ r and µ r. (b) The structures that compose a metamaterial, although larger than the molecular and atomic-scale effects of a real material, can, in many circumstances, be used to compute an averaged electromagnetic response and the assignment of an effective ɛ r and µ r. 1.2 Early Examples of Microwave Metamaterials Although academic metamaterial research is a recent phenomenon within the last 1-15 years, there are more isolated examples much earlier. Wire-mesh plasma media, which can be considered a precursor of modern metamaterials, were introduced as grids for antenna beam collimation by W. E. Kock in A group at the Stanford Research Institute [5] designed and constructed a Luneburg lens-type device from a wire-mesh structure in Many metamaterial applications grew out of advances in artificial photonic band-gap (PBG) structures; although photonic or electromagnetic band-gap (EBG) materials are generally not considered to be metamaterials, there are similarities in their design and construction. The first implementation of a material with a negative index of refraction by Smith et. al. [6] demonstrated behaviors that were initially predicted in 1968 by Veselago [7]. The existence of a so-called left-handed material that could not be found in nature

16 4 was met by a combination of skepticism and excitement, and sparked many more metamaterial research initiatives. The rush to replicate Smith s results and to produce better negative-index structures at different wavelengths, along with the introduction of Transformation Optics (TO) [8], added an additional impetuous to the invention and implementation of better metamaterials. 1.3 The Advent of Magnetic Metamaterials The Split Ring Resonator (SRR) was introduced by Pendry et. al. [9] and almost immediately seized upon by the metamaterial community at large as a valuable building block for the creation of magnetic metamaterials, including negative-index (left-handed materials) [6, 1] and zero-index materials [11]. The original SRR design in Fig. 1.2(a) used two coplanar nested rings, but the asymmetry of this configuration has significant bianisotropic properties. Many authors have contributed to the design of SRR elements with additional points of symmetry for reduced bianisotropic effects [12, 13, 14, 15, 16, 17], such as those designs in Fig. 1.2(c,e,f). (a) (b) (c) (d) (e) (f) Fig Various SRR geometry designs from the literature (non-exhaustive) (a) Original Pendry SRR [9, 6] (b) Schurig SRR [18] (c) Marques SRR [12] (d) Smith SRR [13, 1, 19] (e) Baena dual-loop dual-gap SRR [15, 16] (f) Baena quad-loop quad-gap SRR [16]

17 5 1.4 Electric Metamaterial Elements The original microwave negative-index material used a periodic wire dipole array for the electric negative-permittivity component, similar to the wire mesh structures introduced earlier. The wire mesh medium behaves like an uniaxial plasma with the optical axis aligned with the axis of the dipoles. The wire dipole grid relies on coupling between the dipoles to achieve the desired material properties, and thus needs a large number of wires to produce a viable metamaterial. There can also be significant edge effects, where the effective properties of the edges of the medium differ from the effective bulk properties. The plasma response of the medium can be used to create a negative-permittivity response below the plasma frequency of the structure. This Drude-type behavior generally does not produce a resonant response with large positive values of permittivity, thus limiting the usefulness of the wire mesh as a general metamaterial building block. Although it has several worthwhile applications and is simple to design, tune, and build, the wire mesh medium is not suitable for this application Electric LC Resonator The most well-known electric metamaterial element is the Electric LC resonator (ELC), introduced by Schurig et. al [2], which has replaced the wire mesh medium for some negative-index material implementations [1]. The ELC is a series RLC resonator that is intended to couple strongly to the electric field along a single axis. The ELC contains inductive and capacitive equivalent microstrip lines arranged to emphasize the electric field coupling from the capacitive gaps, and to minimize the magnetic coupling by having two anti-directional current loops. Two variants of the ELC are illustrated in Fig The ELC has many advantages over a wire mesh medium for metamaterial purposes. The ELC is a self-contained resonator that can be easily tuned by adjusting the geometrical dimensions of capacitive and inductive components within the unit cell. By emphasizing the response of the material on the field interactions within a unit cell, instead of requiring interactions with neighboring unit cells for correct operation, the bulk ELC material parameters do not suffer from strongly degraded performance near the material boundaries. The ELC can be a very convenient structure to accompany a SRR for the creation

18 6 of a cubic magneto-electric metamaterial, although there are caveats to successful integration. There are fewer examples of the ELC in the literature than of the SRR, which is by far the most commonly referenced metamaterial molecule. (a) (b) Fig (a) Single-capacitor ELC design [2] (b) Dual-capacitor ELC design [1] Complementary Split-Ring Resonator Falcone et. al. [21] introduced the CSRR element as one of the first new electric metamaterial particles, prior to the introduction of the ELC resonator by Schurig et. al. in 26 [2]. The CSRR is defined as the complementary sheet of the Splitring resonator whose behavior is predicted through the use of the Babinet Principle. A slot with the same shape as the SRR is cut into a sheet of PEC, as illustrated in Fig As a dual element to the SRR, the CSRR should demonstrate an electric polarizability for normally-oriented electric fields. The CSRR has been proposed for use as a microstrip element, with the claim that the CSRR responds to the electric field in the same way as the SRR responds to the magnetic field. (a) (b) Fig (a) Pendry SRR (b) Original Complementary SRR (CSRR)

19 Babinet s Principle Babinet s Principle is a theorem in optics that states that when the field behind a screen with an opening is added to the field of a complementary structure, the sum is equal to the field when there is no screen. 1 The scattering from a PEC patch object represents the complementary field to the scattering from an infinite sheet in which a slot is cut with the same shape and dimensions as the patch. Falcone et al. demonstrated that, based on the complementary scattering property derived from Babinet s Principle, if the SRR behaves as a magnetic dipole particle, then the CSRR will behave approximately as an electric dipole element [21]. The correspondence would be exact if the sheet was infinitely large, very thin, and composed of PEC with no adjacent lossy dielectrics. Babinet s Principle does not consider polarization; for some applications, the polarization performance is not critical End-loaded dipoles A new electric metamaterial proposed in this thesis is the end-loaded dipole (ELD) unit cell. The ELD extends the wire mesh medium by adding spiral-connected endloading arms at the gaps in the wires as illustrated in Fig. 1.5, adding additional capacitance and inductance to the circuit. The added impedance transforms each unit cell into a self-resonant circuit, making the ELD less sensitive to edge effects than the wire mesh medium. The ELD may be used for the same applications as the ELC, and has some advantages over the existing ELC metamaterial design. (a) (b) Fig (a) Unit cell of a simple wire-mesh plasma medium (b) Unit cell of a 2D ELD metamaterial 1 C.A. Balanis, Antenna Theory, Analysis and Design, 697

