DIELECTRIC RESONATOR FABRICATION AND ASSEMBLY METHODS FOR TERAHERTZ METAMATERIALS BY

JAMES MARK LELAND CRAMER B.S. UNIVERSITY OF CALIFORNIA, RIVERSIDE (2003) M.S. UNIVERSITY OF MASSACHUSETTS LOWELL (2006)

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS UNIVERSITY OF MASSACHUSETTS LOWELL

Signature of Author

Signature of Dissertation Chair: Dr. Andrew J. Gatesman Adjunct Professor, Department of Physics

Signatures of Dissertation Committee Members:

Dr. Robert H. Giles

Dr. Viktor Podolskiy

Dr. Richard Stimets

DIELECTRIC RESONATOR FABRICATION AND ASSEMBLY METHODS FOR TERAHERTZ METAMATERIALS BY

JAMES MARK LELAND CRAMER

ABSTRACT OF A DISSERTATION SUBMITTED TO THE FACULTY OF DEPARTMENT OF PHYSICS IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY PHYSICS UNIVERSITY OF MASSACHUSETTS LOWELL 2011

Dissertation Supervisor: Dr. Andrew J. Gatesman Adjunct Professor, Department of Physics

Abstract The goal of this research was the design and manufacture of a negative index of refraction metamaterial having an isotropic negative permeability for terahertz frequencies, specifically the far-infrared region. In this research, a lattice of high-permittivity resonators to provide a negative permeability and a grid of metallic wires that provided negative permittivity were to be combined to create a negative index of refraction material for terahertz frequencies. In the process, several metamaterials were developed, such as a microwave frequency negative refractive index metamaterial based on yttria stabilized zirconia spheres, and terahertz metamaterials based on an array of custom synthesized micro-spheres as well as an array of lithium tantalate micro-rods. An attempt was made to fabricate a negative index of refraction metamaterial from a cubic lattice of lithium tantalate micro-cubes combined with a micro-wire grid. The lattice of lithium tantalate micro-cubes was predicted to have a negative permeability at the micro-cube’s first Mie resonance at approximately 0.390 THz. A grid of copper micro-wires provided a negative permittivity at frequencies below the wire grid’s plasma frequency at approximately 0.626 THz. Theoretically, it is shown that, when the micro-cube array and wire grid are combined, a metamaterial with an isotropic negative permeability and a uniaxial negative permittivity results. All materials fabricated were analyzed with effective medium theory and numerical simulations that were compared with the measured transmittance. Additional analysis was performed on constituent materials to help determine both the success of fabrication and to further explain the electromagnetic behavior of the materials.

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Acknowledgments I would like to acknowledge all the fabulous facilities and funding that the Submillimeter Wave Technology Lab has offered me all these years courtesy of Dr. Robert Giles and Dr. Andrew Gatesman, the use of the clean room at Dr. Goodhue’s center for Photonics And Optical Devices, as well as the support and suggestions of all the staff an employees thereof. Also, to thank my friends including but not limited to: Brooke T, Julia M, Elizabeth S, Shawn M, and Karyl T.; as well as others that I have met along the way, such as: Peach, Claudia, Bianca, and Mellisa. Also, my parents Mark and Lynn and my sister Marcella for all the support, help, and suggestions.

Time is the school in which we learn, Time is the fire in which we burn. – Delmore Schwartz

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TABLE OF CONTENTS LIST OF TABLES

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LIST OF FIGURES

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I INTRODUCTION The purpose of a negative refractive index material . . . . . . . . . . . . . . . . 1.1 Terahertz Frequency Resonator Based Metamaterials . . . . . . . . . . . . 1.1.1 Split-Ring Resonator Metamaterials . . . . . . . . . . . . . . . 1.1.2 Coupled Wire Structure Metamaterials . . . . . . . . . . . . . . 1.1.3 Dielectric Material Resonator Metamaterials . . . . . . . . . . . 1.2 Fab. Methods For Terahertz Frequency Dielectric Resonator Metamaterials 1.2.1 Microwave Frequency Sphere Resonator Metamaterial . . . . . 1.2.2 Micro-Sphere THz Resonator Metamaterial . . . . . . . . . . . 1.2.3 Micro-Rod THz Resonator Metamaterial . . . . . . . . . . . . . 1.2.4 Micro-Cube THz Resonator Metamaterial . . . . . . . . . . . . 1.2.5 Combined Micro-Cube THz Resonators And Wire Grid Metamaterial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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II THEORY 2.1 The Permittivity And Permeability Of Materials . . . . . . . . . . . . . . 2.1.1 Simple Polynomial Model . . . . . . . . . . . . . . . . . . . 2.1.2 Single-Resonance Harmonic Oscillator: Frequency Form . . . 2.1.3 Multiple-Resonance Harmonic Oscillator: Wavenumber Form 2.1.4 Four Parameter Semi-Quantum Multiple-Resonance Model . . 2.1.5 Ferroelectric Materials . . . . . . . . . . . . . . . . . . . . . 2.1.6 Negative Refractive Index From Negative Permittivity And Negative Permeability . . . . . . . . . . . . . . . . . . . . . . 2.2 Effective Medium Theories . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 A Cubic Lattice Of Spheres . . . . . . . . . . . . . . . . . . . 2.2.2 A Cubic Lattice Of Cubes . . . . . . . . . . . . . . . . . . . . 2.2.3 An Array Of Wires . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Combined Effective Medium Theories . . . . . . . . . . . . . 2.3 Reflection, Transmission, And Refraction Equations . . . . . . . . . . . . 2.3.1 Coordinate System And Wave Propagation Convention . . . . 2.3.2 Isotropic Materials . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Uniaxial Materials . . . . . . . . . . . . . . . . . . . . . . . .

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~k Vectors And Dispersion Relations . . . . . . . . . . . . . . . 32 Poynting Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 34 Reflection And Transmission Coefficients . . . . . . . . . . . . 36

III METHODOLOGY 3.1 Material Modeling And Simulation . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Analytic Modeling Of Material Behavior . . . . . . . . . . . . . 3.1.2 Numerical Simulation Of Material Behavior . . . . . . . . . . . 3.2 Powder Diffraction Measurement Of Crystal Structure . . . . . . . . . . . 3.3 Measurements At Microwave Frequencies . . . . . . . . . . . . . . . . . . 3.4 Measurements At Terahertz Frequencies . . . . . . . . . . . . . . . . . . . 3.5 Determination Of Permittivity And Permeability . . . . . . . . . . . . . . . 3.6 Construction Of Microwave Frequency Metamaterials . . . . . . . . . . . . 3.6.1 Embedded Sphere Samples For Waveguide . . . . . . . . . . . . 3.6.2 Embedded Wire Grid Samples For Waveguide . . . . . . . . . . 3.6.3 Sphere Lattice Sample For Free-Space . . . . . . . . . . . . . . 3.6.4 Combined Sphere Array And Wire Grid Sample For Free-Space 3.7 Construction Of Micro-Sphere Metamaterials For Terahertz Frequencies . . 3.7.1 Yttria-Stabilized Zirconia Micro-Spheres . . . . . . . . . . . . . 3.7.2 Synthesis Of Titanium Dioxide Micro-Spheres . . . . . . . . . . 3.7.3 Photolithographic Processing Of SU-8 . . . . . . . . . . . . . . 3.7.4 Titanium Dioxide 10-Micron-Dia. Micro-Sphere Arrangement . 3.7.5 Titanium Dioxide 36-Micron-Dia. Micro-Sphere Arrangement . 3.7.6 Micro-Sphere Patterning And Fabrication . . . . . . . . . . . . 3.8 Construction Of Lithium Tantalate Micro-Rods . . . . . . . . . . . . . . . 3.9 Construction Of Lithium Tantalate Micro-Cubes . . . . . . . . . . . . . . . 3.10 Construction Of The Micro-Cube Groups And Combined Wire Grid Samples

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IV RESULTS 4.1 Titanium Dioxide Micro-Sphere Synthesis . . . . . . . . . . . . . . . . . 4.1.1 10-Micron-Diameter Titanium Dioxide Micro-Spheres . . . . 4.1.2 36-Micron-Diameter Titanium Dioxide Micro-Spheres . . . . 4.2 Measurement Of The Microwave Samples . . . . . . . . . . . . . . . . . 4.2.1 Embedded Sphere Waveguide Samples . . . . . . . . . . . . . 4.2.2 Embedded Wire Array Waveguide Samples . . . . . . . . . . 4.2.3 Yttria Stabilized Zirconia Sphere Sample For Free-Space . . . 4.2.4 Combined Yttria Stabilized Zirconia Sphere And Wire Array Sample For Free-Space . . . . . . . . . . . . . . . . . . . . . 4.3 Measurement Of Material Samples At Terahertz Frequencies . . . . . . .

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4.3.1 Transfer Adhesive . . . . . . . . . . . . . . . . 4.3.2 PFA . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Mylar . . . . . . . . . . . . . . . . . . . . . . 4.3.4 SU-8 . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Lithium Tantalate . . . . . . . . . . . . . . . . Measurement Of The Micro-Sphere Samples . . . . . . . 4.4.1 Yttria Stabilized Zirconia Micro-Spheres . . . . 4.4.2 Titanium Dioxide Micro-Spheres . . . . . . . . Measurement Of The Lithium Tantalate Micro-Rods . . . Simulation Of The Lithium Tantalate Micro-Cubes . . . . Measurement Of The Wire Grid Samples . . . . . . . . . Simulation Of The Combined Micro-Cube And Wire Grid

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V DISCUSSION 108 5.1 Permittivity And Permeability Of The Microwave Samples . . . . . . . . . 108 5.1.1 Embedded Spheres For Waveguide . . . . . . . . . . . . . . . . 108 5.1.2 Embedded Wire Array For Waveguide . . . . . . . . . . . . . . 109 5.1.3 Yttria-Stabilized Zirconia Sphere Grid For Free-Space . . . . . . 109 5.1.4 Yttria Stabilized Zirconia Sphere And Wire Array For Free-Space109 5.2 Permittivity And Permeability Of The Micro-Sphere Samples . . . . . . . . 110 5.2.1 Yttria Stabilized Zirconia Micro-Spheres . . . . . . . . . . . . . 111 5.2.2 Titanium Dioxide Micro-Spheres . . . . . . . . . . . . . . . . . 112 5.3 Permittivity And Permeability Of The Micro-Rod Samples . . . . . . . . . 114 5.4 Predictions and Fabrication Of The Micro-Cube Samples . . . . . . . . . . 115 5.5 Permittivity Of The Wire Grid Samples . . . . . . . . . . . . . . . . . . . 116 5.6 Permittivity And Permeability Of The Micro-Cube And Wire Grid Samples 116

VI CONCLUSION 6.1 Microwave Frequency Macro-Sphere Material . . . . . . . . 6.2 THz Micro-Sphere Materials . . . . . . . . . . . . . . . . . 6.2.1 Yttria-Stabilized Zirconia Micro-Sphere Material 6.2.2 Titanium Dioxide Micro-Sphere Material . . . . . 6.3 Micro-Rod Material . . . . . . . . . . . . . . . . . . . . . . 6.4 THz Micro-Cubes and Wire Grid Negative Index Material .

VII RECOMMENDATIONS

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VIII REFERENCES

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IX APPENDIX 9.1 Sources Of Materials Used . . . . . . . . . . . . . . . 9.2 Photolithographic Processing Details . . . . . . . . . . 9.2.1 Sacrificial Layer Of Polystyrene . . . . . . 9.2.2 SU-8 Negative Resist Processing Details . . 9.3 Procedure For Synthesis Of TiO2 Micro-Spheres . . . 9.4 HFSS Electromagnetic Simulation Details . . . . . . . 9.4.1 Waveguide Simulation Setup . . . . . . . . 9.4.2 Free-Space Floquet Port Simulation Setup . 9.4.3 Free-Space Incident Wave Simulation Setup 9.5 Abbreviated Biography Of The Author . . . . . . . . .

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LIST OF TABLES 1-1 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 4-9 4-10 4-11 4-12 4-13 4-14 9-1 9-2

Metamaterials built for this research. . . . . . . . . . . . . . . . . . . . . Parameters for transfer adhesive in equation 2-1. . . . . . . . . . . . . . . Parameters for PFA in Equation 2-1. . . . . . . . . . . . . . . . . . . . . Parameters for Mylar in Equation 2-1. . . . . . . . . . . . . . . . . . . . Parameters for SU-8 in Equation 2-2. . . . . . . . . . . . . . . . . . . . . Parameters for lithium tantalate in Equation 2-3. . . . . . . . . . . . . . . Parameters for polycrystalline titanium dioxide used in Equation 2-4. . . . Parameters for effective medium model of the air and crystaline compound composing the titanium dioxide micro-spheres. . . . . . . . . . . . . . . Parameters for effective medium model of the 10-micron-diameter embedded titanium dioxide micro-sphere metamaterial. . . . . . . . . . . Parameters for effective medium model of the air and crystaline compound composing the titanium dioxide micro-spheres. . . . . . . . . . . . . . . Parameters for effective medium model of the 36-micron-diameter patterned titanium dioxide micro-sphere metamaterial. . . . . . . . . . . . . . . . . Parameters for effective medium model of lithium tantalate micro-rods. . Parameters for effective medium model of lithium tantalate micro-cubes. . Parameters for wire grid in equation 2-11. . . . . . . . . . . . . . . . . . Parameters for effective medium model of the combined lithium tantalate micro-cubes and wire grid. . . . . . . . . . . . . . . . . . . . . . . . . . Sources of materials used. . . . . . . . . . . . . . . . . . . . . . . . . . Materials for sacrificial template synthesis of micro-spheres. . . . . . . .

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LIST OF FIGURES 1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8 1-9 2-1 2-2 2-3 2-4 2-5 2-6 3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 3-10 3-11 3-12 3-13 3-14 3-15 3-16 3-17 3-18 3-19 4-1

Diagram of a spit-ring resonator. . . . . . . . . . . . . . . . . . . . . . . . Diagram of a coupled wire resonator. . . . . . . . . . . . . . . . . . . . . . Diagram of a dielectric resonator. . . . . . . . . . . . . . . . . . . . . . . . Conceptual diagram of availability by size of high permittivity particles. . . Diagram of microwave sphere resonators. . . . . . . . . . . . . . . . . . . Diagram of embedded sphere resonators. . . . . . . . . . . . . . . . . . . . Diagram of micro-rod resonators. . . . . . . . . . . . . . . . . . . . . . . . Diagram of a micro-cube resonator. . . . . . . . . . . . . . . . . . . . . . Diagram of micro-cubes and wire grid. . . . . . . . . . . . . . . . . . . . . Conceptual diagram of ion displacement in a lattice. . . . . . . . . . . . . . Coordinate system used in this research. . . . . . . . . . . . . . . . . . . . Behavior of positive-refractive-index and negative-refractive-index materials. ~k vector diagrams for the TE and TM modes. . . . . . . . . . . . . . . . . S~ vector diagrams for the TE and TM modes. . . . . . . . . . . . . . . . . Reflectance and transmittance diagram of wave through a plane parallel NIM. X-band waveguide measurement equipment. . . . . . . . . . . . . . . . . . The 27.8 - 40 GHz frequency range free-space measurement equipment. . . Sample holders used for FTIR measurements. . . . . . . . . . . . . . . . . Waveguide metamaterials of yttria-stabilized zirconia spheres in Task 4 resin. Waveguide metamaterial of 24-gage wire array embedded in polycarbonate. Yttria-stabilized zirconia sphere array free-space metamaterial. . . . . . . . Yttria-stabilized zirconia sphere and wire array free-space metamaterial. . . yttria-stabilized zirconia micro-spheres patterned using a number 70 sieve. . Synthesis procedure for titanium dioxide micro-spheres. . . . . . . . . . . Side view of the patterned 10-micron-diameter titanium dioxide microspheres. Top view of the patterned 10-micron-diameter titanium dioxide microspheres. Top view of the patterned 36-micron-diameter titanium dioxide microspheres. A lithium tantalate micro-rod sample on mylar carrier. . . . . . . . . . . . Construction steps of lithium tantalate micro-rod samples. . . . . . . . . . Construction steps of lithium tantalate micro-cube samples. . . . . . . . . . Top view of partially etched lithium tantalate micro-cubes. . . . . . . . . . View of partial etched lithium tantalate micro-cubes. . . . . . . . . . . . . Photograph of the copper wire grid. . . . . . . . . . . . . . . . . . . . . . Single layer assembly diagram of lithium tantalate micro-cubes and wire grid. 30-micron-diameter titanium dioxide and polymer micro-spheres before calcining. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2 10-micron-diameter titanium dioxide micro-spheres after calcining. . . . .

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4-3 Scanning electron microscope image of a 10 micron titanium dioxide microsphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4 X-ray powder diffractogram of 10-micron-diameter titanium dioxide microspheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5 120-micron-diameter titanium dioxide and polymer micro-spheres before calcining. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6 36-micron-diameter titanium dioxide micro-spheres after calcining. . . . . 4-7 Scanning electron microscope image of a 36 micron titanium dioxide microsphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8 X-ray powder diffractogram of 36-micron-diameter titanium dioxide microspheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9 Transmittance of embedded sphere waveguide samples. . . . . . . . . . . . 4-10 Reflectance of embedded sphere waveguide samples. . . . . . . . . . . . . 4-11 Permittivity and permeability of embedded sphere waveguide samples. . . . 4-12 Transmittance and reflectance of the embedded wire array sample. . . . . . 4-13 Permittivity and permeability of the embedded wire array sample. . . . . . 4-14 Transmittance of the yttria-stabilized zirconia sphere array. . . . . . . . . . 4-15 Reflectance of the yttria-stabilized zirconia sphere array. . . . . . . . . . . 4-16 Permittivity and permeability of the yttria-stabilized zirconia sphere array. . 4-17 Transmittance of the yttria-stabilized zirconia sphere array and wire array with the electric field polarized parallel with the with the wires. . . . . . . . 4-18 Reflectance of the yttria-stabilized zirconia sphere array and wire array with the electric field polarized parallel with the wires. . . . . . . . . . . . . . . 4-19 Permittivity and permeability of the yttria-stabilized zirconia sphere array and wire array with the electric field polarized parallel with the wires. . . . 4-20 Permittivity of Mylar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-21 Permittivity of SU-8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-22 Permittivity of lithium tantalate along the ordinary ( k ) axis. . . . . . . . . 4-23 Transmittance of the yttria-stabilized zirconia micro-sphere metamaterial. . 4-24 Permittivity and permeability of the yttria-stabilized zirconia micro-sphere metamaterial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-25 Permittivity of polycrystalline rutile titanium dioxide without any air cavities. 4-26 Permittivity of the bulk titanium dioxide used to fabricate the 10 micron micro-spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-27 Transmittance of the 10 micron titanium dioxide micro-sphere metamaterial. 4-28 Permittivity and permeability of 10 micron titanium dioxide micro-sphere metamaterial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-29 Permittivity of the bulk titanium dioxide used to fabricated the 10 micron micro-spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-30 Transmittance of the 36 micron titanium dioxide micro-sphere metamaterial. x

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4-31 Permittivity and permeability of 36 micron titanium dioxide micro-sphere metamaterial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-32 Permittivity and permeability of lithium tantalate micro-rods for the electric field polarized parallel to the rods. . . . . . . . . . . . . . . . . . . . . . . 4-33 Transmittance of lithium tantalate micro-rods for the electric field polarized perpendicular to the rods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-34 Transmittance of lithium tantalate micro-rods for the electric field polarized parallel to the rods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-35 Transmittance of the lithium tantalate micro-cubes. . . . . . . . . . . . . . 4-36 Permittivity and permeability of lithium tantalate micro-cubes. . . . . . . . 4-37 Transmittance of copper wire grid. . . . . . . . . . . . . . . . . . . . . . . 4-38 Permittivity of copper wire grid. . . . . . . . . . . . . . . . . . . . . . . . 4-39 Transmittance of lithium tantalate micro-cubes and wire grids. . . . . . . . 4-40 Permittivity and permeability of the lithium tantalate micro-cubes and wire grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1 Diagram of the electromagnetic fields of the micro-rod sample at resonance. The solid black lines indicate the outline of the rod. . . . . . . . . . . . . . 5-2 Diagram of the electromagnetic fields at resonance of the micro-cube sample. The solid black lines indicate the outline of the rod. . . . . . . . . . . . . . 6-1 Conceptual diagram of availability by size of high permittivity particles including materials used, fabricated, and attempted in this work. . . . . . . 9-1 SU-8 2002 thickness versus spin speed. . . . . . . . . . . . . . . . . . . . 9-2 SU-8 2005 thickness versus spin speed. . . . . . . . . . . . . . . . . . . . 9-3 SU-8 2010 thickness versus spin speed. . . . . . . . . . . . . . . . . . . . 9-4 SU-8 2025 thickness versus spin speed. . . . . . . . . . . . . . . . . . . . 9-5 Diagram of HFSS waveguide simulation setup. . . . . . . . . . . . . . . . 9-6 Diagram of HFSS free-space Floquet port simulation setup. . . . . . . . . . 9-7 HFSS spherical coordinate system with electric field polarization. . . . . .

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1

I.

INTRODUCTION

As the ability to tailor the dielectric and magnetic properties of materials has improved, so have devices that control electromagnetic waves. For hundreds of years, various materials have been mixed to create materials with new properties. The traditional combining of materials relied on inherent properties of the inclusion’s material; currently, new materials that have effects related to a sub-wavelength-sized inclusion’s shape are being explored with great interest. These engineered structures are called metamaterials. This recent interest is due partly because an array of the inclusions can be designed such that the inclusion’s combined response has a negative effective permittivity or permeability. At frequencies where both the permittivity and permeability are negative, the refractive index is said to be negative; these materials are referred to as negative index materials (NIMs). The refractive index can have negative values that allow NIMs to refract electromagnetic waves differently than standard materials [1–3]. Snell’s law, written to account for different signs of the refractive index, sign(n1 )|n1 | sin θ1 = sign(n2 )|n2 | sin θ2 , shows that at the boundary of materials having opposite signs, the transmitted wave is refracted to the same side of the normal as the incident wave. Metamaterials and NIMs have the potential to enhance the behavior of standard optical components. These new optics could provide a larger choice of lens designs [4–8], better focusing capability [9,10], targeted absorption peaks [11], narrow band-pass filters, field redirection and warping [12–16], and near-field lenses [9].

2 Resonator based metamaterials, which are required for NIMs, are now being developed for terahertz frequencies1, as they have been for microwave and near-infrared frequencies, since terahertz technologies are rapidly being developed for medical imaging, concealed object detection, and material spectroscopy. THz resonator structures are typically based on either split-rings or coupled wires, but design and fabrication limitations generally produce anisotropic metamaterials that operate for only one polarization with limited working angles. Truly isotropic designs would allow for polarization and angular independent optics which include lenses and materials with field warping properties such as cloaking devices. Designs based on resonant cubes or spheres are expected to be isotropic. It is unlikely that a single lattice of dielectric resonators, at any frequency, could provide simultaneous negative permittivity and permeability using known materials. Therefore, a background material with negative permittivity [17,18] such as an array of wires [19–26], a second resonant inclusion [27–33], or multiple layered spheres [34] in addition to the resonators must be used to construct a dielectric NIM. The motivation for this work was to develop the dielectric resonators necessary to have an isotropic negative permeability at resonance in the THz frequency range. These resonators were to be combined with a wire grid to form a NIM with an isotropic permeability and uniaxial permittivity. Although the goal of fabricating a THz NIM was not achieved, a number of necessary techniques for the fabrication and analysis were successfully developed resulting in several unique metamaterials. The novel microwave and terahertz metamaterials constructed in this work are described in Section 1.2. 1

The terahertz freqencies are those of the far-infrared that range approximately from 300 GHz to 6 THz.

