Microwave Metamaterials James A

Home , Victor Veselago

Advances In

MICROWAVE

METAMATERIALS

James A. Wigle

B.S.E.E, University of Maryland, 1991

M.S.E.E, Johns Hopkins University, 1996

A dissertation submitted to the Graduate Faculty of the

University of Colorado at Colorado Springs

in partial fulfillment of the requirements for the degree of Doctor of Philosophy

College of Engineering & Applied Science

Department of Electrical and Computer Engineering

2011 P a g e | ii

The author, James A. Wigle, copyrights all text, ideas, figures and photographs of this document. Any form of copying, reprinting, or publishing of this document, in any portion, is not permitted without the author‟s explicit, written, and prior permission in any form or medium. Contact the author for this written permission.

Advances in Microwave Metamaterials James A. Wigle

This dissertation for Doctor of Philosophy degree by

James A. Wigle

has been approved for the

Department of Electrical and Computer Engineering

______

John Norgard, Chair

______

Hoyoung Song, Co-Chair

______

Thottam S. Kalkur, Chair ECE

______

Tolya Pinchuk, Dept. of Physics

______

Zbigniew Celinski, Dept. of Physics

______

Date P a g e | iv

Abstract

Metamaterials are a new area of research showing significant promise for an entirely new set of materials, and material properties. Only recently has three-fourths of the entire electromagnetic material space been made available for discoveries, research, and applications.

This thesis is a culmination of microwave metamaterial research that has transpired over numerous years at the University of Colorado. New work is presented; some is complete while other work has yet to be finished. Given the significant work efforts, and potential for new and interesting results, I have included some of my partial work to be completed in the future.

This thesis begins with background theory to assist readers in fully understanding the mechanisms that drove my research and results obtained. I illustrate the design and manufacture of a metamaterial that can operate within quadrants I and II of the electromagnetic material space (Ɛr > 0 and µr > 0 or Ɛr < 0 and

µr > 0, respectively). Another metamaterial design is presented for operation within quadrant III of the electromagnetic material space (Ɛr < 0 and µr < 0).

Lorentz reciprocity is empirically demonstrated for a quadrant I and II metamaterial, as well as a metamaterial enhanced antenna, or meta-antenna. Using this meta-antenna I demonstrate improved gain and directivity, and illuminate how the two are not necessarily coincident in frequency. I demonstrate a meta-lens which provides a double beam pattern for a normally hemispherical antenna, which also provides a null where the antenna alone would provide a peak on boresight.

The thesis also presents two related, but different, novel tests intended to be used to definitively illustrate the negative angle of refraction for indices of refraction less than zero. It will be shown how these tests can be used to determine most bulk electromagnetic material properties of the material under test, for both right handed and left handed materials, such as Ɛr , µr , δloss, and n.

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The work concluding this thesis is an attempt to derive modified Fresnel Coefficients, for which I actually believe to be incorrect. Though, in transposing I have corrected a few mistakes, and now I can no longer find the conundrum. I have included this work to illuminate the need for modified Fresnel coefficients for cases of negative indices of refraction, identifying all disparate cases requiring a new set of equations, as well as to assist others in their efforts through illumination of the potential erroneous path chosen.

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Acknowledgements

This work would not be complete, if not for the help of others. My mentor Dr. John Norgard, and my in situ mentor Dr. Hoyoung Song, provided wonderful advice and guidance in my University of

Colorado efforts. Indispensible, and much appreciated, technical assistance and theory were provided by

Dr. Tolya Pinchuk of The University of Colorado Physics Department and Dr. Victor Gozhenko of The

National Aviation University, Ukraine (Віктор Гоженко, Національний Aвіаційний Університет,

Україна). I sincerely aspire to continue our relationships and expanding our knowledge of physics and electromagnetics. Mr. James Vedral significantly helped with modeling and technical discussions. It is with sincere appreciation that I thank all of you, my wife and family, and many others for the support required to complete this work. In the modified words of Sir Isaac Newton, my personal hero, efforts such as this are easy while standing on the shoulders of giants.

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Table of Contents

Abstract ...... iv

Acknowledgements ...... vi

Table of Contents ...... vii

List of Tables ...... xiv

List of Figures ...... xv

Chapter 1. Introduction...... 1

1.1 Purpose and Chapter Descriptions ...... 1

1.2 What is a Metamaterial? ...... 2

1.3 Some Metamaterial History ...... 2

1.4 General Metamaterials Information ...... 3

Chapter 2. Metamaterial Artifacts and Curiosities ...... 7

2.1 Proposed and Theorized Metamaterial Uses...... 7

2.2 The Perfect Lens ...... 9

2.3 Cloaking Success ...... 10

2.4 Directive Emission Using Metamaterials ...... 12

2.5 The Cherenkov Detector ...... 13

2.6 Negative Radiation Pressure ...... 13

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Chapter 3. Metamaterial Theory ...... 15

3.1 Introduction to Metamaterial Theory ...... 15

3.2 Law of Refraction (Snell‟s Law) and Refractive Index ...... 15

3.3 Index of Refraction‟s Forced Radical Sign ...... 16

3.4 Snell‟s Law with a Metamaterial Twist ...... 18

3.5 Permittivity ...... 20

3.6 Split Ring Resonators ...... 21

3.7 Narrow Frequency Bandwidths ...... 22

3.8 Plasma Frequency ...... 23

3.9 Physical Description of Right vs. Left Handed Materials ...... 24

3.10 Proof of Snell‟s Law Radical Sign Result ...... 26

3.11 Bulk Plasma Frequency ...... 32

3.12 Surface Plasma Resonance ...... 38

Chapter 4. Quadrant II, 10.5 GHz Metamaterial ...... 48

4.1 Introduction ...... 48

4.2 Design Theory ...... 48

4.3 The Design ...... 52

4.4 The Model ...... 53

4.5 Model Results ...... 55

Chapter 5. Quadrant III, 4.5 GHz Metamaterial ...... 58

5.1 Introduction ...... 58

5.2 Design Theory ...... 58

5.3 The Planned Design ...... 59

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5.4 The Model ...... 60

5.5 Model Results ...... 61

Chapter 6. Metamaterial Reciprocity for Quadrant II ...... 63

6.1 Introduction ...... 63

6.2 Test Setup for Metamaterial Reciprocity ...... 63

6.3 Metamaterial Reciprocity Test Results ...... 65

Chapter 7. Patch Antenna ...... 68

7.1 Introduction ...... 68

7.2 Patch Antenna Design ...... 68

7.3 Patch Antenna Measurements ...... 72

7.4 Conclusions ...... 78

Chapter 8 Improved Directivity...... 79

8.1 Introduction ...... 79

8.2 Measuring Antenna Directivity ...... 80

8.3 Directivity Graphs ...... 81

8.4 Interpreting the Results ...... 85

8.5 Conclusions ...... 86

Chapter 9. Improved Gain ...... 87

9.1 Introduction ...... 87

9.2 Directivity & Gain Test Setup and Measurements ...... 87

9.3 Frequency Dependency of Directivity versus Gain ...... 90

9.4 Gain Calculations ...... 91

9.5 Improved Gain Results ...... 97

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9.6 Conclusions ...... 98

Chapter 10. Meta-Antenna Reciprocity ...... 100

10.1 Introduction ...... 100

10.2 Test Setup for Meta-Antenna Reciprocity ...... 101

10.3 Test Results ...... 102

10.4 Conclusions ...... 104

Chapter 11. Multifaceted Meta-Lens ...... 105

11.1 Introduction ...... 105

11.2 Meta-Lens Construction ...... 105

11.2 Double Beam Meta-Lens ...... 106

11.2.1 Test Setup ...... 106 11.2.2 Results ...... 107 11.2.3 Conclusions ...... 109 11.3 Double Beam Gains ...... 110

11.3.1 Test Setup ...... 110 11.3.2 Gain Calculations ...... 112 11.3.3 Results ...... 115 11.3.4 Conclusions ...... 115 11.4 Summary...... 116

Chapter 12. Infrared Test to Determine Metamaterial Properties...... 117

12.1 Introduction ...... 117

12.2 Theory...... 118

12.2.1 The Crux of the Matter...... 118 12.2.2 Ray Trace Geometry ...... 118 12.2.2.1 Displacement Solution for Positive Indices of Refraction ...... 119 12.2.2.2 Displacement Solution for Negative Indices of Refraction ...... 121

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12.2.3 Solutions for n, r, r , µr, and δloss ...... 123 12.2.3.1 Positive Indices of Refraction Solution ...... 123 12.2.3.2 Negative Indices of Refraction Solution ...... 124 12.2.4 Interesting Results ...... 125 12.3 Test Design ...... 126

12.4 Test Results ...... 128

12.5 A Test at Optical Frequency ...... 131

12.6 Conclusions ...... 132

Chapter 13. Microwave Test to Determine Metamaterial Properties ...... 133

13.1 Introduction ...... 133

13.2 Test Setup ...... 133

13.3 Results ...... 135

13.4 Conclusions ...... 135

Chapter 14. Fresnel Coefficient Matrix ...... 136

14.1 Work Left to More Capable Hands...... 136

14.2 Fresnel Coefficient Matrix ...... 137

14.3 rh and rh ...... 138

14.3.1 rh ...... 140

14.3.2 rh ...... 143

14.3.3 rh ...... 144

14.3.4 rh ...... 147

14.4 lh and lh...... 149

14.4.1 lh...... 151

14.4.2 lh ...... 154

14.4.3 lh ...... 156

14.4.4 lh ...... 159

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14.5 Right Handed Scenario ...... 161

H 14.6 rh ...... 164

H 14.7 rh ...... 166

H H 14.8 rh and rh Relationship ...... 167

H H 14.9 rh and rh Limit Checks ...... 168

E 14.10 rh ...... 169

E 14.11 rh ...... 172

E E 14.12 rh and rh Relationship ...... 173

E E 14.13 rh and rh Limit Checks ...... 173

14.14 Left Handed Scenario ...... 175

H 14.15 lh ...... 177

H 14.16 lh ...... 180

H H 14.17 lh and lh Relationship ...... 181

H H 14.18 lh and lh Limit Checks ...... 181

E 14.19 lh ...... 183

E 14.20 lh ...... 186

E E 14.21 lh and lh Relationship ...... 187

E E 14.22 lh and lh Limit Checks ...... 188

Chapter 15. Summary ...... 191

Chapter 16. Future Investigations...... 192

Reference List ...... 194

Appendix A – Peer Reviewed Paper ...... 199

Appendix B – Metamaterial Calculator ...... 214

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Index ...... 232

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List of Tables

Table 1. Quadrant II Metamaterial Design Parameters ...... 53 Table 2. 11 GHz Patch Antenna Characteristics ...... 73 Table 3. Meta-Patch Directivity vs. Gain Comparison ...... 90 Table 4. Meta-Patch Gain Test Results ...... 98 Table 5. Meta-Lens Gain Results ...... 115 Table 6. Metamaterial Properties for IR Test...... 131 Table 7. Fresnel Coefficient Matrix ...... 137

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List of Figures

Figure 1. Electromagnetic Material Space ...... 4 Figure 2. Duke Metamaterial, Source = Science Oct 2006 ...... 11 Figure 3. Duke Metamaterial Model, Source = Science Oct 2006 ...... 11 Figure 4. Fish in n<0 material, Source = Jason Valentine of Berkeley ...... 19 Figure 5. Illustration of Snell's Law with Negative Index of Refraction ...... 19 Figure 6. Low Frequency E-Field Effects upon Metals ...... 24 Figure 7. Plasma Frequency E-Field Effects upon Metals ...... 24 Figure 8. Vector Directions vs. Index of Refraction ...... 26 Figure 9. Right Handed Orientation ...... 28 Figure 10. Wave Orientation for N > 0 ...... 29 Figure 11. Wave Orientation for N < 0 ...... 30 Figure 12. Left Handed Orientation ...... 31 Figure 13. Electron Orientation for Metal with an E-Field ...... 32 Figure 14. E-Field & Charged Plate Analogy ...... 33 Figure 15. Effective Permittivity vs. Energy ...... 37 Figure 16. Metamaterial Example ...... 37 Figure 17. E-Field Spheres, Super-positioned ...... 41 Figure 18. Distances and Labels ...... 43 Figure 19. Surface Plasma Resonance...... 44 Figure 20. Quadrant II Metamaterial ...... 48 Figure 21. Thin wire metamaterial structure ...... 51 Figure 22. Proto Circuit Board Machine ...... 52 Figure 23. Quadrant II Metamaterial Model, Source = James Vedral of University of Colorado . 55 Figure 24. Quadrant II Metamaterial Model Results ...... 56 Figure 25. Proposed SSRR Design...... 59 Figure 26. Quadrant III Metamaterial Model, Source = James Vedral of University of Colorado 61 Figure 27. SSRR Model Results ...... 62 Figure 28. Metamaterial Reciprocity Test Setup ...... 64 Figure 29. Metamaterial Reciprocity, same polarization ...... 66 Figure 30. Metamaterial Reciprocity, cross polarized ...... 67 Figure 31. Patch Antenna Schematic ...... 71 Figure 32. Patch, Front Side ...... 72 Figure 33. Patch, Element ...... 72

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Figure 34. Patch, Ground Plane ...... 72 Figure 35. Patch, Feed ...... 72

Figure 36. 11 GHz Patch, S11 ...... 74 Figure 37. 11 GHz Patch, Impedance ...... 75 Figure 38. 11 GHz Patch Antenna, E & H Plane Radiation Patterns ...... 77 Figure 39. Metamaterial with Patch ...... 80 Figure 40. Metamaterial with Patch ...... 80 Figure 41. Normalized Meta-Antenna Patterns over 9.8-10.0 GHz ...... 83 Figure 42. Normalized Meta-Antenna Patter over 11-12 GHz ...... 84 Figure 43. Normalized Meta-Ant Best 3 ...... 85 Figure 44. Normalized Meta-Ant Enhancement ...... 85 Figure 45. Meta-Patch Gain Baseline Test Schematic ...... 88 Figure 46. Meta-Patch Gain Test Schematic ...... 88 Figure 47. Meta-Patch, Normalized E-Plane Pattern at 11.45 GHz ...... 94 Figure 48. Patch Antenna, Normalized dB E-Plane Pattern at 11.45 GHz ...... 95 Figure 49. Bi-Ridged Flared Horn Antenna, Normalized E-Plane Pattern at 11.45 GHz ...... 96 Figure 50. Flared Waveguide Horn Antenna, Normalized E-Plane Pattern at 11.45 GHz ...... 97 Figure 51. Meta-Antenna Reciprocity Test Setup ...... 101 Figure 52. Meta-Antenna Reciprocity, Normalized dB E-Plane ...... 103 Figure 53. Meta-Antenna Reciprocity, Normalized dB H-Plane ...... 104 Figure 54. Meta-Lens Construction ...... 105 Figure 55. Meta-Lens Pattern Test Setup ...... 107 Figure 56. Double Beam Meta-Lens, 10.83 GHz ...... 108 Figure 57. Double Beam Meta-Lens, 10.75 GHz ...... 109 Figure 58. Meta-Lens Gain Test, Baseline Test Setup ...... 111 Figure 59. Meta-Lens Gain Test Setup ...... 111 Figure 60. Ray Trace Geometry ...... 118 Figure 61. Maximum Δ Displacement vs. Angle of Incidence (rads), Source = Victor Gozhenko of Univ. of Colorado ...... 119 Figure 62. n > 0 Ray Trace ...... 119 Figure 63. n < 0 Ray Trace ...... 121 Figure 64. Material Properties Test Schematic, IR Sensor ...... 127 Figure 65. IR Detector Material...... 127 Figure 66. IR Material Properties Test ...... 129 Figure 67. 17 GHz Material Displacement w/Meta ...... 129 Figure 68. IR Material Properties Test ...... 129 Figure 69. 17 GHz Material Displacement w/o Meta ...... 129 Figure 70. 10.8 GHz Material Displacement w/Meta ...... 130 Figure 71. 10.8 GHz Material Displacement w/o Meta ...... 130 Figure 72. Optical Test Schematic ...... 131 Figure 73. Microwave Material Properties Test Schematic ...... 134 Figure 74. Vector Orientations ...... 139

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Figure 75. Right Handed Material Fresnel Coefficients, Scenarios 1-3 ...... 162 Figure 76. Left Handed Material Fresnel Coefficients, Scenario 4 ...... 176

Advances in Microwave Metamaterials James A. Wigle

Chapter 1. Introduction

1.1 Purpose and Chapter Descriptions

his thesis is a doctor of philosophy dissertation for the Electrical and Computer T Engineering Department at the University of Colorado, directed by Dr. John Norgard and co-directed by Dr.‟s Hoyoung Song and Tolya Pinchuk. This dissertation will introduce numerous new areas of research within microwave metamaterials. Given the microwave focus, research was carried out in the macro-scale, or centimeter scale, and does not detail any research on the nanometer scale, which is so common within the metamaterials field.

Chapters one and two, and to some extent chapter three, provide information regarding literary searches. These chapters provide background information while describing the extent of my literary search relating to this dissertation topic. It clarifies what has been accomplished, or current areas of metamaterials research. Chapter three provides extensive detailed metamaterial theory. It is believed this background is required to better understand my work, assumptions, reasons for various research directions, and the like.

Remaining chapters convey my new and unique work, which has transpired over numerous years at the

University of Colorado, since Dr. Pinchuk first addicted me on the subject back in 2008. Appendix A simply reproduces collaborative work submitted for publishing and peer review, which has yet to be published. It is reproduced in unmodified form, and was submitted for publication in Physical Review B, in

January 2011, and later submitted to the Journal of Applied Physics. Appendix B provides my metamaterial calculator Perl computer code. A new paper, based on my meta-lens work detailed here, is to soon be submitted for publishing within a physics journal, but it is a bit premature to include that work within this document, along with a National Science Foundation proposal recently submitted based on this related work. Chapter 1. Introduction | P a g e | 2

1.2 What is a Metamaterial?

A definition of metamaterial is in order, as most works lack the definition or ill define the concept.

A metamaterial is a composite material producing macro-scale permittivities, permeabilities, and resulting indices of refraction not found within its constituent materials. Note that my definition does not exclude positive or negative values for any of these material parameters: Permittivity, permeability, or index of refraction. The more commonly researched metamaterials are those with both negative permittivity and negative permeability, also called double negative materials, which can produce quite startling electromagnetic effects. Currently, most research involves man-made metamaterials. However, this metamaterials definition does not exclude naturally occurring composite materials from being a metamaterial with differing, from constituent components, electromagnetic properties on the macro-scale; as they are most likely certain to naturally exist in relative obscurity, or at least without the fanfare of man- made metamaterials. An example of an unintended man-made metamaterial would be microwave cooking oven doors, which should have a plasma resonance at some determinant frequency; though, most likely not improved directivity at 2.45 GHz where they operate in frequency (hopefully not so). As an interesting side note, it would be curious to model a microwave cooking oven door. Naturally occurring metals suspended within material would constitute a metamaterial, such as metal ions in sea water (though maybe only slightly „metamaterialish‟), or metal distributed within earth on a larger scale than the ore itself, chicken wire, metal fencing, etc.