20 8 1.5 Isotropy and Magneto-electric Coupling Most materials in common usage are isotropic, that is, EM radiation propagates identically in any direction within the material. The material constants for an isotropic material are represented as scalar constants, and the constitutive electric and magnetic field relations as vector equations. D = ɛ E (1.1) B = µ H (1.2) In an anisotropic material, the propagation characteristics of electromagnetic waves within the medium depend on the direction of propagation with respect to the orientation of the material. Anisotropic materials exist naturally in the form of crystals and polarized plasmas. A physically realizable arbitrarily anisotropic material has material parameters ɛ and µ specified as 3-dimensional second-order Hermitian symmetric tensors, changing the constitutive field relations to tensor equations. D = ɛ E B = µ H D x D y D z B x B y B z ɛ = xx ɛ xy ɛ xz ɛ yx ɛ yy ɛ yz ɛ zx ɛ zy ɛ zz µ = xx µ xy µ xz µ yx µ yy µ yz µ zx µ zy µ zz E x E y E z (1.3) H x H y H z (1.4) Anisotropic materials can be classified as uniaxial, biaxial, birefringent, or gyrotropic media, among other classifications, where the form of the material parameter tensors determine the classification. Uniaxial is a particularly useful classification for use with metamaterials. A uniaxial medium is isotropic in two dimensions, with different propagation characteristics in the third normal axis, which is represented by a diagonal tensor with two equal values. The optical axis for a uniaxial medium is the direction of the different material properties. Propagation along

21 9 the optical axis is called extraordinary propagation, while ordinary propagation occurs in the plane normal to the optical axis. Anisotropic ɛ xx ɛ xy ɛ xz ɛ = ɛ yx ɛ yy ɛ yz ɛ zx ɛ zy ɛ zz Uniaxial ɛ ɛ = ɛ ɛ z Biaxial ɛ x ɛ = ɛ y ɛ z Gyrotropic ɛ 1 +jg z ɛ = jg z ɛ 1 ɛ 2 Many metamaterial particles possess anisotropic properties unless specially designed to be isotropic in two or three dimensions. Isotropy is related to the symmetry of the metamaterial construction; in general, a higher degree of symmetry in the metamaterial geometry corresponds to a better approximation of an isotropic medium [16]. For example, completely asymmetric patterned structures will have different responses with different directions of wave incidence, while a completely symmetric metamaterial cube will behave similarly under excitation from any direction, and will thus be approximately isotropic. Materials may also be categorized as bi-isotropic, which is a measure of coupling between the electric and magnetic fields. D = ɛ E + ξ H (1.5) B = µ H + ζ E (1.6) Bi-isotropic materials may be generalized to the case when ɛ, µ, ξ, or ζ are tensors and the material becomes known as bi-anisotropic. Magnetoelectric coupling is generally undesirable when building a metamaterial structure as it leads to nonzero cross-polarized responses, and can be minimized through proper elec-

22 1 tromagnetic analysis and design of the metamaterial molecules Symmetry Groups and coupling Classification of symmetry in a 2-dimensional unit cell can be divided into groups, which are illustrated in Table 1.1. Unit cells with at least D 2h symmetry have been shown [16] to not possess any magneto-electric coupling, while lower-order symmetry unit cells may have bi-anisotropic behavior. For example, the original Pendry SRR with nested loops belongs to the D 1h symmetry group with 2-fold symmetry and exhibits significant bi-anisotropic effects; modified SRR designs have been suggested to counter that disadvantage. The ELC possesses four-fold symmetry in the D 2h group and has no magneto-electric coupling. As an example, the ELC can be analyzed intuitively to show that an electric excitation will not produce a net magnetic response, and vice versa. A z-oriented electric field excitation generates counter-rotating currents in the two current loops, producing magnetic dipole moments that are equivalent but opposite and canceling. Similarly, a normally-oriented magnetic field will induce currents in the same direction around the loops, so that no net current flow is created and no dipole moment is induced. When designing metamaterial unit cells, it is wise to maintain strong symmetry conditions in order to reduce the potential for magneto-electric coupling. Asymmetric bi-anisotropic unit cells can be used to create isotropic cubic metamaterials [16], but intentionally anisotropic metamaterials will be sensitive to any bianisotropy of the constituent molecules.

23 11 Table 1.1. Illustrations of symmetry groups described by Schöenflies notation. Group Symmetry type Geometrical Symmetry representation illustration C 1h No Symmetry D 1h Two-fold reflection symmetry C 2h Two-fold rotational symmetry D 2h Four-fold reflection symmetry C 4h Four-fold rotational symmetry D 4h Eight-fold reflection symmetry

24 Chapter 2 Metamaterial Requirements The design and specification of the collimating near-zero index material (ZIM) lens were provided as an input to this project; the development of the metamaterial and performance prediction of the final lens are the primary objects of this research. Therefore, the dimensions, material requirements, and configuration of the lens and antenna are pre-determined and were not chosen during this research. 2.1 Lens Layout, Material, and Dimensions The metalens was designed using a genetic algorithm to optimize the material parameters, spacing, lens size, and antenna position for operation in the rough range from 8-1 GHz. The dimensions and geometric configuration are illustrated in Fig A square lens composed of cubic metamaterial unit cells is positioned above a ground plane and fed by a circularly-polarized crossed-dipole antenna. The outer faces of the lens are metallic, to prevent undesired field leakage. The lens material is a uniaxial low-index material, with µ and ɛ specified in (2.1). The low-index condition along the direction of desired radiation allows the phase of the radiated energy to be approximately equal across the top surface of the lens, creating high gain. The ground plane and the low-index lens form a cavity that results in highly collimated radiation from the top surface of the lens. Preliminary simulations using homogeneous materials with the desired permittivity and permeability showed a good collimating effect with low return loss back into the feed antennas.

25 13 1 ɛ r = µ r = 1 (2.1).2 5 mm Anisotropic Meta-lens PEC Edges 6 mm 9 mm 7.6 mm Ground Plane Fig Metalens design dimensions and geometry. This lens can be implemented by an array of cubic metamaterial elements that meet the size and performance characteristics. A single layer of 12x12 5x5x5mm unit cells would satisfy the size specification of the lens, giving a constraint for the unit cell dimensions. A common manufacturing technique for microwave metamaterials is to use printed circuit board (PCB) technology to create the patterned metallic traces of each unit cell. Three-dimensional cubic unit cells can be created by sandwiching a vertical grid of perpendicular strips of unit cells between panels of horizontal unit cells. With appropriate notches and holes in each board, the vertical unit cells are assembled in a wine-crate design, as in Fig The PCB construction imposes constraints on the geometric parameters of the unit cells. The minimum dielectric thickness is determined by available substrates, and the trace thickness and tolerances are determined by the capabilities of the PCB vendor. Table 2.1 gives the minimum trace dimensions used for all metamaterial designs in this work.