3

1.1

Terahertz Frequency Resonator Based Metamaterials

The three main types of resonators for NIMs are split-ring resonators, coupled wire sets, and dielectric resonators, described later. Other than the metamaterials described below, metamaterials described or fabricated in this work are based on sub-wavelength resonators. Three other non-resonator NIM designs that will be described briefly are magnetic material in resonance, anisotropic materials, and photonic crystals. The first type, magnetic material NIMs, use naturally magnetic materials that are driven into resonance by an external magnetic field [35–38]. However, natural magnetism cannot be used at terahertz frequencies due to the decline in the effect of permeability from atomic magnetism on an electromagnetic wave. The second type, anisotropic NIMs, use the anisotropy of the material to produce a negative refraction without simultaneously negative permittivity and permeability. The requirement for negative refraction is that either the permittivity or the permeability be negative in the direction perpendicular to the material interface [39–41]. There are also materials with both positive permittivity and permeability that, within a narrow range of incident angles, refract waves negatively [42]. The third type, photonic crystals, exhibit interesting effects such as band-gaps and negative refraction without a negative refractive index [43], but are not homogeneous materials at their operational wavelength so their permittivity and permeability are undefined.

4 1.1.1

Split-Ring Resonator Metamaterials

Split-ring resonator NIMs use an open loop of conductive material for the resonator [21,23,44–47]. Figure 1-1 is a diagram of a split-ring resonator. Split-ring resonators produce a negative permeability just above their resonant frequency. In addition, some of these structures are bianisotropic exhibiting coupled electric and magnetic

Figure 1-1. Diagram of a spit-ring resonator.

effects [48]. A negative refractive index transmission pass-band occurs at the resonant frequency of the split-ring when wires that provide a negative permittivity [19,20,49] are aligned with their axis parallel to the plane of the loop. The first resonator-type NIM used the split-ring design, and split-rings have since been used in various configurations, from microwave to near-infrared frequencies [50–52]. At terahertz frequencies the magnetic, electrical, and half-wave2 responses of the split ring have also been demonstrated [53–62]. Additionally, split-ring resonators including extra structures that can switch the resonance of the split ring on or off [63–67], or with optional additional material to tune the resonance location [68–70] have been fabricated. Multiple sub-structures for multiple resonances [71–73] and flexible structures [60,61,74] have also been developed. As versatile as split-ring resonators are, and despite advances in fabricating multi-dimensional terahertz frequency split-ring resonators [75–79], the structures are inherently planar and require patterning of multiple split-rings for the metamaterial be isotropic [80], and were therefore not chosen for use in this research. 2

The half-wave response is where the legs of split-ring act as a simple antenna.

5 1.1.2

Coupled Wire Structure Metamaterials

Coupled wire structured NIMs use pairs of conducting wire strips, or other shapes, that are spaced apart in the direction of wave propagation at a distance much smaller than the wire’s width. Figure 1-2 is a diagram of a coupled wire resonator. Pattern alignment is critical in this type of design because the strength of the mutual

Figure 1-2. Diagram of a coupled wire resonator.

electromagnetic coupling is easily changed. The electric field interacts with linear elements of the structure leading to negative permittivity. The magnetic field interacts with the small gaps between conducting elements, exciting anti-parallel currents in the open loop of the two conducting structures, leading to negative permeability [81–89]. Coupled wire structures can be made either for use as birefringent materials or rotationally symmetric for use as polarization independent materials. Coupled wire structured NIMs have been built at terahertz frequencies as a birefringent structure [90], a polarization independent structure [91], and an anisotropic structure [92]. While coupled wire structures can be built as a bulk metamaterial, since the desired properties are achieved face-on, the structure is planar by design. Thus the axis perpendicular to the material face has a different permittivity and permeability than do the axes parallel to the material face. Even though the working angle of the metamaterial is broad, the refraction angles of different polarizations will eventually decouple because the structure is uniaxial.

6 1.1.3

Dielectric Material Resonator Metamaterials

Dielectric material resonator NIMs use a non-conductive particle with high permittivity, compared to that of the surrounding material, to create negative permeability. The shape of the inclusions determine the resonant frequencies as well as the level of anisotropy. Dielectric resonators can be isotropic because of their three-

Figure 1-3. Diagram of a dielectric resonator.

dimensional shape. Figure 1-3 is a diagram of a dielectric resonator. The resonances are Mie resonances3 for cubes and spheres: the first resonance is magnetic; the second resonance is electric [93–98]. Several forms of high dielectric material resonators and dielectric resonator NIMs have been built for microwave frequencies. These shapes include disks [99–104], cylinders [18,104–107], cubes [108–112], and spheres [32,33,98,113,114]. These structures were built for waveguide or coaxial cable. Dielectric resonators for terahertz frequencies are not as easily produced as those for microwave frequencies as conventional machining methods are not typically usable. A conventional machining method that has been used to cut solid materials was laser dicing, and was painstaking used, to create a single layer of square cylinders from strontium titanate [115–117] and a single layer of cubes from titanium dioxide [118]. The structures had a negative permeability at resonance but have not been combined with a negative permittivity structure or used to make a NIM. 3

Named after Gustav Mie for his theory of the electromagnetic wave scattering of spheres.

7

1.2

Fabrication Methods For Terahertz Frequency Dielectric Resonator Metamaterials

An alternate method to machining is to synthesize the resonators in the desired sizes. Nanometer-sized particles are required for frequencies higher than terahertz [119–121]. By comparison, the size of dielectric resonators required for terahertz frequencies is in the tens of microns. Though the literature is replete with examples of various high permittivity materials being synthesized in sub-micron sizes [122–130], many of which are also available commercially, the much larger micron-sized particles are non-existent commercially, hard to grow, difficult to work with, rarely reported in the literature, made of composite materials [131], or are deformed [132]. A conceptual diagram of the commercial and reported availability of high permittivity materials versus size is shown in Figure 1-4. These fabrication difficulties have meant that dielectric resonators for terahertz frequencies are rare and therefore dielectric-resonator based NIMs have not yet been reported.

Figure 1-4. Conceptual diagram of availability by size of high permittivity particles.

8 In developing the construction and analysis techniques needed for a dielectric-resonator based THz NIM, a number of fabrication methods were extended for use with the needed geometries. Some metamaterials were also useful in verifying theoretical predictions both of established theories and those modified for use in this work. The main metamaterials developed include a yttria-stabilized zirconia sphere NIM for microwave frequencies, and for terahertz frequencies, a lattice of titanium dioxide high-permittivity sphere resonators, and lithium tantalate rods with a transmission stop-band or pass-band at resonance depending on polarization. Additional fabrication of an array of micron-sized cubes of lithium tantalate was also attempted. A list of the metamaterials built for this work is given in Table 1-1; the principle metamaterials are described next. Detailed fabrication techniques, results, and discussion are in the sections listed with each metamaterial below. Table 1-1. Metamaterials built for this research. Metamaterial Component

Resonance Frequency

Sections

Negative Permeability

YSZ Spheres: waveguide 28 GHz — 3.6.1 4.2.1 5.1.1 yes Wire Grid: waveguide — — 3.6.2 4.2.2 5.1.2 — YSZ Spheres 28 GHz 1.2.1 3.6.3 4.2.3 5.1.3 yes YSZ Spheres and Wires 28 GHz — 3.6.4 4.2.4 5.1.4 yes ............................................................................................................................................................... YSZ Micro-Spheres 0.50 THz — 3.7.1 4.1 / 4.4.2 5.2.2 no TiO2 Micro-Spheres 1.6 THz 1.2.2 3.7.2 4.1 / 4.4.2 5.2.2 no TiO2 Micro-Spheres 3.7 THz 1.2.2 3.7.2 4.1 / 4.4.2 5.2.2 no LiTaO3 Micro-Rods 0.33 THz 1.2.3 3.8 4.5 5.3 yes LiTaO3 Micro-Cubes 0.39 THz 1.2.4 3.9 4.6 5.4 predicted Micro-Wire Array — — 3.10 4.7 5.5 — LiTaO3 Micro-Cubes 0.39 THz 1.2.5 3.10 4.8 5.6 predicted and Wire Array

9 1.2.1

Microwave Frequency Sphere Resonator Metamaterial

As part of the verification of the effective medium theories used to predict the electromagnetic wave behavior of a material, described in Section 2.2, a metamaterial composed of a grid of 2 mm spheres was combined with wire array to form a free-space NIM operating at microwave frequencies. Figure 1-5 is a diagram of the microwave dielectric resonators. The

Figure 1-5. Diagram of microwave sphere resonators.

spheres alone were found to have a transmission stop-band at which the permeability was negative, shown in Section 4.2.3. When combined with a wire array, a transmission passband was observed at the same frequency range, where the permittivity and permeability were both negative, shown in Section 4.2.4. This NIM, while not the first metamaterial built for microwave frequencies that was composed of spheres and a wire grid [113], was the first, and so far only, dielectric-resonator metamaterial built for free-space.

10 1.2.2

Micro-Sphere THz Resonator Metamaterial

In order to fabricate a THz NIM based on spherical inclusions it was necessary to fabricate resonators in the desired size. Titanium dioxide seemed like a promising material based on having a high permittivity and the potential of being fabricated in the desired size. Other candidate materials had significantly less or no potential for being grown as large spheres. Two sizes of spheres,

Figure 1-6. Diagram of embedded sphere resonators.

10 microns and 36 microns in diameter, were synthesized using polydivinylbenzene sacrificial beads on which the precursor to titanium dioxide was converted to titanium dioxide. The resulting heterogeneous spheres were calcined4 to form titanium dioxide micro-spheres in the rutile crystalline phase. The method used in this research is a direct extension to larger template spheres from those found the literature [133]. For the size range, these micro-spheres are unique in their uniformity of size and shape [131,132]. The 10-micron-diameter micro-spheres were embedded in a square arrays of pockets formed in photoresist. Figure 1-6 is a diagram of the embedded micro-sphere resonators. The 36-micron-diameter micro-spheres were adhered to a thin layer of photoresist. While both sets of spheres exhibited a transmission stop-band at resonance, material losses were found to dampen the resonance, thus precluding the negative permeability required for a NIM, shown in Section 4.4.2. 4

To calcine is to heat a substance to initiate phase transitions and burn off organic products.

11 1.2.3

Micro-Rod THz Resonator Metamaterial

Cubic-like resonator shapes have a similar Mie resonance to that of spheres. Both strontium titanate [115–117] and titanium dioxide [118] have been used as the resonator material at THz frequencies. patterned by laser cutting.

Both materials were

However, the materials

are expensive and laser machining is costly and time consuming because it is a serial production process.

Figure 1-7. Diagram of micro-rod resonators.

Lithium tantalate was explored as an alternate high permittivity and low loss material for use as the resonator material. A proof-of-concept metamaterial made of lithium tantalate micro-rods was created through mechanical dicing. Figure 1-7 is a diagram of the micro-rod resonators, which were 85 microns thick and 195 microns wide. This size is near the current limits of mechanical dicing. The rods exhibited a Mie resonance when the electric field was perpendicular to the rod’s axis; a transmission stop-band was observed when the permeability was negative, shown in Section 4.5. When the electric field was parallel to the rod’s axis, a transmission pass-band unrelated to resonance was observed. A new effective medium theory for cubes was developed for this work because the resonance frequency and the effective permittivity and permeability of an array of cubicshaped resonators is not obtained by simple substitution of a sphere by either equivalent volume or equivalent radius [134]. The effective medium theory, described in Section 2.2.2, accurately predicted the transmittance and resonance compared with measurement.

12 1.2.4

Micro-Cube THz Resonator Metamaterial

Having shown that a cubic cross section of lithium tantalate could be used a dielectric resonator for a metamaterial with an effectively negative permeability, a method to produce cubes on smaller scales was sought. Micro-cube resonators of lithium tantalate were to be created by wet etching using a photolithographically produced mask, that would allow for arbitrary sized

Figure 1-8. Diagram of micro-cube resonator.

cubes. The micro-cubes were to be 100 microns on each side. Figure 1-8 is a diagram of the micro-cube dielectric resonators. The cubes were predicted to exhibit a transmission stop-band where the permeability was negative, shown in Section 4.6. Difficulties with the fabrication technique are described later in Section 5.4.

1.2.5

Combined Micro-Cube THz Resonators And Wire Grid Metamaterial

In an attempt to fabricate a THz NIM based on cubic inclusions, a wire grid to provide a negative permittivity, was to be combined with an array of lithium tantalate micro-cube resonators, as shown in Figure 1-9. The wires of the commercially obtained wire grid had a trapezoidal cross-section being 13 microns thick, 28 microns wide at the base, and 12 microns wide at the top. The wire-to-

Figure 1-9. Diagram of micro-cubes and wire grid.

wire spacing was 150 microns center to center to match the spacing of the lithium tantalate

13 micro-cube array. The wire grid alone exhibited negative permittivity at all frequencies below 0.590 THz including those of the predicted micro-cube resonant frequency at approximately 0.390 THz, shown in Section 4.7. The transmittance of the lithium tantalate micro-cube array combined with the wire grid was predicted to have a transmission pass-band where the effective permeability and permittivity were simultaneously negative, shown in Section 4.8.

14

II.

THEORY

The interaction of electromagnetic waves with the various components of a metamaterial must be known to fully understand and predict the electromagnetic behavior of the assembled metamaterial. These properties include the permittivity and permeability of the individual materials, and the scattering of inclusions in a host material.

2.1

The Permittivity And Permeability Of Materials

The permittivity and permeability describe the response of a material to an electromagnetic wave. The permittivity () describes the reaction of the electric field component, and the permeability (µ) describes the reaction of the magnetic field component. The permittivity and permeability of a material are frequency dependent quantities, but often the permittivity and permeability can be considered constant, especially over a small frequency range. However, the permittivity and permeability cannot be assumed to be constant near electric or magnetic resonances, in very dispersive materials, or over large frequency ranges. Factors that affect a material’s permittivity and permeability are molecular and crystal vibrational modes, molecular rotational modes, electron transitions, and phonons. The behavior of the permittivity and permeability are often represented by frequency dependant phenomenological models. The material permeability was unity for the non-metamaterials used in this work.

15 2.1.1

Simple Polynomial Model

The simple polynomial representation of a material’s permittivity represents a frequency region where a homogeneous material does not have any resonances or absorption peaks. The permittivity is continuous as a function of frequency and is modeled with Equation 2-1, where f is the frequency in Hz, and η and α are fitting parameters. A third-order polynomial was sufficient for materials in this work. It should be noted that the permittivity cannot be extrapolated to frequencies higher than the frequency region being modeled over.   ( f ) = η0 + η1 ( 2πc f )1 + η2 ( 2πc f )2 + η3 ( 2πc f )3   + i α0 + α1 ( 2πc f )1 + α2 ( 2πc f )2 + α3 ( 2πc f )3 (2-1)

2.1.2

Single-Resonance Harmonic Oscillator: Frequency Form

The frequency form of the single-resonance harmonic oscillator model represents a frequency region where there is an absorption peak from a resonance mode. This single-resonance oscillator model was combined with an additional polynomial that was required for materials used with this model. Equation 2-2 is the combined model, where high is the high frequency permittivity, low the low frequency permittivity, γ the resonance width, f0 the resonance frequency, f the frequency in Hz, and α is a fitting parameter [135, p.220]. ( f ) = high +

  low − high 1 2 3 2π 2π 2π + i α + α ( f ) + α ( f ) + α ( f ) 0 1 2 3 c c c 1 − f 2 / f02 − iγ f / f02

(2-2)

16 2.1.3

Multiple-Resonance Harmonic Oscillator: Wavenumber Form

The wavenumber form of the multiple-resonance harmonic oscillator model represents a frequency region where there are absorption peaks from several resonance modes. Equation 2-3 is the wavenumber form of the single oscillator, where ν is the wavenumber, ∞ is the high frequency permittivity, and for the j th mode: S j is the strength of the resonance, Γj is the resonance width, and ν j is the resonance wavenumber [136,137]. The wavenumber (ν) in this equation is the spectroscopic wavenumber (cm−1 ). (ν) = ∞ +

2.1.4

n X

S j ν2j

j=1

ν2j − ν2 − iνΓj

(2-3)

Four Parameter Semi-Quantum Multiple-Resonance Model

The four parameter semi-quantum multi-resonator model can be used when the permittivity from multiple longitudinal and transverse lattice modes is not well modeled by classic dispersion theory. The model is given in Equation 2-4 [138], where ν is the wavenumber, high is the high frequency permittivity, and j is the mode number. For the longitudinal mode: νLO is the wavenumber at resonance, and γLO is the resonance damping term. For the transverse mode: νTO is the wavenumber at resonance, and γTO is the resonance damping term. The wavenumber (ν) in this equation is the spectroscopic wavenumber (cm−1 ).

(ν) = high

2 n ν2 Y LO j − ν + iνγLO j

ν2 − ν2 + iνγTO j j=1 TO j

(2-4)

17 2.1.5

Ferroelectric Materials

Ferroelectric materials have higher permittivities than other materials.

These large

permittivity values are due to lattice displacement of the charged ions, as is drawn conceptually in Figure 2-1. In the relaxed state it is energetically favorable for the ions to shift slightly from their charge equilibrium position in the lattice leading to a remanent electrical polarization. The farther the displacement of the ions, the larger the net electric dipole in the lattice [139,140]. The displaced ions have different vibrational modes which depend on the elemental composition of the lattice, the lattice structure, and on other dynamics. The resonance frequency of these vibrational modes is called the relaxation frequency. At frequencies higher than the relaxation frequency, the material permittivity is much lower than at frequencies far below the relaxation frequency. The lattice relaxation frequency and maximum permittivity is in the terahertz frequency range for several of these ferroelectric materials.

Figure 2-1. Conceptual diagram of ion displacement in a lattice.

18 2.1.6

Negative Refractive Index From Negative Permittivity And Negative Permeability

The index of refraction is related to the permittivity () and permeability (µ). This relation is given in Equation 2-5, where n is the refractive index and κ is the extinction coefficient. p n + iκ = ± (re µre − im µim ) + i(re µim + im µre )

(2-5)

The real and complex parts of Equation 2-5 can be algebraically separated leading to Equation Set 2-6. The conventionally chosen sign for the radicals is positive for both n and κ. However, in a NIM the negative sign is chosen for n in Equation 2-6a and the positive sign is chosen for κ in Equation 2-6b [1]. 1 n=±√ 2

r

1 κ=±√ 2

r

+(re µre − im µim ) +

−(re µre − im µim ) +

q

(re µre − im µim )2 + (re µim + im µre )2

q

(re µre − im µim )2 + (re µre + im µre )2

(2-6a)

(2-6b)

The physical argument for the negative sign of n in a NIM is that, at the interface of a positive-refractive-index material and a negative-refractive-index material, a wave does not cross the axis normal to the surface but refracts to the same side as the incident beam. This behavior is the same as if the refractive indices in Snell’s law had opposite signs. The mathematical argument treats n and κ as coupled parts of the propagation constant (k) where k =

ω(n + iκ) . When the extinction coefficient κ in a material has a sign contrary c

to that required for a damped passive material, then the negative sign of Equation 2-5 is used leading to a negative refractive index n [2].

19

2.2

Effective Medium Theories

As compared to models for uniform materials, effective medium theories are used to predict the electromagnetic response of a composite material or metamaterial. Mixing one material into another causes the interaction of an electromagnetic wave with the materials to be altered. The composite is effectively homogeneous when the wavelength is greater than about six times the size of any structure in the composite material. A large ratio of wavelength to structure size improves the accuracy of effective medium theories [141]. For many effective medium theories the permittivity and permeability of the composite material are obtained by averaging the detailed electromagnetic fields over a volume element determined by the density of the inclusions. The effective medium theories used were for a cubic lattice of spheres, a cubic lattice of cubes, an array of wires, and a combination of the wires with either the spheres or cubes.

2.2.1

A Cubic Lattice Of Spheres

The electromagnetic scattering of a spherical shaped material is important because the solutions to the wave equations are applicable to areas such as mixed materials, colloidal solutions, electromagnetic wave absorbers, and atmospheric transmission. There are a number of effective medium theories for spherical inclusions that can be tailored to specific situations or modified for possibilities such as elliptically shaped particles. This research used the effective medium theory for a cubic lattice of finite-sized spheres originally described by Lewin [95].

20 The polarizability of a sphere as a function of frequency is not constant but is described by Mie scattering parameters. These scattering parameters were first developed to explain the behavior of gold colloidal solutions [93,94]. It is necessary to use Mie parameters to solve for the scattering of high-permittivity sphere near resonance because the wavelength inside the sphere is on the order the particle’s size. Lewin’s theory for a cubic lattice of spheres uses the first-order Mie coefficients for a sphere and sums over the response of all the spheres to obtain an effective permittivity and permeability. In the limit of infinitely small sphere Lewin’s solutions reduce to the classical Maxwell-Garnet mixing theory, which was derived for negligibly small spheres in a host material [142]. Lewin’s effective medium theory for a cubic lattice of high-permittivity spheres in a low-permittivity medium was used in this work. The theory assumes that the spheres are physically small with respect to the wavelength in the surrounding medium, that the dipole resonance is dominant, and that the resonance modes are independent so the electromagnetic fields of individual spheres are not strongly coupled. When the embedded spheres have a high permittivity, the electrical size is large enough for resonance modes to be excited in the spheres [95,97,98]. The frequency of the first resonance of a sphere can be found from Equation 2-7 where r s is the radius of the sphere [143]. f sphere =

resonance

√

c

2r s  sphere µ sphere

(2-7)

21 The effective permittivity and permeability of a cubic lattice of spheres are obtained from Equation Set 2-8 where l is the lattice spacing, r s is the radius of the sphere, k0 is the wavenumber in vacuum,  sphere is the permittivity of the sphere, host is the permittivity of the host material, µ sphere is the permeability of the sphere, µhost is the permeability of the host material, and v f is the volume fraction of the spheres in the material. The resonances may exhibit a negative effective permeability or permittivity in low-loss spheres [97,98]. The spheres in this work had a high permittivity and a permeability of unity, for which, the first resonance was magnetic exhibiting a resonant permeability. Higher order resonances were significantly weaker.    − vf 

(2-8a)

    − vf 

(2-8b)

b = host / sphere

(2-8c)

bµ = µhost /µ sphere

(2-8d)

2(sin Θ − Θ cos Θ) − 1) sin Θ + Θ cos Θ

(2-8e)

√ Θ = k0 r s  sphere µ sphere

(2-8f)

4πr3s 3l3

(2-8g)

eff [spheres] = host

µeff [spheres]

  1 + 

  = µhost 1 +

F[Θ] =

3v f F[Θ]+2b F[Θ]−b

3v f F[Θ]+2bµ F[Θ]−bµ

(Θ2

vf =

22 2.2.2

A Cubic Lattice Of Cubes

The polarizability and resonance frequencies of a cube are different than those of a sphere. The equations in Set 2-8 of Section 2.2.1 were modified to account for the different polarizabilities and volume fractions of cubes. The resonance frequency of cubes is related to its size parameter just as with spheres. p The size parameter for a sphere this was 2r s  sphere µ sphere and is both the denominator of Equation 2-7 and implicit in Equation 2-8f [93,94,143]. The resonance frequencies of a high-permittivity cube are given by Equation 2-9 where m are the mode indices [107,144]. .p p The size parameter of a cube is then sc cube µcube (ma )2 + (mb )2 + (mc )2 where sc is the length of the side of the cube. f cube resonance

=

c

√ 2 cube µcube

r

ma 2  mb 2  mc 2 + + W H L

(2-9)

23 The effective permittivity and permeability for a lattice of cubes are given in Equation Set 2-10. The size parameter of a sphere was replaced with that of a cube in Equation 2-10f to use the effective medium theory with cubes. The volume fraction of a cube in a cubic lattice cell was used in Equation 2-10g [110].