„Left handed materials‟ and „metamaterials‟ are used interchangeably within numerous articles and texts. As will be discussed later, I claim left handed materials are a subset of metamaterials, as are right handed materials; both of which render the complete metamaterial set.

1.3 Some Metamaterial History

Metamaterial mania can be traced back to Victor G. Veselago‟s 1967 paper[32], in which he announced the theoretical possibility of manufacturing materials with negative indices of refraction. A

Advances in Microwave Metamaterials James A. Wigle Chapter 1. Introduction | P a g e | 3 fraction over three decades later in 1999, Sir John B. Pendry used Veselago‟s work to develop new concepts, such as the thin wire model to obtain negative permittivity indices, as they should exhibit resonant frequency responses similar to plasma media[33]. Pendry also later described materials that could produce negative permeabilities, using split ring resonators (SRRs) acting like magnetic dipoles. In March

2000, David R. Smith and Sheldon Schultz used Pendry‟s work to create the first double negative metamaterial, within the microwave range of frequencies[29]. Since Smith et. al. had demonstrated the first double negative metamaterial, it was at this point in time a flurry of new research began to expand knowledge of these metamaterials. Though the flurry of new metamaterial research continues, relatively little has been explored since Veselago and Smith have essentially opened up three quarters of the electromagnetic materials space. Invisibility cloaks continue to be one of the most researched topics, regarding metamaterial implementations. In October 2006, David Smith et. al. demonstrated the first

„invisibility cloak‟, within the microwave frequency range[34]. Since then, other uses have been theorized or produced in later years, which will be later explored within this thesis.

1.4 General Metamaterials Information

The vast majority of materials reside within quadrant one of the electromagnetic material space

(see Figure 1). However, some natural materials do exist, under unique conditions, within the second and fourth quadrants. For example, a number of metals such as silver and gold reside within the second quadrant, within the ultraviolet frequency range. However, no natural material has been found within the third quadrant.

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Quadrant II Quadrant I Ɛ < 0 and µ > 0 Ɛ > 0 and µ > 0 N > 0 or N < 0 ??? N > 0 Left handed materials. ??? Right handed materials. Very few natural materials at specific frequencies Almost all natural materials. (e.g. metals at optical frequencies). Large freq. portions transparent to EM radiation. Opaque to EM radiation (huge attenuation). E&M wave propagation. No wave propagation. Ɛ

Quadrant III Quadrant IV Ɛ < 0 and µ < 0 Ɛ > 0 and µ < 0 N < 0 N > 0 or N < 0 ??? Left handed materials. Left handed materials. ??? No natural materials; man-made metamaterials only. Very few natural materials at specific frequencies. Small freq. portions semi-transparent to EM radiation. Opaque to EM radiation (huge attenuation). E&M wave propagation with significant attenuation. No wave propagation.

Figure 1. Electromagnetic Material Space

According to Wikipedia[4], materials residing within the second and fourth quadrants are opaque to electromagnetic radiation. This source also indicates all transparent media resides within the first quadrant of the electromagnetic material space, which shall be proved later. Wikipedia[4] also writes that Ɛ and µ do not have to be simultaneously negative to provide a negative index of refraction, which can be seen in

Depine and Lakhtakia[14]. In essence, negative radical solutions for the index of refraction result if the permittivity and permeability are both less than zero. Some debate continues regarding whether or not it can be demonstrated that the negative radical is also forced if only one of these two parameters is less than zero.

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Hoa and Mittra indicate wave transmission occurs only when both parameters have the same sign[26]. This means that transmission only occurs within quadrants I and III of the electromagnetic material space (see Figure 1). As to why this is so will be detailed later.

One point to keep in mind is that Ɛ and µ are the only relevant electromagnetic macro-scale material parameters used to describe electromagnetic wave interactions with a material. Though this interaction may become extremely complex for inhomogeneous materials, it does hold for homogeneous materials, or materials that are homogeneous to less than a single order approximation. This is one reason why metamaterial inclusions should be much less than the wavelength for which they work. The popular inclusion size appears to be less than an order of magnitude, or a factor of 0.1, of the wavelength for which it is to function, as macro scale approximations are much simplified.

An interface from a regular medium to a metamaterial medium does not imply that Snell‟s law provides a negative index of refraction. Also, metamaterial effects are not always the result of having negative values for both Ɛ and µ, as these parameters can have any number of values and mathematical signs. Rightfully so, the authors of [15] illuminate an obvious and most likely forgotten issue. The interesting portion of metamaterials is not the medium itself, but the resulting refraction on the interface between a regular material and a metamaterial, especially a left handed material.

Hapgood‟s article[17] points out some interesting artifacts of metamaterials. Living within a silvery bubble, anyone inside a visible spectrum metamaterial cloaking device would not be seen, nor would they be able to see outside their metamaterial bubble. However, the 2011 Third International Topical Meeting on Nanophotonics and Metamaterials indicated this is not so, and unidirectional effects could be seen1.

Another effect would be ripples within a pond of metamaterial might flow inward toward the site of impact with a rock[17]. Much like fish swimming above a metamaterial, I find this difficult to visualize.

How would the ripples begin far away from the impact site, and then ripple toward the impact zone? I noticed the article used the subjunctive tense, with the word „might‟. In any event, this phenomenon and others yet to be realized, provide a tantalizing curiosity of the largely unexplored materials space.

1 WED1o plenary talk 3 by Vladimir Shalaev.

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A metamaterial obstacle to overcome is the absorption of radiation as it passes through a metamaterial; γ in Drude‟s formula. In order to experience a visible cloaking metamaterial properly, the radiation must pass through the material before it is significantly absorbed. This appears to be a substantial problem to overcome. Therefore, researchers strive to obtain metamaterials that are broader in bandwidth, as well as reduce absorption of the radiation it affects.

It should be noted that an electromagnetic wave undergoes a wavelength change when traveling from one index of refraction to another different index of refraction. As a side, and something to remember, this gives rise to dispersion in the visual light spectrum of different wavelengths. The frequency on both sides of the media must be the same, so time coincidence is maintained. However, due to the differing velocities, the wavelengths will be different[13].

Advances in Microwave Metamaterials James A. Wigle

Chapter 2. Metamaterial Artifacts and Curiosities

2.1 Proposed and Theorized Metamaterial Uses

y far, the most popular postulated use of metamaterials involves cloaking devices to make B things invisible, particularly within the visual portion of the spectrum2. University of Maryland‟s Clark Labs was the first to manufacture and prove the concept within the infrared spectrum, on the nano-scale[23]. New successful strides toward visible spectrum invisibility cloaks occur almost monthly.

Berkley has been one of the most successful entities broadening the bandwidth of their invisibility cloaks, again on the nano-scale within the infrared spectrum[6]. Recently, the University of California in Los

Angles has successfully manufactured a cloaking device on the centimeter scale, measuring two feet long by four inches wide[17]. This scale is orders of magnitude larger than previous successes, but functions only within a portion of the microwave range.

The 3 March 2009 Science Daily article[16] reported a significant breakthrough, using metamaterial technology. Padilla et. al. controlled a wide range of teraHertz beam radiation using a metamaterial structure. This is significant in that it demonstrates an entirely new method of controlling radiation using a solid state, electronically fast, and wide bandwidth method. This may have significant implications for the traveling wave tube (TWT) and associated industries. This opens up modulation methods for the teraHertz range of radiation, which has historically not been possible on an electronically fast scale.

Hapgood[17] mentions identifying specific molecules using metamaterial enabled spectrometers. I assume this would be enabled via sub-wavelength optical resolution. Radiation shields for specific items were mentioned, such as performing surgery with a metamaterial scalpel while using MRI machine

2 Recently, more and more articles and texts have been loosely defining „invisibility‟ to include non-visible spectra as well, e.g., infrared and ultraviolet. Chapter 2. Metamaterial Artifacts & Curiosities | P a g e | 8 imaging. The metamaterial principles, not metamaterials themselves, could be used to manipulate other waves. Examples of this include controlling water waves from storms around oil rigs and tidal waves, or producing acoustic quiet spots within noisy environments.

The paper by Lindell et. al.[15] indicates that reverse Poynting vectors are already known to exist in some structures, prior to the metamaterials thrust. One such structure is the traveling wave tube (TWT) for certain frequencies. This is a result of a periodic structure within the TWT. Thus, one could argue that this structure, in itself, may be thought of as a metamaterial structure; even if it pre-existed Veselago‟s 1967 introduction of metamaterials.

A unique scenario of electromagnetic parameters provides a unique effect. If 1 = - 2 and µ1 = -

µ2, then there will be no reflections, as I prove later in the Fresnel Coefficients Matrix chapter. If there are no reflected waves when the condition µ1 = -µ2 and Ɛ1 = -Ɛ2 is satisfied, this can be applied to make a simple matched focusing lens; i.e., the „perfect lens‟ (almost perfectly lossless). A lot of metamaterials research is dedicated to this „perfect lens‟, which will be later detailed further. I will soon explain how resolution beyond the diffraction limit is possible, as has already been demonstrated by Berkeley and The

University of Toronto.

Veselago‟s paper[32] predicts right handed versus left handed electromagnetic wave propagation, reverse Doppler effects, reverse Cherenkov radiation, negative or positive permeabilities, negative or positive permittivities, negative radiation pressure (or positive tension), reverse bi-conical and bi-convex lens roles, negative Snell angles, and negative indices of refraction. We have only scratched the surface of material use possibilities. Three fourths of the entire electromagnetic material space has recently been opened. Who knows what will come of this new area? I‟ve detailed only recent tangible successes below, not theoretical or future possibilities. We have entered the era of metamaterials, nano-photonics, or plasmonics.

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2.2 The Perfect Lens

The „perfect lens‟ (also known as „super lens‟) describes a Sir Pendry idea that a metamaterial can be manufactured which acts as a lens, with special properties to include no reflections and no theoretical resolution limit[28]. Refractive index variations allow traditional lenses to focus electromagnetic waves to a single point, but no smaller than a square wavelength. Unfortunately, evanescent waves decay too rapidly, and no traditional mechanism allows perfect phase reconstruction, even if the evanescent wave amplitudes were restored. Thus, the traditional diffraction limit is born. The maximum resolution within an image will not be less than one wavelength, regardless of lens imperfections and aperture size. Evanescent waves are those near field, non-propagating electromagnetic wave components.

Using double negative metamaterials, however, one can manufacture a lens unrestricted by the bounds of the diffraction limit, mainly because evanescent waves are reconstructed. Therefore, all source information is contained within the image created by this perfect lens. If the permittivity and permeability match that of the transmitting medium, but are negative in mathematical sign, then the impedance will be positive and perfectly matched, and thus no reflections will occur. The equation for the reflection coefficient solves to zero (0% reflection), while the equation for the transmission coefficient solves to one (100% transmission), see chapter 14. Moreover, the medium does not attenuate evanescent waves, and evanescent phase is not distorted. Actually, the evanescent waves are amplified via this medium transmission. One might hastily assume a conservation of energy violation, but no energy is transmitted via evanescent waves.

√ √

Given the amplification of evanescent waves, normal propagating and evanescent waves both contribute to image reconstruction. Discounting manufacturing limitations of the metamaterial lens, there is no theoretical image resolution limit. One last item to note is that this perfect lens is a planar slab of metamaterial, and no curved surfaces are required, given that electromagnetic waves bend toward the normal; i.e. no curvature is required, like that of optical glass lenses. Though not fully understood, Smith

Advances in Microwave Metamaterials James A. Wigle Chapter 2. Metamaterial Artifacts & Curiosities | P a g e | 10 et. al. indicate that these metamaterial slabs will not focus electromagnetic waves from infinity (assumed planar waves are referenced)[25]. This may simply state that the perfect lens cannot be located at infinity due to evanescent wave proximity concerns, as opposed to planar waves being focused by the perfect lens.

Just a note to indicate Dr. Pinchuk and I had difficulty solving for the no-reflection situation for the perfect lens situation. Therefore, it is left out of this thesis.

Within the same article, Pendry indicates that the γ = 0 approximation within the Drude formula is fine for numerous metals, specifically calling out silver, gold, and copper. He indicates that this is because these metals behave much like perfect plasmas.

The first perfect lens (resolution below the diffraction limit), at microwave frequencies, was demonstrated at the University of Toronto in 2004. In 2005, Zhang of Berkeley demonstrated the first optical perfect lens, with a resolution several times better than the best optical microscope. Though, Zhang enhanced evanescent modes via surface plasmon coupling, not negative indices of refraction.

2.3 Cloaking Success

As this is of obvious interest to many people, I will detail various successes in this regard, but note that cloaking breakthroughs occur often. Many other cloaking devices have been very successful. Given the sheer number of successes, only the first two breakthroughs are detailed.

The first cloaking device, resonant within the microwave frequency range, was developed at Duke

University by Smith et. al. in 2006. It cloaked a copper cylinder at the resonant microwave frequency.

Advances in Microwave Metamaterials James A. Wigle Chapter 2. Metamaterial Artifacts & Curiosities | P a g e | 11

Figure 2. Duke Metamaterial, Source = Science Oct 2006

Figure 3. Duke Metamaterial Model, Source = Science Oct 2006

Advances in Microwave Metamaterials James A. Wigle Chapter 2. Metamaterial Artifacts & Curiosities | P a g e | 12

The University of Maryland‟s Clark School of Engineering claims to have created the world‟s first true invisibility cloak[23]. The cloaked object is only 10 nanometers wide, for a narrow frequency range, but proves the concept. The cloak is comprised of a two dimensional configuration of gold concentric rings, coated in polymethyl methacrylate (plastic). The entire configuration lies on a gold surface. This work was reported in New Scientist and in Discover Magazine‟s “Top 100 Science Stories of 2007”.

As written previously, a significant metamaterial issue to overcome is the lossy material problem.

It would be amazing to design and manufacture an invisibility cloak, but it would not do much good if the invisibility cloak attenuated light so much as to appear as a dark cloud moving around. In [28], Smith et. al. indicate no material has demonstrated low loss in combination with a negative permittivity. However, numerous claims since then, to include the Nanometa 2011 conference in Seefeld Austria3, indicate that low loss double negative materials have been achieved.

2.4 Directive Emission Using Metamaterials

Enoch et. al.[27] demonstrated that emission within a metamaterial can be used to direct microwave energy. Their paper shows that emissions within close to zero, but positive, relative permittivity metamaterial slabs produce emissions close to the normal upon exiting the slab, even for very small incidence angles at the interface.

To demonstrate this effect, the authors modeled and manufactured a six layered metamaterial, with an additional ground plane. Their feed element was a monopole placed between the third and fourth layers.

They measured the system‟s directivity as 372. It matched decently close to the model, which used the method of moments, thin wire approach. For their frequency, 14.65 GHz, their design approximates the plasma frequency using the Drude formula with the γ = 0 approximation. Their design also uses copper wire mesh on printed circuit boards, sandwiched between foam boards.

3 Nanometa 2011, Third International Topical Meeting on Nanophotonics and Metamaterials, 3-6 January 2011 in Seefeld, Austria.

Advances in Microwave Metamaterials James A. Wigle Chapter 2. Metamaterial Artifacts & Curiosities | P a g e | 13

Another group, Li et. al.[10], used circular waveguides with a double layer metamaterial. They also experienced directive emission improvement. eff was designed using the γ = 0 Drude formula approximation. They indicated an 8.3 dBi improvement.

This work will later illustrate several new examples and methods of directive emission enhancement.

2.5 The Cherenkov Detector

Cherenkov radiation is radiation induced by charged particles moving through a medium, at speeds faster than the speed of light within that medium. Cherenkov radiation detection has a significant place within high energy physics, such as that used within particle accelerators. Antipov and Spentzouris envision using metamaterials to enhance Cherenkov radiation detection within particle accelerators[35].

Veselago‟s paper[32] predicts reverse Cherenkov radiation. Antipov and Spentzouris intend to use this phenomenon to collect this radiation in a cleaner environment, than traditional forward scattering. The radiation propagates in the opposite direction to accelerator particles and propagating waves. Thus, the double negative metamaterial should enjoy a cleaner environment in which to collect this radiation. They have produced the appropriate metamaterial, but have yet to perform the accelerator experiment.

2.6 Negative Radiation Pressure

During the 2011 Nanometa conference, Dr. Henri Lezec of The National Institute of Standards and

Technology (NIST) provided a presentation experimentally demonstrating Veselago‟s predicted negative radiation pressure, or radiation tension4. He experimentally confirmed his results using both a blue and green 1 mW laser. His first experiment used an ultra-tiny left handed material cantilever within a scanning electron microscope. The right handed material exhibited a radiation pressure as expected, and the left handed material exhibited a radiation tension. Another experiment was later performed using an ultra-tiny

4 THU6s breakthrough 4 talk by Henri Lezec of NIST-MD, at Nanometa 2011, Third International Topical Meeting on Nanophotonics and Metamaterials, 3-6 January 2011 in Seefeld, Austria.

Advances in Microwave Metamaterials James A. Wigle Chapter 2. Metamaterial Artifacts & Curiosities | P a g e | 14 left handed material slab, within an evacuated chamber. This slab lifted up toward the laser, and the top of the housing, when the laser was turned on.

Dr. Lezec also briefly mentioned how the discrete photon momentum change occurs at the positive index of refraction to the negative index of refraction interface, via “Photon assisted reverse

Doppler shift force and the Lorentz force”; all without violating conservation of energy or conservation of momentum.

Advances in Microwave Metamaterials James A. Wigle

Chapter 3. Metamaterial Theory

3.1 Introduction to Metamaterial Theory

or a better understanding of mechanisms behind these investigations, this chapter F details the processes behind how metamaterials influence and manipulate electromagnetic fields. This chapter should assist the reader in better understanding efforts detailed within this work.

As the reader will see, material permittivity or permeability will be modified using inclusions. It will later be evident that these modifications of permittivity and permeability are accomplished without employing magnetic or ferrous materials. In reality most modifications are accomplished employing copper patterns.

3.2 Law of Refraction (Snell’s Law) and Refractive Index

Snell’s law gives us:

( ) ( ) and

⁄

Where: Nx = Index of refraction, or refractive index, per subscript material.

ϴx = Angle of electromagnetic wave incidence, per subscript material. Chapter 3. Metamaterial Theory | P a g e | 16

c = Speed of electromagnetic wave propagation, within a vacuum.

vp = Speed of electromagnetic wave phase front, within material, not the speed of energy transfer.

ω = 2πf = Radian frequency.

k = Wave number.

f = Frequency.

λ = Wavelength.