26 14 Fig Single layer of stacked wine-crate metamaterial concept. Table 2.1. PCB fabrication constraints for microwave metamaterials. trace/clearance Standard Producible Achievable (higher cost) Cu Thickness [mils] [mm] [mils] [mm] [mils] [mm] 1/4 oz 3/3.762/.762 2/2.58/.58 1/ /.381 1/2 oz 3/ /.889 3/3.762/.762 2/2.58/ Periodic Unit Cell Analysis Unit cells are modeled and analyzed as infinite arrays of identical unit cells for reduced computational requirements. Approximating the finite-sized array of any real implementation as an infinite array is valid for relatively large finite arrays, where the contribution of mutual coupling between elements to the overall response is small. Although the mutual coupling makes no difference for the infinitely-periodic design simulations, a smaller contribution of mutual coupling, with stronger self-resonance, decreases the parameter degradation at the boundaries of the finite medium, improving the homogeneity of the material. This is a

27 15 loose approximation, but has been accepted as appropriate practice for metamaterial design and simulation. 2.3 Material Parameter Extraction Metamaterials can be designed to meet desired permittivity and permeability dispersion characteristics, or to meet constraints on reflection and transmission or index of refraction and material impedance. Permittivity and permeability are considered to be fundamental properties of real media, such that homogeneous media can be said to possess some relative permittivity and permeability based on their crystalline, molecular, atomic, or sub-atomic interactions with the electromagnetic field. A metamaterial is clearly an inhomogeneous structure composed of multiple materials with different electromagnetic characteristics, which would, strictly speaking, invalidate the assignment of a single number to represent the wave interactions of a group of such structures. However, for unit cells that are much smaller than the wavelength (less than λ/1 is a common rule-of-thumb), the fields within the structures can be homogenized through averaging to obtain effective permittivity and permeability [23, 24]. Effective material parameters describe the average behavior of the wave within the metamaterial, and do not indicate that the waves propagating in the metamaterial structure follow Maxwell s equations according to the assigned homogeneous effective parameters at every point within the material slab. This homogenization procedure also applies for inhomogeneous materials such as sand and soil, which are inhomogeneous on a small scale, but can be assumed to be homogeneous for observations on a large scale for the assignment of effective electromagnetic properties. Practically speaking, no material, natural or otherwise, is truly homogeneous at all scales; materials that are inhomogeneous at the molecular level are still considered in common usage to be homogeneous for all practical purposes. If a slab of metamaterial is assumed to be represented by some effective material parameters, then the plane wave scattering coefficients from the slab can be easily computed using the Fresnel scattering coefficients. Full-wave EM simulation packages can be used to compute the scattering parameters (Reflection and Transmission) from an infinite periodic array of metamaterial devices, but the material

28 16 parameters (ɛ, µ) are the quantities that must be matched to the desired values. The reflection (R) and transmission (T) coefficients computed by the numerical solver may be inverted into permittivity and permeability or index of refraction and intrinsic impedance by simulating a thin slab of the metamaterial composed of one or two layers, and solving the Fresnel scattering coefficient equations for the material parameters [23, 24]. The inversion algorithm is numerically sensitive, and must be applied carefully in order to obtain meaningful and useful results. A single unit cell, or a stack of two unit cells, is modeled in 3D; commercial solvers such as HFSS, CST, and FEKO support the application of periodic boundary conditions, which simulate the effects of an infinite-sized array. Excitation by linear plane waves from waveguide ports above and below the unit cell allows the homogenized scattering parameters to be computed for the infinite array. The procedure of drawing the geometry objects for one particular unit cell, simulating, extracting the frequency-dependent scattering coefficients, and computing the dispersive material parameters should be scripted, parameterized, and automated for efficient and reproducible simulations Inversion Theory The Fresnel scattering coefficients relate the permittivity and permeability of a slab of homogeneous material to the field strength of the reflected and transmitted waves at a single interface. Γ = η 2 η 1 η 2 + η 1 (2.2) T = 2η 2 η 2 + η 1 (2.3) Computing the scattering from a slab with two interfaces requires the solution of a linear equation to determine the total transmitted and reflected energy. t 1 = [ cos(nβd) j ( z + 1 ) ] sin(nβd) e jβd (2.4) 2 η t = te jβd (2.5)

29 ( r = t 2 j z 1 ) sin(nβd) (2.6) z Where n is the effective index of refraction, η is the effective wave impedance, d is the slab thickness, and β = ω/c is the propagation constant of the incident wave. The expressions for the total scattering from a thin slab may be solved to obtain expressions for n and η, which in turn may be used to find ɛ and µ. 17 cos(nβd) = 1 [ ] 1 (r 2 t 2 ) (2.7) 2t { } 1 cos(nβd) = R 1 t 2 t 2 [A 1r + A 2 t ] (2.8) η = ± (1 + r) 2 t 2 (1 r) 2 t 2 (2.9) A 1 = r t + t r (2.1) A 2 = 1 r 2 t 2 (2.11) Both n and η have multiple solutions available in the complex plane, where the correct branch must be chosen based on the material description. For passive materials, the correct sign for the impedance may be chosen by the constraint Rη >. The solution for the index of refraction has multiple complex branches, with parameter m that must be chosen to meet the expected material parameters, where the sign is chosen such that In >. Using very thin slabs improves the accuracy of the inversion by creating widely separated alternate branches. ( [ n = ± cos 1 1 2t 1 (r 2 t 2 ) ]) + 2πm βd βd (2.12) Implementing these equations to use the simulated reflection and transmission data of a single layer of a metamaterial produces the estimates for n and η. Finally, the effective averaged permittivity and permeability may be found from n and η. µ = nη (2.13) ɛ = n η (2.14)

30 18 These equations were derived [23, 24] using the assumptions of approximately symmetric unit cells in the propagation direction and small metamaterial unit cells Inversion Issues The plane-wave scattering coefficient inversion algorithm is numerically unstable, especially for simulations or measurements with low loss. The numerical instability is exhibited as regions of abrupt sign inversions in the extracted material parameters, where the permittivity, permeability, and index of refraction change sign within some band. The instability can be reduced through the introduction of lossy materials into the simulations, often by allowing the airbox surrounding a unit cell to have some small but nonzero loss tangent (tan(δ).1). Simulations with lossy materials may not need additional loss components Anisotropic Material Parameter Extraction The inversion algorithm computes a single field in the diagonal ɛ and µ tensors for each polarization and incidence angle. For example, an HFSS simulation with incoming wave incident from the positive z-axis that computes the scattering data for both TE and TM-mode incidence plane waves will compute ɛ xx and µ yy for the TE polarization, and ɛ yy and µ xx for the TM polarization. A single simulation is not sufficient to characterize the anisotropic material under consideration. In order to compute ɛ zz and µ zz, another simulation must be performed at a different angle of incidence. A simple technique is to rotate the entire unit cell about either the x or y axis such that a z-incident wave will impinge on either the Y or X face of the unit cell. Performing three simulations with incident waves from the X, Y, and Z axes provides two estimates for each material parameter tensor element and allows comparison and verification of simulation results. The simulation setup for the three unit cell orientations is demonstrated in Fig