   − vf 

(2-10a)

    − vf 

(2-10b)

b = host /cube

(2-10c)

bµ = µhost /µcube

(2-10d)

2(sin Θ − Θ cos Θ) (Θ2 − 1) sin Θ + Θ cos Θ

(2-10e)

p k0 sc cube µcube Θ= p ma 2 + mb 2 + mc 2

(2-10f)

s3c l3

(2-10g)

eff [cubes] = host

  1 + 

µeff [cubes] = µhost

  1 + 

F[Θ] =

3v f F[Θ]+2b F[Θ]−b

3v f F[Θ]+2bµ F[Θ]−bµ

vf =

24 2.2.3

An Array Of Wires

An array of high conductivity wires has three operating frequency regions: the longwavelength region, the effective medium region, and the scattering region. In the longwavelength limit an array of wires becomes a polarizer in which the transmittance is dependant only on the angle between the wires’ axis and the polarization of the electric field of an incident wave. At the other extreme, in the scattering region, the wire array is no longer a homogeneous material and the wires are strong scatters that can exhibit photonic crystal stop-bands. In the effective medium region, between the two limits, the wire array is considered homogeneous while the transmittance depends on the number of layers, structure of the array, and the wire size. The effective medium region is the working region of the wire arrays used in this research and to which Equations 2-11 and 2-12, shown later, apply. For a wire array medium to be treated as homogeneous, both the wire lattice spacing compared to the wavelength in the host material, and the wire radius compared to the wire lattice spacing, must be small. An array of wires, aligned parallel to a wave’s electric field, can be used to create a material in which the effective permittivity is negative. At frequencies below the effective plasma frequency of the wire array, the effective permittivity is negative for a lossless wire medium in air.

25 The permittivity of the wire array medium along the direction of the axis of the wires is obtained from Equation Set 2-11, where l is the wire lattice spacing, rw is the wire radius, σ is the conductivity of the wires, host is the permittivity of the host material, γ is a loss term, and f p is the effective plasma frequency of the wire array [19–26]. Axes orthogonal to the wires’ axis have a permittivity approximately equal to the host material’s permittivity.

eff [wires] = host −

f p2 f 2 + i( f γ)

(2-11a)

µeff [wires] = µhost

(2-11b)

γ=

0 l2 f p2 (2π)πrw2 σ

(2-11c)

!  c 2 2π/l2 = 2π ln(l/2πrw ) + F(a/b)

(2-11d)

! π a +∞ a X coth(πm( b )) − 1 b F = − ln + + b b m 6 m=1

(2-11e)

f p2

a

a

when

a=l then F (1) ' 0.5275 b=l

26 Spatial Dispersion Of Wire Array Media As with polarizers, the permittivity of the wire medium depends on the direction of propagation of the wave in the medium and is considered spatially dispersive where even a three dimensional array of wires is not truly isotropic. The spatial dispersion is due to highly localized fields around each wire and a longitudinal mode.1 The effective permittivity given in Equation 2-12 is a modified form of Equation 2-11a, where ~k is the wave vector of the wave and nˆ is the axis of the wires in the array being considered [26,145].

nˆ :eff [wires] = host −

f p2  2     ~k · nˆ     + i( f γ) f 2 1 −   |~k|

(2-12)

For this research the spatial dispersion of the wire array media was considered negligible for four reasons. First, due to the fabrication methods used, the negative refraction property is only experienced by transverse electric incident waves where the wire grid is in the plane of the electric field and therefore does not experience spatial dispersion. Second, the spatial dispersion is minimal for small incident angles because the relationship between the propagation direction and wire axis is a function of cosine squared. Third, spatial dispersion is weak near the wire plasma frequency which is the frequency range the materials used in this research were designed for. Fourth, the absorption of the media is unaffected by spatial dispersion [24]. 1

The longitudinal mode is a transmission line wave along the wire.

27 2.2.4

Combined Effective Medium Theories

In order to determine the effective permittivity of the combined structure of a lattice of resonators and wire array, two effective medium theories were used sequentially. Either the resonant inclusion medium was taken to be the host medium for wire array, or the wire array medium was taken to be the host medium for the lattice of resonant inclusions medium. Some inaccuracy is to be expected when ignoring the contribution to the host permittivity from the dielectric inclusions when calculating eff [wires] and neglecting the wires in the mixing relation used to obtain eff [spheres/cubes] . In the case of a resonant inclusion host medium, the effective eff [spheres/cubes] was determined from Equation 2-8a or 2-10a where host was the permittivity of the support material without wires. Then eff [spheres/cubes] from Equation 2-8a or 2-10a was substituted as host into Equation 2-11a of the wire array theory to give a combined material response. In the reverse case of a wire array host medium, the effective eff [wires] of the wires was determined from Equation Set 2-11 where host was the permittivity of the support material without spheres. Then eff [wires] from Equation 2-11 was substituted as host in Equation Set 2-8 or 2-10 of the resonant inclusion theory to give a combined materials response.

28

2.3

Reflection, Transmission, And Refraction Equations

The reflection, transmission, and refraction of an electromagnetic wave at the boundary of two materials can be described using ~k vectors, Poynting vectors, and continuity conditions at the interface. The ~k vectors determine the direction of the wave phase. ~k vectors also help clarify the behavior of a negative refractive index material. The Poynting vector determines the actual direction and power flow of a wave. The reflection and transmission coefficients at the interface of the material can be determined from the ~k vectors and the material’s permittivity and permeability. The reflectance and transmittance are determined from the sum of waves reflected by and transmitted through the material. 2.3.1

Coordinate System And Wave Propagation Convention

The coordinate system and the assumed time convention in the solution to the wave equation are given below to avoid confusion between differing coordinate and wave propagation conventions. The sign of some values in later equations depend on the direction of propagation and assumed time convention. Coordinate System The coordinate system used is shown in Figure 2-2. The coordinate system, for this research, is a combination of the Cartesian and spherical coordinate systems. The origin is common to both coordinate systems. The first surface of the material is at the origin and in the x − y plane; the z axis is perpendicular to the material surface where the positive direction points into the material. The φ angle is in the x − y plane and is the angle from the x axis increasing toward the y axis. The θ angle is the direction rˆ points from the z axis.

29

Figure 2-2. Coordinate system used in this research. Wave Propagation Convention The assumed time dependence of a wave traveling in the positive z direction is given in Equation 2-13 where the extinction coefficient κ takes positive values in lossy media. i (kz−ωt)

e

=e

i


ω(n+iκ) z−ωt c

(2-13)

The wavenumber (k) is defined in Equation 2-14 where the permittivity () and permeability (µ) are relative to the vacuum permittivity (0 ) and permeability (µ0 ). √ ω µ 2π √ √ k= = ω µ 0 µ0 = λ c 2.3.2

(2-14)

Isotropic Materials

In homogeneous isotropic materials the constitutive equations describe the interrelation of ~ the electric field (E), ~ the electromagnetic field components of the electric displacement (D), ~ and the magnetic flux ( B). ~ The constitutive relations for an isotropic the magnetic field (H), material are given in Equation set 2-15.

30 The path of a refracted electromagnetic wave is also determined by Snell’s Law, given in Equation 2-16. The sign of the refractive index (n) is important in NIMs. For example, at the interface of one material with a positive refractive index with n > 0 and another material also with n > 0, the path of the refracted electromagnetic wave crosses the axis perpendicular to the interface as shown in Figure 2-3a. At the interface of one material with n > 0 and a negative refractive index material with n < 0, the path of the electromagnetic wave does not cross the axis perpendicular to the interface plane. Instead the electromagnetic wave refracts to the same side of the axis perpendicular to the surface as the incident wave as shown in Figure 2-3b. ~ =  E~ D

(2-15a)

~ = µH ~ B

(2-15b)

sign(n2 )|n2 | sin(θ1 ) = sign(n2 )|n2 | sin(θ2 )

(2-16)

(a) n1 > 0 and n2 > 0

(b) n1 > 0 and n2 < 0

Figure 2-3. Refraction at positive refractive index material with (a) positive refractive index and (b) negative refractive index materials.

31 2.3.3

Uniaxial Materials

For non-isotropic materials, the refractive index (n) can change as a function of polarization and incident angle. The reflection, transmission, and refraction can be determined using the wave-vectors (~k) regardless of the level of anisotropy or of the sign of the permittivity or permeability of the medium [39]. In a uniaxial material one of the three axes, the optical axis, has a permittivity or permeability different than the other two axes. The resonant-cube negative refractive index material designed during this research had an isotropic permeability (µ) and a uniaxial permittivity () where the optical axis was oriented along the z axis. The permittivity was uniaxial because the wire grid ran only in two of the axes. Anisotropy is a consequence of the wire grid dimensionality but was not the cause of the negative refraction in this work. The constitutive equations for this material are given in Equation set 2-17 [146–148]. Neither reciprocal rotation nor non-reciprocal Faraday rotation occurred indicated by the constitutive matrices having only diagonal terms.     1 0 0     ~ ~  ¯ D =  · E =  0 1 0  · E~     0 0 2 

(2-17a)

    µ1 0 0    ~ = µ¯ · H ~ =  0 µ 0  · H ~ B 1      0 0 µ1 

(2-17b)

32 2.3.4 ~k Vectors And Dispersion Relations The scattering of an electromagnetic wave at the interface of two materials can be determined from the equations for the dispersion relationship, power flow, and reflection and transmission coefficients. The dispersion relations are determined by solving for k as a function of ω or k0 in Maxwell’s equations. For the constitutive relations in Equation set 2-17 the dispersion relationship for the transverse electric2 (TE) wave is given by Equation 2-18a and for the transverse magnetic3 (TM) wave by Equation 2-18b, where k x , ky , and kz are the wavenumbers along the x, y, and z axes, respectively [39,40,146]. TE :

ky2 kz2 k2x + + = k2 1 µ1 1 µ1 1 µ1

(2-18a)

TM :

ky2 kz2 k2x + + = k2 2 µ1 2 µ1 1 µ1

(2-18b)

Equations 2-18a and 2-18b expressed as a function of the incident angle (θ), with respect to the z axis, are given in Equations 2-19a and 2-19b respectively [39,40,146].

2 3

TE : k = k0

r

TM : k = k0

r

cos2

µ1 1 √ = k0 µ1 1 2 θ + (µ1 /µ1 ) sin θ

µ1  1 cos2 θ + (1 /2 ) sin2 θ

Transverse Electric: where the incident electric field is parallel to the material surface. Transverse Magnetic: where the incident magnetic field is parallel to the material surface.

(2-19a)

(2-19b)

34 2.3.5

Poynting Vectors

The direction of power flow for a wave is determined from its Poynting vector (S~ ). For the constitutive relations of a uniaxial material the time-averaged Poynting vector of the TE wave is given in Equation Set 2-20 and of the TM wave in Equation set 2-21 [39,40] [146, p.120].  D E   ~i = 1 (+ˆzkzi + xˆk x ) |E0 |2  S (2-20a)    2ωµ  0       DS~ E = 1 (+ˆzk + xˆk ) |R|2 |E |2 (2-20b) TE :  zr x 0 r  2ωµ0    !   D E  1 kzt kx  2 2  ~  (2-20c)   S t = 2ωµ +ˆz µ + xˆ µ |T | |E0 | 0

              TM :             

1

1

1 (+ˆzkzi + xˆk x ) |H0 |2 2ω0 D E 1 (+ˆzkzr + xˆk x ) |R|2 |H0 |2 S~r = 2ω0 ! D E 1 kzt kx ~ St = +ˆz + xˆ |T |2 |H0 |2 2ω0 1 2 D E S~i =

(2-21a) (2-21b) (2-21c)

For both TE and TM waves, the sign of kzt is set by the requirement that the zˆ term in D E S~t must remain positive because power is flowing away from the interface into the material. D E D E The k x wave vector in both S~r and S~t is the same as the incident wave, again, because of phase-matching requirements at the material interface. The refracted wave direction, and D E thus sign or the refractive index, is determined from the xˆ term in S~t .

36 2.3.6

Reflection And Transmission Coefficients

The reflection (R) and transmission (T ) coefficients at the material’s surface, as given in Equation sets 2-22 and 2-23, can be determined once the ~k vectors at the interface are known [146, p.119].     1−       Ri →r =      1+    TE :           T i →t =       1+      1−       Ri →r =      1+    TE :           T i →t =       1+ 

µ0 kzt µ0 µ1 k z µ0 kzt µ0 µ1 k z 2 µ0 kzt µ0 µ1 kz 0 kzt 0 1 kz 0 kzt 0 1 kz 2 0 kzt 0 1 kz

(2-22a)

(2-22b)

(2-23a)

(2-23b)

37 For a single layer material, the total reflectance and transmittance are determined from the summation of multiple internal reflections taking into account the phase delay between the two surfaces of the material. The reflectance (R) and transmittance (T ) are given in Equation set 2-24, where d is the material thickness and the reflection and transmission coefficients are defined as in Figure 2-6 [149]. R=

R12 + R23 e2ikz d 1 + R12 R23 e2ikz d

(2-24a)

T =

T 12 T 23 eikz d 1 + R12 R23 e2ikz d

(2-24b)

Figure 2-6. Reflectance and transmittance diagram of wave through a plane parallel NIM.

38

III.

METHODOLOGY

This section presents the methods used to design, fabricate, and measure metamaterials for terahertz frequency NIMs. The metamaterials built were designed using a combination of analytic modeling and an electromagnetic wave numerical simulation program. For microwave frequencies, the metamaterials built include sphere and wire grid waveguide samples, and were successfully combined to create a free-space NIM. Microwave frequency transmittance and reflectance measurements of the microwave metamaterials were acquired using a network analyzer and verified negative refractive index behavior. For terahertz frequencies, metamaterials built to explore materials for creating the desired negative permeability included: patterned yttria-stabilized zirconia spheres, synthesized and patterned titanium dioxide micro-spheres, and lithium tantalate micro-rod arrays. Fabrication of a terahertz NIM made from a combination of lithium tantalate micro-cube lattice and wire array was attempted. Terahertz frequency transmittance measurements were taken using a Fourier transform infrared spectrometer. Measurement data were analyzed using the same modeling and simulation programs used in the prediction of material’s electromagnetic response to understand and confirm metamaterial behavior.

3.1

Material Modeling And Simulation

Analytic modeling and numerical simulations were used to design the metamaterials and predict transmittance and reflectance before construction. The analytic modeling uses theories that average over the electromagnetic response of sub-wavelength details. By

39 comparison, the numerical modeling used an exact representation of the metamaterial’s structure, including sub-wavelength details, and therefore able to predict transmittance and reflectance beyond the region of effective medium theory. 3.1.1

Analytic Modeling Of Material Behavior

The graphical programming software LabView [150] was used to obtain the permittivity and permeability from the material models and effective medium theories described in the Theory section. The transmittance and reflectance of the materials was then predicted using a modified form of Equation Set 2-24. 3.1.2

Numerical Simulation Of Material Behavior

The electromagnetic finite-element solver HFSS [151] was used for numerical simulation of the materials’ electromagnetic behavior. The simulations were used to verify analytically predicted transmittance and reflectance. The simulations are described in Appendix 9.4.

3.2

Powder Diffraction Measurement Of Crystal Structure

The crystal phase of the synthesized micro-spheres, shown later in the results section, was measured by X-ray powder diffraction. This was done to verify the results of the fabrication method, and the expected permittivity which depends the crystal phase. Measurements were taken using an InXitu model BTX. The crystal phases were determined by comparing measurements with known crystal diffraction patterns from the American Mineralogist Crystal Structure Database (AMSCD) [152] using Xpowder software [153].

40

3.3

Measurements At Microwave Frequencies

An Agilent Technologies model E8363B network analyzer was used to measure the transmittance and reflectance of samples designed for microwave frequencies. Both waveguide and free-space measurements were performed. For waveguide measurements the network analyzer was connected to adapters specific to each waveguide band. For free-space measurements, the network analyzer was connected to two spot-focusing horns facing each other. Waveguide Measurement Setup A Through-Reflect-Line method was used to calibrate the waveguide measurements. The ‘Through’ was done by connecting the waveguides directly together without the sample measurement waveguide piece. The ‘Reflect’ was done by shorting the end of the waveguide by connecting a solid piece of brass across the face of the waveguide. The ‘Line’ was done by inserting a precision waveguide length that was one quarter the wavelength of the geometric mean frequency. Figure 3-1 shows the X-band waveguide equipment where the black waveguide is the sample measurement holder.

Figure 3-1. X-band waveguide measurement equipment.

41 Free-Space Measurement Setup A Through-Reflect-Line method was also used to calibrate the free-space measurements. The ‘Through’ was done with the spot-focusing horns at their zero positions and an empty measurement area. The ‘Reflect’ was done by inserting a 2 foot by 2 foot square aluminum plate of precision thickness in the sample measurement holder. The aluminum plate was translated by its thickness on a moving stage between calibration reflection measurements so that after calibration, the zero phase planes from horns 1 and 2 were at the same place. The ‘Line’ was done by translating each horn apart half of the total separation, which was one quarter the wavelength at the geometric mean frequency. Radiation from the spot-focusing horns was linearly polarized, and samples could be measured in either polarization by rotating the sample. Figure 3-2 shows the free-space measurement equipment for the 27.8 40 GHz frequency range where the blue-colored horns shown were those used.

Figure 3-2. The 27.8 - 40 GHz frequency range free-space measurement equipment.

42

3.4

Measurements At Terahertz Frequencies

A Fourier transform infrared spectrometer (FTIR) was used to measure transmittance at terahertz frequencies. Bruker Optics models IFS66vs and a Vertex 80v were used. Different frequency ranges were measurable using different beam splitters, bolometers, and bolometer cold filters. The beam splitters cover different frequency regions and have different transmission nulls. Liquid helium cooled silicon-composite bolometers were used as detectors. Samples were mounted in holders during measurement, two of which are shown in Figure 3-3. The sample holders were placed into measurement fixtures that were either in the FTIR sample chamber or external to the FTIR. The sample chamber was under vacuum during all measurements. For non-isotropic materials, a polarizer was used to measure separate polarizations.

(a) 1 inch circle holder

(b) 1 cm square holder

Figure 3-3. Sample holders used for FTIR measurements.

43

3.5

Determination Of Permittivity And Permeability

There are two common methods used to determine the permittivity () and permeability (µ) of a material. One method is to fit the predicted transmittance and reflectance to the measured reflectance and transmittance, and the other, is to invert the measured transmittance and reflectance data to determine the complex index of refraction and complex impedance. Fitting Methods When using curve-fitting methods, the permittivity and permeability can be either obtained from a theoretical model and then the predicted transmittance and reflectance compared with the measured data. The permittivity and permeability could also be determined by inserting test values and comparing predicted transmittance and reflectance with the measured data and improving the values iteratively [154,155]. Models of permittivity and permeability exist for both embedded spheres [95,97] as well as wires [19–26] and were used to predict the response of those types of metamaterials. By filling in the known material properties and dimensions, the transmittance and reflectance of the model were compared with measured data. Differences between the measurements and the predicted responses were from either invalid theoretical assumptions, or from construction and material defects.

44 Inversion: Nicolson Ross Weir Method Inversion of the measured transmittance and reflectance data has been used for some time to determine the complex index of refraction (n + iκ) and complex impedance (Z), and then deriving the permittivity () and permeability (µ) from those values [156–158]. The inversion method was later expanded for materials with both negative permittivity and negative permeability [46]. There are three main problems with the inversion approach: firstly, normal incidence complex reflectance and transmittance data are needed; secondly, the solution fails at frequencies where the sample thickness is a multiple of half the wavelength; and thirdly, there are multiple solutions for the refractive index (n). The solution to the first problem requires measurement equipment other than an FTIR. The second problem of having inaccurate solutions at certain frequencies is specific to the inversion of the equations and does not have a solution. For the third problem, there are several ways to determine the refractive index uniquely. One way is to use a sample thin enough so that the wavelength in the sample (λ) is greater than half the sample thickness, which allows for only one solution for the refractive index. Another way is to measure samples of different thickness and requiring that they have the same refractive index. One can also compare the measured group velocity with the calculated group velocity.

45

3.6

Construction Of Microwave Frequency Metamaterials

Samples scaled up in size to work for microwave frequencies, where high-permittivity materials are more available, were built to verify the expected material behavior and compare the measured results with effective medium theory predictions and numerical simulations. Samples of an embedded lattice of spheres and an embedded wire array were built for measurement in both waveguide and free-space. The knowledge obtained from the fabrication of microwave metamaterials and the ability for effective medium theories to predict their behavior were used in the design and fabrication of THz metamaterials and the attempted fabrication of a THz NIM.

3.6.1

Embedded Sphere Samples For Waveguide

Microwave metamaterials of a cubic lattice of embedded spheres were made using 1.88 mm diameter yttria-stabilized zirconia spheres embedded in Task 4, a castable polyurethane resin. The waveguide samples in Figure 3-4 are, from largest to smallest, for the X, Ku, K, and Ka waveguide bands.

Figure 3-4. Waveguide metamaterials of yttria-stabilized zirconia spheres in Task 4 resin.

46 The sphere lattice samples were constructed by casting the Task 4 around the patterned spheres. The spheres were positioned by a metal template, at the bottom of an open-top rectangular mold, so that the spheres were at the vertical center of the sample and the host material poured around the spheres. After curing, excess host material was machined away. The metal template was removed and the host material containing spheres was placed on the bottom of the mold with the embedded spheres facing up for the second cast. The sample was milled to fit tightly into the waveguide sample holders.

3.6.2

Embedded Wire Grid Samples For Waveguide

Embedded wire array samples were built using 24-gauge round copper wire with a diameter of 0.511 mm that were spaced 0.2 inches apart in a host material. All samples were built to fit X-band waveguide. The sample shown in Figure 3-5 was made from copper wire embedded in polycarbonate. The wire array samples were constructed by threading the wire through the holes, coating exposed wire with cyanoacrylate glue, and then pulling the glue-coated wire into the holes in the host material. Excess lengths of wire and host material were then machined down together to the desired height. It was necessary to coat the sample sides in conductive paint silver paint, that was thinned with lacquer thinner and sprayed on, to ensure an electrical connection of the wire ends to the waveguide walls.