The index of refraction can also be calculated via:

√

Where: N = Refractive index of material.

Ɛr = Relative permittivity of material.

µr = Relative permeability of material.

Lastly,

Where: = Group velocity, speed of energy transfer.

3.3 Index of Refraction’s Forced Radical Sign

Victor Veselago predicted a double negative material would produce a negative index of refraction[32]. With the exception of very few materials in unique circumstances (e.g., certain frequencies),

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 17 all materials have a positive index of refraction. For centuries, scientists and engineers assumed the positive radical result. A negative radical does indeed result for the index of refraction, if Ɛr and µr are negative, as shown below for the purely real parameter case. This describes quadrant III of the electromagnetic material space, see Figure 1.

√

Where: N = Refractive index of material.

Ɛr = Relative permittivity of material.

µr = Relative permeability of material.

Now let‟s display the variable signs, explicitly, and solve for N again.

We know Ɛr < 0 and µr < 0, so we can rewrite this as:

√( ) ( )

√( )√( )√

√

As normally used for centuries from empirical results found in nature, a positive radical results for the index of refraction, if the Ɛr and µr components are both positive. This describes quadrant I of the electromagnetic material space, see Figure 1. However, what if one of the two electromagnetic material parameters is less than zero, while the other is positive? These two circumstances describe quadrants II and

IV of the electromagnetic material space, see Figure 1. As you can see below, the index of refraction becomes purely imaginary, again for the purely real parameter case, and does not propagate or transfer energy.

√( ) ( )

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 18

√( )√

√

3.4 Snell’s Law with a Metamaterial Twist

Explanations regarding the physical effects of a negative index of refraction are sometimes difficult to imagine. Figure 5 illustrates the physical situation, which has been verified experimentally, and two new methods of measurement are presented later in this work.

Articles provide examples of how straight poles from air into a metamaterial with a negative index of refraction would appear to bend above the interface, as well as any fish swimming within this metamaterial would also appear to swim above this metamaterial[5, 6 part 1, 6 part 2, 17]. See Figure 4 for an illustration of fish swimming above a double negative material.

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 19

Figure 4. Fish in n<0 material, Source = Jason Valentine of Berkeley5

These phenomena are believed to be the result of „mental Line normal to interface tricks‟, much like that of watching a fish swimming in water from a stream bank. To the shore observer, the fish appears to swim n1 Material1 in a location where it is not Material2 n2 actually located. Referring to

Figure 5, the fish‟s location is

ϴ2 ϴ2 along the n2 > 0 solid red line Angle of Angle of refraction refraction trace, but to the observer it‟s for for

n2 < 0 n2 > 0 located along the dashed black „ray Figure 5. Illustration of Snell's Law with Negative Index of Refraction

5 Use authorized 29 March 2011 by Sarah Yang, Office of Public Affairs, Berkeley University.

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 20 trace of the incidental ray‟ since the observer‟s normal visual experiences usually never involve refraction.

Therefore, a double negative material is theorized to produce like effects; poles bending back above the surface, and fish swimming above the interface surface. I continue to truly find it difficult to visualize this negative index effect, and wonder how it would truly appear to an observer, especially when a fish swimming above a metamaterial is also mentally non-intuitive. I anxiously await such a double negative metamaterial.

3.5 Permittivity

One of the many uses of metamaterials is to reduce a material’s plasmon frequency. Altering effective permittivity is one method used, as was used for the quadrant I and II metamaterial described within this thesis. This subsection describes the theory regarding how this is achieved.

( ) 6 7

Equation 1. Drude Formula.

And

Where: Ɛeff = Effective permittivity.

ω = Radian frequency.

ωp = Plasma frequency.

γ = Scattering rate.

η = Scattering time.

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 21

Where: ωp = Plasma frequency.

neff = Effective electron density, of single electron.

e = Electron charge.

Ɛo = Permittivity of free space.

meff = Effective electron mass, of single electron.

Note that Ɛeff is negative (metamaterial criteria via some articles) when ω < ωp . Though this may be a poor approximation, a few articles[10, 27, 28, 29] took γ = 0 for long wavelengths. Given this approximation, the Drude formula becomes:

( )

This γ = 0 approximation of the Drude formula did appear to work for Li et. al. in their antenna array work [10]. They actually manufactured the modeled array, and both the model and manufactured antenna array achieved fairly close results. For the metamaterials designed and manufactured within this thesis, the γ = 0 Drude formula approximation was also successfully used.

3.6 Split Ring Resonators

Split ring resonators (SRRs), and variants thereof, have played a significant role in producing double negative materials. Sir Pendry was the first to predict their potential use for obtaining negative permeabilities. David Smith was the first to actually design and manufacture a double negative material

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 22 using split ring resonators. Smith et. al.[29] did derive an equation for their specific split ring resonator‟s effective permeability, µeff.

Where: µeff = Effective permeability of material.

F = Fractional area of unit cell occupied by the interior of the split ring.

ω = Radian frequency.

ωo = Technically unanswered, but assumed free space radian frequency.

Γ = Dissipation factor.

Smith et. al.‟s article goes further providing another approximation for the plasma frequency. It does not provide the derivation, but gives the approximation as:

(for high conductivity values) √

Where: ωp = Radian frequency.

d = technically unanswered, but may be trace diameter.

L = Self-inductance per unit length.

⁄ Free space permittivity.

3.7 Narrow Frequency Bandwidths

One significant item to note is the narrow band characteristics of metamaterial enhanced items. Li et. al.[10] illustrate this using two graphs within their paper. The first graph illustrates antenna gain versus

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 23 mesh spacing. This demonstrates just how narrowband the metamaterial functional frequency is, which is due solely to the wire mesh spacing. The 3 dB mesh spacing is ≈ 3 mm. The second graph plots antenna gain versus frequency. The 3 dB frequency bandwidth is ≈ 4.2% (500 MHz at 12 GHz). Again this illustrates just how narrowband the metamaterial usable frequency is. It appears that new approaches will be required to overcome the inherent narrow frequency bandwidth issue. Of course, narrow frequency bandwidths can be very advantageous, depending upon the application.

Indeed this appears to be one major issue to overcome for metamaterials. A lot of energy and work is focused on making wider bandwidth metamaterials; especially with respect to optical frequencies to manufacture optical cloaking devices. Research, thus far, indicates that not much progress has been made in this regard. The best success appears to be from Berkley‟s work on wire meshes, which obtains a negative refractive index between 1.5 µm to 1.8 µm, which is within the infrared spectrum[6]. However, recently Vladimir Shalaev of Purdue University claims to have developed a theoretical concept that allows much wider bandwidths for double negative materials[35], as described at the Nanometa 2011 conference6.

3.8 Plasma Frequency

The plasma frequency (also known as the plasmon frequency) definition pertains to free electrons within metals, not the traditionally thought of ionized gas (i.e., plasma). Alternating electric fields, via the electromagnetic force, compel electrons to switch from side to side, almost instantaneously within metals, as illustrated in Figure 6. When alternating electric fields begin to oscillate fast enough, electron mass momentum issues begin to have affect. Normally these frequencies occur around ultra violet frequencies.

The frequency at which electrons oscillate at the same frequency, but opposite to normally expected charge repulsion, is called the plasma frequency, ωp, also known as the plasmon frequency. This condition is illustrated in Figure 7.

6 Nanometa 2011, Third International Topical Meeting on Nanophotonics and Metamaterials, 3-6 January 2011 in Seefeld, Austria.

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 24

+ + + + + + + + + + + + + + + + + + + + +

Metal Ē ------Low Frequency Effects

------

Metal Ē + + + + + + + + + + + + + + + + + + + + +

Figure 6. Low Frequency E-Field Effects upon Metals

+ + + + + + + + + + + + + + + + + + + + +

Metal Ē ------

Plasma Frequency Effects ------

Metal Ē + + + + + + + + + + + + + + + + + + + + +

Figure 7. Plasma Frequency E-Field Effects upon Metals

3.9 Physical Description of Right vs. Left Handed Materials

There are actually two solutions to Snell‟s equation; a positive radical result, and a negative one.

Traditionally, the positive radical is used, as it matches the observable naturally existing materials, with empirical perfection. The normal equation modeled solution is illustrated below. Note that right handed materials describe materials normally found in nature, and left handed materials are not thought of as existing in nature; though, I would argue double negative metamaterials probably do exist in relative obscurity. Both are subsets of metamaterials (composite materials), and furthermore, comprise the entire electromagnetic material set.

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 25

√ (Snell‟s Law)

Where: N = Refractive index of material.

Ɛr = Relative permittivity of material.

µr = Relative permeability of material.

⃑ ⃑⃑ ⃑⃑ (Energy flux, or Poynting vector)

Where: ⃑ = Poynting vector.

⃑⃑ = Electric field vector.

⃑⃑ = Magnetic field vector.

The Poynting vector, , and the wave vector, ⃑⃑ , are right handed if Ɛr > 0 and µr > 0. The wave vector, ⃑⃑ , is left handed and the Poynting vector, , is right handed if Ɛr < 0 and µr < 0. By right and left handed, I refer to the normal three finger position representing directions for the traditional power transfer

(thumb), the electric field (index finger), and the magnetic field (middle finger). Figure 8 illustrates the overall results of the various vectors within various index of refraction materials. I show later why some vectors reverse direction. One thing to note is the Poynting vector, or direction of energy movement, continues in the same direction across media boundaries, irrespective of the K vector and index of refraction sign. This is a point of consequence for conservation of energy.

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 26

N > 0 N < 0 N > 0

K K K

Vphase Vphase Vphase

Vgroup Vgroup Vgroup

S S S

Where: N = Index of refraction. K = Wave vector, or K vector.

Vphase = Phase velocity vector.

Vgroup = Group velocity vector. S = Poynting vector, or S vector.

Figure 8. Vector Directions vs. Index of Refraction

3.10 Proof of Snell’s Law Radical Sign Result

Again, the „choice‟ of the radical sign result for the index of refraction is really forced, depending upon the signs of the medium‟s permittivity and permeability. The index of refraction is positive for positive permittivity and positive permeability, which produces right handed materials. The index of refraction is negative for both negative permittivity and negative permeability, which produces left handed materials.

Sir Pendry indicates a mix of positive and negative permittivity and permeability also produces left handed materials[18]. Thus, according to Pendry[18], the only right handed quadrant of the electromagnetic material space is quadrant I; see Figure 1. However, this continues to be debated for quadrants II and IV, even within our physics team. I will provide an example. Sir Pendry, a non-disputed and knighted innovator and leader within the metamaterials industry, indicates quadrants II and IV provide left-handed materials, or materials with indices less than zero[18]. Why then does the iconic Handbook of

Optical Constants of Solids7 continue to be valid, before and after metamaterial mania, even for those

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 27 relatively few materials exhibiting right-handed material properties within quadrants II and IV, such as gold and silver at specific frequencies?

I attempt to show, more heuristically, why the corresponding radical signs result for a double negative, or left handed, material. For this effort, the scenario uses a region that is charge free with linear isotropic media. Therefore the following relationships hold true:

⃑⃑ ⃑⃑

⃑⃑

Where: ρ = Volume charge density.

⃑⃑ = Electric flux density vector.

Relative permittivity, or dielectric constant, of medium.

⁄ Free space permittivity.

⃑⃑ = Electric field vector.

⃑⃑ = Magnetic flux density vector.

Relative permeability of medium.

⁄ Free space permeability.

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 28

⃑⃑ = Magnetic field vector.

= Current density vector.

= Conductivity of the medium.

The point form of Maxwell‟s free space equations:

⃑⃑ ⃑⃑ ( )

⃑⃑ ⃑⃑ E

Right OrientationHanded ⃑⃑ ( )

⃑⃑

K & S

Figure 9. Right Handed Orientation With the original assumptions of the region noted above, and time dependence of for both ⃑ and ⃑⃑ , two of Maxwell‟s free space equations become:

⃑⃑ ⃑⃑

⃑⃑ ⃑⃑ and

⃑⃑ ⃑ ⃑⃑ ⃑⃑

⃑⃑ ⃑⃑ ⃑⃑

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 29

⃑ ⃑⃑ ⃑⃑

Using the right hand rule for cross product, this time, yields the electromagnetic right hand rule for wave propagation, as shown in Figure 9. This information and orientation is well known and understood.

Keep in mind that the right hand rule for mathematical cross product is defined with the right hand and does not change, regardless of disparate physics definitions such as left or right handed electromagnetic vector orientations.

Now if the scenario uses negative values for permittivity and permeability, then things change with respect to orientation. This will be shown using well known and understood electromagnetic boundary conditions. I will continue using the same region assumptions and relationships defined in the beginning. I will use Figure 10 and Figure 11 to illustrate electromagnetic field orientations, which is critical to this proof.

En1 E1

E t1 > 0 H1 µ1 > 0 K 1 & S1 n1 < n2

( > ) 2 > 0 2 1

µ2 > 0

n1 < n2 En2

Et2

Figure 10. Wave Orientation for N > 0

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 30

En1 E1

> 0 E t1 µ > 0 H1 1 n < n K1 & S1 2 1

n2 < 0

2 < 0

µ2 < 0

n2 < n1

n2 < 0

Et2

En2

Figure 11. Wave Orientation for N < 0

Well understood electromagnetic boundary conditions dictate that tangential electric field components, relative to the interface, do not change from the passage of one material into the next. It is usually stated that the electric field‟s tangential component is continuous across the boundary. As well, this holds for normal magnetic field components, for current free conditions, again relative to the interface.

Proofs for electromagnetic boundary conditions may be found in [19, 20, and 21].

Boundary conditions:

Et1 = Et2

r1En1 = r2En2

Ht1 = Ht2 (Current free conditions)

µr1Hn1 = µr2Hn2

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 31

Critical to note for this proof are a few things. Please refer to Figure 10 and Figure 11. The only component that changes is the normal component of the electric field. The tangential component of the electric field does not change, due to well understood boundary conditions. As for the magnetic field, the orientation chosen has the entire magnetic field oriented in the tangential direction, with respect to the interface. Thus, the magnetic field is continuous across the boundary, assuming current free conditions.

Again, this is via well understood boundary conditions. There is no normal magnetic field component, which would have changed across the interface.

The same analogous argument can be made when the entire electric field component is tangential to the interface. In this case, the tangential magnetic component is continuous across the interface boundary. The normal component of the magnetic field is the only component that changes, and does so in the inverse direction just like the normal component of the electric field in the original scenario.

Using the same equations in the beginning of this proof, but now for Ɛr < 0 and µr < 0, we obtain:

E Left Handed Orientation ⃑⃑ ⃑ ⃑⃑ | | ⃑⃑

⃑⃑ ⃑⃑ | | ⃑⃑ K S

⃑ ⃑⃑ ⃑⃑ H

Figure 12. Left Handed Orientation

Using the right hand rule for mathematical cross product, one can now understand the Figure 12 results, and thus, why ⃑⃑ alters direction.

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 32

3.11 Bulk Plasma Frequency

Einternal Refer to the Plasma Frequency subsection of the - - - - + + + + Metamaterial Theory chapter, for a higher level description - - - - Bulk + + + + - - - - + + + + of plasma frequency. That subsection should assist in - - - - Metal + + + + - - - - + + + + understanding this more detailed subsection. As the reader goes through this derivation, it will become evident that this Eexternal is a resonant effect if either permittivity or permeability is X direction less than zero Figure 13. Electron Orientation for Metal with an E-Field

An electric field, ⃑ field, surrounding bulk metal produces a force upon the metal‟s electrons, since they have charge. Figure 13 displays the configuration of bulk metal electrons within the presence of an electric field.

For ease of understanding, the single dimensional electron movement case will be detailed. From general Newtonian Physics:

⃑ ⃑ ( ̈) ̂

Equation 2. Newton's 2nd Law

Where: ⃑ = Force vector.

m = Average mass of electrons.

⃑ = Acceleration of electrons.

̂ = Unit direction of accelerated electrons.

x = Average position of electrons.

̈ = Double derivative of average electron positions.

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 33

Also:

⃑ ⃑⃑

Equation 3. Force on Electron within an Electric Field

Where: ⃑ = Force vector.

e = Magnitude of electron charge.

⃑⃑ = Applied electric field vector.

Combining Equation 2 and Equation 3 for the single dimensional, magnitude only case, yields:

Equation 4. Electron Movement within E-Field

Now I will obtain a substitute for E, using a very analogous capacitor-like case.

⃑⃑ ̂ (free space between charges) E

-σ +σ

Figure 14. E-Field & Charged Plate Analogy

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 34

Equation 5. Modified Newton's 2nd Law & Electric Field Force

Where: ⃑⃑ = Applied electric field vector.

ζ = Surface charge density.

⁄ Free space permittivity.

̂ = Normal vector.

= Applied electric field magnitude.

Q = Charge.

A = Area containing charges.

ρ = Volume charge density.

V = Volume containing charges.

x = Length in x-dimension, or x position, since zero-based.

n = Electron density.

e = Magnitude of electron charge.

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 35

Combining Equation 4 and Equation 5 yields:

Using the standard mathematical differential equation solution provides:

Where: x = Position.

ω = 2πf = Radian frequency.

t = Time.

Thus,

(bulk plasma frequency)

(bulk plasma frequency for SI units)

(bulk plasma frequency for Gauss units)

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 36

The plasma frequency, ωp, is the natural resonance of the material to an applied electric field. This kind of excitation was solved for, and is applied to, bulk metal materials. Therefore, ωp is often referred to as the „bulk plasma frequency‟.

To take things a bit further and find the energy and plasma wavelength for a couple metals:

Where:

h = Planck‟s constant, 6.625 x 10-34 Joule-Seconds.

ωp = Plasma frequency.

For the metals of gold and silver, the energy is about 9.1 eV (electron Volts). This corresponds to plasma frequencies within the ultra violet range (λp (Au) ≈ 180 nm, λp (Ag) ≈ 410 nm).

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 37

Єeff I{Єeff}

Re{Єeff}

Energy (ħω)

Transparent

Opaque

Figure 15. Effective Permittivity vs. Energy

Most metals have an electron density on the order of n ≈

23 2 10 /cm . Thus, ωp is usually within the ultra violet region of the

„visible‟ spectrum. It is usually advantageous to lower this Metal with large n frequency, especially when invisibility cloaks are of interest.

Given the equation for bulk plasma Dielectric with low n frequency, there is really only one

Figure 16. Metamaterial Example

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 38

parameter we can change to lower ωp. This may not be readily apparent since 4, π, e, and m are all constants. That only leaves the electron density, n, which also appears to be a constant. n really is a constant for homogeneous materials, but we can indeed modify this number, within reason. In order to lower ωp, we need to lower n. Engineers lower the macro-scale value of n by integrating sub-wavelength inclusions having much lower values of n. Dielectrics have been used to do this. Thus, the overall macro- scale electromagnetic wave interaction is with an overall lower value for n. This is the basic premise behind metamaterials with lower relative permittivities, which provide composites with overall lower effective values of n.