31 19 Fig Three unit cell orientations for anisotropic equivalent material parameter extraction. (a) Normal Z-incidence (Actual metamaterial orientation) (b) Parallel X- incidence (c) Parallel Y-incidence

32 Chapter 3 Metamaterial Particle Development Given the goals and constraints outlined above, the development of the metamaterial to implement the collimating meta-lens is the next step. The minimum feature size and unit cell sizes are fixed, leaving the choice of the patterned metal structures as the remaining design parameter for the metamaterial. The cubic metamaterial unit cell with PCB construction can be visualized as a hollow cube, with walls that are half the actual substrate thickness and metallic patches resting on the interior faces of the cube. Since planar magnetic unit cells primarily interact with the normal magnetic field, and planar electric unit cells primarily interact with the tangential electric field, the magnetic particles will be on the top and bottom faces, and the electric particles on the four side faces of the cube for a uniaxial magneto-electric metamaterial. 3.1 Magnetic Particle Split ring resonators are the reasonable choices for use as a magnetic metamaterial, due to their tractability in tuning and analysis. Other configurations, such as the Jerusalem Cross, also result in magnetic resonances, but the SRRs are designed to interact primarily with the magnetic field with minimal electric field coupling.

33 Split Ring Resonators The first step towards the optimization of a metamaterial particle is to choose the geometry and perform a parameter study. For a well-behaved particle like the SRR, where geometry alterations can be easily linked to the corresponding change in the response, altering parameter values by small degrees allows the system to be easily tuned. This well-behaved property is quite convenient, as it allows rapid, manual tuning without the construction and execution of a global stochastic optimization routine, as is required for many frequency-selective surface (FSS) and infrared (IR) patterned metamaterial elements SRR Equivalent Circuit Analyzing an individual SRR as a microstrip circuit, it may be decomposed into an equivalent resonant LC or RLC circuit, as illustrated in Fig Although the microstrip possesses distributed inductance, capacitance, and resistance throughout the structure, the geometry of the SRR emphasizes different behavior. The microstrip loop is primarily inductive, giving the circuit an equivalent inductance. The break in the loop together with the parallel arms create a primarily capacitive effect, with some equivalent capacitance. With highly conductive microstrip, the resistive component of the RLC equivalent circuit may be largely ignored for most circumstances. This circuit approximation can aid in understanding the change in metamaterial response due to a change in material geometry. (a) (b) (c) Fig The SRR may be analyzed as a Series LC resonant circuit (a) Single-SRR geometry (b) Equivalent LC circuit diagram (c) Equivalent RLC circuit including source The resonant frequency of an RLC circuit is given by (3.1), where L and C are the equivalent inductance and capacitance of the SRR.

34 22 1 f c = 2π (3.1) LC The exact capacitance and inductance may be approximated for a single unit cell from the simulation results, but are not expected to be terribly accurate due to the effects of mutual coupling between adjacent unit cells. Analytic expressions exist for the inductance of a microstrip loop or spiral [25], but microstrip capacitance is more troublesome, because the stray capacitance of the microstrip traces is a significant contribution to the total capacitance, complicating a computation from first principles. Therefore, a combined technique that uses simulation results and the analytic value of the loop inductance to compute the loop capacitance using equation 3.1. This procedure assumes that the resonant frequency of the infinitely periodic array in the simulation is not substantially different from the resonant frequency of a single loop. First, an SRR design with specific parameters is simulated to find the location of the primary resonance. Then, the inductance is found with equation 3.2, and the capacitance from equation 3.1 and the simulated resonant frequency and predicted inductance. L = 37.5µ n 2 a 2 22r 14a (3.2) a r n: number of loops r: outer radius a: average radius Fig Planar spiral inductor geometry and parameters. Even without exact knowledge of the L and C parameters for a particular SRR structure, the equivalent circuit representation can be used to predict and explain the effects of geometric parameter changes. For example, increasing either capacitance or inductance decreases the resonant frequency of the equivalent RLC circuit. Capacitance is increased with longer capacitive arms, or a narrower capacitive gap. Inductance is increased with additional current loops or enclosing more area inside

35 23 the loop. Both capacitance and inductance may be increased through the appropriate addition of lumped circuit elements to the microstrip circuit. All of the SRR designs are tuned by considering the required change in resonant frequency and making geometry changes accordingly to modify either L or C. In addition to the approximation of ignoring the mutual coupling effects, the desired operation of the SRR is not necessarily at the resonant frequency; for a ZIM/LIM metamaterial, the operational band is 1% higher than the resonant frequency. The location of the ZIM/LIM band depends on the shape and amplitude of the resonance curve, which can vary with the position of the resonance. For this reason, tuning of a ZIM/LIM metamaterial must be performed incrementally to allow control over both the position of the resonance and its amplitude and shape in order to achieve the desired behavior with an acceptable bandwidth Geometric Design and Parameters Pendry s original SRR in Fig. 3.3(a) with the two concentric rings was compact and had significant capacitive coupling between the two rings for a strong response, but also has bi-anisotropic characteristics from the different-sized rings and the resulting asymmetry in the unit cell. Since the meta-lens should be polarizationinsensitive and every element of the anisotropic permittivity and permeability tensors controlled to a fixed value, it is undesirable in the extreme for magnetoelectric coupling to occur. The Marques SRR [12] was modified to eliminate the bi-anisotropy by using two identical concentric resonators offset and rotated about the normal axis, as in Fig. 3.3(b). The two rings are still strongly coupled, but without the possibility for a magnetic excitation to create an electric dipole response. Two stacked SRR loops are appropriate for applications with PCB construction, where the two corresponding rings may be printed on opposite sides of the substrate. Introducing parallel arms at the capacitive gap adds more capacitance to the circuit, decreases the resonant frequency, and gives an additional tuning parameter to the design. Combining this Schurig SRR with the Marques concept for stacked resonators as in Fig. 3.3 gives a strongly-coupled pair of resonators with tuning capability. For the vertically-stacked single-split with arms SRR, the primary parameters with respect to tuning the magnetic resonance are the unit cell size, the split width,

36 24 (a) (b) (c) (d) (e) Fig (a) The Pendry SRR has significant bi-anisotropy. (b) Using stacked, concentric resonators as suggested by [12] removes some of the asymmetry from the SRR and decreases the magneto-electric coupling that leads to bi-anisotropy. (c) With the addition of arms to the capacitive gaps, the Schurig SRR has a wider tuning range of possible behaviors. (d) The actual unit cell for simulations places the SRR elements on the inside of a hollow cube of dielectric slabs. (e) adjacent unit cells are combined to model the full-thickness substrates. and the arm length. The trace width and gap between the edge of the resonator and the edge of the unit cell are additional parameters, but these are more suitable to be chosen at the outset. For the situation where the unit cell size is already chosen, the only two parameters are the arm length and split width. Figure 3.4 Figures 3.5 through 3.7 show the results of varying the gap width and arm length for 5 mm periodic unit cells SRR Simulation Results for 5% arms and.3 mm gaps Fig. 3.5 shows a z-oriented magnetic resonance at about 2.75 GHz with an associated narrow ZIM band at about 3 GHz. For the unit cell size of 5 mm, this makes the relative SRR size to be λ/2, which more than satisfies the small unit