47

Figure 3-5. Waveguide metamaterial of 24-gage wire array embedded in polycarbonate. 3.6.3

Sphere Lattice Sample For Free-Space

A microwave metamaterial for free-space was also made using a lattice of yttria-stabilized zirconia spheres with diameters from 1.86 mm to 2.00 mm. The spheres were not embedded in a host material this time but were suspended in the air by adhering the spheres to a very thin layer of paper-backed transfer adhesive. Patterning was done using a metal plate template in which a square array of pockets, separated 0.100 inches apart, was drilled with a #46 drill. The paper-backed transfer adhesive was stretched over an extruded metal frame with the backing against the frame. The frame and adhesive were placed adhesive side down over the patterned spheres and pressure applied to the paper backing to create contact with as many spheres as possible. Figure 3-6 shows the yttria-stabilized zirconia sphere sample.

48

Figure 3-6. Yttria-stabilized zirconia sphere array free-space metamaterial. 3.6.4

Combined Sphere Array And Wire Grid Sample For Free-Space

A wire array was added to the sample after measurements were taken on the sphere array. The wires were 24-gauge round copper wire with a diameter of 0.511 mm that was also used in the waveguide samples. The wires were aligned along one axis of the sphere array. The wires were added in individual lengths prepared from a continuous spool of wire. To remove any kinks and the natural curl that the spooled wire had, the wire was unspooled out in lengths and attached to a post at one end then pulled until its tensile strength was reached. The wire was then attached to the substrate by placing one end of the wire on the adhesive and laying the wire onto the adhesive. Figure 3-7 shows the yttria-stabilized zirconia spheres and 24-gage wires attached to the transfer adhesive and paper substrate.

49

Figure 3-7. Yttria-stabilized zirconia sphere and wire array free-space metamaterial.

3.7

Construction Of Micro-Sphere Metamaterials For Terahertz Frequencies

Micro-spheres were first attempted for use as terahertz frequency resonators. The microsphere metamaterials were built from commercial yttria-stabilized zirconia micro-spheres as well as custom synthesized titanium dioxide micro-spheres. Micro-spheres, while much smaller than those used for the microwave samples, were amenable to similar patterning techniques but had unique challenges.

50 3.7.1

Yttria-Stabilized Zirconia Micro-Spheres

A potentially high permittivity material that was available commercially in an appropriate size was yttria-stabilized zirconia. A sample was constructed using micro-spheres that had diameters from 180 microns to 212 microns. The micro-spheres were arranged in a square array with a center-to-center spacing of 338 microns by using a U.S.-standard number 70 precision sieve. For patterning, the micro-spheres were poured over the sieve, and shaken side to side, depositing the spheres onto a film of transfer adhesive on a support ring that was set back with a spacer while held rigidly against the sieve. Figure 3-8 shows the yttria-stabilized zirconia micro-spheres attached to transfer adhesive.

Figure 3-8. yttria-stabilized zirconia micro-spheres patterned using a number 70 sieve.

51 3.7.2

Synthesis Of Titanium Dioxide Micro-Spheres

As described later in Section 4.4.1, the yttria-stabilized zirconia micro-spheres were not usable as dielectric resonators so it was necessary to synthesize other high-permittivity materials for use as dielectric resonators. Titanium dioxide (TiO2 ) spheres are commonly grown in nanometer sizes for colloidal solutions, but not in micrometer sizes, therefore, it was necessary to synthesize titanium dioxide spheres on the appropriate size. A solgel sacrificial template method was used to construct the micron sized spheres where the titanium dioxide was formed by the hydrolysis of titanium isopropoxide [133], the chemical reaction for which is given in equation 3-1. Ti{OCH(CH3 )2 }4 + 2 H2 O −−→ TiO2 + 4 ( CH3 )2 CHOH

(3-1)

For the sol-gel procedure, two sizes of porous polydivinylbenzene spheres (polybeads) were used as structural templates: Source 30 RCM from GE Health Sciences which were 30micron-diameter and monodisperse1 that resulted in 10-micron diameter micro-spheres, and Amberchom CG300C from Dow Specialty Materials which were 120 microns in diameter and polydisperse that resulted in 36-micron average diameter micro-spheres. An overview of the synthesis steps for the titanium dioxide micro-spheres using the polybeads is shown in Figure 3-9. These steps are described next. Another technique in which titanium dioxide micro-spheres are formed directly from droplets of titanium isopropoxide was unsuccessful because titanium isopropoxide reacts readily with water and is miscible in many fluids, preventing droplet formation. 1

Particles having a small size distribution, essentially all the same size.

52

Figure 3-9. Synthesis procedure for titanium dioxide micro-spheres.

53 A quick reference of the micro-sphere fabrication procedure is given in Appendix 9.3, and the detailed steps are as follows. First: The stock solvent and residual water moisture were removed by drying the template polybeads in an oven for 48 hours at 60°C . Second:

The dried polybeads were placed into a nitrogen atmosphere dry box along with

titanium isoproproxide, the titanium dioxide precursor. An equal amount by weight of dry template polybeads and titanium isoproproxide, which was added dropwise, were combined in an air-tight container. The nitrogen dry box and sealed container prevented the premature hydrolysis of titanium isoproproxide to titanium dioxide. Third: The container with the mixture was gently vibrated for 24 hours by placing it on top of a warm agitator, where slight warming at around 30°C lowered the viscosity of the titanium isoproproxide, which improved the filling of the porous polybeads. Fourth: The coated template polybeads were poured into a large dish of water. 100 mL of water was used for every 2 grams of the precursor filled polybeads. The solution was stirred for 24 hours at room temperature. Fifth: The polybeads infilled with amorphous titanium dioxide were dried for 24 hours at 60°C. The micro-spheres were then calcined in atmosphere for 3 hours at 871°C (1600°F) to convert the amorphous titanium dioxide to rutile titanium dioxide. The complete transition to the rutile phase, which has the highest permittivity and density of the titanium dioxide crystal phases, occurs near 850°C [159,160].

54 3.7.3

Photolithographic Processing Of SU-8

The photoresist SU-8, a negative-image permanent epoxy resin that is polymerized by ultraviolet light, was used as the host material for the titanium dioxide micro-spheres. The processing steps for SU-8 photoresist were as follows: spin the resist on a wafer; soft-bake the resist; expose the resist to ultraviolet light through the desired mask; post-exposure bake the resist; and finally develop the resist pattern. These processing details are given in appendix 9.2.2. Different thicknesses of SU-8 resist were achieved by using various viscosity formulations and spin speeds. The thickness, as a function of spin speed, was measured on different wafer substrates because the thickness also depended on the surface of the substrate. The method used to obtain the resist height versus spin speed was to spin-coat SU-8 on the desired substrate and use an aluminum foil covered glass slide to mask the area where SU-8 was to be removed. The step height for the resists was measured using a KLA Tencor model Alpha-Step IQ surface profilometer. Graphs of the expected thickness versus spin speed of different SU-8 resists on given substrates are shown in appendix 9.2.2. The a data was fit using an inverse root function of: height(speed) = p where a is a fitting speed parameter [161]. Measured thicknesses of the SU-8 2025 did not fit well with the inverse root function and instead were fit with an offset exponential. The spin control for SU-8 2010 and 2025 was performed manually, so heights of the resist were measured for each sample when possible. Thick samples of SU-8 resist were made for determination of the material’s permittivity that were made by spin-coating multiple consecutive layers of SU-8. The permittivity

55 was needed because SU-8 was used as the host material for the resonators of several metamaterials. The sample’s edges were adhered to a plastic support ring for mounting during measurement as was described in Section 3.4. The patterning of SU-8 and of the micro-spheres required a smooth carrier during fabrication, and so inexpensive silicon wafers were used as backing. A sacrificial layer was used to remove the fabricated samples from the silicon wafers. AZ 1512, a positive-image resist, was tried as the sacrificial layer but was incompatible with SU-8 being spun on top of it. PMMA (poly[methyl methacrylate]) was tried as a sacrificial layer which was removed by dissolving the PMMA in acetone. This method was abandoned because PMMA etched poorly, leading to the destruction of many completed samples. The acetone also caused the SU-8 resist to curl and become brittle. The third, and most successful, sacrificial layer material used was polystyrene dissolved in toluene [162]. The polystyrene layer led to fewer defects in the resist layer spun on top of it than the previous methods had and could be dissolved overnight in toluene with no degradation of the SU-8 resist. Polystyrene was, therefore, used whenever a sacrificial layer for SU-8 was needed.

3.7.4

Titanium Dioxide 10-Micron-Diameter Micro-Sphere Arrangement

The synthesized 10-micron-diameter titanium dioxide microspheres were too small to be arranged into a pattern with standard wire-mesh screens; instead, a template pattern was formed in a host material. The titanium dioxide micro-spheres were arranged by directly spreading the micro-spheres on a an array of square cavities developed by photolithography in SU-8 photoresist.

56 Several interdependent factors made direct spreading of the micro-spheres the most successful assembly procedure. These factors were the stability of the solution, density of the spheres, density of the solution, cavity size, capillary forces, concentration of the spheres, surface charge, hydrophilicity of the spheres, hydrophilicity of the substrate, and flow or evaporation rate of the solvent. The stability of the solution was determined by the density of the micro-spheres compared with the density of the solvent, the surface charge on the spheres, the concentration of the spheres, and the hydrophilicity of the spheres. The stability of the solution determines the strength of the surface interaction effects. In this work the micro-sphere solution was unstable and the spheres sedimented readily in the solution. For comparison, when the solution is stable the spheres settle at the bottom very slowly or not at all. In stable solutions, when the solution de-wets from the surface, a density of spheres is built up at the meniscus. If the surface is hydrophobic then, meniscus curvature tends to lift the spheres away from the template preventing filling. If the surface is moderately hydrophilic, then meniscus curvature pins the spheres to the surface. Additionally, the spheres are pulled into template cavities and are able to be swept from unpatterned areas of the surface by capillary forces [163–167]. But, as mentioned, in unstable solutions the spheres sedimented readily in the solvent. When the hydrophilic behavior of patterned substrate is changed to aid the capillary forces in selective deposition [168–171], the spheres still need to have some buoyancy in the solution to be pulled from the template surface. The titanium dioxide micro-spheres had effectively no buoyancy in the solution and so were always in contact with the surface. Tilting the template so that gravity aids in spreading the micro-spheres across the surface was only marginally successful in this

57 work. The technique adopted was to deposit the micro-spheres on top of a level template. A dilute suspension of micro-spheres was deposited onto the template and was then spread over the pattern by using a plastic dropper as a wand to spread the solution. Excess water was removed and the template left to dry. When dry, the corner of lint-free wipe was used to wipe the spheres across the template, breaking sphere clumps, and further filling the cavities. After the template was filled, a thin layer of SU-8 was spun over the spheres to ensure that the micro-spheres were not removed in lift-off of the template from its support. Figures 3-10 and 3-11 are side and top views, respectively, of the patterned titanium dioxide microspheres. The side view indicates that the spheres were covered by a thin layer of SU-8 but their cavities were not completely filled with the resist.

Figure 3-10. Side view of the patterned 10-micron-diameter titanium dioxide microspheres.

58

Figure 3-11. Top view of the patterned 10-micron-diameter titanium dioxide microspheres. 3.7.5

Titanium Dioxide 36-Micron-Diameter Micro-Sphere Arrangement

The 36-micron-diameter micro-spheres were arranged in a square array with a center-tocenter spacing of 76 microns by using a U.S.-standard number 325 precision sieve. For patterning, the micro-spheres were poured over the sieve, and shaken side to side, depositing the spheres onto an un-cured film of SU-8 pressed directly onto the sieve. The SU-8 was then cured. The spheres were not top-coated. Figure 3-12 is a top view of the patterned titanium dioxide microspheres. The microspheres are adhered to a thin layer of SU-8 photoresist. A small portion of the each individual micro-sphere is embedded into the substrate but they are not covered with resist like the 10-micron micro-spheres were.

59

Figure 3-12. Top view of the patterned 36-micron-diameter titanium dioxide microspheres. 3.7.6

Micro-Sphere Patterning And Fabrication

The patterning of micro-spheres through a mechanical template, like a sieve, and into a template, like an array of cubic cavities, are both useful techniques. The sieve patterning only requires the availability or construction of the necessary sieve and precision placement of the receiving substrate. The patterning of spheres into a template has been shown to be viable for all sphere sizes, so long as an appropriate method of deposition is used. The use of polybeads as structural supports to form spheres in sizes of tens-of-microns of high permittivity materials was shown to be usable with polybeads of much larger sizes than the original technique as is shown in Section 4.1.

60

3.8

Construction Of Lithium Tantalate Micro-Rods

The fabricated micro-spheres were somewhat smaller than desired and, as described in Section 4.4.2, had too much loss at the frequency of resonance to have an effectively negative permeability. Thus fabricating larger resonators directly from a bulk material with lower material loss was attempted. The viability of using mechanical dicing was tested. Micro-rod resonators do not have an isotropic effective permeability, but the polarization of an incident electromagnetic wave with the electric field perpendicular to the rod’s axis will excite a magnetic resonance. A proof-of-concept set of micro-rods was built from lithium tantalate before investing in more intricate fabrication methods required to produce micro-cubes. Boston Piezo Optics in Bellingham, Massachusetts performed all machining. A lithium tantalate wafer was thinned by mechanical lapping and polishing, after which cutting was attempted using a dicing saw. On dicing the wafer it was found that the material would crumble while attempting to cut 100-micron-wide rods, so micro-rods with a target width of 200 microns were cut instead. The resulting rods averaged 190 microns wide by 85 microns thick and were approximately 1 cm long. The space of the material removed between the rods varied from 106 to 110 microns wide.

61 The micro-rods were received adhered by wax to a glass carrier. The wax was removed from the micro-rods with trichlorethylene and cleaned with acetone while the micro-rods were still on the carrier. The rods were not self-supporting so a substrate made of 49micron-thick transfer adhesive backed with 3-micron-thick mylar was used. The micro-rods were lifted directly from the carrier using the adhesive. The patterns, as manufactured, transferred to the substrate successfully shown in Figure 3-13. Figure 3-14 is a diagram of the construction process.

Figure 3-13. A lithium tantalate micro-rod sample on mylar carrier.

62

Figure 3-14. Construction steps of lithium tantalate micro-rod samples.

63

3.9

Construction Of Lithium Tantalate Micro-Cubes

In order to provide an isotropic effective negative permeability an attempt was made to fabricated cubes of 100 x 100 x 100 microns photolithographically from lithium tantalate, as a magnetic resonance for cubes is excited by both incident polarizations as opposed to only one polarization by micro-rods. The limits of mechanical dicing prevented small structures from being fabricated either in the required size or with the dimensional tolerances required to create resonators operating at higher terahertz frequencies. A method used to create small waveguides in lithium niobate (LiNbO3 ) [172] was adapted for lithium tantalate. The crystal structures of lithium tantalate at the faces of a z-cut wafer are different. The z-cut wafers used had a +z face with a positive remanent polarization and a −z face with a negative remanent polarization. The −z face is readily etched while the +z face is not. A combination of photolithographic and wet etching techniques allowed for micro-cube resonators to be fabricated and patterned in a single step with the desired dimensions. An overview of the fabrication steps for the lithium tantalate micro-cube samples is shown in Figure 3-15, and is described in detail next.

64

Figure 3-15. Construction steps of lithium tantalate micro-cube samples.

65 The fabrication steps for lithium tantalate micro-cubes are as follows. First:

Lithium tantalate 16 mm squares were diced from 3-inch wafers by Innovative

Fabrication in Lowell, Massachusetts. The squares were thinned to 100 microns by mechanical lapping and polishing by Aptek in San Jose, California. Second:

The lithium tantalate squares was lithographically patterned for etching. The

etch mask was a square array of 100 micron squares with a center-to-center spacing of 150 microns. Each grouping was 11 mm by 11 mm on each a with a 2 mm outside frame for support; the whole pattern was 15 mm square. To create the lift-off layer for the metal deposition, AZ 5214 negative image photoresist was spun-coated on the −z face, patterned, and developed. Once patterned, a 100 nm layer of chrome was vapor deposited by Research Services in Wilmington, Massachusetts. Then the photoresist and unwanted metal were lifted-off in an acetone spray. The chrome etch mask was annealed at 320°C for 3 hours to improve contact between the lithium tantalate and etch mask to prevent under etching [172]. The hot plate was ramped up to, and down from the annealing temperature, because it was found that test samples placed directly onto the hotplate at temperature displayed a large amount of visible electrical arcing across the material and often cracked. This electrical discharge occurred because lithium tantalate, being pyroelectric, would develop an electrical potential of several kilovolts across the material surface [173]. Third: The squares were re-poled to remove damage to the local domains at the −z face that resulted from polishing. This was done by placing the squares between two rubber gaskets in a fixture having two liquid cavities. A solution of graphite and salt mixed in

66 deionized water, which had a very low resistivity, was used to fill the cavities to provide uniform conductive contact across the surface. A voltage of 3.6 kV, equivalent to an electric field of 36 kV per mm, was applied at the two faces for 10 seconds and then reduced to 0 V over 2 seconds. The +z face was connected the negative side of the power supply. Fourth: In order to support the cubes after etching, a layer of PFA, a type of plastic, was bonded to the +z face of the lithium tantalate squares by melting a 20 micron sheet of PFA at 320°C onto the surface. The squares were clipped to high density polyethylene blocks for etching. Fifth: The lithium tantalate was etched using 48 percent hydrofluoric acid (HF)2 . The chemical reaction of HF with lithium tantalate at the −z face is given by equation 32 [174–176]. There was virtually no etching of the +z face because the lithium and tantalum have stronger bonds to the oxygen on the +z face. The ±x and ±y crystal faces also etch at rates far lower than the −z face and any under-etching was not noticeable [172,174,177–179]. LiTaO3 + 6 HF −−→ LiF(aq) + Ta (V) F5 (aq) + 3 H2 O Sixth: The chrome was removed with chrome etchant

2

Hydrofluoric acid can be lethal. Take all necessary precautions before beginning any work.

(3-2)

67 The lithium tantalate cubes were not etched satisfactorily and are further discussed in Section 5.4. Figure 3-16 is a composite microscope image of partially etched cubes. Figure 3-17 shows the entire pattern, back illuminated, in the sample holder.

Figure 3-16. Top view of partially etched lithium tantalate micro-cubes.

68

Figure 3-17. View of partial etched lithium tantalate micro-cubes.

3.10

Construction Of The Micro-Cube Groups And Combined Wire Grid Samples

The lithium tantalate micro-cube lattice was to be combined with an interleaved copper wire grid to create a simultaneously negative permittivity and negative permeability. The wire grid was constructed using electroplating and purchased commercially from Precision Eforming in Cortland, New York, and supplied in 3-inch square sections. Figure 3-18 shows a section of the wire grid. The wires in the grid had trapezoidal cross sections having sloped side walls. The wires were 13 microns thick, 28 microns wide at the base, and 12 microns at the top. The center-to-center spacing of the wires was 150 microns to match the spacing of the lithium tantalate micro-cube array. The grids were cut into 11 mm squares using a plastic template as a scalpel guide.

69

Figure 3-18. Photograph of the copper wire grid. Due to fabrication difficulties discussed in Section 5.4, a THz metamaterial with a negative permeability using micro-cube resonators was unsuccessful. The copper wire grid was to be placed between the cubes and separated from the substrate and held in place using thin wire spacers. Figure 3-19 shows a diagram of the planned micro-cube and wire grid assembly. This combined structure would then have a simultaneously negative permeability and negative permittivity to produce a THz NIM. The predicted response is shown later in the results section.

70

Figure 3-19. Single layer assembly diagram of lithium tantalate micro-cubes and wire grid.

71

IV.

RESULTS

This section presents the synthesis results of the titanium dioxide micro-spheres; construction results of other samples was shown at the end of the their respective methodology sections. Measurements are shown for the microwave metamaterials, as well as the terahertz micro-sphere, and micro-rod metamaterials; also, predicted transmittance for the micro-cube metamaterial and combined micro-cube and wire-grid negative index of refraction metamaterial.

4.1

Titanium Dioxide Micro-Sphere Synthesis

Titanium dioxide (TiO2 ) spheres were synthesized using a sacrificial template as described in Section 3.7.2 using both 30-micron-diameter Source 30 RCM polymer micro-spheres and 120-micron-diameter Amberchom CG300C polymer micro-spheres. For the finished micro-spheres the desired diameter was tens of microns and that they be of the rutile phase. The micro-spheres were examined with a light microscope, a scanning electron microscope, and by X-ray powder diffraction. 4.1.1

10-Micron-Diameter Titanium Dioxide Micro-Spheres

Figure 4-1 is a photograph of the Source 30 RCM polymer micro-spheres mixed with titanium dioxide before calcining that had a diameter of 30 microns. Figure 4-2 is a photograph of the titanium dioxide micro-spheres after calcining, which had an average diameter of 10 microns. Figure 4-3 is a scanning electron microscope image of a titanium dioxide micro-sphere after calcining.

72

Figure 4-1. 30-micron-diameter titanium dioxide and polymer micro-spheres before calcining.

Figure 4-2. 10-micron-diameter titanium dioxide micro-spheres after calcining.

73

Figure 4-3. Scanning electron microscope image of a 10 micron titanium dioxide microsphere.

74 X-ray Powder Diffraction X-ray powder diffraction measurements of the 10 micron titanium dioxide micro-spheres indicated that the spheres had been converted entirely to the rutile crystal phase. The diffractogram was matched to AMSCD card 0005164 Rutile [180]. Figure 4-4 is the X-ray powder diffractogram for the 10-micron spheres and shows a strong rutile peak at 27.5°, a complete absence of an anatase phase peak at 25.3°, and also a low amorphous content.

Figure 4-4. X-ray powder diffractogram of 10-micron-diameter titanium dioxide microspheres.

75 4.1.2

36-Micron-Diameter Titanium Dioxide Micro-Spheres

Figure 4-5 is a photograph of the Amberchom CG300C polymer micro-spheres mixed with titanium dioxide before calcining that had an average diameter of 120 microns. Figure 4-6 is a photograph of the titanium dioxide micro-spheres after calcining, which had an average diameter of 36 microns. Figure 4-7 is a scanning electron microscope image of a 36 micron titanium dioxide micro-sphere after calcining.

Figure 4-5. 120-micron-diameter titanium dioxide and polymer micro-spheres before calcining.

76

Figure 4-6. 36-micron-diameter titanium dioxide micro-spheres after calcining.

Figure 4-7. Scanning electron microscope image of a 36 micron titanium dioxide microsphere.

77 X-ray Powder Diffraction The diffractogram of the 36 micron titanium dioxide micro-spheres shown in Figure 4-8 was matched to AMSCD card 0005164 Rutile [180]. The crystal phase of the micro-spheres was 100 percent rutile by weight.