3.12 Surface Plasma Resonance

Surface plasma resonance (SPR) is very similar to bulk plasma frequency, or resonance. This is a local phenomenon and is more used on the nano-scale, than the bulk plasma frequency. As the reader goes through this derivation, it will become evident that this too is a resonant effect if either permittivity or permeability is less than zero. SPR is acquired via a different, but similar, method. The proof begins in the same manner as that for bulk plasma resonance, again using a single dimension for electron movement.

⃑ ⃑ ( ̈) ̂

Equation 6. Newton's 2nd Law

Where: ⃑ = Force vector.

m = Average mass of electrons.

⃑ = Acceleration of electrons.

̂ = Unit direction of accelerated electrons.

x = Average position of electrons.

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 39

̈ = Double derivative of average electron positions.

Also:

⃑ ⃑⃑

Equation 7. Force on Electron within an Electric Field

Where: ⃑ = Force vector.

e = Magnitude of electron charge.

⃑⃑ = Applied electric field vector.

Combining Equation 6 and Equation 7 for the single dimensional, magnitude only case, yields:

Equation 8. Electron Movement within E-Field.

Now Gauss‟ Law is used to obtain the electric field intensity. Gauss‟ Law dictates that the charge enclosed by a surface is equal to the electric flux density vector integrated over the differential surface vector.

∯ ⃑⃑ (Gauss‟ Law)

Since ⃑⃑ ⃑⃑ , in free space, Gauss‟ Law can be modified to:

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 40

∯ ⃑⃑

Where: Qenc = Q = Charge enclosed by a surface.

⃑⃑ = Electric flux density vector.

= Differential surface element vector.

⁄ Free space permittivity.

⃑⃑ = Applied electric field vector.

For a spherical surface, the equation now becomes:

Where: r = Radius of spherical surface enclosing charge.

E = Electric field intensity.

If we now expand out Q using volume charge density and the volume of a sphere, we obtain:

(simplified)

(since ρv = ne)

Where: ρv = Volume charge density.

n = Electron density.

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 41

e = Magnitude of electron charge.

Since we assume a spherical surface for the Gaussian surface, we will use two spheres having opposite charges and model them oscillating in a single dimension, using the superposition principle. This is illustrated within Figure 17.

++++++++++++ ++++++++++++ ++++++++++++E ++++++++++++1 +++ ++++++++++++ +++++++++

E = E1 + E2

------Via Superposition ------. ------E ------2------

Figure 17. E-Field Spheres, Super-positioned

Via superposition, and taking the negative charge sign into account, the prior equation is modified to become:

⃑⃑ ⃑⃑ ⃑⃑ ( )

( )

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 42

⃑⃑

Using this solution with Equation 8 of this subsection, and x = l if we choose the origin correctly, we then obtain:

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 43

Observation r1 Point l r

Figure 18. Distances and Labels

⃑⃑⃑ ⃑⃑⃑

| |

Where: ⃑⃑⃑ = Distance vector from sphere x to an observer.

= Distance vector between two sphere centers.

Making a simple definition will create the following equation:

Using the standard mathematical differential equation solution provides:

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 44

Where: x = Position.

Ω = Ωf = Surface plasma frequency.

t = Time.

Since (bulk plasma frequency for SI units), we obtain:

(surface plasma frequency) √

The surface plasma frequency, Ωf, is a local natural resonance phenomenon of the material within an applied electric field. This is normally used to describe interactions in the nano-scale.

SPR

n Absorptio

Ω Energy (ħω)

Figure 19. Surface Plasma Resonance

To take this a bit further and find the drift velocity:

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 45

⃑ ⃑⃑ ⃑ ⃑ ⃑

Where: ⃑ = Force vector.

m = Average mass of electrons.

⃑ = Acceleration of electrons.

e = Magnitude of electron charge.

⃑⃑ = Applied electric field vector.

and

⃑⃑⃑⃑

Where: = Current density vector.

n = Electron density.

e = Magnitude of electron charge.

⃑⃑⃑⃑ = Drift velocity vector.

In one dimension, and since :

( ) 4 5

Where: J = Current density magnitude.

n = Electron density.

e = Magnitude of electron charge.

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 46

m = Average mass of electrons.

E = Applied electric field vector.

η = Scattering time.

and

Where: ζ = Conductivity.

and

(Ohm‟s Law)

( ) 4 5 ( )

Where: γ = Scattering rate.

and

Where: Vd = Drift velocity.

a = Acceleration induced from E-field.

Advances in Microwave Metamaterials James A. Wigle Chapter 3. Metamaterial Theory | P a g e | 47

e = Magnitude of electron charge.

E = Applied electric field intensity magnitude.

m = Average mass of electrons.

η = Scattering time.

Advances in Microwave Metamaterials James A. Wigle

Chapter 4. Quadrant II, 10.5 GHz Metamaterial

4.1 Introduction

his chapter illuminates the quadrant II T metamaterial design. The metamaterial structure was designed and manufactured using available resources. Although available resources did not afford preferred quality controlled tolerances, the design achieved desired results. See Figure 20 for an image of the final product. Figure 20. Quadrant II Metamaterial

4.2 Design Theory

Sir Pendry‟s thin wire model was used to obtain the desired plasmon resonance[33]. This particular metamaterial was designed for quadrant II of the electromagnetic material space, but realize it can be used within both quadrants I and II, depending upon its frequency. Actually, this wire mesh metamaterial was operated and tested within quadrants I and II, as detailed within the various chapters of this thesis.

Therefore, only the effective permittivity was modified for the desired plasmon frequency. The relative permeability has a value of one, given the materials used8.

One significant design issue to note is the metamaterial‟s largest inclusion is close to 0.25λo, which does not meet the rule-of-thumb criteria of 0.1λo. Being more conservative, I chose the inclusion size to be the periodicity, rather than the wire diameter. Unfortunately, this is simply a manufacturing

8 FR-4, single sided 0.5 ounce copper, material was used in this design. µr = 1.0, and εr = 4.34 at 1 GHz. Chapter 4. Quadrant II, 10.5 GHz Metamaterial | P a g e | 49

constraint with the materials on hand. I believe this issue is most likely the cause of the low Q plasmon resonances. In the end, however, the desired effects were demonstrated.

Also note that this design has an almost three dimensional periodicity, which affords plane wave interactions in almost any direction. This is usually not the case for a number of metamaterials, though the quest is on for the perfect „Harry Potter‟ invisibility cloak from all directions.

( ) 6 7

This is called the Drude Formula.

And

Where: Ɛeff = Effective permittivity.

ω = Radian frequency.

ωp = Plasma frequency.

γ = Scattering rate.

η = Scattering time.

and

Where: ωp = Plasma frequency.

neff = Effective electron density, of single electron.

Advances in Microwave Metamaterials James A. Wigle Chapter 4. Quadrant II, 10.5 GHz Metamaterial | P a g e | 50

e = Electron charge.

Ɛo = Permittivity of free space.

meff = Effective electron mass, of single electron.

9 Note that Ɛeff is negative when ω < ωp . Though this may be a poor approximation, a number of articles use γ = 0 for long wavelengths. γ is the scattering rate within the Drude formula, as described within the

Metamaterial Theory chapter, of this thesis. Given this approximation, the Drude formula becomes:

( )

After some more equations and math, Pendry‟s thin wire model is obtained. Using this design, the

„Pendry classical formula for the plasma frequency of thin wire structures‟ was used. See Figure 21 for an illustration of this construction. The model:

. /

Where: ωp = Plasma frequency.

π = 3.141592.

co = Speed of light within a vacuum.

a = Square lattice periodicity.

r = Metal thickness.

9 References [9, 10, 11, 12].

Advances in Microwave Metamaterials James A. Wigle Chapter 4. Quadrant II, 10.5 GHz Metamaterial | P a g e | 51

r, or wire radius a

a, or grid spacing periodicity

Figure 21. Thin wire metamaterial structure

2 . / 2 . /

. /

2 8 9

Given that „a‟ cannot be solved explicitly, I solicited help10 to go a few steps further and use

MatLab 2006b‟s solver to obtain a numerically approximated explicit solution for „a‟. My wire mesh metamaterial calculator solution uses “Lambert‟s W function” (Lw), which is a Newtonian-like numeric approximation. The solution obtained is detailed below.

8 6 79 2

10 Mathematician Mr. James King of Denver, Colorado.

Advances in Microwave Metamaterials James A. Wigle Chapter 4. Quadrant II, 10.5 GHz Metamaterial | P a g e | 52

4.3 The Design

Circuit CAM software was used for the computer aided drawing (CAD) design layout. The resultant CAD file drove LPKF Laser & Electronics AG‟s

ProtoMat S62 proto circuit board maker to mill the design into a 9” x 12”, FR-4 material. 0.5 ounce, single sided,

11 Figure 22. Proto Circuit Board Machine FR-4 material was used . This material has a εr of 4.34 at

1 GHz, and a µr of 1.0. The proto circuit board maker could not make use of the entire FR-4 board.

Therefore, the final design is actually 19.6 cm x 19.6 cm.

Unfortunately, one of the variables that could not be controlled was the permittivity at our resonant frequency around 11 GHz. FR-4 material is not specified to operate above 2 GHz, and Rogers material, which meets this specification, was not available. Experiments that follow assume any permittivity change is negligible, to well within an order of magnitude. A non-rigorous experiment was performed, which indicates the relative permittivity is near 4.2 at 11 GHz, which supports this assessment12. See the Patch Antenna Measurements subsection, of the Patch Antenna chapter, for further details of this assessment.

Another uncontrolled variable is the ProtoMat‟s tolerance. The proto board maker should provide tolerances less than 0.1 mm. However, I found the actual measurements to be significantly off from the design values (26%). These dimensions were measured using microscopes, which should provide measured accuracies less than 0.05 mm. Given this manufacturing issue, the measured values are displayed, along with the design values. The calculated plasmon resonance is shown for each.

11 FR-4 material uses epoxy resin bonded glass fabric (ERBGF), ours with a single sided copper superstrate. 12 Antenna design dependence upon substrate permittivity, performed 26 Oct 2009.

Advances in Microwave Metamaterials James A. Wigle Chapter 4. Quadrant II, 10.5 GHz Metamaterial | P a g e | 53

Table 1. Quadrant II Metamaterial Design Parameters

a r Calculated ωp (mesh period in mm) (wire diameter in mm) (plasma freq. in GHz) Designed: 7.20 0.60 10.54 Manufactured: 7.39 ±0.1 0.44 ±0.051 9.66

As will be shown in much more detail later, this design demonstrated a gain improvement at 10.75

GHz, and a directivity improvement at 10.8 GHz. This is 2.0% and 2.5% off from the theoretical design, respectively; and 11.3 % and 12.8% off the calculated manufactured plasma frequency, respectively. Later,

I found a second resonance at a higher frequency, which actually may prove to be the better resonance.

The gain improvement was found at 11.45 GHz, and the directivity improvement was found at 11.34 GHz.

These are 8.6% and 7.6% off the theoretical design, respectively, and 18.5% and 17.4% off the calculated manufactured design, respectively. Keep in mind the manufacturing tolerances are a bit „sloppy‟.

These values are actually to be expected since directivity or gain improvement occurs not at the plasmon frequency, but just slightly higher than the plasmon frequency. I will go into much more detail later, regarding the difference between gain and directivity, as well as their frequency dependences. One could easily contend that the theoretical design matches the measured results, and required a 2% off from the plasmon frequency to improve the gain. The original target for improved gain was 5% above the plasmon frequency, so the remaining 6% (of 11%) would most likely be explained by relative permittivity and inclusions size relative to wavelength issues described earlier. Thus, the design appears to match well with theory and measurements.

4.4 The Model

A model of the quadrant II mesh metamaterial was created to assist in determining the proper plasmon frequency, as well as to assist in determining which frequency may provide the quadrant I directivity improvement for a patch antenna (detailed within the Improved Directivity chapter).

Advances in Microwave Metamaterials James A. Wigle Chapter 4. Quadrant II, 10.5 GHz Metamaterial | P a g e | 54

A few reasons drove the choice of a finite element analysis modeler. The university‟s finite element analysis modeler is very accurate (full wave modeler), employs a trick to allow free space field modeling (a critical need for this work), and is free (another critical need of graduate students). Method of moments modelers were available for free as well, and provide the critical free space field modeling, but fail miserably to model material parameters such as permittivity and permeability, which is another critical need of this model. Physical optics, uniform theory of diffraction, and geometrical modelers all compromise fidelity too much for the needs of this model, and are usually required of very complex in situ scenarios. This model required a full wave modeler. Again, method of moments is great for antennas, but will not incorporate material parametric properties, an essential need for this model. Thus, finite element or finite difference modelers were the only choices, of which the ANSYS Incorporated‟s finite element modeler‟s use is free of charge.

ANSYS Incorporated‟s HFSS modeler is a finite element analysis modeler, which allows material parametric modeling. HFSS also uniquely employs a trick to model free space electromagnetic fields, for items like antennas13. In order to save computation time, as a proper model could employ a serious computer server for a week at a time, this model used a subsection of the entire wire mesh structure. Mr.

James Vedral modeled two layers, each with four full and four half periodic structures. However, through the use of HFSS‟ master and slave boarders, this structure was infinitely extended within the X and Y planes. The excitation was an infinite plane wave directed in the ̂ direction. See Figure 23 for an image of exactly what was modeled, with these caveats in mind.

13 HFSS employs a material box (be it vacuum or air) outside the material of interest, in order to calculate free space electromagnetic propagation.

Advances in Microwave Metamaterials James A. Wigle Chapter 4. Quadrant II, 10.5 GHz Metamaterial | P a g e | 55

Figure 23. Quadrant II Metamaterial Model, Source = James Vedral of University of Colorado

4.5 Model Results

Model results include a graph of two scattering parameters, S11 and S21, over non-radian frequencies. Figure 24 displays the scattering parameters output for this model.

Note the total reflectance, and significantly reduced transmission, before the first resonance. This indicates that this wire mesh metamaterial is within quadrant II, and therefore does not propagate.

Propagation only occurs within quadrant I of the electromagnetic material space, see Figure 1. Frequencies greater than 7 GHz display transmittance with reduce reflection, indicating quadrant I operation. In summary, frequency operation less than 9 GHz lies within quadrant II, and greater than 7 GHz lies within quadrant I.

Advances in Microwave Metamaterials James A. Wigle Chapter 4. Quadrant II, 10.5 GHz Metamaterial | P a g e | 56

Figure 24. Quadrant II Metamaterial Model Results

I purposely left an undetermined range of frequencies, as there continues to be debate as to where the measured plasmon frequency truly resides. Some industry icons indicate just when transmission occurs, and others define it at the peak of the transmittance curve. However, no one debates theoretically where this is; when ωp transitions from negative to positive, or vice versa. Within the near future, I intend to demonstrate this transition, experimentally, using one of the new techniques detailed within this thesis.

See the Metamaterial Theory chapter for more details.

As for various measured results within this thesis, experiments agree quite well with the model.

The graph of Figure 24 shows a resonance around 11 GHz, which should provide directivity improvement, gain improvement, and optimal meta-lens operation. This matches almost perfectly with the patch antenna design frequency, even before this model was created. Experimental results displayed resonances around

Advances in Microwave Metamaterials James A. Wigle Chapter 4. Quadrant II, 10.5 GHz Metamaterial | P a g e | 57

10.7 GHz and 10.8 GHz. The slight differences, of less than 1% frequency bandwidth, can be readily explained through compromised construction tolerance, as the metamaterial was disassembled and reassembled on numerous occasions. One experiment I performed14, not detailed within this thesis, found that minor construction variances did have a measureable effect on the resonance, around 1% frequency bandwidth.

14 29 December 2010 test using Agilent‟s E8364A vector network analyzer.

Advances in Microwave Metamaterials James A. Wigle

Chapter 5. Quadrant III, 4.5 GHz Metamaterial

5.1 Introduction

hough most laws of physics are reversible with respect to time, they do not create nor T reverse the direction of time. And as such, time did not permit manufacturing another metamaterial before the publication of this dissertation. A new metamaterial was designed and modeled, but not yet manufactured. Manufacturing should take place this summer. This is a completely different design, which incorporates below zero modifications in both permeability and permittivity. This SSRR design, therefore, should reside within quadrant three of the electromagnetic material space, see Figure 1.

Given this resides within quadrant III, left handed propagation should be evident. Using one of the two new measurement techniques, see the Microwave Test to Determine Material Properties and the Infrared

Test to Determine Material Properties chapters, the negative index of refraction and associated angle should be visibly and measurably evident.

5.2 Design Theory

An S-shaped split ring resonator (SSRR) design was chosen this time, to reduce complexity in obtaining a metamaterial with quadrant III of the electromagnetic material space, see Figure 1. It was suggested a combination of a quadrant IV metamaterial, a split ring resonator (SRR), and the existing quadrant II wire mesh metamaterial should provide a quadrant III metamaterial. While this is true, I believe construction tolerances would be much preferable using an SSRR, which in itself is a quadrant III metamaterial. Moreover, the associated nasty attenuation would be far less with the much thinner SSRR. Chapter 5. Quadrant III, 4.5 GHz Metamaterial | P a g e | 59

Much of this design theory originates from Chen et. al., [37], and I will not belabor the reader with theory repeated from the source. The „S‟ shape allows electric and magnetic field coupling via pattern repetition and internal capacitances and inductances. The particular parameters allow frequency regions with both the electric plasma frequency and the magnetic plasma frequency to occur simultaneously. Thus, this is a quadrant III metamaterial, with both a negative permittivity and negative permeability.

Unlike the wire mesh metamaterial, this SSRR design structure is only periodic in two dimensions. Therefore, for desired metamaterial results, plane waves must interact almost face-on. One of my future investigations hopes to take this a bit further using flexible circuit board material.

5.3 The Planned Design

The plan is to leverage the proto circuit board maker, LPKF Laser & Electronics AG‟s ProtoMat

S62, or an outside circuit board manufacturer if not cost prohibitive. Regular double sided FR-4 material15 will be used to produce the S, and inverted S, patterns on each side of the board. Lexan sheets16 will then be employed as spacers between the FR-4 boards. These alternating layers could then be easily clamped together, making disassembly and reassembly easy, as well as possessing consistent reconstruction results.

I estimated that a set of three to ten boards would suffice. Figure 25 illustrates the proposed design.

Finances allowing, the FR-4 material will be FR-4 replaced by an outside circuit board manufacturer using Lexan

Rogers 4003 substrate, on much larger boards. This should afford much better trace tolerances, known and Copper „S‟ or inverted „S‟ lower permittivity at our frequency of interest, larger Figure 25. Proposed SSRR Design overall material so fringing effects are less of an issue, and should reduce the material‟s loss tangent. Our design also includes periodicities near 0.1λ, which should significantly reduce inclusion diffraction effects.