37 25 p w g a u Fig SRR dimensions and variable parameters. cell requirement for the application of the effective medium theorem. Note that the z-oriented permittivity and the x- and y-oriented permeability are at free-space values for the band surrounding the resonance. The tangential (x and y) permittivity is higher than desired, however, with ɛ x = ɛ y = 4 within the resonance band. The tangential electric and magnetic resonances at 7 and 1 GHz do not affect the response at the ZIM band and can be ignored. The electric resonances are due to the linear dipole-mode excitation of the SRR. The difference in frequency of the two electric resonances is due to the asymmetric split in the SRR s, which creates longer dipoles in the y-direction than the x-direction, which has a split that changes the resonance performance. The asymmetric performance at higher frequencies is the result of the lack of complete symmetry in the unit cell, but the lack of symmetry is acceptable as long as the response is isotropic in the X-Y plane near the ZIM band SRR Simulation Results for 25% arms and.3 mm gaps In order to test the tuning capability of the SRR, the arms are shortened from 5% of the SRR diameter to 25% of the ring diameter, reducing the capacitance and increasing the resonant frequency. The effect is small, but noticable, in the z-oriented permeability in Fig. 3.6(c); the resonance and associated ZIM band has been shifted by.5 GHz. The permeability remains below.5 after the resonance, giving a wide Low-index material band.

38 26 1 mil 12 mil.3 mm 5% 5 mm Permittivity Permeability ε zz ' ε zz '' (a) µ zz ' µ zz '' (c) ε zz ' µ zz ' ε xx ' 2 ε yy ' ε xx ' 1-1 ε yy '' (b) ε xx '' 1.5 µ xx ' 1 µ yy '.5 µ xx '', µ yy '' (d) µ xx ' ε yy ' µ yy ' ε zz '' µ zz '' Fig Extracted material parameters for 1-SRR with 5% arm length (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (c) Tangential permeability ε xx '' µ xx '' ε yy '' µ yy '' 1 mil 12 mil.3 mm 25% 5 mm Permittivity Permeability ε zz ' ε zz '' (a) µ zz ' µ zz '' (c) ε zz ' µ zz ' ε xx ' 2 ε yy ' 1 ε xx '' ε xx ' -1 ε yy '' -2 (b) 1 µ yy ' 5 µ xx ' µ xx '' µ yy '' (d) µ xx ' ε yy ' µ yy ' ε zz '' µ zz '' Fig Extracted material parameters for 1-SRR with 25% arm length (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (c) Tangential permeability ε xx '' µ xx '' ε yy '' µ yy ''

39 27 1 mil 12 mil.6 mm 25% 5 mm Permittivity Permeability ε zz ' ε zz '' (a) µ zz ' µ zz '' (c) ε zz ' µ zz ' ε xx ' 2 1 ε yy ' ε xx '' ε xx ' ε yy '' -1-2 (b) 1 µ yy ' 5 µ xx ' µ xx '' µ yy '' (d) µ xx ' ε yy ' µ yy ' ε zz '' µ zz '' Fig Extracted material parameters for 1-SRR with 25% arm length and.6 mmgaps. The ZIM frequency has shifted higher relative to the smaller gap and longer arms. (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability ε xx '' µ xx '' ε yy '' µ yy '' SRR Simulation Results for 25% arms and.6 mm gaps Increasing the split width while maintaining the 25% arms is the next method of increasing the resonant frequency, again through reduction of the capacitance of the equivalent RLC circuit. The resonance and ZIM band in Fig. 3.7(c) has shifted by about.25 GHz from the previous case Summary Based on the three sets of simulation results in this section, it is clear that this SRR geometry will not give a good response at the desired 8 GHz simply by tuning the gap width and arm length parameters. Additional geometry changes must be made.

40 Multilayer Split Ring Resonator The previous three resonator designs used 5mm unit cells and produced resonances near 3 GHz. If a lower resonant frequency is required, or a smaller unit cell for the same frequency, then either the capacitance or inductance of the equivalent circuit must be increased. Increasing the number of wire loops in the resonator by using vias to reach the back of the PCB is one possible method of increasing the inductance. An illustration of such SRR geometries is shown in Fig Fig Multilayer SRR structures using vias for lower resonant frequencies without lumped elements. (a) one loop (b) two loops (c) three loops Increasing the inductance in this fashion does indeed decrease the resonant frequency of the device, but also decreases the bandwidth of the ZIM/LIM band. The asymmetries introduced by the requirements for the via and loop placement will create higher cross-polarized scattered fields and cause trouble for use in an isotropic metamaterial. These proposed designs are also undesirable from a manufacturing standpoint as the addition of vias increases expense and complication, and prohibits the use of multiple resonators placed on the front and back of each board. The multilayer SRR is not useful for this project. Note that multiple inductive loops can be effectively added to the SRR without using vias, as in the Baena spiral SRR structures in Fig. 1.2(e-f). These designs were not evaluated for this project SRR with Lumped Elements The addition of lumped capacitance or inductance to the equivalent circuit of the split-ring resonator will change the operating frequency of an SRR metamaterial.

41 29 (a) (b) (c) Fig (a) Single gap SRR (b) Dual-gap SRR (c) Four-gap SRR These techniques can be applied to reduce the resonant frequency of a circuit and therefore make the unit cells smaller with respect to the operating wavelength. Simulations were performed to explore the use of lumped elements to reduce the ZIM/LIM frequency of small 2mm SRR unit cells. Lumped inductor elements were applied opposite the split. The addition of the lumped inductance decreased the resonant frequency, but also reduced the bandwidth of the ZIM/LIM band compared to the bandwidth available without the inductor. When changing the resonant frequency of the equivalent circuit by increasing the inductance, the magnetic coupling of the circuit to the incident H-field was not altered, resulting in a weaker overall coupling strength and weaker resonance, finally giving a narrow ZIM/LIM band. The decreased bandwidth and increased manufacturing cost of a metamaterial constructed using lumped elements make such a design unsuitable for this application, when other options are possible for lower cost and larger bandwidth Multi-Split-Ring Resonator The capacitance of a set of identical series-connected capacitors is smaller than the capacitance of one of the capacitors. Decreasing either the inductance or capacitance in the SRR will increase the resonant frequency. So, introducing more splits into the SRR, as illustrated in Fig. 3.9, will increase the resonant frequency of the resonators.