Figure 4-8. X-ray powder diffractogram of 36-micron-diameter titanium dioxide microspheres.

78

4.2

Measurement Of The Microwave Samples

This section presents the transmittance and reflectance of microwave samples measured using the network analyzer described in Section 3.3. These measurements were performed to verify the effective medium theories on macroscopic meta-materials. 4.2.1

Embedded Sphere Waveguide Samples

The transmittance of the yttria-stabilized zirconia array of spheres embedded in Task 4, fabricated as described in Section 3.6.1, is shown in Figure 4-9 and the reflectance in Figure 4-10. Figure 4-11 is the corresponding permittivity and permeability determined using effective medium theory. The permeability was effectively negative from 26 to 27 GHz. Although the structure size of the material is comparable to the wavelength and the permittivity and permeability no longer accurate above 20 GHz the permittivity and permeability were still phenomenologically correct. Cuts in the continuity of the graph are from using different samples for each waveguide band.

Figure 4-9. Transmittance of embedded sphere waveguide samples.

79

Figure 4-10. Reflectance of embedded sphere waveguide samples.

Figure 4-11. Permittivity and permeability of embedded sphere waveguide samples.

80 4.2.2

Embedded Wire Array Waveguide Samples

For the copper wire array embedded in polycarbonate metamaterial, fabricated as described in Section 3.6.2, the transmittance and reflectance are shown in Figure 4-12. Figure 4-13 shows the corresponding permittivity of the sample as determined using effective medium theory. The permittivity was negative below 11 GHz; the permeability was unity.

Figure 4-12. Transmittance and reflectance of the embedded wire array sample.

Figure 4-13. Permittivity and permeability of the embedded wire array sample.

81 4.2.3

Yttria Stabilized Zirconia Sphere Sample For Free-Space

For the yttria-stabilized zirconia array of spheres metamaterial for free-space, built as described in Section 3.6.3, the transmittance is shown in Figure 4-14 and the reflectance in Figure 4-15. The host material was effectively air so the ratio of wavelength to the size of the sphere was large enough that effective medium theory could be used to determine the permittivity and permeability through the resonance region around 27 GHz. After 30 GHz effective medium theory was no longer accurate. Figure 4-16 shows the corresponding permeability and permittivity of the sample as determined using effective medium theory. The permeability was negative from 28 to 30 GHz.

Figure 4-14. Transmittance of the yttria-stabilized zirconia sphere array.

82

Figure 4-15. Reflectance of the yttria-stabilized zirconia sphere array.

Figure 4-16. Permittivity and permeability of the yttria-stabilized zirconia sphere array.

83 4.2.4

Combined Yttria Stabilized Zirconia Sphere And Wire Array Sample For Free-Space

The yttria-stabilized zirconia sphere array for free-space could be treated like an effective medium through the resonance region, and so was combined with 24-gage copper wires. When the electric field was polarized parallel with the wires an effective negative permittivity would result. The transmittance for this polarization is shown in Figure 4-17 and the reflectance in Figure 4-18. Figure 4-19 is the corresponding permittivity and permeability as determined using effective medium theory. The permeability was negative from 28 to 30 GHz and the permittivity negative below 29 GHz. The metamaterial had a negative index of refraction from simultaneously negative permeability and negative permittivity between 28 to 29 GHz.

Figure 4-17. Transmittance of the yttria-stabilized zirconia sphere array and wire array with the electric field polarized parallel with the with the wires.

84

Figure 4-18. Reflectance of the yttria-stabilized zirconia sphere array and wire array with the electric field polarized parallel with the wires.

Figure 4-19. Permittivity and permeability of the yttria-stabilized zirconia sphere array and wire array with the electric field polarized parallel with the wires.

85

4.3

Measurement Of Material Samples At Terahertz Frequencies

The measurements at microwave frequencies demonstrated that the material models and effective medium theories can be used to predict and analyze the behavior of meta-materials. The same techniques can be used at THz frequencies once the electromagnetic behavior of the materials is known. Materials were measured using a FTIR as described in Section 3.4. The relative permeability was unity for all materials in this section. 4.3.1

Transfer Adhesive

The transfer adhesive was modeled using the simple polynomial model, Equation 2-1. The transfer adhesive was found to have an effectively constant permittivity of 1.94 + i0.11 from 500 GHz to 3 THz. The fitting values are given in Table 4-1. Table 4-1. Parameters for transfer adhesive in equation 2-1. η0 η1 η2 η3 4.3.2

1.94 0 0 0

α0 α1 α2 α3

+0.11 0 0 0

PFA

PFA (Perfluoroalkoxy) plastic was modeled using the simple polynomial model, Equation 21. PFA was found to have an effectively constant permittivity of 2.07 + i0.00 from 200 GHz to 3 THz [181]. The fitting values are given in Table 4-1. Table 4-2. Parameters for PFA in Equation 2-1. η0 η1 η2 η3

2.07 0 0 0

α0 α1 α2 α3

0 0 0 0

86 4.3.3

Mylar

Mylar was modeled with the simple polynomial model, Equation 2-1. The fitting values are given in Table 4-3. The permittivity as a function of frequency is shown in Figure 4-20. Table 4-3. Parameters for Mylar in Equation 2-1. η0 η1 η2 η3

3.15 −3.0 × 10−06 (m) 0 0

α0 α1 α2 α3

+0.03 +3.4 × 10−06 (m) −2.5 × 10−11 (m2 ) 0

Figure 4-20. Permittivity of Mylar.

87 4.3.4

SU-8

SU-8 was modeled using the frequency form of the single-resonance harmonic oscillator, Equation 2-2. There was a resonant absorption peak at 8.7 THz. The fitting values are given in Table 4-4. The permittivity as a function of frequency is plotted in Figure 4-21 and was consistent with previously reported literature values [182–184]. Table 4-4. Parameters for SU-8 in Equation 2-2. low high f0 γ

2.75 + i0 2.70 + i0 8.7 THz 180 GHz

α0 α1 α2 α3

+0.205 −3.2 × 10−07 (m) −5.5 × 10−12 (m2 ) +2.1 × 10−17 (m3 )

Figure 4-21. Permittivity of SU-8.

88 4.3.5

Lithium Tantalate

Lithium tantalate is uniaxial with the optical axis along the z axis, and was modeled using the wavenumber form of the single-resonance harmonic oscillator, Equation 2-3. The permittivity for the ordinary-axis is plotted in Figure 4-22. The fitting values are given in Table 4-5 and are consistent with previously reported literature values [137,185]. Table 4-5. Parameters for lithium tantalate in Equation 2-3. Ordinary Axis ( k )

Extraordinary Axis ( ⊥ )

dc ∞ S0 ν0 Γ0

dc ∞ S0 ν0 Γ0

41.4 13.4 28 142 (cm−1 ) 11 (cm−1 )

40.4 8.4 32 200 (cm−1 ) 45 (cm−1 )

Figure 4-22. Permittivity of lithium tantalate along the ordinary ( k ) axis.

89

4.4

Measurement Of The Micro-Sphere Samples

Micro-sphere-based metamaterials for THz frequencies were constructed as described in Section 3.7. This section presents measurements using a FTIR as described in Section 3.4.

4.4.1

Yttria Stabilized Zirconia Micro-Spheres

The transmittance of the yttria-stabilized zirconia micro-sphere metamaterial is shown in Figure 4-23, and the permittivity and permeability in Figure 4-24, which was determined using effective medium theory. The dip in the transmission around 0.5 THz indicates a resonance, although the permittivity did not reach negative values as discussed in Section 5.1.3.

Figure 4-23. Transmittance of the yttria-stabilized zirconia micro-sphere metamaterial.

90

Figure 4-24. Permittivity and permeability of the yttria-stabilized zirconia micro-sphere metamaterial.

91 4.4.2

Titanium Dioxide Micro-Spheres

To determine both the permittivity of the bulk micro-sphere material the sphere permittivity was estimated from effective medium theory by assuming air spheres as inclusions in homogenous titanium dioxide. First, the permittivity of polycrystalline rutile titanium dioxide was determined using the multiple resonance model, Section 2.1.4, using fitting parameters obtained from the literature, given in Table 4-6 [138]. Then, the average permittivity of titanium dioxide was obtained by weighting the contribution of randomly oriented axes for a uniaxial material, given by Equation 4-1 [138]. The permittivity for randomly oriented polycrystalline rutile titanium dioxide is shown in Figure 4-25. Lastly, effective medium theory for spheres, Section 2.2.1, was used. Air spheres having a complex permittivity were assumed to be mixed into homogeneous polycrystalline titanium dioxide. The air cavities in the micro-spheres lowered the effective permittivity and increased loss.

 = 32 c⊥ + 13 c k

(4-1)

Table 4-6. Parameters for polycrystalline titanium dioxide used in Equation 2-4. Axis

j

c⊥

1 2 3

ck

Transverse Mode ν jT O γ jT O −1 (cm ) (cm−1 )

Longitudinal Mode ν jLO γ jLO −1 (cm ) (cm−1 )

189 377 491

40 20 40

821 60 353 10 428 40 high (c ⊥) = 6.843

178±4

30

797 30 high (c k) = 8.427

92

Figure 4-25. Permittivity of polycrystalline rutile titanium dioxide without any air cavities. Once the permittivity of the titanium dioxide micro-spheres was determined the effective medium theory for spheres was again used to model the embedded sphere metamaterial. The permittivity of the micro-spheres was also confirmed by using the modeled permittivity in HFSS simulations, and comparing the simulated results to measurements. The results from the effective medium theory show that not only can the permittivity of the embedded spheres can be recovered accurately, but that the model can be successfully compounded. The results for the 10-micron-diameter and 36-micron-diameter micro-spheres are shown next.

93 10-Micron-Diameter Spheres The resulting permittivity of just the titanium dioxide bulk sphere material of the 10 micron spheres, not the metamaterial, is shown in Figure 4-26. The parameters for the effective medium theory are given in Table 4-7.

Figure 4-26. Permittivity of the bulk titanium dioxide used to fabricate the 10 micron micro-spheres. Table 4-7. Parameters for effective medium model of the air and crystaline compound composing the titanium dioxide micro-spheres. η0 η1 η2 η3 diameter

Fitting parameters of air inclusions. 1.01 α0 0 0 α0 0 0 α2 0 0 α3 −2.0 × 10−14 (m3 ) Air inclusion parameters. 0.015 (nm) spacing 0.0145 (nm)

94 Transmittance measurements of the 10 micron titanium dioxide micro-sphere metamaterial are shown in Figure 4-27. The permittivity and permeability, as determined using effective medium theory, are shown Figure 4-28. The permeability does not reach negative values as is discussed in Section 5.2.2. The parameters used in the effective medium theory are given in Table 4-8, where the values for the host SU-8 were adjusted to compensate for the square pockets.

Figure 4-27. Transmittance of the 10 micron titanium dioxide micro-sphere metamaterial.

95

Figure 4-28. Permittivity and permeability of 10 micron titanium dioxide micro-sphere metamaterial.

Table 4-8. Parameters for effective medium model of the 10-micron-diameter embedded titanium dioxide micro-sphere metamaterial. low high f0 γ diameter

Fitting parameters of host SU-8. 2.4 + i0 α0 0 2.35 + i0 α1 +1.5 × 10−06 (m) 8.7 THz α2 0 180 GHz α3 0 Micro-sphere parameters. 10 (um) spacing 20 (um)

96 36-Micron-Diameter Spheres The resulting permittivity of just the titanium dioxide bulk sphere material of the 36 micron spheres, not the metamaterial, is shown in Figure 4-29. The parameters for the effective medium theory are given in Table 4-9.

Figure 4-29. Permittivity of the bulk titanium dioxide used to fabricated the 10 micron micro-spheres. Table 4-9. Parameters for effective medium model of the air and crystaline compound composing the titanium dioxide micro-spheres. η0 η1 η2 η3 diameter

Fitting parameters of air inclusions. 1 α0 12 0 α0 0 0 α2 0 0 α3 0 Air inclusion parameters. 0.101 (nm) spacing 0.091 (nm)

97 Transmittance measurements of the 36 micron titanium dioxide micro-sphere metamaterial are shown in Figure 4-30. The permittivity and permeability, as determined using effective medium theory, are shown Figure 4-31. The permeability does not reach negative values as is discussed in Section 5.2.2. The parameters used in the effective medium theory are given in Table 4-10, the lattice spacing of the spheres was not the spacing of the template sieve because measurements were taken in a poorly patterned region. Also neither HFSS nor the effective medium theory take into account the polydispersity of the spheres. Unlike the 10 micron sized spheres the transmission predicted by HFSS did not match those of the effective medium theory, which is most likely from the whole structure not being well approximated by the effective medium theory.

Figure 4-30. Transmittance of the 36 micron titanium dioxide micro-sphere metamaterial.

98

Figure 4-31. Permittivity and permeability of 36 micron titanium dioxide micro-sphere metamaterial.

Table 4-10. Parameters for effective medium model of the 36-micron-diameter patterned titanium dioxide micro-sphere metamaterial. low high f0 γ diameter

Fitting parameters of host SU-8. 1.24 + i0 α0 0.012 1.19 + i0 α1 0 8.7 THz α2 −1.0 × 10−11 (m2 ) 180 GHz α3 0 Micro-sphere parameters. 36 (um) spacing 50 (um)

99

4.5

Measurement Of The Lithium Tantalate Micro-Rods

The lithium tantalate micro-rod metamaterial exhibited a resonance around 340 GHz when the electric field was polarized perpendicular to the rod axis. For this polarization the permittivity, which is negative from 0.33 to 0.38 THz, is shown in Figure 4-32 and the transmittance in Figure 4-33. For comparison, Figure 4-34 shows the transmittance for the electric field polarized parallel to the rod axis where no resonance is exhibited. The parameters for the effective medium theory, Section 2.2.1, are given in Table 4-11. Table 4-11. Parameters for effective medium model of lithium tantalate micro-rods. thickness 78 (um) width 195 (um) height 1000 (um)

lattice thickness 200 (um) lattice width 300 (um) lattice height 1000 (um)

thickness mode 1.1 width mode 1.1 height mode 0

Figure 4-32. Permittivity and permeability of lithium tantalate micro-rods for the electric field polarized parallel to the rods.

100

Figure 4-33. Transmittance of lithium tantalate micro-rods for the electric field polarized perpendicular to the rods.

Figure 4-34. Transmittance of lithium tantalate micro-rods for the electric field polarized parallel to the rods.

101

4.6

Simulation Of The Lithium Tantalate Micro-Cubes

An attempt to was made to fabricate micro-cubes of lithium tantalate as described in Section 3.9. Because no complete sample was fabricated, predictions for the transmittance and the permittivity and permeability are presented. Figure 4-35 shows the predicted transmittance and measured results and Figure 4-36 the corresponding the permittivity and permeability. The micro-cubes should exhibit a resonance at 390 GHz, where the permeability would be negative from 0.385 to 0.44 THz. The parameters for the effective medium theory for cubes, Section 2.2.2, are given in Table 4-12. Table 4-12. Parameters for effective medium model of lithium tantalate micro-cubes.

thickness 100 (um) width 100 (um) height 100 (um)

Fitting parameters of micro-cubes. lattice thickness 150 (um) thickness mode 1.0 lattice width 150 (um) width mode 1.0 lattice height 150 (um) height mode 1.0

Polynomial fitting parameters of host material. η0 1.14 α0 0.00

102

Figure 4-35. Transmittance of the lithium tantalate micro-cubes.

Figure 4-36. Permittivity and permeability of lithium tantalate micro-cubes.

103

4.7

Measurement Of The Wire Grid Samples

Shown in Figure 4-37 the transmittance of the copper wire grid described in Section 3.10. Figure 4-38 depicts the permittivity, which was negative below 0.590 THz, and permeability, which was unchanged from unity. The effective medium parameters for the wire grid are given in Table 4-13. According to the effective medium theory, the wire grid acts as if it had a high resistivity (300 Siemens per meter) compared to that of bulk copper (59.6 × 106 Siemens per meter), which also disagreed with the conductivity from HFSS (20,000 Siemens per meter). The low conductivity of the wire increases the loss of the wire grid. The peak around 0.480 THz is from a Fabry-Perot resonance between the parallel wire grids, where changing the grid separation changes the peak position.

Figure 4-37. Transmittance of copper wire grid.

104

Figure 4-38. Permittivity of copper wire grid.

Table 4-13. Parameters for wire grid in equation 2-11. Wire Radius σ

16 microns 300 Siemens per meter

Wire Spacing 150 microns Plasma Frequency 0.626 THz

105

4.8

Simulation Of The Combined Micro-Cube And Wire Grid

Lacking a complete micro-cube sample, predictions for the transmittance, permittivity, and permeability of the combined micro-cube and wire grid, Section 3.10, are presented. Figure 4-39 shows the predicted transmittance and Figure 4-40 the permittivity and permeability of the combined lithium tantalate micro-cubes and wire grid. The differences between predictions is likely from the wire grid structure not being well represented in the effective medium theory. The transmission pass-band was expected near 0.390 THz resulting from a simultaneously negative permittivity and negative permeability. Although the permittivity was not negative over the resonance region, a negative refractive index was obtained, which is from im µre in Equation 4-2 being both negative and having a magnitude larger than re µim leading to the negative sign in the index of refraction because a positive κ is required in a lossy medium. The parameters for the combined effective medium model are given in Table 4-14, where the results of the medium theory for cubes was used in the effective medium theory for wires. p n + iκ = ± (re µre − im µim ) + i(re µim + im µre )

(4-2)

106

Figure 4-39. Transmittance of lithium tantalate micro-cubes and wire grids.

Figure 4-40. Permittivity and permeability of the lithium tantalate micro-cubes and wire grids.

107 Table 4-14. Parameters for effective medium model of the combined lithium tantalate micro-cubes and wire grid. Lithium tantalate cube parameters. thickness 100 (um) lattice thickness 150 (um) thickness mode 1.0 width 100 (um) lattice width 150 (um) width mode 1.0 height 100 (um) lattice height 150 (um) height mode 1.0 Wire Radius σ η0

Wire grid parameters. 16 (um) Wire Spacing 150 (um) 300 Siemens per meter Polynomial fitting parameters of host material. 1.14 α0 0.00

108

V.

DISCUSSION

The results of the sample measurements beginning with the microwave samples are discussed. Then the terahertz frequency measurement and simulation results of the microsphere, micro-rod, micro-cube, wire-grid, and negative refractive index sample of the combined cube and wire grid are discussed.

5.1

Permittivity And Permeability Of The Microwave Samples

The fabricated microwave samples demonstrated successful application of theory, numerical simulation, and scalability of the metamaterials.

5.1.1

Embedded Spheres For Waveguide

The yttria stabilized zirconia embedded sphere samples built for waveguide, described in Section 4.2.1, showed that the effective medium theory for a regular array of spheres can provide accurate results when the size of the spheres is sufficiently small compared to the wavelength in the surrounding material. Indeed, in the long-wavelength limit the permittivity and permeability can be determined using the Clausius-Mossotti relation for macroscopically-sized spheres arranged in a cubic lattice [186,187]; however, Lewin’s effective medium theory, described in Section 2.2.1, must be used as the wavelength in the material approaches the resonance wavelength of the spheres, as is the case in this work.

109 5.1.2

Embedded Wire Array For Waveguide

The embedded copper wire structures for waveguide, Section 4.2.2, show that the effective medium theory for a wire array, Section 2.2.3, can provide accurate results when the assumptions of the theory are met. The results also show that the theory is accurate and that an effective negative permittivity results even when the wires are surrounded by material with a relative permittivity different from unity. 5.1.3

Yttria-Stabilized Zirconia Sphere Grid For Free-Space

The yttria-stabilized zirconia sphere metamaterial for free-space, Section 4.2.3, used a sphere array in air. This model showed that effective medium theory is accurate through the resonance region when the wavelength in the surrounding medium is sufficiently long compared to the electrical size of the spheres. The permittivity of the yttria stabilized zirconia spheres was found to be 33 + i 0.01 at 30 GHz, which is consistent with values reported in the literature [188,189]. 5.1.4

Yttria Stabilized Zirconia Sphere And Wire Array For Free-Space

Since the yttria stabilized zirconia sphere array could be treated as an effective medium when suspended in air, the spheres were combined with a wire array to provide a negative refractive index at resonance, Section 4.2.4. The effective medium theory was reasonably accurate through the resonance region and the determined permittivity and permeability were both negative from approximately 26 to 28 GHz as shown in Figure 4-19. This is the first demonstration of a dielectric-resonator based NIM in free-space at micro-wave frequencies.

110 Differences between the measured and effective medium theory predicted transmittance and reflectance are from several causes. Firstly, the structure-size to compared the wavelength was reached near 30 GHz. In addition, interaction effects between the spheres and wires, because of the close packing in this sample, was not accounted for in the effective medium theory. Also, interaction of the wires with the spheres and the wires with the transfer adhesive paper required that the modeled plasma frequency of the wire be lowered to 42 GHz, from an expected 47 GHz, to adequately model the sample. This also has the effect of raising the wires’ permittivity. The result of the sphere effective medium theory was used as the host material in the wire effective medium theory. The transmittance and reflectance measurements differed from the HFSS simulations due to sample imperfections such as a bowed surface, varying sphere size, and missing spheres important at higher operational frequencies. Lastly, the highest attainable reflectance should be 0 dB after calibration. There were a few places in spectra, however, where the measured reflectance is above 0 dB. This is because a partially bowed surface can focus the electromagnetic wave energy compared to a flat plate, which does not have this effect.

5.2

Permittivity And Permeability Of The Micro-Sphere Samples

The micro-sphere samples demonstrated that, given the right fabrication techniques, high permittivity spheres could be used as resonators at terahertz frequencies.

111 5.2.1

Yttria Stabilized Zirconia Micro-Spheres

Yttria stabilized zirconia micro-spheres that were ten times smaller than those used at microwave frequencies were arranged into a square lattice, Section 4.4.1. If the permittivity of the yttria-stabilized zirconia was the same as those used at microwave frequencies then the resonance frequency would be ten times higher than the microwave sample because the resonance frequency is inversely proportional to the size of the sphere. However, the permittivity of the micron sized spheres was lower than that of the millimeter sized spheres, and was found to be too low to create a THz material with a simultaneously negative effective permittivity and permeability. The resonance location of the spheres was unaffected by patterning disorder so the determination of the permittivity of the spheres at resonance was unaffected. The permittivity of the micro-spheres was found to be 10+i 0.38 at 0.5 THz, which is not consistent with the permittivity of yttria-stabilized zirconia at terahertz frequencies [190,191] nor at microwave frequencies [188,189]. The permittivity depends on the amount of yttria doping. So X-ray diffraction was performed on the micro-spheres to help determine the reason. The spheres were determined to be seventy percent in the monoclinic state, matching AMSCD card 018731 Baddel, and the remaining amount in the cubic phase, matching AMSCD card 021723 Tazher. The monoclinic phase has a lower permittivity than the yttria-stabilized cubic phase [188]. The mixed phases indicated inhomogeneous doping, and the mixed phases is most likely the cause of the low permittivity observed.