15 FR-4, double sided 0.5 ounce copper, µr = 1.0, εr = 4.34 at 1 GHz. 16 Polycarbonate sheet, µr = 1.0, εr = 2.89 at 1 GHz.

Advances in Microwave Metamaterials James A. Wigle Chapter 5. Quadrant III, 4.5 GHz Metamaterial | P a g e | 60

5.4 The Model

A model of the quadrant III SSRR metamaterial was created to assist in determining the proper plasmon frequency. A few reasons drove the choice of a finite element analysis modeler. The university‟s finite element analysis modeler is very accurate (full wave modeler), employs a trick to allow free space field modeling (a critical need for this work), and is free (another critical need of graduate students).

Method of moments modelers were available for free as well, and provide the critical free space field modeling, but fail miserably to model material parameters such as permittivity and permeability, which is another critical need of this model. Physical optics, uniform theory of diffraction, and geometrical modelers all compromise fidelity too much for the needs of this model, and are usually required of very complex in situ scenarios. This model required a full wave modeler. Again, method of moments is great for antennas, but will not incorporate material parametric properties, a critical need for this model. Thus, finite element or finite difference modelers were the only choices, of which the ANSYS Incorporated‟s finite element modeler‟s use is free of charge.

ANSYS Incorporated‟s HFSS modeler is a finite element analysis modeler, which allows material parametric modeling. HFSS also uniquely employs a trick to model free space electromagnetic fields, for devices like antennas17. In order to save computation time, as a proper model could employ a serious computer server for a week at a time, this model used a subsection of the entire SSRR mesh structure. Mr.

James Vedral modeled a single layer, with three full periodic structures. However, through the use of

HFSS‟ infinite plane structures, this modeled an infinitely extended plane within X and Y18. See Figure 26 for an image of exactly what was modeled, with these caveats in mind.

17 HFSS employs a material box (be it vacuum or air) outside the material of interest, in order to calculate free space electromagnetic propagation. 18 Well documented method via R. W. Ziolkowski. Design Fabrication and Testing of Double Negative Metamaterials. IEEE Transactions. Antennas and Propagation, Volume 51, Number 7, July 2003.

Advances in Microwave Metamaterials James A. Wigle Chapter 5. Quadrant III, 4.5 GHz Metamaterial | P a g e | 61

Figure 26. Quadrant III Metamaterial Model, Source = James Vedral of The University of Colorado

5.5 Model Results

Our model results produced a graph of two scattering parameters, S11 and S21, over non-radian frequency. Respectively, these relate to the reflectance and transmittance of the metamaterial. Figure 27 displays this graph.

Advances in Microwave Metamaterials James A. Wigle Chapter 5. Quadrant III, 4.5 GHz Metamaterial | P a g e | 62

Figure 27. SSRR Model Results

Note that there is no clear frequency below which transmission does not occur. Unlike the wire mesh metamaterial, there is no transition from quadrant II (no propagation) into quadrant I (propagation).

This SSRR metamaterial resides within quadrant III, for all the frequencies displayed within the graph.

Quadrant III affords propagation.

There are two decent regions of transmission with reduced reflection, good for taking advantage of quadrant III propagation characteristics, for example a negative index of refraction. A narrow propagation region appears to exist near 1.25 GHz and a wider region centered around 4.45 GHz. Experimental verification, unfortunately, will need to wait a few months while this SSRR metamaterial is manufactured.

Advances in Microwave Metamaterials James A. Wigle

Chapter 6. Metamaterial Reciprocity for Quadrant II

6.1 Introduction

hough this may seem intuitive, I never found a single reference to metamaterial reciprocity, T theoretical or empirical. Electromagnetic reciprocity is defined here, as that called The Lorentz Reciprocity Theorem, which requires a linear isotropic medium, but not necessarily homogeneous.

Pages 144 to 150 of [38] provide an excellent description of this theorem. In essence, reciprocity here describes identical wire mesh metamaterial interactions with electromagnetic radiation, from either face direction. I.e., either transmitting or receiving through the metamaterial is identical, using the same physical test setup space. However, any direction should provide reciprocity, given this metamaterial‟s symmetry.

In this chapter, I detail an experiment that indeed empirically supports Lorentz reciprocity of the quadrant I wire mesh metamaterial. This is illustrated for quadrant I only, since quadrant II does not afford propagation; unless, of course, one considers total reflection upon transmission and nil upon receive, in either direction, as empirical support for quadrant II reciprocity. Lorentz reciprocity may hold for non- propagating radiation as well, but this was not measured and reported in this work.

6.2 Test Setup for Metamaterial Reciprocity

Of course there are many more test setup details, but the overall test setup schematic is shown in

Figure 28. This test was actually performed on many occasions over the years, but this latest test was performed in a much more rigorous manner using the university‟s microwave anechoic chamber. Chapter 6. Metamaterial Reciprocity for Quadrant II | P a g e | 64

Metamaterial

S21

S12

Vector Network Port 1 Analyzer Port 2

Microwave Anechoic Chamber

Figure 28. Metamaterial Reciprocity Test Setup

Though it should not matter for reciprocity, antennas and other equipment employed were verified to be within operating specifications. This was simply carried out to ensure nothing were awry, and was easy enough to execute. The test was performed in the middle of the chamber‟s test, or quiet, zone, with all absorber cones in place and the vector network analyzer properly calibrated.

Both antennas were determined to be in the far field, for proper plane wave propagation, though this should not matter for Lorentz reciprocity. The far field here is defined using Stutzman‟s three criteria as detailed below[39]. And for completeness, I performed the same experiment with the antennas cross polarized.

Where: R = Range from antenna phase center.

Advances in Microwave Metamaterials James A. Wigle Chapter 6. Metamaterial Reciprocity for Quadrant II | P a g e | 65

D = Longest physical dimension of antenna.

λ = Operational wavelength.

Two network scattering parameters were used, S21 and S12, since the transmission and reception ports are reversed without test setup changes. This method is quick, convenient, and removes test setup variables. If my hypothesis is correct, then both S21 and S12 plots over frequency should be identical in nature and amplitude.

6.3 Metamaterial Reciprocity Test Results

Lorentz reciprocity test results are displayed within the graphs below.

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Figure 29. Metamaterial Reciprocity, same polarization

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Figure 30. Metamaterial Reciprocity, cross polarized

Both network scattering parameters, for both graphs, are essentially identical. Thus, this empirically demonstrates what was hypothesized; that metamaterial reciprocity holds for the wire mesh quadrant I metamaterial. Note the Lorentz reciprocity appears to hold for like polarizations, as well as for cross polarized transmitter and receiver stations.

Advances in Microwave Metamaterials James A. Wigle

Chapter 7. Patch Antenna

7.1 Introduction

n antenna was required to drive and test the wire mesh metamaterial close to, and on either A frequency side, of the plasma frequency. I designed and manufactured this patch antenna for this sole purpose. The patch antenna was constructed years ago and is used extensively within this multifaceted metamaterials research effort. As the reader will later encounter, this antenna was used to determine any novel patch antenna gain and directivity improvements, helped in determining the plasma frequency, obtained transmittance and reflectance coefficients, attempted construction of two novel permittivity measurement techniques, as well as constructed a novel multifaceted meta-lens.

7.2 Patch Antenna Design

I desired a resonant type antenna, so antenna and metamaterial influences were limited in frequency range. The belief was that this would facilitate measuring characteristics and influences, especially given metamaterials‟ usual narrow frequency band characteristics.

I chose a patch antenna design for numerous reasons. Patches are resonant type antennas, and thus have narrow operating frequency bandwidths, usually around 3% to 5%. Patch antennas are well understood, and easy to characterize. They are also easy to manufacture, especially when one has access to a proto circuit board machine and double sided PC boards. Lastly, mating the flat PC board patch to the metamaterial would be fairly straightforward.

The metamaterial was designed for a plasma resonance at 10.54 GHz. Therefore, the patch antenna was designed to operate at 11.0 GHz, which is 4.4% higher than the predicted plasma resonance of Chapter 7. Patch Antenna | P a g e | 69

the metamaterial. As the reader recalls from [27] and the Metamaterial Theory chapter, Bulk Plasma

Frequency subsection, this frequency was chosen since directivity improvement should exist when the permittivity is positive, but close to zero, or the plasma resonance. 5% above the plasmon resonance should be optimal, and the antenna was anticipated to have greater than 3% bandwidth. The metamaterial used for this experiment can operate within the first and second quadrants of the electromagnetic material space. However, electromagnetic propagation does not exist within quadrant II, which does not make for an exciting, or traditionally useful, antenna. See the Index of Refraction’s Forced Radical Sign subsection, of the Metamaterial Theory chapter, for more detail.

The operating frequency is also compatible with existing university (UCCS) materials. One of the criteria I used was at least a 3λ ground plane margin on all sides of the illuminated patch. With this criteria, and only 9” x 12” double sided PC boards, I chose to use the entire circuit board, as opposed to cutting the ground plane to a smaller size. However, the usable portion ended up providing an 8” x 10” (20.5 cm x

25.5 cm) ground plane. This provides a very adequate ground plane of close to 15.6λ (7.5λo) in its smaller dimension, and 19.5λ (9.3λo) in its larger dimension, both calculations using of 4.4 and λo being the free space wavelength of the patch‟s operating frequency.

Unfortunately, the relative permittivity, εr, of the PC board‟s dielectric material was unknown, as well as the manufacturer and model information of the PC board. By word of mouth, I was told it is about

4.4, which seemed very reasonable for these boards. Therefore, εr = 4.4 is what I used in my calculations, though later I did find the material specifications indicating a εr of 4.34 at 1 GHz. I later performed a rough permittivity estimate at the operating frequency, which identified it as 4.2. Equations further below detail the design parameters.

A square patch design was employed. The equations are fairly straightforward. Note that all parameters are basically symmetric, with the exception of only one dimension for the feed point. This is an artifact of the patch‟s current distribution. Note by boundary conditions, that the currents must go to zero while the voltages are maximum at the edges. In the symmetric center, they are opposite, so that the currents are maximum while the voltages are minimum. Therefore, to obtain the objective 50 Ω

Advances in Microwave Metamaterials James A. Wigle Chapter 7. Patch Antenna | P a g e | 70

impedance, we must make the voltage divided by the current a specific ratio of 50. To do this requires the feed point be offset in one direction, labeled ρ.

⁄ 2

⁄

√ √ √ ( ⁄ ) ( ⁄ )

2 2 ( ⁄ )

( ⁄ )

2 2 √ √

FR-419 material was used to manufacture the patch antenna. LPKF Laser & Electronics AG‟s proto circuit board machine, ProtoMat S62, milled one side of a two layer board to leave the patch antenna.

The other FR-4 side was left as is, for the ground plane. A copper isolated hole was drilled for the patch feed point. An SMA type connector was simply and appropriately soldered on both sides of the board. The figures below display the end product square patch antenna, as well as the schematic diagram.

19 FR-4, double sided 0.5 ounce copper, material was used in this design. µr = 1.0, and εr = 4.34 at 1 GHz.

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One side of PC board ϕ

≥3λ

l ρ

≥3 λ

Figure 31. Patch Antenna Schematic

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Figure 32. Patch, Front Side Figure 34. Patch, Ground Plane

Figure 33. Patch, Element Figure 35. Patch, Feed

7.3 Patch Antenna Measurements

Once the patch antenna was complete, the patch antenna characteristics were measured using the university‟s microwave anechoic chamber, along with a plethora of other equipment. A vector network analyzer was employed to obtain the 50 Ω impedance match (S11), its true characteristic impedance, its resonant frequency, as well as its 3 dB frequency bandwidth. The square patch antenna performed remarkably well and proved amazingly accurate to design criteria.

Advances in Microwave Metamaterials James A. Wigle Chapter 7. Patch Antenna | P a g e | 73

The university‟s microwave anechoic chamber instrumentation has a characteristic impedance of

50 Ω, as one would expect. Therefore, this was the design criteria for the physical location of the patch antenna feed. The vector network analyzer measured the characteristic impedance of the patch antenna‟s feed port. This was accomplished using a direct impedance measurement, as well as measuring the S11 network scattering parameter indicating how closely matched it was to the vector network analyzer‟s

NIST20 calibrated 50 Ω, port one, input impedance.

The S11 network scattering parameter also indicates the antenna‟s useful frequency range. S11 indicates how much power actually makes it into the antenna, i.e., how well it is matched to 50 Ω

(maximum energy transfer to antenna). Note that this does not indicate how much of that power is actually radiated, which is more a function of the antenna‟s efficiency. Most engineers use the value of -9 dB or -10 dB for S11, since at these levels almost all the power is transferred to the antenna. I chose to use the more conservative -10 dB level. Figure 36 and Figure 37 show the measured results, and Table 2 collects all measured results.

Table 2. 11 GHz Patch Antenna Characteristics

Parameter Value Comments Resonant Frequency 11.01 GHz Zo 50.10 Ω At 11.01 GHz S11 -53.1 dB At 11.01 GHz Frequency Bandwidth 8.56% Gain 5.9 dBi At 11.0 GHz Polarization Isolation 21.9 dB At 11.0 GHz

It should be understood that the antenna actually radiates during the S11 measurements. Therefore, it is important the antenna be well positioned during this measurement, lest a large piece of metal near and in front of the antenna provide erroneous results for normal usage. The university‟s microwave anechoic chamber was used for these measurement setups. Thus, this should be a non-issue.

20 National Institute of Standards and Technology.

Advances in Microwave Metamaterials James A. Wigle Chapter 7. Patch Antenna | P a g e | 74

Figure 36. 11 GHz Patch, S11

Advances in Microwave Metamaterials James A. Wigle Chapter 7. Patch Antenna | P a g e | 75

Figure 37. 11 GHz Patch, Impedance

Advances in Microwave Metamaterials James A. Wigle Chapter 7. Patch Antenna | P a g e | 76

The E-plane and H-plane antenna radiation patterns used a more complex setup. I will spare the reader a plethora of details, in this regard. Suffice it to write here that this test setup operated equipment within their specifications, using the far field, microwave anechoic chamber. All anechoic chamber test zones resided within the antennas‟ far fields, as described by Stutzman, for proper plane wave propagation.

Stutzman‟s three far field criteria are detailed below[39].

Where: R = Range from antenna phase center.

D = Longest physical dimension of antenna.

λ = Operational wavelength.

Figure 38 displays the 11 GHz square patch antenna normalized decibel radiation pattern at 11.0

GHz. As the reader can see via the undulations, some anechoic chamber multipath did exist, but the reader can ascertain the overall pattern result is as expected for a well behaved patch antenna.

The 11 GHz patch antenna gain measurements were obtain via another test, described with the

Directivity & Gain Test Setup & Measurements subsection of the Improved Gain chapter. I go into much greater detail there, since this provides new material, as opposed to well-known radiation pattern tests.

Advances in Microwave Metamaterials James A. Wigle Chapter 7. Patch Antenna | P a g e | 77

170 -170 160 0 -160 150 -150 -5 140 -140 130 -10 -130 120 -15 -120 -20 110 -110 -25 100 -100 -30 90 -35 -90

80 -80

70 -70

60 -60 50 -50 40 -40 30 -30 20 -20 E-Plane 10 0 -10 H-Plane deg

Figure 38. 11 GHz Patch Antenna, E & H Plane Radiation Patterns

Using the antenna design parameters (refer to subsection Patch Antenna Design of this chapter), a rough estimate of FR-4 permittivity at operational frequency was generated. Though not a rigorous method of permittivity measurement, I obtained 2 at 11.0 GHz. This provided a „sanity check‟ regarding wild permittivity fluctuations for FR-4 material, well above the specified 2 GHz frequency limit. This antenna design dependence upon substrate permittivity was performed 26 Oct 2009.

Advances in Microwave Metamaterials James A. Wigle Chapter 7. Patch Antenna | P a g e | 78

7.4 Conclusions

The patch antenna design went smoothly. The measured results were very close to the design criteria chosen. The end-state measured characteristics were excellent. This design is ready for characterization with the incorporated wire mesh metamaterial. The hope is to see improved gain and directivity, just above the plasmon frequency. Later, I use this antenna to evaluate a novel multifaceted meta-lens.

Advances in Microwave Metamaterials James A. Wigle

Chapter 8 Improved Directivity

8.1 Introduction

euristically, directivity improvement will be shown, when employing the wire mesh H metamaterial with the 11 GHz patch antenna. It is important to understand exactly what I claim as “improved directivity”. Directivity is very difficult and cumbersome to measure, and realistically is never accomplished in a direct manner. Therefore directivity improvement is illustrated via numerous pattern measurements, which will also convey just how frequency sensitive the wire mesh metamaterial is, or how this is a resonance result.

The wire mesh metamaterial was driven within quadrant I of the electromagnetic material space.

Just above the plasma frequency is where one would expect this directivity improvement. The metamaterial was mounted ≈ 3 cm directly in front of the patch antenna, as shown in Figure 39 and Figure

40.

This experiment differs from [27] in that improved directivity was a result of positioning a metamaterial directly in front of a patch antenna element, much like a radome, not embedding a source within the metamaterial. Via ( ), [27] also assumes radiation only within the main lobe.

My experiments heuristically assume improved directivity, not the actual value, via a narrower main lobe than the radiating antenna without metamaterial enhancement.

Chapter 8. Improved Directivity | P a g e | 80

Figure 39. Metamaterial with Patch Figure 40. Metamaterial with Patch

8.2 Measuring Antenna Directivity

Since

Where: g = Antenna gain.

k = Antenna efficiency.

d = Antenna directivity.

one must know the efficiency of the antenna, if they only measure gain. Antenna efficiencies are an inherent component of the antenna. Thus, the Directivity & Gain Test Setup & Measurements subsection of the Improved Gain chapter describes how to measure gain, and not directivity, in a direct manner anyway. This test setup is used for both directivity and gain measurements. In order to measure directivity, one must measure all power the antenna radiates, in all spatial dimensions, not the power transferred to the antenna. This means the power density, everywhere on a spherical surface enclosing the antenna, must be measured and summed. This process is extremely cumbersome, and expensive in time

Advances in Microwave Metamaterials James A. Wigle Chapter 8. Improved Directivity | P a g e | 81 and cost. Moreover, few institutions have proper equipment to perform this type of three dimensional measurement with decent uncertainties (accuracy and precision). I have been the more particular regarding the directivity measurement section, since this continues to be a point of consternation for the physic team, as well as with me as I read article after article claiming directivity measurements, when their authors most likely have measured gain. As I will show later, the two may not be frequency coincident, as may differ greatly due to metamaterial efficiencies.

Given the complexity and equipment required for a proper directivity measurement, I measured and compared 3 dB spatial beam widths, with and without the metamaterial mounted on the same 11 GHz patch antenna. This certainly is not rigorous, and only provides a general sense of improved directivity, as explained within the paragraph that follows. I make no claim, however, to know the actual directivity value.