42 3 1 mil 12 mil.3 mm 25% 5 mm Permittivity Permeability ε zz ' ε zz '' (a) µ zz ' µ zz '' (c) ε zz ' µ zz ' ε xx ' ε xx ' ε yy ' ε xx '' ε yy '' (b) µ xx ' µ yy ' µ xx '' µ yy '' (d) µ xx ' ε yy ' µ yy ' ε zz '' µ zz '' Fig Extracted material parameters for Dual-split SRR with 25% arms (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability ε xx '' µ xx '' ε yy '' µ yy '' Dual-split SRR The dual-split SRR introduces an additional split opposite to the original gap in the SRR, reducing the capacitance in the circuit by a factor of two compared to the original SRR simulations. The same two parameters are available for tuning, the gap width and the arm length. With the additional split, the arm length is constrained to be less than 5% to prevent electrical contact between the two sub-rings Four-split SRR Four splits reduce the capacitance in the ring by a factor of four relative to the original SRR simulations, pushing the magnetic resonance much higher in frequency. Two simulations, one with 25% arms and the other with no arms, show the frequency range of possible resonances. The tangential electric and magnetic material parameters show a much higher degree of isotropy compared to earlier simulations, due to the completely symmetric unit cell, which now belongs to the D 4h symmetry group.

43 31 1 mil 12 mil.3 mm 5 mm Permittivity Permeability ε zz ' ε zz '' (a) µ zz ' µ zz '' (c) ε zz ' µ zz ' ε xx ' ε yy ' ε xx ' ε xx '' (b) ε yy '' µ yy ' µ xx ' µ xx '' µ yy '' (d) µ xx ' ε yy ' µ yy ' ε zz '' Fig Extracted material parameters for Dual-split SRR with higher resonant frequency due to arm length reduction (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability µ zz '' ε xx '' µ xx '' ε yy '' µ yy '' With long arms, the four-split SRR can be considered to be four small closelycoupled SRRs combined in the space of a single resonator Selected Design The best candidate for a 5 mm unit cell with an operational band near 8 GHz was the dual-split SRR. The final tuning must take place in combination with the electric metamaterial molecule to account for mutual coupling between the electric and magnetic components, but the two cases shown in Fig. 3.1 and Fig both demonstrate resonances near the desired band, with only simple parameter adjustments required to reach the exact desired performance.

44 32 1 mil 12 mil.3 mm 25% 5 mm Permittivity Permeability ε zz ' ε zz '' -.5 (a) 5 µ zz ' µ zz '' (c) ε zz ' µ zz ' ε xx ' 2 1 ε xx ' ε yy ' ε xx '' ε yy '' -1-2 (b) 1 µ xx ' 5 µ yy ' µ xx '', µ yy '' (d) µ xx ' ε yy ' µ yy ' ε zz '' µ zz '' Fig Extracted material parameters for 4-SRR with 25% arm length.(a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability ε xx '' µ xx '' ε yy '' µ yy '' 1 mil 12 mil.3 mm 5 mm Permittivity Permeability ε zz ' ε zz '' (a) µ zz ' µ zz '' (c) ε zz ' µ zz ' ε xx ' 5 ε xx ' ε yy ' ε xx '' ε yy '' (b) 5 µ xx ' µ yy ' µ xx '', µ yy '' (d) µ xx ' ε yy ' µ yy ' ε zz '' µ zz '' Fig Extracted material parameters for 4-SRR without arms (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability ε xx '' µ xx '' ε yy '' µ yy ''

45 Electric Particle As the most prominent of the existing electric metamaterial particles, the ELC is the first choice for the electric component of the metamaterial Electric LC Resonators Like the SRR, the ELC also has a series RLC equivalent circuit, with the same relationships between capacitance, inductance, and the resonant frequency. Thus, the same concepts of modifying the capacitance or inductance of the device to adjust the resonant frequency and thus, the ZIM/LIM band, can be applied. Fig contains the geometric parameters for the ELC structures. Both styles, (2L, 1C and 1L, 2C) exhibit similar responses, but simulation results for both are presented here for comparison. w p w p u c g u c g (a) (b) Fig (a) Single-Capacitor ELC structure and parameters. (b) Dual-Capacitor ELC structure and parameters Single-Capacitor Dual-Inductor (1C 2L) ELC Simulations of a single ELC in a cubic unit cell demonstrate the basic performance of the resonator, without the added complexity of mutual coupling from four identical copies in the same unit cell. The material parameter curves in Fig show sharp z-directed permittivity resonances at 1 GHz, with the zero-crossing and low-index band at about 1.5 GHz. The resonance and ZIM band are very narrow, with fractional percentage bandwidth where the material satisfies the low-index condition. Note the asymmetric tangential permittivity and permeability, which is justified by the lack of symmetry for the x- and y-axes.

46 34 14 mil 12 mil.3 mm 1.5 mm 5 mm Permittivity ε zz ' (a) ε zz '' ε yy ' (b) ε yy '' ε xx ' ε xx '' Permeability µ zz ' µ zz '' µ xx '', µ yy '' µ yy ' µ xx ' (c) ε zz ' µ zz ' ε xx ' (d) µ xx ' ε yy ' µ yy ' ε zz '' µ zz '' Fig Extracted material parameters for Single 1C 2L ELC (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability ε xx '' µ xx '' ε yy '' µ yy '' The actual metamaterial must be at least roughly isotropic in the x-y plane, which requires either two or four ELC structures for each unit cell. A good homogeneous response is achieved by using four ELCs, one on each interior vertical face of the cubic unit cell. To demonstrate the difference between a single ELC element and multiple elements per unit cell, Fig shows the effective parameters for metamaterials with four tiled copies of the same ELC design. The primary resonance location for the 4-ELC unit cell does not change relative to the single ELC unit cell simulation, but additional resonances are introduced by the additional coupling effects between resonators. Changing the capacitive gap size from.3 mm to.1 mm, with results in Fig. 3.17, decreases the resonant frequency and ZIM band location slightly, down to 9 and 9.5 GHz, respectively Dual-Capacitor Single-Inductor (2C 1L) ELC The same simulations are repeated for the 2C 1L ELC structures with the extracted material parameters plotted in Fig Fig. 3.2; the results are very similar to the 1C 2L ELC structures. There is a frequency shift in the resonances of these

47 35 14 mil 12 mil.3 mm 1.5 mm 5 mm Permittivity Permeability ε zz ' ε zz '' (a) µ zz ' µ zz '' (c) ε zz ' µ zz ' ε xx ' ε yy ' ε xx ' ε xx '' ε yy '' (b) µ yy ' µ xx ' µ xx '', µ yy '' (d) µ xx ' ε yy ' µ yy ' ε zz '' µ zz '' Fig Extracted material parameters for Four 1C 2L ELC (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability ε xx '' µ xx '' ε yy '' µ yy '' 14 mil 12 mil.1 mm 1.5 mm 5 mm Permittivity Permeability ε zz ' ε zz '' (a) µ zz ' µ zz '' (c) ε zz ' µ zz ' ε xx ' ε xx ' ε yy ' ε xx '' ε yy '' (b) µ xx ' µ yy ' µ xx '', µ yy '' (d) µ xx ' ε yy ' µ yy ' ε zz '' µ zz '' Fig Extracted material parameters for Four 1C 2L ELC with smaller capacitive gap (g =.1) for reduced ZIM frequency (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability ε xx '' µ xx '' ε yy '' µ yy ''