112 5.2.2

Titanium Dioxide Micro-Spheres

The micro-spheres of titanium dioxide were made using a polysphere template, Section 3.7.2. To obtain the highest permittivity from titanium dioxide it was important that the micro-spheres be of the rutile crystal phase. For comparison the average permittivity of rutile is 100 [159], brookite is 78 [192], and anatase is 38 [193], and the density of rutile is around 4.26 g/cc, brookite is 4.17 g/cc, and anatase is 3.84 g/cc [194, p.4-113]. By calcining the micro-spheres at temperatures above the crystal phase transition point the micro-spheres could be converted from an amorphous phase to the desired crystal phase. Calcining removed the polysphere template and increased the density of the entire sphere. 10-Micron-Diameter Titanium Dioxide Micro-Spheres The 10-micron-diameter titanium dioxide spheres fabricated using polysphere templates had residual pores, Section 4.1, that decreased the real part of the permittivity from that of homogeneous rutile titanium dioxide. Additionally the polycrystalline nature [138] of the micro-spheres and the porosity of the spheres increased the extrinsic scattering.1 In the effective medium theory for spheres an average diameter of 15 nanometers was assumed for the air-pockets. Additionally, while technically a violation of the effective medium theory model requiring non-connected spheres, a lattice spacing larger than the average diameter was chosen to represent the fact the air spheres were interconnected. The validity of the results of the model was confirmed by using the modeled permittivity of the spheres both in the effective medium for spheres applied to the metamaterial and in HFSS simulations. 1

As opposed to intrinsic scattering which is a function of the material itself.

113 The 10 micron titanium dioxide micro-spheres were found to have a permittivity of 62 + i 20 at the resonance position of 3.65 THz, Section 4.4.2. While the real part of the permittivity was sufficiently high to be an effective medium material, the imaginary part damped out the resonance inhibiting a negative permeability. 36-Micron-Diameter Titanium Dioxide Micro-Spheres The 36-micron-diameter titanium dioxide micro-spheres were modeled the same way as the 10-micron-diameter spheres using the embedded medium for spheres and having air pockets inside homogeneous titanium dioxide. Using the combined effective medium theory approach, the 36 micron spheres were found to have a permittivity of 25.5 + i 10 at the resonance position of 1.55 THz, Section 4.4.2. For unknown reasons the permittivity of these spheres is significantly lower than the smaller 10-micron-diameter spheres of titanium dioxide. The 36 micron spheres have larger pores and may have a large fraction of air pockets despite appearing very smooth on the surface. The air pockets were also required to a have large permittivity value to fit the measured data, which may be a result of choosing the wrong size pockets. The resonance was also damped by losses, however, it is the lower real part of the permittivity that prevented a negative permeability in this case.

114

5.3

Permittivity And Permeability Of The Micro-Rod Samples

The lithium tantalate micro-rods exhibited a resonance based on the polarization of the electric field with respect to the cylinder axis, Section 4.5. For the sample constructed, the resonance occurred when the electric field was perpendicular to the cylinder axis [18,94, 96,117,195]. The electromagnetic field intensity, plotted using HFSS, of the resonance is shown in Figure 5-1, where the solid black lines indicate the outline of the rod.

(a) Magnetic field intensity

(b) Electric field intensity

Figure 5-1. Diagram of the electromagnetic fields of the micro-rod sample at resonance. The solid black lines indicate the outline of the rod.

The majority of rod sets had their resonance at 0.330 THz, but the resonance was sensitive to changes in rod thickness [117]. When the electric field was polarized perpendicular to the rod axis the resonance was strong enough to create a negative permeability as was shown in Figure 4-32. However, a negative index of refraction material was not built because the structure of the material did not allow interleaving metal wires to provide a simultaneously negative permittivity. When the electric field was polarized parallel to the rod axis no resonance was observed. The increase in transmission around 0.330 THz is from the standard Fabry-Perot behavior of a plane parallel sample.

115

5.4

Predictions and Fabrication Of The Micro-Cube Samples

The predicted transmittance, permittivity, and permeability of the lithium tantalate microcubes, Section 4.6, was found to exhibit an effectively negative permeability from 0.390 THz to 0.440 THz. The electromagnetic field intensity, plotted using HFSS, of the resonance is shown in Figure 5-2, where the solid black lines indicate the outline of the rod.

(a) Magnetic field intensity

(b) Electric field intensity

Figure 5-2. Diagram of the electromagnetic fields at resonance of the micro-cube sample. The solid black lines indicate the outline of the rod. Unfortunately it was not possible to satisfactorily fabricate lithium tantalate microcubes despite various efforts. In a typical case much of the chrome was lifted from the surface by either under-etching the lithium tantalate, directly by etching underside of chrome itself. Using a titanium adhesion layer for the gold was equally ineffective, and using gold without an underlayer did not stay adhered to the lithium tantalate for very long. It was also found that the polishing done by one company, Aptek in San Jose, CA, caused the polished −z face to become very resistant to etching from inversion of the local crystal

116 domains from polishing damage, where as those polished by another company, Boston Piezo Optics in Bellingham, MA, did not exhibit this effect. This problem could be seen during etching, when many small negative domain areas were eventually trapped by nearby positive domains. The one semi-cubic sample produced had cubes near 50 microns tall instead of the desired 100 microns, so the measured resonance (not shown) was higher than desired. Additionally the structure showed a great deal of absorbance, probably due to large variances in the structures.

5.5

Permittivity Of The Wire Grid Samples

The copper wire grids had an negative effective permittivity up to 0.590 THz, Section 4.7. Whether due to the shape of the individual wires or the method of fabrication, the copper wire grid had a lower conductivity than bulk copper, which increased the effective permittivity. Due to this increase in permittivity an alternate wire grid may have to be fabricated for use with THz resonators.

5.6

Permittivity And Permeability Of The Micro-Cube And Wire Grid Samples

The lithium tantalate micro-cubes if combined with a wire grid, Section 3.10, should have a negative refractive index from 0.380 THz to 0.440 THz. The refractive index n is expected to be −1 at approximately 0.395 THz where the loss factor κ would be 2.2; the material’s n figure of merit, defined as , would be 0.45. κ

117

VI.

CONCLUSION

In attempting to produce a dielectric-resonator based NIM for THz frequencies a number of fabrication and patterning techniques were exploited to create new metamaterials. Figure 6-1 shows the size and type of resonators produced in this work in comparison to relative availability of high dielectric materials as a function of frequency.

Figure 6-1. Conceptual diagram of availability by size of high permittivity particles including materials used, fabricated (—), and attempted (- - -) in this work.

6.1

Microwave Frequency Macro-Sphere Material

A microwave frequency NIM for free-space was fabricated from commercially available yttria-stabilized zirconia spheres. The real part of the permittivity was high enough and the imaginary part low enough to provide an effectively negative permeability at resonance. When combined with a wire array, a negative index of refraction was demonstrated. These spheres have the advantage of being readily available and inexpensive.

118

6.2

THz Micro-Sphere Materials

Commercially produced yttria-stabilized spheres and custom synthesized titanium dioxide spheres were used to produce metamaterials with magnetic resonances at THz frequencies.

6.2.1

Yttria-Stabilized Zirconia Micro-Sphere Material

Yttria-stabilized zirconia has a permittivity high enough to produce a negative permeability. The micro-spheres obtained, however, had a low permittivity and could not be used to produce a negative permeability resonance, Section 4.4.1. Should micro-spheres with the expected high permittivity be available this would be a candidate material for THz dielectric resonators.

6.2.2

Titanium Dioxide Micro-Sphere Material

Two sizes of micro-spheres of titanium dioxide were produced, Section 3.7.2. The synthesis was done using a sacrificial porous polymer micro-sphere onto which a precursor to titanium dioxide was coated. The precursor was hydrolyzed to create titanium dioxide. The mixed spheres were calcined, leaving compacted rutile phase titanium dioxide spheres. The smaller 10-micron-diameter micro-spheres were arranged using a template photolithographically patterned into the host material. The solution of spheres was left to dry on the template. When dry the micro-spheres were spread with a lint-free towel, which broke apart clumps and further filled the cavities. These spheres were found to have a magnetic resonance at 3.65 THz, Section 4.4.2. Losses in the spheres damped out the resonance preventing a negative permeability, however.

119 The larger 36-micron-diameter micro-spheres were arranged onto an adhesive substrate using a precision metal sieve. These spheres were found to have a magnetic resonance at 1.5 THz, Section 4.4.2. Losses in these micro-spheres also damped out the resonance. The micro-spheres produced are unique and not readily comparable to similar work, as spheres fabricated using this method were smaller, of the anatase phase, and used in catalytic studies. Spheres in similar size ranges typically have other problems associated with them such as: varying sizes, deformities, being hollow, or not produced in sufficient quantities. If the air cavities in the produced spheres were filled-in then a higher permittivity would be obtained, allowing the use of titanium dioxide spheres as THz dielectric resonators.

6.3

Micro-Rod Material

Micro-rods of lithium tantalate were fabricated by thinning a crystalline wafer to the desired height and dicing the strips using a wafer saw. The micro-rods were found to have a resonance that depended on the polarization of the electric field with respect to the rod axis. When the electric field was polarized perpendicular to the rod axis, a magnetic resonance exhibiting an effectively negative permeability was observable at 0.330 THz. Therefore lithium tantalate is a good candidate material for dielectric THz resonators; however, the rod structure does not allow interleaving wires to provide an effectively negative permittivity required for creating a NIM. The fragility of the material and the width of the saw blade prevent producing higher frequency resonators using mechanical dicing.

120

6.4

THz Micro-Cubes and Wire Grid Negative Index Material

A negative index of refraction material utilizing a lattice of high permittivity resonators and electrically conductive wires was designed to function at far-infrared frequencies. An attempt was made to fabricate a lattice of micro-cubes to provide a negative permeability at the first Mie resonance. The pattern and micro-cubes were to be obtained from lithium tantalate by etching fully polled lithium tantalate wafers. A chrome etch mask would be photolithography patterned and the wafer wet-etched in hydrofluoric acid. Various difficulties during each fabrication step often ruined entire batches of test samples, so only partially etched cubes were obtained. It was demonstrated that a copper wire grid, obtained commercially, provided a negative permittivity. The wire grid when combined with the cubes was predicted to create a NIM at terahertz frequencies ranging from 0.390 THz to 0.440 THz. This would be a dielectricresonator based metamaterial with an isotropic negative permeability and a uniaxial negative permittivity.

121

VII.

RECOMMENDATIONS

Recommendations for further work for the various metamaterials that are expected to prove the most beneficial are suggested. Titanium Dioxide Micro-Spheres The main problem with the titanium dioxide micro-spheres produced is with the porosity of the spheres. A method that reduces or removes the pores would increase the dielectric constant and usability of the spheres. This might be done by infilling complete spheres, filling a hollow-polymer sphere, or the most direct, but so far unsuccessful, method of producing the spheres directly from droplets of precursor. The size range of future spheres could be tailored by fabricating the polymer template spheres in the desired sizes, and tuning the ratio of precursor used to the weight polymer spheres used at each size range. Lithium Tantalate Micro-Cubes It was found that many substrates do not stay adhered to lithium tantalate during wet-etching. If a strongly adhering material is found, it has to meet the following restrictions: is resistant to hydrofluoric acid, does not peel from the lithium tantalate after exposure to hydrofluoric acid, and is transparent to electromagnetic radiation at terahertz frequencies. An alternate approach to the substrate may be to use a thicker lithium tantalate piece during etching and then back-grinding to expose and separate the cubes.

122 The problem with the etch mask is not well understood, as the method of using chrome or chrome and gold etch masks was demonstrated to work on lithium niobate, a very similar material. It is assumed that either residual photoresist under the deposited metal, and the inability to etch the polished surface contribute to the problems observed. Other observations suggest that if the surface is not smooth on the nanometer level and a good oxygen layer is not available for the chromium to bond to, then poor adhesion results between the chrome mask and the substrate. Re-poling the polished pieces would re-enable the etching of the polished face. To avoid an etch mask completely, the lithium tantalate could be selectively poled such that parts of the +z face become −z faces. Since the +z faces do not appreciably etch, they are essentially their own etch masks. The lithium tantalate in this case must be thick enough that the reverse domains on the opposite side do not etch through. The micro-cubes would be exposed and separated by back-grinding.

123

VIII.

REFERENCES

[1] V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of  and µ,” Soviet Physics Uspekhi, vol. 10, no. 4, pp. 509–513, Jan. 1968. [2] D. Felbacq and A. Moreau, “Direct evidence of negative refraction at media with negative  and µ,” Journal of Optics A: Pure and Applied Optics, vol. 5, pp. L9–L11, Mar. 2003. [3] G. V. Eleftheriades and K. G. Balmain, eds., Negative Refraction Metamaterials: fundamental properties and applications. Hoboken, New Jersey: John Wiley and Sons Inc, 2005. [4] C. G. Parazzoli, R. B. Greegor, J. A. Nielsen, M. A. Thompson, K. Li, A. M. Vetter, M. H. Tanielian, and D. C. Vier, “Performance of a negative index of refraction lens,” Applied Physics Letters, vol. 84, no. 17, pp. 3232–3234, Apr. 2004. [5] R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, D. C. Vier, S. Schultz, D. R. Smith, and D. Schurig, “Microwave focusing and beam collimation using negative index of refraction lenses,” IET Microwaves Antennas and Propagation, vol. 1, no. 1, pp. 108 – 115, Feb. 2007. [6] T. Driscoll, D. N. Basov, A. F. Starr, P. M. Rye, S. C. Nemat-Nasser, D. Schurig, and D. R. Smith, “Free-space microwave focusing by a negative-index gradient lens,” Applied Physics Letters, vol. 88, no. 8, p. 081101, Feb. 2006. [7] R. B. Greegor, C. G. Parazzoli, J. A. Nielsen, M. A. Thompson, M. H. Tanielian, and D. R. Smith, “Simulation and testing of a graded negative index of refraction lens,” Applied Physics Letters, vol. 87, p. 91114, Aug. 2005. [8] D. Bao, W. Qin, J. Tong, X. M. Yang, Q. Cheng, and T. J. Cui, “Gradient Index Metamaterials Based on Dielectric Disks,” in 2008 International Workshop on Metamaterials, vol. 3, (Nanjing, China), pp. 394–396, IEEE, Nov. 2008. [9] J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Physical Review Letters, vol. 85, no. 18, pp. 3966–3969, Oct. 2000. [10] D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Applied Physics Letters, vol. 82, no. 10, pp. 1506–1508, Mar. 2003. [11] N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect Metamaterial Absorber,” Physical Review Letters, vol. 100, no. 20, p. 207402, May 2008.

124 [12] D. R. Smith, “How to Build a Superlens,” Science, vol. 308, no. 5721, pp. 502–503, Apr. 2005. [13] J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science, vol. 312, no. 5781, pp. 1780–1782, June 2006. [14] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science, vol. 314, no. 5801, pp. 977–980, Nov. 2006. [15] U. Leonhardt, “Notes on conformal invisibility devices,” New Journal of Physics, vol. 8, no. 118, pp. 1–16, July 2006. [16] U. Leonhardt, “Optical Conformal Mapping,” Science, vol. 312, no. 5781, pp. 1777– 1780, June 2006. [17] B.-J. Seo, T. Ueda, T. Itoh, and H. Fetterman, “Isotropic left handed material at optical frequency with dielectric spheres embedded in negative permittivity medium,” Applied Physics Letters, vol. 88, no. 16, p. 16112, Apr. 2006. [18] L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental Observation of Left-Handed Behavior in an Array of Standard Dielectric Resonators,” Physical Review Letters, vol. 98, no. 15, p. 157403, Apr. 2007. [19] J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Physical Review Letters, vol. 76, no. 25, pp. 4773–4776, June 1996. [20] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” Journal of Physics: Condensed Matter, vol. 10, no. 22, pp. 4785–4809, June 1998. [21] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Physical Review Letters, vol. 84, no. 18, pp. 4184–4187, May 2000. [22] R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamarial,” Applied Physics Letters, vol. 78, no. 4, pp. 489–491, Jan. 2001. [23] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science, vol. 292, no. 5514, pp. 77–79, Apr. 2001.

125 [24] M. Hudliˇcka, J. Mach´acˇ , and I. S. Nefedov, “A Triple Wire Medium As An Isotropic Negative Permittivity Metamaterial,” Progress In Electromagnetics Research, vol. 65, pp. 233–246, 2006. [25] P. A. Belov, C. R. Simovski, and S. A. Tretyakov, “Two-dimensional electromagnetic crystals formed by reactively loaded wires,” PHYSICAL REVIEW E, vol. 66, p. 36610, 2002. [26] P. A. Belov, R. Marqu´es, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire median in the very large wavelength limit,” Physical Review B, vol. 67, no. 11, p. 113103, Mar. 2003. [27] I. Vendik, O. Vendik, and M. Odit, “Isotropic Artifical Media With Simultaneously Negative Permittivity and Permeability,” Microwave and Optical Technology Letters, vol. 48, no. 12, pp. 2553–2556, Dec. 2006. [28] O. G. Vendik and M. S. Gashinova, “Artificial Double Negative (DNG) Media Composed by Two Different Dielectric Sphere Lattices Embedded in a Dielectric Matrix,” in 34th European Microwave Conference, (Amsterdam), pp. 1209–1212, Oct. 2004. [29] V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” Journal of Physics: Condensed Matter, vol. 17, no. 25, pp. 3717–3734, June 2005. [30] L. Jylha, I. Kolmakov, S. Maslovski, and S. A. Tretyakov, “Modeling of isotropic backward-wave materials composed of resonant spheres,” Journal of Applied Physics, vol. 99, no. 4, p. 43102, Feb. 2006. [31] I. A. Komakov, L. Jylha, S. A. Tretyakov, and S. Maslovski, “Lattice Of Dielectric Particles With Double Negative Response,” in Union Radio-Scientifique Interational, Proceedings of the General Assembly, pp. 1–4, 2005. [32] I. B. Vendik, O. G. Vendik, and M. A. Odit, “An isotropic metamaterial formed with ferroelectric ceramic spherical inclusions,” Physics of the Solid State, vol. 51, no. 8, pp. 1590–1594, Aug. 2009. [33] I. B. Vendik, M. A. Odit, and D. S. Kozlov, “3D isotropic metamaterial based on a regular array of resonant dielectric spherical inclusions,” Metamaterials, vol. 3, no. 3-4, no. 3-4, pp. 140–147, 2009. [34] M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Physical Review B, vol. 73, no. 4, no. 4, p. 45105, 2006.

126 [35] G. Dewar, “A thin wire array and magnetic host structure with n < 0,” Journal of Applied Physics, vol. 97, no. 10, p. 10Q101, May 2005. [36] Y. He, P. He, V. G. Harris, and C. Vittoria, “Role of Ferrites in Negative Index Materials,” IEEE Transactions on Magnetics, vol. 42, no. 10, pp. 2852–2854, Oct. 2006. [37] Y. He, P. He, S. D. Yoon, P. V. Parimic, F. J. Rachfordd, V. G. Harrisa, and C. Vittoria, “Tunable negative index metamaterial using yttrium iron garnet,” Journal of Magnetism and Magnetic Materials, vol. 313, no. 1, pp. 187–191, June 2007. [38] H. Zhao, J. Zhou, Q. Zhao, B. Li, L. Kang, and Y. Bai, “Magnetotunable left-handed material consisting of yttrium iron garnet slab and metallic wires,” Applied Physics Letters, vol. 91, no. 13, p. 131107, Sept. 2007. [39] I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW Media - Media With Negative Parameters, Capable of Supporting Backward Waves,” Microwave and Optical Technology Letters, vol. 31, no. 2, pp. 129–133, Oct. 2001. [40] P. A. Belov, “Backward Waves and Negative Refraction in Unixial Dielectrics with Negative Dielectric Permittivity Along The Anisotropy Axis,” Microwave and Optical Technology Letters, vol. 37, no. 4, pp. 259–263, May 2003. [41] A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nature Materials, vol. 6, pp. 946–950, Dec. 2007. [42] X. Chen, M. He, Y. Du, W. Wang, and D. Zhang, “Negative refraction: An intrinsic property of uniaxial crystals,” Physical Review B, vol. 72, no. 11, pp. 1–4, Sept. 2005. [43] C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Physical Review B, vol. 65, no. 20, p. 201104(R), May 2002. [44] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from Conductors and Enhanced Nonlinear Phenomena,” IEEE Transactions on Microwave Theory and Techniques, vol. 47, no. 11, pp. 2075–2804, Nov. 1999. [45] D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, “Direct calculation of permeability and permittivity for a left-handed metamaterial,” Applied Physics Letters, vol. 77, no. 14, pp. 2246–2248, Oct. 2000. [46] D. R. Smith, S. Schultz, P. Markoˇs, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Physical Review B, vol. 65, no. 19, p. 195104, Apr. 2002.

127 [47] D. R. Smith, P. Rye, D. C. Vier, A. F. Starr, J. J. Mock, and T. Perram, “Design and Measurement of Anisotropic Metamaterials that Exhibit Negative Refraction,” IEICE Transactions on Electronics, vol. E87-C, no. 3, pp. 359–370, Mar. 2004. [48] R. Marqu´es, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Physical Review B, vol. 65, no. 14, p. 144440, Apr. 2002. [49] W. Rotman, “Plasma Simulation by Artificial Dielectrics and Parallel-Plate Media,” IRE Transactions on Antennas and Propagation, vol. 10, no. 1, pp. 82–95, Jan. 1962. [50] S. O’Brien and J. B. Pendry, “Magnetic activity at infrared frequencies in structured metallic photonic crystals,” Journal Of Physics: Condensed Matter, vol. 14, no. 25, pp. 6383–6394, June 2002. [51] S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science, vol. 306, no. 5700, pp. 1351–1353, Nov. 2004. [52] J. Zhou, T. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the Magnetic Response of Split-Ring Resonators at Optical Frequencies,” Physical Review Letters, vol. 95, no. 22, p. 223902, Nov. 2005. [53] H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz Response of a Microfabricated Rod–Split-Ring-Resonator Electromagnetic Metamaterial,” Physical Review Letters, vol. 94, no. 6, p. 63901, Feb. 2005. [54] B. D. F. Casse, H. O. Moser, L. K. Jian, M. Bahou, O. Willhelmi, B. T. Saw, and P. D. Gu, “Fabrication of 2D and 3D Electromagnetic Metamaterials for the Terahertz Range,” Journal of Physics: Conference Series, vol. 34, pp. 885–890, May 2006. [55] T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz Magnetic Response from Artifical Materials,” Science, vol. 303, no. 5663, pp. 1494–1496, Mar. 2004. [56] N. Katsarakis, G. Konstantinidis, A. Kostopoulos, R. S. Penciu, T. F. Gundogdu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Magnetic response of split-ring resonators in the far-infrared frequency regime,” Optics Letters, vol. 30, no. 11, pp. 1348–1350, June 2005. [57] W. J. Padilla, M. T. Aronsson, C. Highstrete, M. Lee, A. J. Taylor, and R. D. Averitt, “Electrically resonant terahertz metamaterials: Theoretical and experimental investigations,” Physical Review B, vol. 75, no. 4, p. 041102(R), Jan. 2007.