Although gain is proportional to directivity, metamaterial losses cloud this relationship to directivity. A fairly solid conjecture can be claimed, however. Metamaterials are well known for their losses. Thus, a metamaterial enhanced antenna most likely has a lower efficiency, k. If what is measured is truly gain, which I claim it to be, then one could claim an improvement in directivity if an improvement in gain is measured. Actually, one may even claim the directivity improvement is much greater than that of gain, given the significant metamaterial losses.

8.3 Directivity Graphs

I chose to graph radiation patterns within the linear scale, as opposed to the dB scale. Given the dB scale is compressive at large value changes, and expansive at small value changes, the linear scale is preferred for illustrating the claimed directivity improvement. I graphed a number of metamaterial modified patch antenna patterns, in order to determine the optimal frequency for directivity improvement.

Although it is a bit of a quagmire, these graphs may be see in Figure 41 and Figure 42, and are presented so the reader may better understand the resonant effect of the wire mesh metamaterial; very profound contrast

Advances in Microwave Metamaterials James A. Wigle Chapter 8. Improved Directivity | P a g e | 82 between frequencies. For lucidity, Figure 43 graphs only the best three frequencies. The resultant directivity improvement, over the unmodified patch antenna, may be seen in Figure 44.

Advances in Microwave Metamaterials James A. Wigle Chapter 8. Improved Directivity | P a g e | 83

Meta-Antenna,1.2 9.8-10.9 GHz Freq's

1 9.8 GHz 9.9 GHz 0.8 10.0 GHz 10.1 GHz

0.6 10.2 GHz 10.3 GHz 10.4 GHz 0.4 10.5 GHz 10.6 GHz 10.7 GHz 0.2 10.8 GHz 10.9 GHz 0

-30 -20 -10 0 deg 10 20 30

-0.2

Figure 41. Normalized Meta-Antenna Patterns over 9.8-10.0 GHz

Advances in Microwave Metamaterials James A. Wigle Chapter 8. Improved Directivity | P a g e | 84

Meta-Antenna,1.2 11.0-12.0 GHz Freq's

11.0 GHz 11.1 GHz 0.8 11.2 GHz 11.3 GHz 11.4 GHz 0.6 11.5 GHz 11.6 GHz 11.7 GHz 0.4 11.8 GHz 11.9 GHz 12.0 GHz 0.2

0 -30 -20 -10 0 deg 10 20 30

Figure 42. Normalized Meta-Antenna Patter over 11-12 GHz

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1.2 1.2 Metamaterial Improvement for 11 GHz Patch Meta-Antenna Radiation Pattern Response Antenna 1 1 10.2 GHz 10.8 GHz 0.8 0.8 11.7 GHz

0.6 0.6 Patch Antenna 11.0 GHz 0.4 0.4

Meta- Antenna at 0.2 0.2 10.8 GHz

0 0 -30 -10 deg 10 30 -40 -20 0 deg 20 40

Figure 43. Normalized Meta-Ant Best 3 Figure 44. Normalized Meta-Ant Enhancement

8.4 Interpreting the Results

As the reader can see from Figure 43 and Figure 44, the best improvement occurred at 10.8 GHz, which is only 2% off the original design goal. If the true plasma frequency of this wire mesh is indeed

10.54 GHz, then this indicates the optimal antenna directivity enhancement occurs about 2.5% above the plasma frequency. This empirical value should provide great utility in our future SSRR metamaterial design.

The 3 dB beamwidth of the 11 GHz patch antenna was measured from a radiation plot to be 88°, which indeed is normal for a patch antenna. The 3 dB beamwidth of the metamaterial enhanced antenna was measured to be 13.8º. This by no means provides an absolute directivity value, since the power radiated in other directions was not accounted, and it is impossible to know how the metamaterial affected

Advances in Microwave Metamaterials James A. Wigle Chapter 8. Improved Directivity | P a g e | 86 the efficiency value. Again, this only provides an intuitive concept of how the directivity of the patch antenna was enhanced via the metamaterial.

8.5 Conclusions

Although a value cannot be determined from these efforts, these experiments appear to demonstrate that indeed the wire mesh metamaterial did provide directivity enhancement, when driven just over the line into quadrant I of the electromagnetic material space. Readers should also have a much better feel for how frequency sensitive this wire mesh metamaterial is, and how multiple resonances do occur.

Advances in Microwave Metamaterials James A. Wigle

Chapter 9. Improved Gain

9.1 Introduction

sing Sir Pendry‟s thin wire model[33], a metamaterial was created to perform within the U first quadrant of the electromagnetic material space, as detailed within the Quadrant II, 10.5 GHz Metamaterial chapter of this thesis. Recall that I operated this just above the plasma frequency, which makes this metamaterial operate within quadrant I. Using this metamaterial within quadrant I, I took this metamaterial a bit further and demonstrated improved directivity and improved gain of an in-house manufactured patch antenna, as exhibited via microwave anechoic chamber testing, which my literary search leads me to believe is unique for patch antennas. These anechoic chamber tests appeared to display improved directivity within the radiation patterns. Improved gain was also achieved, and was verified using the three unknown antenna technique, an industry standard. This experiment shows gain improvement using a metamaterial placed much like a radome, not directivity improvement as reference

[27] shows for a source embedded within a metamaterial.

9.2 Directivity & Gain Test Setup and Measurements

It does not serve positive purpose to clutter and cloud results with minutia, so of course there is a surfeit of details beyond this thesis, especially when this experiment was repeated on numerous occasions using different test arrangements. However, this subsection provides the basis of the experiment demonstrating how the directivity and gain measurements were produced, using the most reliable test setup and methods. The baseline test schematic is shown in Figure 45. These provide the test points and losses used to calculate all antenna gains. The basic gain measurement test schematic is shown in Figure 46. Chapter 9. Improved Gain | P a g e | 88

Baseline Setup: TP TP4 Meta-Patch Gain Test

PP1 C2 C6 C3 C4 PP2

Microwave Anechoic Chamber

C6 only used for baseline setup Loss1

Loss2

SA C C 5 1 Cn = Cables

PPn = Patch Panels

TPn = Test Points TP5 TP2 Lossn = Segment Losses

Figure 45. Meta-Patch Gain Baseline Test Schematic

Test Setup: Meta-Patch Gain Test

G G

t r

PP PP1 C2 C3 C4 2

Microwave Anechoic Chamber

Cn = Cables PP = Patch C n 1 SA C5 PanelsTPn = Test Points

TP1

Figure 46. Meta-Patch Gain Test Schematic

Advances in Microwave Metamaterials James A. Wigle Chapter 9. Improved Gain | P a g e | 89

Loss1 was a loss dependent only upon passive devices, for which the loss should be static, over a single frequency. Therefore, I averaged all sub-test Loss1 values to achieve the value I used for gain calculations. Note that I averaged the linear losses, not the decibel losses (which would be inappropriate).

Loss2, however, relied upon an active device (amplifier), which is sometimes prone to gain drift. Thus, I chose to calculate Loss2 for each sub-test, and respective calculation. The reader should note that Loss2 is actually a gain. Therefore, the Loss2 loss value is less than one, linear, or a less than zero in dB.

( )

This test setup operated equipment within their specifications, using the far field microwave anechoic chamber. All anechoic chamber test zones resided within the antennas‟ far fields, as described by

Stutzman, for proper plane wave propagation. Stutzman‟s three far field criteria are detailed below[39].

Where: R = Range from antenna phase center.

D = Longest physical dimension of antenna.

λ = Operational wavelength.

Advances in Microwave Metamaterials James A. Wigle Chapter 9. Improved Gain | P a g e | 90

9.3 Frequency Dependency of Directivity versus Gain

Prior to any measurements being taken, I ran quite a number of tests to determine where the peak directivity was, given reconstruction tolerance issues I have experience before with this quadrant II wire mesh metamaterial. This actually required two days of solid frustrating testing before I slowly realized that peak directivity and peak gain may not necessarily frequency collocated.

I tackled this issue by creating a log of best directivity and best gain, or best receive power on boresight. Soon after beginning to create this log, it became evident that both metamaterial enhanced antenna gain and directivity indeed were not frequency collocated. My focus then turned to completing this log, and determining the best compromise of directivity and gain. I found this gain versus directivity compromise at 11.45 GHz. My log results are displayed in Table 3. Note how the metamaterial becomes more transparent (more received power), as the metamaterial further enters quadrant I of the electromagnetic material space (higher frequency).

Table 3. Meta-Patch Directivity vs. Gain Comparison

Boresight Frequency Power How (GHz) (dBm) Directive?

11.20 -59.01 --- 11.25 -54.72 - 11.30 -49.43 +++ 11.34 -45.97 +++ 11.35 -45.13 +++ 11.37 -45.32 ++ 11.40 -41.59 ++ 11.43 -40.35 ++ 11.45 -38.38 ++ 11.46 -39.32 ++ 11.48 -37.93 ++ 11.50 -37.44 + 11.52 -37.66 + 11.55 -38.45 -- 11.58 -39.07 ---

Advances in Microwave Metamaterials James A. Wigle Chapter 9. Improved Gain | P a g e | 91

I began contemplating the ramifications of this finding, since I have not read literature indicating this may be so. One could achieve the best directivity, but a metamaterial directivity-enhanced antenna would not be of much use if the efficiency is so low that hardly any radiation is harvested at the receiver.

On the other hand, a metamaterial gain-enhanced antenna may have a decent efficiency, but not be directive enough for a particular use. Gain is related to directivity, but I write about the case where the radiation pattern may be so non-uniform to make it essentially useless. Obviously, most metamaterial enhanced antennas would desire both enhanced gain and directivity.

Obviously I would like to further pursue the directivity and gain frequency dependence issues in future work.

9.4 Gain Calculations

All antenna gains, to include the meta-lens gains associated with each lens face, were calculated using the industry standard three unknown antenna technique. This accurate technique, also employed by the National Institute of Standards and Technology (NIST), does not require a priori knowledge of any antenna‟s gain used in the test procedure. Thus, this technique does not require a standard gain horn antenna, or two identical antennas.

The technique essentially uses a modified Friis21 equation to obtain three equations and three unknowns (i.e. the gains). The test generated more equations than unknowns, simply because antenna patterns of all antennas used were desired. The calculations are shown below in detail, so the reader can acquaint themselves with the technique, as well as adjudicate its validity via this peer review.

Below is the modified version of Friis‟ equation, to which I referred.

21 Friis was Danish, and is therefore pronounced like “Freece” (frēs).

Advances in Microwave Metamaterials James A. Wigle Chapter 9. Improved Gain | P a g e | 92

Where: pr = Power receive at the antenna feed.

pt = Power transmitted at the antenna feed.

gt = Gain of the transmitting antenna.

gr = Gain of the receiving antenna.

SL = Spreading loss, or space loss.

For the five tests performed, labeled A through E, the resulting equations are listed below.

Superscripts indicate the specific test.

( )

( ) ( )

Now combining equations:

Advances in Microwave Metamaterials James A. Wigle Chapter 9. Improved Gain | P a g e | 93

* +

√

All antenna patterns are displayed here to demonstrate that all patterns appear normal, especially with regards to the patch antenna, which is close to its frequency edge and is exhibiting lower gain, probably due to reduced efficiency. All graphs are E-plane cuts of the antenna radiation patterns. The metamaterial enhanced graph is normalized linear and rectangular, since it is pointless to display beyond the edges of the metamaterial slab, as diffraction is not a desired effect for the enhancement being sought.

Thus, this graph only displays ±45° in azimuth. The three remaining graphs are normal 360° azimuth displays, plotted in the normalized dB scale.

Advances in Microwave Metamaterials James A. Wigle Chapter 9. Improved Gain | P a g e | 94

Meta-Patch1.2 at 11.45 GHz

0.8

0.6

0.4

0.2

0 -50 -40 -30 -20 -10 0 deg 10 20 30 40 50

Figure 47. Meta-Patch, Normalized E-Plane Pattern at 11.45 GHz

Advances in Microwave Metamaterials James A. Wigle Chapter 9. Improved Gain | P a g e | 95

11 GHz Patch, E-Plane -178 174178 -174-170 166170 -166-162 158162 0 -158 154 -154 150 -150 146 -146 142 -5 -142 138 -138 134 -134 130 -10 -130 126 -126 122 -15 -122 118 -118 114 -20 -114 110 -110 106 -25 -106 102 -102 98 -30 -98 94 -94 90 -35 -90 86 -86 82 -82 78 -78 74 -74 70 -70 66 -66 62 -62 58 -57.9 54 -54 50 -50 46 -45.9 42 -42 38 -38 34 -34 30 -30 26 -26 22 -22 18 -14-18 14 10 6 -6 -10 2 -2 deg

Figure 48. Patch Antenna, Normalized dB E-Plane Pattern at 11.45 GHz

Advances in Microwave Metamaterials James A. Wigle Chapter 9. Improved Gain | P a g e | 96

HP11966E, E-Plane -178 170174178 -174-170 166 -166-162 158162 0 -158 154 -154 150 -150 146 -146 142 -10 -142 138 -138 134 -134 130 -20 -130 126 -126 122 -122 118 -30 -118 114 -114 110 -110 -40 106 -106 102 -102 98 -50 -98 94 -94 90 -60 -90 86 -86 82 -82 78 -78 74 -74 70 -70 66 -66 62 -62 58 -58 54 -54 50 -49.9 46 -46 42 -42 38 -38 34 -34 -30 30 -26 26 -22 2218.1 -18 14 10 -6 -10-14 6 2 -2 deg

Figure 49. Bi-Ridged Flared Horn Antenna, Normalized E-Plane Pattern at 11.45 GHz

Advances in Microwave Metamaterials James A. Wigle Chapter 9. Improved Gain | P a g e | 97

"E34" Antenna w/Narda 601A Feed, E-Plane -178 170174178 -174-170 166 -166-162 158162 0 -158 154 -154 150 -150 146 -146 142 -10 -142 138 -138 134 -134 130 -20 -130 126 -126 122 -122 118 -30 -118 114 -114 110 -110 -40 106 -106 102 -102 98 -50 -98 94 -94 90 -60 -90 86 -86 82 -82 78 -78 74 -74 70 -70 66 -66 62 -62 58 -58 54 -54 50 -49.9 46 -45.9 42 -42 38 -38 34 -34 30 -30 26 -26 22 -22 18 -14-18 14 10 -6 -10 6 2 -2 deg

Figure 50. Flared Waveguide Horn Antenna, Normalized E-Plane Pattern at 11.45 GHz

9.5 Improved Gain Results

All four antennas had gains near or within expected ranges. The metamaterial enhanced patch antenna did experience a gain improvement of 4.4 dB, a significant improvement. All results are displayed in Table 4.

Advances in Microwave Metamaterials James A. Wigle Chapter 9. Improved Gain | P a g e | 98

Table 4. Meta-Patch Gain Test Results

Antenna Gain Gain Expected Manufacture Description Value Specifications (linear) (dBi) (dBi) (dBi) HP11966E, bi-ridge 7.669 8.85 10-15 ≈12.122 Meta-Patch 2.572 4.10 2-10 N/A Patch (no metamaterial) 0.927 -0.33 1-4 N/A E34, flared waveguide horn 64.976 18.13 12-17 N/A

9.6 Conclusions

Patch antenna gain improvement was demonstrated using the quadrant II metamaterial, driven within quadrant I. The wire mesh metamaterial improved the patch-alone gain by 4.4 dB. This appears to correlate well with the meta-lens improvement of 3.2 dB (at -48°), see the Multifaceted Meta-Lens chapter and Double Beam Gains subsection for more details.

Again, the patch antenna was driven near the frequency limits of that antenna23. As one could see, the radiation pattern is not adversely affected, though the gain appeared to suffer. Again, this issue is irrelevant, as long as the patch parameters remain constant at this frequency, which they did. The metamaterial gain improvement is irrespective of any well behaved antenna, which this is at this frequency, as demonstrated via the radiation pattern and parameters throughout the entire test period.

I was surprised to discover optimal directivity and gain are frequency independent, though they are decently close in frequency. Within the realm of each other, the best power reception occurred at 11.50

GHz, while the best boresight pattern directivity appeared to occur at 11.34 GHz. Again, directivity was not measured directly, and this was only a rough estimate from the radiation pattern. This difference was

1.4% in frequency. This is not significant in number percent, but this 1.4% frequency difference had a

22 G(dBi) = 20log10(freq in MHz) – Ant. Factor (dB) – 29.78 dB, reference [44]. 23 Patch antenna parameters  fc = 11.01 GHz, BW = 8.56%, S11 = -53.1 dB, Zo = 50.10 Ω, Z = 58.73 Ω & S11 = -9.81 dB at f = 11.47 GHz.

Advances in Microwave Metamaterials James A. Wigle Chapter 9. Improved Gain | P a g e | 99

profound effect upon the radiation pattern and receive power level. My directivity versus gain compromise appears to have worked well for this experiment.

In summary, it appears as though metamaterial antenna enhancement for gain, as well as directivity, have a place in antenna engineering. Metamaterial resonance and antenna parameter effects can be used to significantly enhance receiver designs. The frequency selective nature of the metamaterial could be used to reject undesired interferers, both spatially as well as in frequency. This inherent meta-antenna filter would enhance signal to noise receiver ratios in systems that would otherwise employ a lossy or noisy front-end filter. Additionally, a metamaterial enhanced antenna would reject undesired out-of-band frequencies, assisting in electromagnetic compatibility and electromagnetic interference compliance.

Advances in Microwave Metamaterials James A. Wigle

Chapter 10. Meta-Antenna Reciprocity

10.1 Introduction

iven material already presented within the Metamaterial Reciprocity for Quadrant II G chapter, it may seem intuitive that a metamaterial enhanced antenna would also experience Lorentz reciprocity. However, literary searches have failed to produce references detailing how metamaterial enhanced antennas may affect antenna reciprocity, a critical criteria for a significant number of antenna roles.

In this chapter, I share novel experimental data providing empirical evidence that the wire mesh metamaterial, used within quadrant I of the electromagnetic material space (see Figure 1), displays such reciprocity behavior as that described by the Lorentz Reciprocity Theorem. Of course, meta-antenna reciprocity will not hold for antennas that do not have reciprocity behavior, such as those employing active elements or ferrites.

Significant differences exist between the test setups of the Metamaterial Reciprocity for

Quadrant II chapter and this one. A vector network analyzer was exploited to determine the S21 and S12 scattering parameters. The metamaterial was positioned between transmit and receive antennas. However, this chapter describes a test setup which uses transmit and receive antenna patterns, which is far more intuitive and fundamental for antenna engineers. It is one thing to believe electromagnetic radiation propagates in the same manner in either direction, but an entirely more compelling demonstration is to employ a metamaterial enhanced antenna in both receive and transmit mode to obtain the same, or very similar, antenna beam patterns. This technique will be illustrated within this chapter.