48 36 14 mil 12 mil.3 mm.75 mm 5 mm Permittivity ε zz ' ε zz '' (a) ε xx '' ε yy '' (b) µ yy ' µ xx ' ε yy ' ε xx ' Permeability µ zz ' µ zz '' µ xx '', µ yy '' (c) ε zz ' µ zz ' ε xx ' (d) µ xx ' ε yy ' µ yy ' ε zz '' µ zz '' Fig Extracted material parameters for Single 2C 1L ELC (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability ε xx '' µ xx '' ε yy '' µ yy '' simulations due to the slightly longer capacitive arms of the modified geometry. Other differences include a frequency increase of the secondary normal electric resonance, and tangential magnetic resonances occurring at about 7-8 GHz that creates the tangential electric antiresonance at the same frequency ELC Summary The differences in the response of the 1C, 2L and 2C, 1L ELC resonators are minor, and do not greatly affect the primary ZIM band. However, both ELCbased metamaterials have very narrow LIM/ZIM bands associated with narrow, sharp resonances at relatively high frequencies, relative to the particle size. Similar to the SRR, lumped inductors or capacitors or additional loops with vias could be added to decrease the LIM/ZIM to a more reasonable frequency, but both of these efforts decrease the LIM/ZIM bandwidth of the ELC even further. For an already low-bandwidth circuit, reducing the bandwidth further is not acceptable. The ELC metamaterial particle is not capable of satisfying the performance requirements for the LIM/ZIM metamaterial lens. A different electric particle

49 37 14 mil 12 mil.3 mm.75 mm 5 mm Permittivity Permeability ε zz ' ε zz '' (a) µ zz ' µ zz '' (c) ε zz ' µ zz ' ε xx ' ε xx ' ε yy ' ε xx '' ε yy '' (b) µ xx ' µ yy ' µ xx '', µ yy '' (d) µ xx ' ε yy ' µ yy ' ε zz '' µ zz '' Fig Extracted material parameters for Four 2C 1L ELC (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability ε xx '' µ xx '' ε yy '' µ yy '' 14 mil 12 mil.1 mm.75 mm 5 mm Permittivity Permeability ε zz ' ε zz '' (a) µ zz ' µ zz '' (c) ε zz ' µ zz ' ε xx ' ε xx ' ε xx '' µ xx '', µ yy '' ε yy ' ε yy '' (b) µ xx ' µ yy ' (d) µ xx ' ε yy ' µ yy ' ε zz '' µ zz '' Fig Extracted material parameters for Four 1C 2L ELC, g =.1 (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability ε xx '' µ xx '' ε yy '' µ yy ''

50 38 must be designed to meet the performance criteria Complementary Split-Ring Resonators The original proposed CSRR, as well as all of the derived examples that have been located in the literature, use Pendry s original SRR incarnation [9] as the CSRR complimentary structure. This SRR consists of dual nested circular annular rings, with oppositely oriented gaps. This design possesses a significant degree of bianisotropy and cross-polarized response due to the lack of symmetry in the design, a feature that is maintained in the derived CSRR particle. An interesting characteristic of the CSRR is that cross-polarization appears to be necessary for operation at normal incidence. Falcone s initial analysis ignores the bianisotropy and cross-polarization and concludes that the CSRR would not demonstrate the interesting electric resonance effects when excited at normal incidence [21]. Instead, a Drude permittivity response is exhibited; this is reasonable and expected due to the presence of a perforated but connected infinite ground plane. The bianisotropic and cross-polarizing behavior is necessary for a transmission passband to exist at normal incidence. Without the transmission passband, the sheet behaves as a reflective ground plane with Drude-type dispersion as the contribution from the apertures. Simulations of a CSRR based on the well-behaved dual-gap SRR structures designed for the ZIM lens do not demonstrate a strong electrical resonance, instead showing the expected Drude behavior for ɛ xx and ɛ yy. Fig shows the CSRR geometry, and Fig the extracted material parameters. There are weak resonance effects visible for normal electric field excitation of the structure, but these effects are small and dwarfed by the Drude response for the tangential axes. There are very weak transmission peaks visible in the scattering parameters, but most of the normally incident energy is reflected from the surface as demonstrated in Fig Most publications that reference the CSRR element use the CSRR as an artificial dielectric for the purpose of negative-index microstrip lines or other planar microwave circuits [26], and no papers that use the CSRR as a general metamaterial element for 3D operation have been located. For left-handed transmission

51 39 (a) (b) Fig (a) Modified dual-gap symmetric SRR (b) Modified Complementary SRR (CSRR) lines and microstrips, the polarization performance of the metamaterial elements are less important, and don t affect the performance in the same way as a transmission lens is affected. Planar metamaterials and frequency-selective surfaces may be less affected by the bianisotropy as well, since the incident polarizations of interest are limited Falcone CSRR In order to make a valid comparison between the two CSRR designs, the CSRR particle in Fig. 1.4 was simulated in HFSS, with extracted material parameters in Fig As was claimed in the introductory paper, transmission passbands do emerge at the resonance for the normal incidence simulation as seen in Fig. 3.25, but only for a single polarization. Comparing the location of the transmission band to the inverted material parameters, the passband corresponds to the frequency band where the permittivity rises above zero,.5 ɛ xx 2. This CSRR is claimed to respond to the Z-oriented electric field, but there are no permittivity responses or resonances visible in ɛ zz over the frequency band of interest for the Falcone CSRR. There are narrow transmission passbands and resonances visible for ɛ xx, but at much higher frequencies than desired for lens operation. The Falcone CSRR does not produce a useful response for the creation of a wideband electric ZIM metamaterial.

52 4 16 mil 12 mil 45%.3 mm 5 mm Permeability Permittivity 2 ε zz ' 1 ε zz '' -1-2 (a) 2 1 µ zz ' µ zz '' -1 (c) ε zz ' µ zz ' ε xx ' 5 5 ε xx '' µ yy '' (d) µ xx ' ε yy ' µ xx ' ε yy '' (b) µ yy ' µ yy ' ε xx ' ε yy ' µ xx '' ε zz '' Fig Extracted material parameters for CSRR (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability µ zz '' ε xx '' µ xx '' ε yy '' µ yy '' CSRR Summary Given the poor performance, the Drude-type dispersion, and the required bianisotropy in order to achieve a normal-incidence passband, the CSRR particle is unlikely to be useful for a metamaterial lens. Many papers have demonstrated the usefulness of the CSRR in microwave circuits and microstrip devices, but these applications have different constraints and requirements for the CSRR operation. The dominant Drude dispersion from the perforated sheet is less useful than the Lorentzian response of the SRR or ELC particles. The Lorentzian resonances in the CSRR are almost negligible when compared to the Drude behavior.