128 [58] W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt, “Dynamical Electric and Magnetic Metamaterial Response at Terahertz Frequencies,” Physical Review Letters, vol. 96, no. 10, p. 107401, Mar. 2006. [59] N. Wongkasem, A. Akyurtlu, J. Li, A. Tibolt, Z. Kang, and W. D. Goodhue, “Novel Broadband Terahertz Negative Refractive Index Metamaterials, Analysis and Experiment,” Progress In Electromagnetics Research, vol. 64, pp. 205–218, 2006. [60] H. Tao, A. C. Strikwerda, K. Fan, C. M. Bingham, W. J. Padilla, X. Zhang, and R. D. Averitt, “Terahertz metamaterials on free-standing highly-flexible polyimide substrates,” Journal of Physics D: Applied Physics, vol. 41, no. 23, p. 232004, Nov. 2008. [61] X. G. Peralta, M. C. Wanke, C. L. Arrington, J. D. Williams, I. Brener, A. C. Strikwerda, R. D. Averitt, W. J. Padilla, E. Smirnova, A. J. Taylor, and J. F. O’Hara, “Large-area metamaterials on thin membranes for multilayer and curved applications at terahertz and higher frequencies,” Applied Physics Letters, vol. 94, no. 16, p. 161113, Apr. 2009. [62] T. Driscoll, G. O. Andreev, D. N. Basov, S. Palit, T. Ren, J. Mock, S.-Y. Cho, N. M. Jokerst, and D. R. Smith, “Quantitative investigation of a terahertz artificial magnetic resonance using oblique angle spectroscopy,” Applied Physics Letters, vol. 90, no. 9, p. 092508, Mar. 2007. [63] H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature, vol. 444, no. 7119, pp. 597–600, Nov. 2006. [64] H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nature Photonics, vol. 2, no. 5, pp. 295–298, Apr. 2008. [65] H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nature Photonics, vol. 3, no. 3, pp. 148–151, Feb. 2009. [66] T. Driscoll, S. Palit, M. M. Qazilbash, M. Brehm, F. Keilmann, B.-G. Chae, S.-J. Yun, H.-T. Kim, S. Y. Cho, N. M. Jokerst, D. R. Smith, and D. N. Basov, “Dynamic tuning of an infrared hybrid-metamaterial resonance using vanadium dioxide,” Applied Physics Letters, vol. 93, no. 2, p. 024101, July 2008.

129 [67] T. Driscoll, H.-T. Kim, B.-G. Chae, B.-J. Kim, Y.-W. Lee, N. M. Jokerst, S. Palit, D. R. Smith, M. Di Ventra, and D. N. Basov, “Memory Metamaterials,” Science, vol. 325, no. 5947, pp. 1518–1521, Sept. 2009. [68] T. Driscoll, G. O. Andreev, D. N. Basov, S. Palit, S. Y. Cho, N. M. Jokerst, and D. R. Smith, “Tuned permeability in terahertz split-ring resonators for devices and sensors,” Applied Physics Letters, vol. 91, no. 6, p. 062511, Aug. 2007. [69] J.-M. Manceau, N.-H. Shen, M. Kafesaki, C. M. Soukoulis, and S. Tzortzakis, “Dynamic response of metamaterials in the terahertz regime: Blueshift tunability and broadband phase modulation,” Applied Physics Letters, vol. 96, no. 2, no. 2, p. 021111, 2010. [70] J. F. O’Hara, R. Singh, I. Brener, E. Smirnova, J. Han, A. J. Taylor, and W. Zhang, “Thin-film sensing with planar terahertz metamaterials: sensitivity and limitations,” Optics Express, vol. 16, no. 3, p. 1786, Feb. 2008. [71] C. M. Bingham, H. Tao, X. Liu, R. D. Averitt, X. Zhang, and W. J. Padilla, “Planar wallpaper group metamaterials for novel terahertz applications,” Optics Express, vol. 16, no. 23, pp. 18565–18575, Nov. 2008. [72] Y. Yuan, C. Bingham, T. Tyler, S. Palit, T. H. Hand, W. J. Padilla, D. R. Smith, N. M. Jokerst, and S. A. Cummer, “Dual-band planar electric metamaterial in the terahertz regime.,” Optics Express, vol. 16, no. 13, pp. 9746–9752, June 2008. [73] Y. Yuan, C. Bingham, T. Tyler, S. Palit, T. H. Hand, W. J. Padilla, N. M. Jokerst, and S. A. Cummer, “A dual-resonant terahertz metamaterial based on single-particle electric-field-coupled resonators,” Applied Physics Letters, vol. 93, no. 19, p. 191110, Nov. 2008. [74] S.-H. Lee, S.-Y. Gee, C. Kang, and C.-S. Kee, “Terahertz Wave Transmission Properties of Metallic Periodic Structures Printed on a Photo-paper,” Journal of the Optical Society of Korea, vol. 14, no. 3, pp. 282–285, Sept. 2010. [75] H. Tao, A. C. Strikwerda, K. Fan, W. J. Padilla, X. Zhang, and R. D. Averitt, “Reconfigurable Terahertz Metamaterials,” Physical Review Letters, vol. 103, no. 14, p. 147401, Oct. 2009. [76] H. Tao, A. C. Strikwerda, K. Fan, W. J. Padilla, X. Zhang, and R. D. Averitt, “MEMS Based Structurally Tunable Metamaterials at Terahertz Frequencies,” Journal of Infrared, Millimeter, and Terahertz Waves, pp. 1–16, May 2010.

130 [77] B. D. F. Casse, H. O. Moser, J. W. Lee, M. Bahou, S. Inglis, and L. K. Jian, “Towards three-dimensional and multilayer rod-split-ring metamaterial structures by means of deep x-ray lithography,” Applied Physics Letters, vol. 90, no. 25, p. 254106, June 2007. [78] V. J. Logeeswaran, M. S. Islam, M. L. Chan, D. A. Horsley, W. Wu, S.-Y. Wang, and R. S. Williams, “Realization of 3D Isotropic Negative Index Materials using Massively Parallel and Manufacturable Microfabrication and Micromachining Technology,” in Materials Research Society Symposium Proceedings (S.-Y. Wang, N. X. Fang, L. Thylen, and M. S. Islam, eds.), vol. 919E, (Warrendale, PA), pp. 0919–J02–01, 2006. [79] H. J. In, W. J. Arora, P. Stellman, S. Kumar, Y. Shao-Horna, H. I. Smith, and G. Barbastathis, “The Nanostructured Origami 3D Fabrication and Assembly Process for nanopatterned 3D structures,” in Proceedings of SPIE (V. K. Varadan, ed.), vol. 5763, (Bellingham, WA), pp. 84–95, SPIE, 2005. [80] J. R. Wendt, D. B. Burckel, G. A. Ten Eyck, A. R. Ellis, I. Brener, and M. B. Sinclair, “Fabrication techniques for three-dimensional metamaterials in the midinfrared,” Journal of Vacuum Science and Technology B, vol. 28, no. 6, p. 4, Nov. 2010. [81] J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Physical Review B, vol. 73, no. 4, p. 041101(R), Jan. 2006. [82] V. P. Drachev, W. Cai, U. Chettiar, H.-K. Yuan, A. K. Sarychev, A. V. Kildishev, G. Klimeck, and V. M. Shalaev, “Experimental verification of an optical negativeindex material,” Laser Physics Letters, vol. 3, no. 1, pp. 49–55, Oct. 2006. [83] A. V. Kildishev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and V. M. Shalaev, “Negative refractive index in optics of metal-dielectric composites,” Journal of the Optical Society of America B, vol. 23, no. 3, pp. 423–433, Mar. 2006. [84] S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Physical Review Letters, vol. 95, no. 13, p. 137404, Sept. 2005. [85] S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Optics Express, vol. 13, no. 13, pp. 4922–4930, June 2005. [86] G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Optics Letters, vol. 31, no. 12, pp. 1800–1802, June 2006.

131 [87] G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial,” Science, vol. 312, no. 5775, pp. 892–894, May 2006. [88] G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Optics Letters, vol. 32, no. 1, pp. 53–55, Jan. 2007. [89] J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature, vol. 455, no. 7211, pp. 376–379, Sept. 2008. [90] C. Imhof and R. Zengerle, “Strong birefringence in left-handed metallic metamaterials,” Optics Communications, vol. 280, no. 1, pp. 213–216, Dec. 2007. [91] O. Paul, C. Imhof, B. Reinhard, R. Zengerle, and R. Beigang, “Negative index bulk metamaterial at terahertz frequencies,” Optics Express, vol. 16, no. 9, pp. 6736–6744, Apr. 2008. [92] H. O. Moser, J. A. Kong, L. K. Jian, H. S. Chen, G. Liu, M. Bahou, S. M. P. Kalaiselvi, S. M. Maniam, X. X. Cheng, B. I. Wu, P. D. Gu, A. Chen, S. P. Heussler, S. bin Mahmood, and L. Wen, “Free-standing THz electromagnetic metamaterials.,” Optics Express, vol. 16, no. 18, pp. 13773–80, Sept. 2008. [93] G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidar Metallosungen,” Annalen Der Physik, vol. 25, no. 3, no. 3, pp. 377–455, 1908. [94] G. Mie, “Contributions on the optics of turbid media, particularly colloidal metal solutions,” tech. rep., Sandia Laboratories, Albuquerque, New Mexico, 1978. [95] L. Lewin, “The Electrical Constants of a Material Loaded With Sphereical Particles,” The Proceedings of the Institution of Electrical Engineers, vol. 94, pp. 65–68, 1947. [96] S. O’Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” Journal of Physics: Condensed Matter, vol. 14, no. 15, pp. 4035–4044, Apr. 2002. [97] C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A Double Negative (DNG) Composite Medium Composed of Magnetodielectric Spherical Particles Embedded in a Matrix,” IEEE Transactions on Antennas and Propagation, vol. 51, no. 10, pp. 2596–2601, Oct. 2003. [98] J. Baker-Jarvis, M. D. Janezic, D. Love, T. M. Wallis, C. L. Holloway, and P. Kabos, “Phase Velocity in Resonant Structures,” IEEE Transactions on Magnetics, vol. 42, no. 10, pp. 3344–3346, Oct. 2006.

132 [99] T. Ueda and M. Tsutsumi, “Left-Handed Rectangular Waveguides Periodically Filled with Ferrite,” Microwave Symposium Digest, 2005 IEEE MTT-S International, pp. 335–338, June 2005. [100] N. Michishita, A. Lai, T. Ueda, and T. Itoh, “Recent Progress of Dielectric ResonatorBased Left-Handed Metamaterials,” in 2007 International Symposium on Signals, Systems and Electronics, (Montreal, Quebec), pp. 143–146, IEEE, July 2007. [101] T. Ueda, N. Michishita, and T. Itoh, “Composite Right/Left Handed Metamaterial Structures Composed of Dielectric Resonators and Parallel Mesh Plates,” in IEEE/MTT-S International Microwave Symposium, 2007, (Honolulu, HI), pp. 1823– 1826, IEEE, June 2007. [102] T. Ueda, A. Lai, and T. Itoh, “Demonstration of Negative Refraction in a Cutoff Parallel-Plate Waveguide Loaded With 2-D Square Lattice of Dielectric Resonators,” IEEE Transactions on Microwave Theory and Techniques, vol. 55, no. 6, pp. 1280– 1287, June 2007. [103] T. Ueda, N. Michishita, A. Lai, M. Akiyama, and T. Itoh, “2.5-D stacked composite right/left-handed metamaterial structures using dielectric resonators and parallel mesh plates,” in Microwave Symposium Digest, 2008 IEEE MTT-S International, (Atlanta, GA), pp. 335–338, IEEE, June 2008. [104] T. Lepetit, E. Akmansoy, and J.-P. Ganne, “All-dielectric metamaterial: a ferroelectricbased scheme in the microwave range,” in Proceedings of SPIE (M. A. Noginov, N. I. Zheludev, A. D. Boardman, and N. Engheta, eds.), vol. 7392, (San Diego, CA), p. 73920H, SPIE, Aug. 2009. [105] O. Acher and M. Ledieu, “Design and measurement of negative permeability metamaterials made from conductor-coated high-index dielectric inclusions,” Metamaterials, vol. 2, pp. 18–25, 2008. [106] T. Lepetit, E. Akmansoy, M. Pat´e, and J. P. Ganne, “Broadband negative magnetism from all-dielectric metamaterial,” Electronics Letters, vol. 44, no. 19, p. 1119, Sept. 2008. [107] T. Lepetit, E. Akmansoy, and J. P. Ganne, “Experimental measurement of negative index in an all-dielectric metamaterial,” Applied Physics Letters, vol. 95, no. 12, p. 121101, Sept. 2009. [108] A. N. Lagarkov, V. N. Semenenko, V. N. Kisel, and V. A. Chistyaev, “Development and simulation of microwave artificial magnetic composites utilizing nonmagnetic inclusions,” Journal of Magnetism and Magnetic Materials, vol. 258-259, pp. 161– 166, Mar. 2003.

133 [109] Q. Zhao, B. Du, L. Kang, H. Zhao, Q. Xie, B. Li, X. Zhang, J. Zhou, L. Li, and Y. Meng, “Tunable negative permeability in an isotropic dielectric composite,” Applied Physics Letters, vol. 92, no. 5, p. 51106, Feb. 2008. [110] Q. Zhao, L. Kang, B. Du, H. Zhao, Q. Xie, X. Huang, B. Li, J. Zhou, and L. Li, “Experimental Demonstration of Isotropic Negative Permeability in a ThreeDimensional Dielectric Composite,” Physical Review Letters, vol. 101, no. 2, p. 27402, July 2008. [111] T. Giannakopoulou, D. Niarchos, and C. Trapalis, “Experimental investigation of electric and magnetic responses in composites with dielectric resonator inclusions at microwave frequencies,” Applied Physics Letters, vol. 94, no. 24, p. 242506, June 2009. [112] J. Kim and A. Gopinath, “Simulations and Experiments with Metamaterial Flat Antenna Lens Using Cubic High Dielectric Resonators,” in 2009 IEEE International Workshop on Antenna Technology, (Santa Monica, CA), pp. 1–4, IEEE, Mar. 2009. [113] X. Cai, R. Zhu, and G. Hu, “Experimental study for metamaterials based on dielectric resonators and wire frame,” Metamaterials, vol. 2, no. 4, pp. 220–226, Dec. 2008. [114] M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Magnetic response in NonMagnetic Materials and Coupled Dipole Interactions,” in Antennas and Propagation Society International Symposium 2008, IEEE, (San Diego), p. 4, IEEE, July 2008. [115] F. Kadlec, H. Nˇemec, P. Kuˇzel, R. Yahiaoui, and P. Mounaix, “Dielectric Tunable Metamaterials with Negative Permeability in Terahertz Range,” in Conference on Lasers and Electro-Optics, (Baltimore, MD), pp. 5–6, IEEE, June 2009. [116] H. Nˇemec, P. Kuˇzel, F. Kadlec, C. Kadlec, R. Yahiaoui, and P. Mounaix, “Tunable terahertz metamaterials with negative permeability,” Physical Review B, vol. 79, no. 24, p. 241108(R), June 2009. [117] R. Yahiaoui, H. Nemec, P. Kuzel, F. Kadlec, C. Kadlec, and P. Mounaix, “Broadband dielectric terahertz metamaterials with negative permeability.,” Optics Letters, vol. 34, no. 22, pp. 3541–3, Nov. 2009. [118] K. Shibuya, K. Takano, N. Matsumoto, and M. Hangyo, “Investigation of Metamaterials with Dielectric Arrays in the Terahertz Region,” tech. rep., Institute of Laser Engineering, Osaka University, Osaka, Japan, 2007. [119] J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric Metamaterials Based on Electric and Magnetic Resonances of Silicon Carbide Particles,” Physical Review Letters, vol. 99, p. 107401, Sept. 2007.

134 [120] Q. Wu, J. H. Lee, J. Ahn, and W. Park, “Metal nanocluster metamaterial,” in Lasers and Electro-Optics, 2008 and 2008 Conference on Quantum Electronics and Laser Science. CLEO/QELS 2008. Conference on, pp. 1–2, May 2008. [121] N. Limberopoulos, A. Akyurtlu, K. Higginson, A.-G. Kussow, and C. D. Merritt, “Negative refractive index metamaterials in the visible spectrum based on MgB2 SiC composites,” Applied Physics Letters, vol. 95, no. 2, p. 023306, July 2009. [122] E. A. Barringer and H. K. Bowen, “High-purity, monodisperse TiO2 powders by hydrolysis of titanium tetraethoxide. 1. Synthesis and physical properties,” Langmuir, vol. 1, no. 4, pp. 414–420, July 1985. [123] E. A. Barringer and H. K. Bowen, “High-purity, monodisperse TiO2 powders by hydrolysis of titanium tetratethoxide. 2. Aqueous interfacial electrochemistry and dispersion stability,” Langmuir, vol. 1, no. 4, no. 4, pp. 420–428, 1985. [124] N. J. Marston, B. Vincent, and N. G. Wright, “The synthesis spherical rutile titanium dioxide particles and their interaction with polystyrene latex particles of opposite charge,” Progress in Colloid and Polymer Science, vol. 109, pp. 278–282, 1998. [125] M. Z.-C. Hu, V. Kurian, E. A. Payzant, C. J. Rawn, and R. D. Hunt, “Wet-chemical synthesis of monodispersed barium titanate particles hydrothermal conversion of TiO2 microspheres to nanocrystalline BaTiO3,” Powder Technology, vol. 110, no. 1-2, pp. 2–14, May 2000. [126] M. Keshmiri, “Synthesis of narrow size distribution sub-micron TiO2 spheres,” Journal of Non-Crystalline Solids, vol. 311, no. 1, pp. 89–92, Oct. 2002. [127] S. Eiden-Assmann, J. Widoniak, and G. Maret, “Synthesis and characterization of porous and nonporous monodisperse colloidal TiO2 particles,” Chemistry of Materials, vol. 16, no. 1, no. 1, pp. 6–11, 2004. [128] J. H. Bang and K. S. Suslick, “Dual Templating Synthesis of Mesoporous Titanium Nitride Microspheres,” Advanced Materials, vol. 21, no. 31, pp. 3186–3190, Aug. 2009. [129] R. A. Caruso, D. Chen, F. Huang, and Y.-B. Cheng, “Mesoporous Anatase TiO 2 Beads with High Surface Areas and Controllable Pore Sizes: A Superior Candidate for High-Performance Dye-Sensitized Solar Cells,” Advanced Materials, vol. 21, no. 21, pp. 2206–2210, June 2009. [130] S. Liu, X. Sun, J.-G. Li, X. Li, Z. Xiu, and D. Huo, “Synthesis of Dispersed Anatase Microspheres with Hierarchical Structures via Homogeneous Precipitation,” European Journal of Inorganic Chemistry, vol. 2009, no. 9, no. 9, pp. 1214–1218, 2009.

135 [131] S. Nagaoka, “Preparation of carbon/TiO2 microsphere composites from cellulose/TiO2 microsphere composites and their evaluation,” Journal of Molecular Catalysis A: Chemical, vol. 177, no. 2, pp. 255–263, Jan. 2002. [132] Z. Baolong, C. Baishun, S. Keyu, H. Shangjin, L. Xiaodong, D. Zongjie, and Y. Kelian, “Preparation and characterization of nanocrystal grain TiO2 porous microspheres,” Applied Catalysis B: Environmental, vol. 40, no. 4, pp. 253–258, Feb. 2003. [133] A. S. Deshpande, D. G. Shchukin, E. Ustinovich, M. Antonietti, and R. A. Caruso, “Titania and Mixed Titania/Aluminum, Gallium, or Indium Oxide Spheres Sol-Gel/Template Synthesis and Photocatalytic Properties,” Advanced Functional Materials, vol. 15, no. 2, pp. 239–245, Feb. 2005. [134] G. Salib, DePalma, “Specular Optical Density of Diffusing Media,” Photographic Science and Engineering, vol. 18, pp. 145–150, 1974. [135] E. D. Palik, ed., Handbook of Optical Constants of Solids. Orlando, Florida: Academic Press, 1st ed., 1985. [136] W. G. Spitzer, R. C. Miller, D. A. Kleinman, and L. E. Howarth, “Far Infrared Dielectric Dispersion in BaT iO3 , S rT iO3 , and T iO2 ,” Physical Review, vol. 126, no. 5, pp. 1710–1721, June 1962. [137] M. Schall, H. Helm, and S. R. Keiding, “Far Infrared Properties Of Electro-Optic Crystals Measured By THz Time-Domain Spectroscopy,” International Journal of Infrared and Millimeter Waves, vol. 20, no. 4, pp. 595–604, Apr. 1999. [138] N. Matsumoto, T. Hosokura, K. Kageyama, H. Takagi, Y. Sakabe, and M. Hangyo, “Analysis of Dielectric Response of TiO2 in Terahertz Frequency Region by General Harmonic Oscillator Model,” Japanese Journal of Applied Physics, vol. 47, no. 9, pp. 7725–7728, Sept. 2008. [139] C. Kittel, Introduction to solid state physics. New York: Wiley, 7th ed., 1996. [140] S. Kasap and P. Capper, eds., Springer Handbook of Electronic and Photonic Materials. New York: Springer, 2006. [141] A. I. Cabuz, D. Felbacq, and D. Cassagne, “Homogenization of Negative-Index Composite Metamaterials A Two-Step Approach,” Physical Review Letters, vol. 98, no. 3, p. 37403, Jan. 2007. [142] M. J. C. Garnett, “Colours in Metal Glasses and in Metallic Films,” Philosophical Transaction of the Royal Society of London, Series A, vol. 203, pp. 385–420, 1904.

136 [143] A.-G. Kussow, A. Akyurtlu, and N. Angkawisittpan, “Optically isotropic negative index of refraction metamaterial,” Physica Status Solidi b, vol. 245, no. 5, pp. 992– 997, Feb. 2008. [144] J. Kim and A. Gopinath, “Simulation of a metamaterial containing cubic high dielectric resonators,” Physical Review B, vol. 76, no. 11, no. 11, p. 115126, 2007. [145] C. R. Simovski and P. A. Belov, “Low-frequency spatial dispersion in wire media,” Physical Review E, vol. 70, no. 4, p. 046616, Oct. 2004. [146] J. A. Kong, Electromagnetic Wave Theory. New York: John Wiley and Sons Inc, 2nd ed., 1990. [147] R. E. Collin, Field Theory of Guided Waves. New York: McGraw-Hill Book Company, 1960. [148] J. Helszajn, “Chapter One, Tensor Permeability of Magnetized Ferrite,” in Nonreciprocal Microwave Junctions and Circulators, p. 349, New York: WileyInterscience, 1975. [149] P. Yeh, Optical Waves in Layered Media. New York: John Wiley & Sons, 1988. [150] National Instruments, “LabView,” 2009. [151] Ansys, “HFSS.” [152] R. T. Downs and M. Hall-Wallace, “The American Mineralogist crystal structure database,” American Mineralogist, vol. 88, pp. 247–250, 2003. [153] J. D. Martin, “Using XPowder - a sofware package for powder X-ray diffraction analysis. D.L. GR-1001/04,” 2004. [154] J. Baker-Jarvis, E. J. Vanzura, and W. A. Kissick, “Improved Technique for Determining Complex Permittivity with the Transmission / Reflection Method,” IEEE Transactions on Microwave Theory and Techniques, vol. 38, no. 8, pp. 1096–1103, Aug. 1990. [155] J. Baker-Jarvis, R. G. Geyer, and P. D. Domich, “A Nonlinear Least-Squares Solution with Causality Constraints Applied to Transmission Line Permittivity and Permeability Determination,” IEEE Transactions on Instrumentation and Measurement, vol. 41, no. 5, pp. 646–652, Oct. 1992. [156] A. M. Nicolson and G. F. Ross, “Measurement of the Intrinsic Properties of Materials by Time-Domain Techniques,” IEEE Transactions on Instrumentation and Measurement, vol. IM-19, no. 4, pp. 377–382, Nov. 1970.