Chapter 10. Meta - Antenna Reciprocity | P a g e | 101

10.2 Test Setup for Meta-Antenna Reciprocity

Testing for meta-antenna reciprocity is just like that of measuring antenna beam patterns, using the university‟s microwave anechoic far-field chamber. Figure 51 displays the schematic of the test and measurement setup. The university‟s automated measurement system garnered the measurements used to construct the transmit and receive beam patterns.

Meta-Antenna Reciprocity Test Setup:

G G

t/r r/t

C PP C C PP 1 1 2 3 2

Microwave Anechoic Chamber

Cn = Cables PP = Patch Panels SA n C5 C4

Figure 51. Meta-Antenna Reciprocity Test Setup

All anechoic chamber test zones were within the antenna‟s far field, as described by Stutzman, for proper plane wave propagation. Stutzman‟s three far field criteria are detailed below[39].

Advances in Microwave Metamaterials James A. Wigle Chapter 10. Meta - Antenna Reciprocity | P a g e | 102

Where: R = Range from antenna phase center.

D = Longest physical dimension of antenna.

λ = Operational wavelength.

10.3 Test Results

Essentially four graphs of the meta-antenna beam patterns were created: E-plane transmit, E-plane receive, H-plane transmit, and H-plane receive. For better comparison purposes, respective transmit and receive patterns were overlaid to produce two plots of the results. These are shown in Figure 52 and Figure

53 below.

Advances in Microwave Metamaterials James A. Wigle Chapter 10. Meta - Antenna Reciprocity | P a g e | 103

Meta-Antenna Reciprocity, E-Plane deg 0 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50.1 60 70 80 90

-5

-10

-15

-20 MetaAnt Receive, E-Plane, Normalized Power (dB) MetaAnt Transmit, E-Plane, Normalized Power (dB) -25

Figure 52. Meta-Antenna Reciprocity, Normalized dB E-Plane

Advances in Microwave Metamaterials James A. Wigle Chapter 10. Meta - Antenna Reciprocity | P a g e | 104

Meta-Antenna Reciprocity, H-Plane deg 0 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50.1 60 70 80 90

-5

-10

-15

-20

-25 MetaAnt Receive, H-Plane, -30 Normalized Power (dB) MetaAnt Transmit, H-Plane, Normalized Power (dB) -35

Figure 53. Meta-Antenna Reciprocity, Normalized dB H-Plane

10.4 Conclusions

The reader can plainly see that meta-antenna reciprocity certainly holds for this scenario, and I submit should for any meta-antenna consisting of non-active, non-ferrous materials; or any meta-antenna where both parts exhibit Lorentz reciprocity behavior. Note that tests were performed from -90º to +90°, which faces sideways to the metamaterial at both ±90º. Reciprocity appears to be valid for even sections of the meta-antenna for which it is not intended, though one can see the beginnings of the deviations on either side.

Advances in Microwave Metamaterials James A. Wigle

Chapter 11. Multifaceted Meta-Lens

11.1 Introduction

hilst contemplating antenna enhancement using metamaterials, I pondered making a W multifaceted meta-lens from the existing wire mesh metamaterial. The thought was that one could break up the six-layer metamaterial into two, three-layer metamaterials. Both wire mesh, quadrant II metamaterials, should have identical properties (Figure 1 illustrates the electromagnetic material space). Furthermore, if operated just within quadrant I, both should provide directivity improvement, and possibly gain improvements as well.

Initial tests were performed to identify if the double beamed meta-lens did indeed provide a double beam. Once this was found to be true, a later test attempted to characterize the gains at both beam peaks.

Unlike the embedded source of [27], this experiment situated the meta-lens in front of the antenna, much like a radome. Directivity and gain were shown to improve, while [27] shows only directivity improvement, using an embedded source.

11.2 Meta-Lens Construction

The existing six-layer wire mesh metamaterial was simply split into two pieces, of three-layers each. Both these pieces were then intersected at a 90° angle from each other. The

Figure 54. Meta-Lens Construction Chapter 11. Multifaceted MetaLens | P a g e | 106 patch antenna was used to drive the meta-lens, and was placed at the open-ended base to form a triangle.

The metamaterial ends required a bit of overlap to maintain the effective electron density, as well as not create a gap, through which electromagnetic radiation could escape and provide a false beam peak. Thus, this triangle was not an equilateral type, and the patch was not centered under the triangle apex as well.

This is apparent in the test results, since the peak centers are off by ≈ 5°. Figure 54 illustrates this end-state construction.

It should be noted that the beam patterns and the gain measurements were made on different days, after the metamaterial had been taken apart and reconstructed. A more proper test would have found the double beam resonant frequency and used the same meta-lens construction. Given the construction tolerances, this could have a significant affect.

11.2 Double Beam Meta-Lens

This section only details the means in determining if indeed the meta-lens has a double beam configuration. For proper peer review, the test setup will be presented, then the results of this test, followed by some of my conclusions obtained from the test data.

11.2.1 Test Setup

The test setup schematic is shown in Figure 55. Test patterns were garnered via measurements from this setup. All measurements were obtained using the university‟s automated system within their microwave anechoic chamber. Prior to any pattern measurements, a quick test of peak gain was performed, given the loose construction tolerances. The frequency of interest was determined to be 10.75 GHz, not the prior 10.83 GHz. However, I performed tests at both frequencies. This test used the existing setup, but swept over frequency, with the spectrum analyzer on “max hold” and the meta-lens with antenna at a 45° angle (one side of the meta-lens was face-on).

Advances in Microwave Metamaterials James A. Wigle Chapter 11. Multifaceted MetaLens | P a g e | 107

Meta-Lens Pattern Test Setup:

G G

t r

C PP C C PP 1 1 2 3 2

Microwave Anechoic Chamber

Cn = Cables PP = Patch Panels SA C n C5 4

Figure 55. Meta-Lens Pattern Test Setup

Measurements were taken with and without the meta-lens, at both frequencies previously mentioned. In every instance, the transmit antenna remained the same. However, the receive antenna was either the meta-lens with the 11 GHz patch antenna configuration, or the patch antenna alone (without the meta-lens).

11.2.2 Results

The four antenna beam patterns (recall reciprocity) were obtained. It was very delightful to see the two peaks near the center of the metamaterial structure, which was not likely caused by leakage. It is more intuitive to overlay the meta-lens and patch antenna patterns. I did so for both frequencies. These are displayed in Figure 56 and Figure 57. It turns out that the prior 10.83 GHz response was better than the

Advances in Microwave Metamaterials James A. Wigle Chapter 11. Multifaceted MetaLens | P a g e | 108

10.75 GHz response, which was the result of my quick experiment. It simply goes to show the reader how sensitive the metamaterial, meta-lens in this case, is to frequency and construction tolerances.

Normalized Power, 10.83 GHz 1.2 Normalized Antenna Power Normalized Meta_Lens Power 1

0.8

0.6

0.4

0.2

0 6

-6

18 12 30 36 42 48 54 66 72 78 84 90

-90 -84 -78 -72 -66 -60 -54 -48 -42 -36 -30 -24 -18 -12 60.1 deg 24.1

Figure 56. Double Beam Meta-Lens, 10.83 GHz

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Normalized Power, 10.75 GHz 1.2 Normalized Meta-Lens Power

Normalized Antenna Power 1

0.8

0.6

0.4

0.2

0 6

-6

18 12 24 30 36 42 48 54 60 66 72 78 84 90

-90 -84 -78 -72 -66 -60 -54 -48 -42 -36 -30 -24 -18 -12 deg

Figure 57. Double Beam Meta-Lens, 10.75 GHz

11.2.3 Conclusions

The reader can see from the plots that indeed there is a significant directivity increase from both faces of the metamaterial covering the patch antenna. Note the beam peaks occur directly in the center of the metamaterial, indicating the peaks are indeed due to the metamaterial, and not convenient leakage. To me, the most striking feature is the null at the center of the patch antenna where there should be a peak without the meta-lens. This provides extremely good empirical evidence that the metamaterial is

Advances in Microwave Metamaterials James A. Wigle Chapter 11. Multifaceted MetaLens | P a g e | 110 performing as expected. Thus, the meta-lens does indeed provide two beam peaks, at each of the metamaterial faces.

11.3 Double Beam Gains

Gains of this Meta-lens were measured for both beam peaks. This section provides information regarding the test setup, shows calculations required, provides the results, and gives my conclusions regarding those results.

11.3.1 Test Setup

I will be more particular with the details of this setup, given the novelty of the concept and results.

Again, there is a surfeit of information not provided within this thesis. The baseline setup schematic for these measurements is displayed in Figure 58. Figure 59, shows the test setup schematic.

Advances in Microwave Metamaterials James A. Wigle Chapter 11. Multifaceted MetaLens | P a g e | 111

Baseline Setup: TP3 TP Meta-Lens Gain Test

PP2 C2 C7 C3 C4 PP1

Microwave Anechoic Chamber

C7 only used for baseline setup Loss1

Loss2

C SA C C 1 6 5

Cn = Cables TP PP = Patch Panels TP2 5 n

TPn = Test Points

Lossn = Segment Losses

Figure 58. Meta-Lens Gain Test, Baseline Test Setup

Test Setup:

G G

Meta-Lens Gain Test t r

PP2 C2 C3 C4 PP1

Microwave Anechoic Chamber

Cn = Cables

PPn = Patch Panels

C SA C C 1 5

TP1

Figure 59. Meta-Lens Gain Test Setup

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( )

Given that TP2 measurements were not taken throughout the test procedure, Loss1 is realistically irrelevant, and the TP3 value was used as the transmitted power. Of course this is valid, but not the purist‟s way forward.

Stutzman, for proper plane wave propagation. Stutzman‟s three far field criteria are detailed below[39].

Where: R = Range from antenna phase center.

D = Longest physical dimension of antenna.

λ = Operational wavelength.

11.3.2 Gain Calculations

Advances in Microwave Metamaterials James A. Wigle Chapter 11. Multifaceted MetaLens | P a g e | 113 the National Institute of Standards and Technology, does not require a priori knowledge of any antenna‟s gain used in the test procedure. The technique essentially uses a modified Friis24 equation to obtain three equations and three unknowns (i.e. the gains). The test generated more equations than unknowns, simply because antenna patterns of all antennas used was desired. Below goes through the calculations in detail, so the reader can acquaint themselves with the technique, as well as adjudicate its validity via this peer review.

I begin with the modified Friis‟ equation, to which I referred.

Where: pr = Linear power receive at the antenna feed.

pt = Linear power transmitted at the antenna feed.

gt = Linear gain of the transmitting antenna.

gr = Linear gain of the receiving antenna.

SL = Linear spreading loss, or space loss.

For the five tests performed, labeled A through E, the resulting equations are listed below.

Superscripts indicate the specific test.

24 Friis was Danish, and is therefore pronounced like “Freece” (frēs).

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Now combining equations:

* +

√

It is critical to note test A‟s receive antenna was the meta-lens. The gain was initially calculated for the peak near +50º, which is one of the two faces of the meta-lens (recall the slight offset due to overlapping meta-lens faces). The calculations were repeated for the peak near -50º. The peaks were actually at +49º and -48°, which is very symmetrical, within the face center, expected, and provides

Advances in Microwave Metamaterials James A. Wigle Chapter 11. Multifaceted MetaLens | P a g e | 115 comfort in the results. I spared the reader another set of equations, both sets of which had super-super scripts with + and - signs.

Since both sets of calculations were performed, two values for three of the antennas were obtained; the fourth being the meta-lens with a value for each face, for which only one set of calculations was performed. It is comforting to know that the second set of calculations provided gain solutions for the three antennas that were within 0.5 dBi of each of the prior results. The following results section average these two results for the three antennas that had two calculated results.

11.3.3 Results

After all calculations are said and done, the results provide gains of the four antennas used during testing, to include both faces of the meta-lens. Table 5 provides the end gain results of all antennas used during this test. As the previous subsection described, calculations that provided two gain results were averaged. Again, all duplicative gain calculation results were within 0.5 dBi ; very close indeed.

Table 5. Meta-Lens Gain Results

Antenna Gain Gain Expected Manufacture Description Value Specifications (linear) (dBi) (dBi) (dBi) HP11966E 13.603 11.34 10-15 12.0425 E34 37.123 15.70 12-17 N/A Patch (no lens) 1.200 0.78 1-4 N/A Meta-Lens at +49° 2.503 3.99 2-10 N/A Meta-Lens at -48° 1.748 2.43 2-10 N/A

11.3.4 Conclusions

The meta-lens did indeed show gain improvement over the patch antenna without the meta-lens,

1.7 to 3.2 dB improvement. All antenna gains were as expected, besides the patch antenna. A less than 0

25 G(dBi) = 20log10(freq in MHz) – Ant. Factor (dB) – 29.78 dB, reference [44].

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dBi gain (1.0 linear) for the patch antenna is not likely at resonance, especially when I have measured the gain to be 5.9 dBi (see Patch Antenna chapter), which is as expected for this type of antenna. However, recall the resonant frequency of the patch antenna is 11.01 GHz with an 8% frequency bandwidth. Thus, I had to use this patch antenna near the edge of its acceptable frequency use, which I am certain compromised efficiency. I did check the patch antenna pattern, which was as expected for this antenna type, so the gain must be lower due to reduced efficiency running near the outer edge of frequency use.

More to the point, this reduced gain is irrelevant for detection of improved gain, as improvement was displayed, regardless of any patch antenna anomaly that may have existed during the duration of this test.

The other test results do appear as expected. It should be recalled that gains do exist below 0 dBi, for example electrically small antennas due to poor efficiencies. However, directivity cannot be ≤ 1.0 or 0 dBi because no isotropic radiators exist in nature, even with metamaterial enhancement (☺).

11.4 Summary

It appears directivity improvement is quite evident, and gain improvement, though not as profound, was evident as well. Thus, meta-lenses have a place within the engineering realm to design lenses that provide extremely tailored antenna responses such as that used to obtain multiple transmitters or null out interferers. Additionally, metamaterials resonant-like response should prove useful as a frequency discriminator, which affords significant signal to noise ratio improvements where filters may be required, adding significant system receiver noise.

Given the results demonstrated here, I am anxious to attempt a meta-lens upon a flexible circuit board to construct a 360° lens, with designable beam patterns. How cool is that?

Advances in Microwave Metamaterials James A. Wigle

Chapter 12. Infrared Test to Determine Metamaterial

Properties

12.1 Introduction

This chapter describes a novel test conceived to demonstrate a metamaterial‟s negative index of refraction, though it can be used for any material with either a positive or negative angle, actually. Given some questions the physics team raised regarding how Dr. David Smith demonstrated the negative angle of refraction, this test‟s intent is to visually demonstrate the negative or positive angles, which would be difficult to refute.

I detail the fundamental theory of how it works, illustrate a specific test design, show the results, and provide my conclusions. A patent may be sought for this method of determining material properties.

A sincere thank you goes to Dr. Victor Gozhenko for working the equations and determining the optimal angle for maximum displacement between positive and negative index of refraction materials, as well as correcting a plethora of mistakes.

As with a lot of research and experimentation, more questions result from the solutions, or demonstrations, of one.

Chapter 12. IR M e t a - Property Test | P a g e | 118

12.2 Theory

12.2.1 The Crux of the Matter

The crux of the test is to measure very slight localized heating of a detector material, after passing through a slab of material positioned at an angle. Given specific angles of incidence and indices of refraction, a displacement could be measured, indirectly indicating material properties. The novelty with this specific test is the use of a sensitive infrared camera, with a detector material having a known resistivity per area. Employing the appropriate test setup and equipment with enough accuracy, one should be able to recover most electromagnetic material properties, such as permittivity, permeability, index of refraction, and loss tangent. Within which side of the normal incidence mark the radiation peak heats, will definitively indicate whether or not a material is exhibiting a negative or positive index of refraction, another key novelty of the test.

12.2.2 Ray Trace Geometry

n1 Again, I thank the great Dr. Gozhenko for the use of his work. Below describes the geometry of n2 the test scenario, and provides the critical equations for the indices of refraction, both for negative and n 1 Δ positive indices. Note the solution works toward δ+ δ measurable parameters, such as α and d, instead of β - or γ. Figure 61 shows Dr. Gozhenko‟s work to obtain the maximum Δ displacement versus angle of Figure 60. Ray Trace Geometry incidence, which is close to 60º; note that the graph is in radians.

Advances in Microwave Metamaterials James A. Wigle Chapter 12. IR M e t a - Property Test | P a g e | 119

Figure 61. Maximum Δ Displacement vs. Angle of Incidence (rads), Source = Victor Gozhenko of Univ. of Colorado

12.2.2.1 Displacement Solution for Positive Indices of Refraction

| |

| | | | ( ) α

n | | ( ) 1 A α d E | | n2 γ ( ) | | β

B C n | | 1 | | δ ( ) ( ) +

Figure 62. n > 0 Ray Trace

Advances in Microwave Metamaterials James A. Wigle Chapter 12. IR M e t a - Property Test | P a g e | 120

( ) ( )

Via a well-known trigonometric identity, ( ) ( ) ( ) ( ) ( )

, ( ) ( ) ( ) ( )- ( )

( ) ( ) ( ) 6 ( ) 7 6 ( ) ( ) 7 ( ) ( )

Via a well-known trigonometric identity, ( ) √ ( )

√ ( ) [ ( ) ( ) ] √ ( )

Via Snell‟s law, ( ) ( ) , the solution now becomes

√ ( ) ( ) ( ) ( )

√ . / ( ) [ ]

√ ( ) ( ) ( )

. / √ . / ( ) [ ]

√ ( ) ( ) ( )

√. / ( ) [ ]

Advances in Microwave Metamaterials James A. Wigle Chapter 12. IR M e t a - Property Test | P a g e | 121

√ ( ) ( )

√. / ( ) [ ]

Please note that this displacement solution is from the incident ray trace to the refracted real ray trace. Since I used a flat surface, normal to the incident ray with the material under test at an angle, my experiment required a bit more geometry to make things measureable26. This is simple geometry, but must be performed to obtain correct parameters.

12.2.2.2 Displacement Solution for Negative Indices of Refraction

| | α | | | | ( ) n1 A E | | ( ) d α n 2 β γ | | ( ) C | | B n1 δ- | | | | ( ) ( ) Figure 63. n < 0 Ray Trace

( ) ( )

Via a well-known trigonometric identity, ( ) ( ) ( ) ( ) ( )

, ( ) ( ) ( ) ( )- ( )

26 ( ) ( )

Advances in Microwave Metamaterials James A. Wigle Chapter 12. IR M e t a - Property Test | P a g e | 122

( ) ( ) ( ) 6 ( ) 7 6 ( ) ( ) 7 ( ) ( )

Via a well-known trigonometric identity, ( ) √ ( )

√ ( ) [ ( ) ( ) ] √ ( )

Via Snell‟s law, ( ) ( ) , the solution now becomes

√ ( ) ( ) ( ) ( )

√ . / ( ) [ ]

√ ( ) ( ) ( )

. / √ . / ( ) [ ]

√ ( ) ( ) ( )

√. / ( ) [ ]

√ ( ) ( )

√. / ( ) [ ]

Once again, note that this displacement solution is from the incident ray trace to the refracted real ray trace. Simple geometry is required to obtain the correct measurements.