53 41 Scattering Power Magnitude [db] TE Polarization (a) TM Polarization Reflection Transmission (b) Absorption Fig (a) TE-mode and (b) TM-mode scattering parameters at normal incidence from a single layer of Modified CSRR resonators. 16 mil 12 mil.2 mm.3 mm 5 mm Permeability Permittivity 2 ε zz ' 1 ε zz '' -1-2 (a) 2 1 µ zz ' µ zz '' -1 (c) ε zz ' µ zz ' ε xx ' 5 ε xx ' ε yy '' ε xx '' ε yy ' (b) 1 5 µ yy ' µ xx ' µ xx '', µ yy '' (d) µ xx ' ε yy ' µ yy ' ε zz '' Fig Extracted material parameters for Falcone CSRR (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability µ zz '' ε xx '' µ xx '' ε yy '' µ yy ''

54 42 Scattering Power Magnitude [db] TE Polarization (a) TM Polarization Reflection Transmission (b) Absorption Fig (a) TE-mode and (b) TM-mode scattering parameters at normal incidence from a single layer of Falcone CSRR resonators.

55 Volumetric End-loaded Dipoles The length of dipole and monopole antennas can be decreased or the resonant frequency tuned through the introduction of a loading impedance at the wire termination at the cost of band-limited performance. These end-loaded dipoles can be created using discrete components, or by modifying the traces of the antenna to produce additional inductance and capacitance [27][28]. A similar approach may be used when attempting to minimize the size of a wire grid medium. A wire grid or wire mesh medium acts as an anisotropic plasma medium whose plasma frequency may be tuned by adjusting the wire spacing and gaps to produce a narrow zero-index band at a desired location. The geometric dimensions of the wire mesh may be reduced while continuing to operate at the same frequency by loading the wire grid with either lumped capacitors and inductors or extending the ends of the traces into the same zig-zag pattern used for conventional end-loaded dipoles. Adding additional end-loading arms to a dipole decreases the frequency at which the device resonates and decreases the size of the unit cell relative to the wavelength. An illustration of the end-loaded dipole (ELD) element is included in Fig (a) (b) Fig (a) Unit cell of a simple wire-mesh plasma medium (b) Unit cell of a 2D ELD metamaterial For use in a 3D metamaterial with controlled electric anisotropy, instead of a 2D FSS structure, multiple ELDs may be combined to form a Volumetric End Loaded Dipole (VELD). The orientation of the dipoles may be changed to meet the anisotropy requirements of the application. Fig. 3.27(a) shows four identical ELDs arranged in the same orientation, to produce a uniaxial Z-oriented permittivity response. Isotropic arrangements using six identical ELDs, illustrated in

56 44 Fig. 3.27(b), are also possible. An arbitrary effective medium could be formed by independently tuning each axis of the VELD cube; inhomogeneous media are possible by making element tuning parameters spatially dependent. (a) (b) Fig (a) Unit cell of a uniaxial volumetric end-loaded dipole (VELD) metamaterial (b) Unit cell of an isotropic VELD metamaterial. From observation, the ELD particle possesses chiral characteristics with 2-fold rotational symmetry, a member of the C 2h group. A chiral response is undesirable for the current application, because the desired metamaterial response should be independent of polarization. Appropriate placement and orientation of the tiled elements, such as those orientations in Fig. 3.27, will correct any chiral responses[16]. The primary reason for using a VELD unit cell, instead of an ELC-based structure, is the capability for smaller unit cells. With self-resonant unit cells, finitesized metamaterial blocks will possess the same or very similar effective material parameters as those extracted from an infinite array of unit cells. The wire grid medium, for example, must be extended over many periods to avoid dramatic degradation in effective performance at the edges of the medium. As described in section 3.2.1, the ELC element must be large compared to the wavelength in order to achieve a low frequency resonance; an SRR design with a ZIM band at f may be as small as λ/1, but an ELC element must be larger than λ/5 to λ/3 in order to demonstrate even a very narrow ZIM/LIM region. The electric resonances from the ELC are quite weak in amplitude, compared to an SRR resonance; this can be explained in terms of the active field coupling regions in each device. In an SRR, the entire interior of the loop will couple with the magnetic field. Only the small capacitive gaps in an ELC will couple with the

57 45 electric field, however. For an ELD, all of the end-loading arms contribute to stronger coupling between the electric field and the device; the stronger coupling can result in a stronger resonance and wider ZIM/LIM band. Fig illustrates the difference in coupling area between the SRR and the two electric metamaterial particles. E H (a) (b) (c) Fig (a) Large coupling magnetic field coupling region (red) for a dual-srr particle. (b) Small electric field coupling region (blue) of an ELC device. (c) Larger electric field coupling regions (blue) of an ELD structure VELD simulations Simulations of the VELD unit cell successfully demonstrated the usefulness of the new design as a metamaterial particle for NIM/ZIM/LIM applications. Response tuning is performed by adjusting the number of end-loading arms, the padding around the edge of the unit cell, and the trace width. Thinner traces and more arms result in a longer electrical path, decreasing the resonant frequency. Increasing the perimeter padding decreases the area available for the sinuous end-loading arms, decreasing the trace length and increasing the ZIM frequency. A smaller perimeter gap will result in increased coupling between adjacent unit cells and more complications during fabrication; trade-offs between inter-unit cell spacing and unit cell size must be made to optimize the inter-element coupling and still achieve resonance and size requirements Selected Design With the unsatisfactory CSRR removed from consideration, the VELD shows a much wider ZIM/LIM bandwidth than the ELC devices, and is the best choice for

58 46 14 mil 6 mil 6 mil 5 mm Permittivity Permeability ε zz ' ε zz '' (a) µ zz ' µ zz '' (c) ε zz ' µ zz ' ε xx ' 1 5 ε yy ' ε xx ' ε xx '' ε yy '' -1 (b) 5 µ xx ' µ yy ' µ xx '', µ yy '' (d) µ xx ' ε yy ' µ yy ' ε zz '' Fig Extracted material parameters for 3-arm VELD (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability µ zz '' ε xx '' µ xx '' ε yy '' µ yy '' 14 mil 6 mil 6 mil 5 mm Permittivity Permeability ε zz ' ε zz '' (a) µ zz ' µ zz '' (c) ε zz ' µ zz ' ε xx ' 1 ε xx ' ε yy ' 5 ε xx '' ε yy '' -1 (b) 5 µ xx ' µ yy ' µ xx '', µ yy '' (d) µ xx ' ε yy ' µ yy ' ε zz '' Fig Extracted material parameters for 6-arm VELD (a) Normal [z] permittivity (b) Tangential [x,y] permittivity (c) Normal permeability (d) Tangential permeability µ zz '' ε xx '' µ xx '' ε yy '' µ yy ''