137 [157] W. B. Weir, “Automatic Measurement of Complex Dielectric Constant and Permeability at Microwave Frequencies,” Proceedings of the IEEE, vol. 62, no. 1, pp. 33–36, Jan. 1974. [158] Hewlett Packard, “Measuring Dielectric Constant with the HP 8510 Network Analyzer, The Measurement of Both Permittivity and Permeability of Solid Materials,” Tech. Rep. Product Note 8510-3, Hewlett Packard, 1985. [159] F. A. Grant, “Properties of Rutile (Titanium Dioxide),” Reviews of Modern Physics, vol. 31, no. 3, pp. 646–674, July 1959. [160] P. S. Ha, H.-J. Youn, H. S. Jung, K. S. Hong, Y. H. Park, and K. H. Ko, “AnataseRutile Transition of Precipitated Titanium Oxide with Alcohol Rinsing.,” Journal of Colloid and Interface Science, vol. 223, no. 1, pp. 16–20, Mar. 2000. [161] P. C. Sukanek, “Dependence of Film Thickness on Speed in Spin Coating,” Journal of the Electrochemical Society, vol. 138, no. 6, no. 6, p. 1712, 1991. [162] C. Luo, A. Govindaraju, J. Garra, T. Schneider, R. White, J. Currie, and M. Paranjape, “Releasing SU-8 structures using polystyrene as a sacrificial material,” Sensors and Actuators A, vol. 114, no. 1, pp. 123–128, Aug. 2004. [163] Y. Yin, Y. Lu, B. Gates, and Y. Xia, “Template-Assisted Self-Assembly A Practical Route to Complex Aggregates of Monodispersed Colloids with Well-Defined Sizes, Shapes, and Structures,” Journal of the American Chemical Society, vol. 123, no. 36, pp. 8718–8729, Aug. 2001. [164] Y. Xia, Y. Yin, Y. Lu, and J. McLellan, “Template-Assisted Self-Assembly of Spherical Colloids into Complex and Controllable Structures,” Advanced Functional Materials, vol. 13, no. 12, pp. 907–918, Dec. 2003. [165] S. Grego, T. W. Jarvis, B. R. Stoner, and J. S. Lewis, “Template-Directed Assembly on an Ordered Microsphere Array,” Langmuir, vol. 21, no. 11, pp. 4971–4975, Apr. 2005. [166] Y. Yin and Y. Xia, “Self-Assembly of Spherical Colloids into Helical Chains with Well-Controlled Handedness,” Journal of the American Chemical Society, vol. 125, no. 8, pp. 2048–2049, Jan. 2003. [167] J. P. Hoogenboom, C. Rtif, E. de Bres, M. van De Boer, A. K. van Langen-Suurling, J. Romijn, and A. van Blaaderen, “Template-Induced Growth of Close-Packed and Non-Close-Packed Colloidal Crystals during Solvent Evaporation,” Nano Letters, vol. 4, no. 2, pp. 205–208, Jan. 2004.

138 [168] E. J. Tull, P. N. Bartlett, and K. R. Ryan, “Controlled Assembly of Micrometer-Sized Spheres: Theory and Application,” Langmuir, vol. 23, no. 14, pp. 7859–7873, June 2007. [169] E. J. Tull and P. N. Bartlett, “The assembly of micron sized glass spheres on structured surfaces by dewetting,” Colloids and Surfaces A: Physicochemical and Engineering Aspects, vol. 327, no. 1-3, pp. 71–78, June 2008. [170] E. J. Tull, P. N. Bartlett, G. S. Murugan, and J. S. Wilkinson, “Manipulating Spheres That Sink: Assembly of Micrometer Sized Glass Spheres for Optical Coupling,” Langmuir, vol. 25, no. 3, pp. 1872–1880, Jan. 2009. [171] M. Nordstrom, R. Marie, M. Calleja, and A. Boisen, “Rendering SU-8 hydrophilic to facilitate use in micro channel fabrication,” Journal of Micromechanics and Microengineering, vol. 14, no. 12, pp. 1614–1617, Aug. 2004. [172] H. Hu, R. Ricken, W. Sohler, and R. B. Wehspohn, “Lithium Niobate Ridge Waveguides Fabricated by Wet Etching,” IEEE Photonics Technology Letters, vol. 19, no. 6, pp. 417–419, Mar. 2007. [173] T. Z. Fullem and Y. Danon, “Electrostatics of pyroelectric accelerators,” Journal of Applied Physics, vol. 106, no. 7, p. 074101, Oct. 2009. [174] C. L. Sones, S. Mailis, W. S. Brocklesby, R. W. Eason, and J. R. Owen, “Differential etch rates in z-cut LiNbO3 for variable HF/HNO3 concentrations,” Journal of Materials Chemistry, vol. 12, no. 2, pp. 295–298, Jan. 2002. [175] Sigma-Aldrich, “Lithium Fluoride MSDS,” tech. rep., Sigma-Aldrich, Saint Louis, Missouri, Jan. 2010. [176] Sigma-Aldrich, “Tantalum Fluoride MSDS,” tech. rep., Sigma-Aldrich, Saint Louis, Missouri, 2010. [177] I. E. Barry, G. W. Ross, P. G. R. Smith, R. W. Eason, and G. Cook, “Microstructuring of lithium niobate using differential etch-rate between inverted and non-inverted ferroelectric domains,” Materials Letters, vol. 37, no. 4-5, pp. 246–254, Nov. 1998. [178] Y. Hiranaga, Y. Wagatsuma, and Y. Cho, “LiTaO3 Recording Media Prepared by Polarization Controlled Wet Etching Process,” Integrated Ferroelectrics, vol. 68, pp. 221–228, 2004. [179] A. B. Randles, M. Esashi, and S. Tanaka, “Etch rate dependence on crystal orientation of lithium niobate.,” IEEE transactions on ultrasonics, ferroelectrics, and frequency control, vol. 57, no. 11, pp. 2372–80, Nov. 2010.

139 [180] E. P. Meagher and G. A. Lager, “Polyhedra Thermal Expansion in the TiO2 Polymorphs: Refinement of the Crystal Structures of Rutile and Brookite at High Temperature,” Canadian Mineralogist, vol. 17, pp. 77–85, 1979. [181] W. R. Folks, S. K. Pandey, and G. Boreman, “Refractive Index at THz Frequencies of Various Plastics,” in Optical Terahertz Science and Technology, OSA Technical Digest Series, p. MD10, 2007. [182] S. Arscott, F. Garet, P. Mounaix, L. Duvillaret, J.-L. Coutaz, and D. Lippens, “Terahertz time-domain spectroscopy of films fabricated from SU-8,” Electronics Letters, vol. 35, no. 3, pp. 243–244, Feb. 1999. [183] S. Lucyszyn, “Comment on “Terahertz time-domain spectroscopy of films fabricated from SU-8”,” Electronics Letters, vol. 37, no. 20, p. 1267, Sept. 2001. [184] C. E. Collins, R. E. Miles, R. D. Pollard, D. P. Steenson, J. W. Digby, G. M. Parkhurst, J. M. Chamberlain, N. J. Cronin, S. R. Davies, and J. W. Bowen, “Millimeter-Wave Measurements of the Complex Dielectric Constant of an Advanced Thick Film UV Photoresist,” Journal of Electronic Materials, vol. 27, no. 6, pp. L40–L42, June 1998. [185] A. Fujii, A. Ando, and Y. Sakabe, “Dielectric Characteristics of Ferroelectric Materials in Submillimeter-Wave Regions,” Japanese Journal Of Applied Physics, vol. 43, no. 9B, pp. 6765–6768, Sept. 2004. [186] R. W. Corkum, “Isotropic Artificial Dielectric,” Proceedings of the IRE, vol. 40, no. 5, pp. 574–587, May 1952. [187] H. T. Ward, W. O. Puro, and D. M. Bowie, “Artificial Dielectrics Utilizing Cylindrical and Spherical Voids,” Proceedings of the IRE, vol. 44, no. 2, pp. 171–174, Feb. 1956. [188] M. T. Lanagan, J. K. Yamamogo, A. Bhalla, and S. G. Sankar, “The Dielectric Properties of Yttria-Stabilized Zirconia,” Materials Letters, vol. 7, no. 12, pp. 437– 440, Mar. 1989. [189] P. A. Smith and L. E. Davis, “Dielectric loss tangent of yttria stabilized zirconia at 5.6 GHz and 77 K,” Electronics Letters, vol. 28, no. 4, pp. 424–425, Feb. 1992. [190] D. Grischowsky and S. Keiding, “THz time-domain spectroscopy of high Tc substrates,” Applied Physics Letters, vol. 57, no. 10, pp. 1055–1057, Sept. 1990. [191] M. Watanabe, S. Kuroda, H. Yamawaki, and M. Shiwa, “Terahertz dielectric properties of plasma-sprayed thermal-barrier coatings,” Surface and Coatings Technology, vol. 205, no. 19, pp. 4620–4626, June 2011.

140 [192] S. Roberts, “Dielectric constants and polarizabilities of ions in simple crystals and barium titanate,” Physical Review, vol. 76, no. 8, pp. 1215–1220, Oct. 1949. [193] R. J. Gonzalez, R. Zallen, and H. Berger, “Infrared reflectivity and lattice fundamentals in anatase TiO2 ,” Physical Review B, vol. 55, no. 11, pp. 7014–7017, Mar. 1997. [194] D. R. Lide, ed., CRC Handbook of Chemistry and Physics. Boston: CRC Press, 71st ed., 1990. [195] K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Applied Physics Letters, vol. 85, no. 4, no. 4, pp. 543–545, 2004.

141

IX. 9.1

APPENDIX

Sources Of Materials Used

Table 9-1 lists the source of essential materials used, excluding standard solvents. Table 9-1. Sources of materials used. Source

Material

Aldrich Chemical Company Titanium(IV) Isopropoxide Milwaukee, Wisconsin Alfa Aesar Calcium D-gluconate Gel Ward Hill, Massachusetts (Hydrofluoric Acid Burn Cream) AZ Electronic Materials AZ 327 MIF Photoresist Developer Branchburg, New Jersey AZ 1514 Photoresist AZ 5412 Photoresist Dow Specialty Materials Philadelphia, Pennsylvania

Amberchom CG300C

Dual Manufacturing Company Number 70 U.S. Standard Sieve Chicago, Illinois Number 325 U.S. Standard Sieve Galomb Inc Allentown, Pennsylvania GE Health Sciences Piscataway, New Jersey

Polystyrene Resin Pellets Source 30 RCM

Grainger Industrial Supply 3M Adhesive Transfer Tape 465 Clear Lake Forest, Illinois Inframat Corporation Manchester, Connecticut

Yttria Stabilized Zirconia 2 mm Grinding Media

Continued on next page . . .

142

Table 9-1 – continued from previous page Source

Material

MicroChem Corp PMMA Newton, Massachusetts SU-8 2002 SU-8 2005 SU-8 2010 SU-8 2025 SU-8 Developer Microspheres-Nanospheres.com Cold Spring, New York MSC Industrial Supply Melville, New York

Yttria Stabilized Zirconia Micro-Spheres 24 Gage ASMT B3 Grade Copper Bus Bar Wire

MTI Corporation Lithium Tantalate Wafers Richmond, California Precision Eforming Cortland, New York

Copper Wire Grid

Satellite City Inc Hot Stuff Super ‘T’ cyanoacrylate glue Simi Valley Saint-Gobain Performance Plastics Valley Forge, Pennsylvania

PFA (Perfluoroalkoxy) sheet

Sigma-Aldrich Chromium Etchant Saint Louis, Missouri Hydrofluoric Acid Smooth-On Incorporated Easton, Pennsylvania UniversityWafer.com Boston, Massachusetts

Task 4 Silicon Wafers

Continued on next page . . .

143

Table 9-1 – continued from previous page Source

Material

W.K. Robson Co. DuPont Silver Paint - Composition 4817N Newark, Delaware laboratory stock

Mylar

144

9.2

Photolithographic Processing Details

Preparing wafers for any work was done by rinsing with trichlorethylene, acetone, methanol, and isopropanol followed by drying with a nitrogen flow.

9.2.1

Sacrificial Layer Of Polystyrene

The solution for the sacrificial layer of polystyrene was made by dissolving polystyrene pellets in toluene [162]. The preparation procedure is: 1. Weigh 5.7 grams of polystyrene pellets for every 50 mL of toluene. 2. Immerse the polystyrene beads in the toluene. 3. Stir gently at 60°C for 24 hours 4. Remove from stirring and heating. All polystyrene should be dissolved. The solution of the polystyrene in toluene was used at room temperature. The sacrificial layer of polystyrene was spin-coated at 3000 r.p.m by dynamically dispensing the solution from a pipette. The layer can be optionally baked at 100°C for 1 minute. The polystyrene layer can be safely cleaned with isopropanol followed by nitrogen drying. The polystyrene sacrificial layer was removed with toluene.

145 9.2.2

SU-8 Negative Resist Processing Details

SU-8 2002 Processing And Development 1. Coat SU-8 2002 is dispensed dynamically requiring less than 1 mL of resist. Spin coating parameters are as follows: • • • •

For spin speed see Figure 9-1 30 seconds Acceleration setting 1 No pre-spin

2. Soft Bake Soft baking prepares the SU-8 for exposure. Place on the hot plate for: • 1:00 minutes at 95°C. • Remove and cool 1 minute before next step. 3. Expose If using a mask, place the mask in full contact with the SU-8 resist using a vacuum chuck. UV expose for 20 seconds. 4. Post Exposure Bake Post exposure baking prepares the SU-8 for developing. Place on the hot plate for: • 2:00 minutes at 95°C. • Remove and cool 1 minute before next step. 5. Develop The entire wafer should be submerged in developer for developing. (a) (b) (c) (d) (e) (f) (g)

Place an adequate amount of SU-8 developer in a wide bottom glassware. Place wafer into developer for 1:00 minutes. Agitate wafer gently. Remove the wafer from the developer. For small features flush with new developer for 10 seconds. Rinse with Isopropanol. Nitrogen flow dry.

Figure 9-1. SU-8 2002 thickness versus spin speed.

146

SU-8 2002 Thickness Versus Spin Speed

147 SU-8 2005 Processing And Development 1. Coat SU-8 2005 is dispensed dynamically requiring less than 1 mL of resist. Take care not to introduce bubbles. Spin coating parameters are as follows: • • • •

For spin speed see Figure 9-2 30 seconds Acceleration setting 1 No pre-spin

2. Soft Bake Soft baking prepares the SU-8 for exposure. For bare silicon wafers: When placing the wafer on the hot plate, immediately cover the substrate with an inverted glass beaker and leave covered until removed from the hot plate. Place on the hot plate for: • 3:00 minutes at 95°C. • Remove and cool 1 minute before next step 3. Expose If using a mask, place the mask in full contact with the SU-8 resist using a vacuum chuck. UV Expose for 30 seconds. 4. Post Exposure Bake Post exposure baking prepares the SU-8 for developing. Place on the hot plate for: (no need to cover the wafer for this step). • 3:30 minutes at 95°C. • Remove and cool 1 minute before next step. 5. Develop The entire wafer should be submerged in developer for developing. (a) (b) (c) (d) (e) (f) (g)

Place an adequate amount of SU-8 developer in a wide bottom glassware. Place wafer into developer for 1:30 minutes. Agitate wafer gently. Remove the wafer from the developer. For small features flush with new developer for 10 seconds. Rinse with Isopropanol. Nitrogen flow dry.

Figure 9-2. SU-8 2005 thickness versus spin speed.

148

SU-8 2005 Thickness Versus Spin Speed

149 SU-8 2010 Processing And Development 1. Coat SU-8 2010 is dispensed by pouring a puddle in center of the wafer directly from the bottle. Take care not to introduce bubbles. Spinning follows the speed chart below, for final speed see Figure 9-3:

2. Soft Bake Soft baking prepares the SU-8 for exposure. Place on the hot plate for: • 4:00 minutes at 95°C. • Remove and cool 1 minute before next step. 3. Expose If using a mask place the mask in full contact with the SU-8 resist using a vacuum chuck. Expose for 45 seconds. 4. Post Exposure Bake Post exposure baking prepares the SU-8 for developing. Place on the hot plate for: • 5:00 minutes at 95°C. • Remove and cool 1 minute before next step. 5. Develop The entire wafer should be submerged in developer for developing. (a) (b) (c) (d) (e) (f) (g)

Place an adequate amount of SU-8 developer in a wide bottom glassware. Place wafer into developer for 3:00 minutes. Agitate wafer gently. Remove the wafer from the developer. For small features flush with new developer for 10 seconds. Rinse with Isopropanol. Nitrogen flow dry.

Figure 9-3. SU-8 2010 thickness versus spin speed.

150

SU-8 2010 Thickness Versus Spin Speed

151 SU-8 2025 Processing And Development 1. Coat SU-8 2025 is dispensed by pouring a puddle in center of the wafer directly from the bottle. Take care not to introduce bubbles. Spinning follows the speed chart below, for final speed see Figure 9-4:

2. Soft Bake Soft baking prepares the SU-8 for exposure. Place on the hot plate for: • 6:00 minutes at 95°C. • Remove and cool 1 minute before next step. 3. Expose If using a mask place the mask in full contact with the SU-8 resist using a vacuum chuck. Expose for 60 seconds. 4. Post Exposure Bake Post exposure baking prepares the SU-8 for developing. Place on the hot plate for: • 7:00 minutes at 95°C. • Remove and cool 1 minute before next step. 5. Develop The entire wafer should be submerged in developer for developing. (a) (b) (c) (d) (e) (f) (g)

Place an adequate amount of SU-8 developer in a wide bottom glassware. Place wafer into developer for 4:00 minutes. Agitate wafer gently. Remove the wafer from the developer. For small features flush with new developer for 10 seconds. Rinse with Isopropanol. Nitrogen flow dry.

Figure 9-4. SU-8 2025 thickness versus spin speed.

152

SU-8 2025 Thickness Versus Spin Speed

153

9.3

Procedure For Synthesis Of TiO2 Micro-Spheres

Titanium dioxide (TiO2 ) micro-spheres were made from a sacrificial template. A porous polymer sphere was used as a template onto which the titanium dioxide precursor was coated. The precursor was reacted with water forming amorphous TiO2 ; the product fills the voids of the polymer sphere. The amorphous spheres were calcined to obtain rutile phase TiO2 spheres. Materials necessary to complete the process are given in Table 9-2. Table 9-2. Materials for sacrificial template synthesis of micro-spheres. Material

Purpose

Polydivinylbenzene porous beads: Amberchom 100C 100 µm polydisperse template microspheres Source 30 RCM 30 µm monodisperse template microspheres Titanium Isoproproxide TiO2 precursor Water Hydrolysis initiator Ceramic Crucible For calcination of spheres Instructions: 1. Dry the spheres supplied in ethanol by heating in a oven for 48 hours at 60°C. 2. In a nitrogen dry box: (a) Weigh an equal amount of dry porous beads and titanium isopropoxide. Suggested 1g of each.

3. 4. 5. 6. 7. 8.

(b) In a reasonably air tight container mix the titanium isopropoxide into the porous beads, close the container. Shake the container with mixture for 24 hours. Pour the mixture into a large dish of water using 100 mL of water for every 2 grams of combined precursor and template beads. Stir for 24 hours. Dry the beads for 24 hours at 60°C. Place the beads in a ceramic crucible. Heat the beads in atmosphere for 3 hours at 871°C (1600°F).

154

9.4

HFSS Electromagnetic Simulation Details

This section is explains the waveguide, free-space Floquet port, and general free-space incident wave setups in HFSS. 9.4.1

Waveguide Simulation Setup

The setup for waveguide used a rectangular region, which defined the waveguide interior. The lateral dimensions of the simulation region used the standard dimensions waveguide band simulated. The length of the simulation region extended beyond the sample a short distance to properly model the measurement setup. Waveports were used to provide the incident electromagnetic wave. The boundaries of the waveguide were assigned to be copper. Figure 9-5 is a diagram of the simulation as viewed in HFSS with markups including the detailed boundary and port parameters.

Figure 9-5. Diagram of HFSS waveguide simulation setup.

155 9.4.2

Free-Space Floquet Port Simulation Setup

The setup for the free-space Floquet port used a rectangular region, which defined location and volume of the sample and surrounding air. The lateral dimensions were those of one lattice spacing. The length of the simulation region extended beyond the sample a short distance. The boundaries were master-slave boundaries that can have a phase delay between equal fields on the opposing boundaries. Floquet ports were used to provide the incident electromagnetic wave incident waves. Figure 9-6 is a diagram of the simulation as viewed in HFSS with markups including the detailed boundary and port parameters. The phase angle definitions are shown in Figure 9-7.

Figure 9-6. Diagram of HFSS free-space Floquet port simulation setup.

156 9.4.3

Free-Space Incident Wave Simulation Setup

The setup for the free-space incident wave used a rectangular region, which defined location and volume of the sample and surrounding air. A plane propagating wave was incident with a given direction and polarization. Figure 9-7 shows the coordinate system.The the electromagnetic wave propagates toward the origin, the Poynting vector was anti-parallel to the ~r vector. The wave polarization was determined from the ‘ETheta’ or ‘EPhi’ components that are the magnitude of the electric field along the θˆ and φˆ directions. When θ and φ were zero then ‘ETheta’ pointed along the x axis in the x − z plane. ‘EPhi’ pointed along the y axis in the y − z plane. The electric field set was so the wave was a TE or TM incident wave. A TM interaction is set-up when ‘ETheta’ is 1 and ‘EPhi’ is 0. A TE interaction is set-up when ‘ETheta’ is 0 and ‘EPhi’ is 1.

(a) TE Polarization

(b) TM Polarization

Figure 9-7. HFSS spherical coordinate system with electric field polarization.

157

9.5

Abbreviated Biography Of The Author James Cramer was born in San Diego, California. He obtained his bachelor of science in physics from the University of California, Riverside. While at Riverside he worked on material surface science in ultra-high vacuum, including water decomposition studies on the surface of CeO2 .

After graduating, James began studies at the

University of Massachusetts Lowell. He completed his master of science in physics having assembled and qualified the calibration routines of a free-space and waveguide measurement system for the determination of the dielectric constant of materials. The next several years were spent upgrading the dielectric measurement system as well as working on negative index of refraction materials for far-infrared frequencies based on dielectric resonators, the subject of this thesis.