Advances in Microwave Metamaterials James A. Wigle Chapter 12. IR M e t a - Property Test | P a g e | 123

12.2.3 Solutions for n, r, r , µr, and δloss

12.2.3.1 Positive Indices of Refraction Solution

Taking the result from the prior section, for δ+ , I manipulate this to obtain n2

( ) ( )

√. / ( ) [ ]

( ) ( )

( ) ( ) √. / ( )

, ( )- √( ) ( ) ( ) ( )

( ) ( ) √( ) ( ) ( )

( ) ( ) ( ) ( ) 6 7 ( )

( ) ( ) ( ) ( ) { ( ) 6 7 } ( )

( ) ( )

√ ( ) 6 7 ( )

Note that the displacement, , is from the incident ray trace to the refracted real ray trace. Since I used a flat surface, normal to the incident ray with the material under test at an angle, my experiment

Advances in Microwave Metamaterials James A. Wigle Chapter 12. IR M e t a - Property Test | P a g e | 124

required a bit more geometry to make things measureable. This is simple geometry, but must be performed to obtain the correct used in this index of refraction solution.

12.2.3.2 Negative Indices of Refraction Solution

Taking the result from the prior section, for δ- , I manipulate this to obtain n2

( ) ( )

√. / ( ) [ ]

( ) ( )

( ) ( ) √. / ( )

, ( )- √( ) ( ) ( ) ( )

( ) ( ) √( ) ( ) ( )

( ) ( ) ( ) ( ) 6 7 ( )

( ) ( ) ( ) ( ) { ( ) 6 7 } ( )

( ) ( )

√ ( ) 6 7 ( )

Advances in Microwave Metamaterials James A. Wigle Chapter 12. IR M e t a - Property Test | P a g e | 125

Note that the displacement, , is from the incident ray trace to the refracted real ray trace. Again simple geometry must be employed to obtain the correct used in this index of refraction solution.

Unfortunately, there is no method in this test that will distinguish material permittivity from permeability, unless one or the other is known, a priori. For quite a number of materials, relative permeability is known, for example 1.0, but this unknown quantity can be a point of consequence for metamaterials within quadrants III and IV.

Loss tangent values can be found in numerous ways. Using this test, however, one can simply use a detector material with a known, per area resistivity (e.g. Kapton paper manufactured by DuPont).

Accurately measuring the heated area can provide information leading to ⃑ -field intensity at the detector material surface. Knowing the ⃑ -field intensity, spatially before and after the material under test, can

[46, 50] provide information leading to the value of the loss tangent, δloss .

12.2.4 Interesting Results

The reader should note that, given a measured displacement, n2 (the material under test) is exactly the same for both the negative and positive index of refraction! Again, the sign of the radical plagues humanity, or should I write clarifies the laws of physics for meaningful purpose? Both displacements may be the same, but the index of refraction has two results, positive and negative. Though this is almost expected, this perpetual radical situation mimics a „race condition‟, within engineering speak.

As I went through some actual measurements, I did notice precision is of the essence. Very slight changes produced semi-significant different indices of refraction. Thus, a user should be weary of precision issues, and not rely upon approximate values.

Advances in Microwave Metamaterials James A. Wigle Chapter 12. IR M e t a - Property Test | P a g e | 126

12.3 Test Design

As the reader can see in Figure 64, the test delivers electromagnetic radiation through the angled material under test. The infrared sensor detects very slight heating of the detector material, situated after the material under test. Accurate measurement of the displacement from the incident beam trace, which is related to Snell‟s law, the angle of interface, and the indices of refraction, is required to reveal material properties, such as the index of refraction, permittivity and permeability of the material at the operational frequency. Furthermore, if a detector material is used with a known resistivity per square meter (e.g., carbon paper), then more material properties can be determined, such as the material‟s loss tangent.

Note that this experiment should be carried out within the far field. Thus, the material under test should be much closer to the detector material, both within the far field. A compromise was necessary, given power levels and material under test size constraints.

I first photographed and measured the center of heating concentration with the metamaterial, or material under test. Next, the material under test was removed, and all other test parameters and equipment remained the same, and in their former positions. This should provide both displacements, with and without the material under test. The difference between the two centers should provide the displacement necessary to calculate material under test properties.

Advances in Microwave Metamaterials James A. Wigle Chapter 12. IR M e t a - Property Test | P a g e | 127

Test Setup: Carbon loaded Neoprene, or Material under test, other absorber such as IR Material Properties Test which can rotate. carbon paper with known resistivity

G t α n > 0

n = 0

PP C D n < 0 1 3 IR camera

TP 1 Anechoic Chamber

Cn = Cables

PPn = Patch Panels

TPn = Test Points

C C 2 Figure 64. Material Properties Test Schematic, IR Sensor

Distance D = 34 ± 1 cm, though this does not influence results.

Figure 65 illustrates the detector material dimensions, so a rough estimate of displacement can be made.

7.5 cm ± 2 mm

6.5 cm ± 3 mm

5.4 cm ± 3 mm

52.0 cm ± 3 mm

Figure 65. IR Detector Material

Advances in Microwave Metamaterials James A. Wigle Chapter 12. IR M e t a - Property Test | P a g e | 128

Purely as a side note, caution must be exercised when working with high powered radiation within a microwave anechoic chamber, the number one reason for microwave anechoic chamber conflagrations[41].

An essential matter for this test is the heating, however small or large, of material. If heating becomes too great for the anechoic chamber‟s microwave absorbing material, a fire could result. Fires are self- combusting and fumes are lethal.

12.4 Test Results

Although I had enough power to induce detectible heating, antenna directivity was not sufficient to ascertain accurate displacement, especially when within the near field. Actually, this is a tradeoff between directivity, detection sensitivity, and the thermal cooling rate of the detection material. The combination was not sufficient to determine accurate displacements. For this particular test, not enough accuracy existed to allow marking a „zero line‟, indicating the normal interface position where the index of refraction would equate to zero value. I also believe this test to be plagued with diffraction affects, mostly due to the material under test size.

The figures below illustrate the results obtained for various frequencies. Realize this experiment should be accomplished within the far field, so „hot and cold spots‟ within the reactive near field or radiating near field regions, dependent upon spatial location, do not adversely affect test results.

Given the time required for sufficient detector material heating, it became fairly obvious where, in frequency, this resonance‟s division between quadrant I and quadrant II resides (around 11.5 GHz). Recall that quadrant II is opaque, or non-propagating, while quadrant I is translucent. This was an unintended artifact of my testing, but lends credit to other results‟ frequency locations.

Advances in Microwave Metamaterials James A. Wigle Chapter 12. IR M e t a - Property Test | P a g e | 129

Figure 66. IR Material Properties Test Figure 68. IR Material Properties Test

Figure 67. 17 GHz Material Displacement w/Meta Figure 69. 17 GHz Material Displacement w/o Meta

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Figure 70. 10.8 GHz Material Displacement w/Meta Figure 71. 10.8 GHz Material Displacement w/o Meta

It appears as though the displacement for 17 GHz is close to 3.8 cm, and the displacement for 10.8

GHz is close to 9.4 cm, given no diffraction affects occurred. Performing the calculations, as those within the Theory subsection of this chapter, yields the results indicated within Table 6. Given constituent material properties, and the lack of a magnetically coupling design, the permeability should have a value of one and is assumed to be so. Given that I could not mark the zero index of refraction line, one cannot definitively determine whether or not the material exhibited a positive or negative index of refraction, for this test. Therefore, I have listed out both possibilities within the table. Though, it is curious that the positive result has an index of refraction less than unity. Furthermore, the lower frequency appears to have a substantially greater displacement, which would correspond to a very high and positive index of refraction, or a negative index of refraction. Again, more accurate results using larger metamaterials are required, to be certain this is not a diffraction affect.

Advances in Microwave Metamaterials James A. Wigle Chapter 12. IR M e t a - Property Test | P a g e | 131

Table 6. Metamaterial Properties for IR Test

Operational Frequency n µr (GHz) 17.0 1.4 2.1 1.0

+0.9 +0.8 10.8 or or +1.0 -0.9 -0.8

Again, these results are a bit suspect, given measurement tolerance issues, and the displacement was found while in the near reactive or near radiated fields. However, this experiment was intended to be a proof of concept to see if this should be pursued further.

12.5 A Test at Optical Frequency

As a proof of concept, I decided to demonstrate the concept at optical frequencies, though no heating was required for this experiment. It is an easy extrapolation to other realms of the electromagnetic spectrum, but the novel experiment would require some sort of detectible heating upon a known resistive surface. The schematic of the test I performed is illustrated in Figure 72.

Top-Down View: Wall Front View:

Air

Tap Water

Vertical Laser Vertical Laser Line Generator Line Figure 72. Optical Test Schematic

Advances in Microwave Metamaterials James A. Wigle Chapter 12. IR M e t a - Property Test | P a g e | 132

I obtained a measured displacement of 5.15 cm. Using solutions within the theory section, with an index of refraction for tap water of 1.3, I obtain a theoretical displacement of 5.03 cm. This is about a 2.4% difference from the measured value, which is within measurement tolerances for this quick test, and is particularly reasonable given the previously mentioned accuracy requirements experienced through calculations of various experiments.

12.6 Conclusions

Test theory is solid, although test results do not display conclusive results. A more directive antenna, or a more sensitive infrared receiver, is required to provide more accurate results at our frequency of interest. A larger test material is also required to eliminate diffraction effect concerns.

I shall repeat these experiments using the SSRR, quadrant III metamaterial with much larger surface area to remove any potential diffraction effects. Moreover, there should be no dispute as to whether or not this quadrant III metamaterial has a negative index of refraction. Hopefully this, along with a higher directive antenna, will provide the clear results expected.

I will also experiment to see if this test can be used in determining the plasma frequency, which is associated with the transition from opaque to translucent medium.

The optical proof of concept experiment did display the desired effect. Thus, the experiment should work at other frequencies as well, using proper equipment. This is not an outlying extrapolation, as this is really the exact setup, although at a different frequency. Furthermore, heating of a detector material continues to be an option for loss tangent measurement, as well as displacement for permittivity and permeability measurements.

Advances in Microwave Metamaterials James A. Wigle

Chapter 13. Microwave Test to Determine Metamaterial

Properties

13.1 Introduction

This test is very similar to the infrared test of the previous chapter, though without heating and sensing detector material. Using a directional sensitive receiver, I simply measure the peak position of microwave energy passing through and beyond the material under test. The desire for this test was also to definitively demonstrate a positive or negative angle for the material under test. Of course, identical to the infrared test, how far displaced from the original trace path, or the path from n = 0 trace path, provides information about the material‟s property. Thus, in the same manner as the previous chapter, one can determine material properties such as permittivity, permeability, and index of refraction. The loss tangent could be measured using very accurate power density measurements, with and without the material under test.

In order to save more trees, I will not repeat the theory and calculations of last chapter. Suffice it to write here, the calculations are the same, and can be found within the Theory subsection of the Infrared

Test to Determine Metamaterial Properties chapter.

13.2 Test Setup

A vector network analyzer (VNA) is used for this test. Although any well designed transmitter and receiver will work, the VNA makes calibration and testing easy and accurate. Figure 73 illustrates the test composition in great detail. Essentially, a highly directional emitter propagates microwave energy, at Chapter 13. µ W a v e M e t a - Property Test | P a g e | 134

the frequency of interest, to the material under test and the receiver, both residing within the antennas‟ far fields. The energy must propagate through the material, which is situated at a known angle. The resulting displacement indirectly provides material under test property information; see the Theory subsection of the last chapter. The displacement is measured by sliding a highly directional receiving antenna, normal to the propagating energy‟s plane-wave Poynting vector, or normal to the direction of energy flow in the far field.

The displacement is determined where the highest energy level is detected. Therefore, it is critical to be within the far field for this test, and not the reactive near or radiated near fields, which could significantly skew results.

Test Setup: µWave Material Properties Test Material & Material under test, receiver in which can rotate. far field

G t α Gr n > 0 n = 0

n < 0 PP1 C2 D

Anechoic Chamber C3

PP Cn = Cables C1 PPn = Patch Panels

C4 Vector Network Port Port 1 Analyzer 2

(S21)

Figure 73. Microwave Material Properties Test Schematic

Advances in Microwave Metamaterials James A. Wigle Chapter 13. µ W a v e M e t a - Property Test | P a g e | 135

13.3 Results

Influences like those for the infrared test were evident within this test as well, and even more so.

Due to low antenna directivities and fringing affects, results were inconclusive. Displacement resolution was measured at various frequencies, using various antennas. Displacement resolutions ranged from 50 cm to 76 cm, near our plasmon frequency, which is not sufficient to ascertain accurate displacement.

13.4 Conclusions

The test method is solid, as exemplified in the optical frequency test within the last chapter, although test results do not display conclusive results. Future research shall include higher gain antennas and larger materials for testing. The SSRR, quadrant III, metamaterial will be a good material to test, when manufactured. Given higher gain antennas, the negative Snell‟s law angle should be evident.

Like the infrared test, illuminating where in the frequency spectrum the metamaterial goes from opaque to transparent should assist in locating the plasma resonance of the metamaterial.

Advances in Microwave Metamaterials James A. Wigle

Chapter 14. Fresnel Coefficient Matrix

14.1 Work Left to More Capable Hands

iven that electromagnetic material parameters can take on any sign, Dr. Pinchuk and I G noticed that the Fresnel coefficients no longer hold for all cases of permittivity and permeability. My literary research concluded that no one had published this fact, expect for Veselago[47] and Marqués et. al[30]. Veselago simply mentioned the fact, and Marqués et. al. only detailed a single left handed scenario, my matrix scenario number four.

On numerous occasions I attempted using Maxwell‟s equations to develop what I called the

Fresnel Coefficient Matrix to cover all values and signs of the index of refraction. This proved to be much more of a challenge than imagined, as I encountered an ongoing stream of mathematical dilemmas, such as sign inconsistencies and a transmission coefficient greater than 1.0 scenario. Later, the physics team and I attempted explanations of the transmission coefficient greater than one by investigating the possibility of amplitude variations of the electric and magnetic fields such that the conservation of energy would be maintained (analogous to power, voltage, impedance, and current in electronics), but nothing succeeded.

Given other new material in this dissertation, I have abandoned this effort and leave it in more capable hands. However, below I did include some of my partial work, in hopes it may help others on their successful journey in this effort.

In order to keep this work under 4,000 pages, I only detailed the RHM matrix scenario (numbers

1-3), used in laying the foundation, and the first of the LHM matrix scenarios (number 4). The remaining scenarios were left out of this thesis, but are easily worked in the same manner, with lots of duplication.

Furthermore, a lot of this is repetitive, but left in for crystal clear direction followed, as well this allows for easier future modifications when one cannot predict where errors and changes will occur. Chapter 14. Fresnel Coefficient Matrix | P a g e | 137

Leveraging Veselago‟s 2007 work[47], Dr. Victor Gozhenko found a much more promising path using wave impedances, rather than the path Dr. Pinchuk and I assumed. Dr. Gozhenko is an extremely capable physicist and I wish him all good luck, and any assistance when I am able.

14.2 Fresnel Coefficient Matrix

The aforementioned Fresnel Coefficient Matrix details all the different scenarios for a single interface, which should be easily expandable to all multiple layered scenarios. This Fresnel Coefficient

Matrix is displayed in Table 7. The matrix is not filled in, for reasons mentioned in the previous subsection. Note that the matrix assumes µ1 = µ1 = 1, though I solved for the µx ≠ 1 scenarios. The more complete solution is left out for table clarity. It has yet to be determined if new, or fewer, scenarios will be required if absolute values of the index of refraction affect solution results.

Table 7. Fresnel Coefficient Matrix

Scenario N1 N2 N1 vs. N2 Status ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ N > 0 N > 0 N < N Known 2 1 1 2 1 2 Γ

N > 0 N > 0 N > N Known 2 2 1 2 1 2 Γ

N > 0 N > 0 N = N Known 2 3 1 2 1 2 Γ

N > 0 N < 0 N > N Recently 2 4 1 2 1 2 Γ Described [31] 5 N1 > 0 N2 = 0 N1 > N2 Previously TBD TBD Unknown 6 N1 = 0 N2 > 0 N1 < N2 Previously TBD TBD Unknown 7 N1 = 0 N2 < 0 N1 > N2 Previously TBD TBD Unknown 8 N1 < 0 N2 < 0 N1 < N2 Previously TBD TBD Unknown 9 N1 < 0 N2 < 0 N1 > N2 Previously TBD TBD Unknown 10 N1 < 0 N2 < 0 N1 < N2 Previously TBD TBD Unknown 11 N1 < 0 N2 < 0 N1 = N2 Previously TBD TBD Unknown 12 N1 < 0 N2 = 0 N1 < N2 Previously TBD TBD Unknown 13 N1 = 0 N2 = 0 N1 = N2 Previously TBD TBD Unknown

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14.3 ⃑⃑ ⃑⃑⃑ rh and ⃑⃑ ⃑⃑ rh

Begin with Maxwell‟s laws, Ampere‟s and Faraday‟s point, or differential, forms:

⃑⃑ ⃑⃑

( , assumes no surface currents )

⃑⃑ ⃑⃑

Where: ⃑⃑ = Magnetic field intensity vector.

⃑⃑ = Electric flux density vector.

t = Time.

= Electric conduction current density vector.

⃑⃑ = Electric field intensity vector.

⃑⃑ = Magnetic flux density vector.

A general form of plane wave propagation will be used:

⃑ ⃑⃑ ⃑ ⃑⃑ ⃑⃑ [ ( )] ⃑⃑

Where: = Radian frequency ( = 2πf, where f = frequency).

= Observation vector.

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⃑ = Propagation vector.

and, ⃑⃑ ⃑⃑ , ⃑ ⃑⃑ , ⃑ ⃑⃑

This general form assumes α = 0 (no amplitude decay). This can be easily added, but is left out for better clarity. This form does not assume any specific component directions for ⃑⃑ , and ⃑⃑ . All vectors are more detailed below, illustrating this direction independence.

⃑⃑ ̂ ̂ ̂

⃑ ̂ ̂ ̂

̂ ̂ ̂

Ho Eo

y Ex k Ey

Figure 74. Vector Orientations

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14.3.1 ⃑⃑ rh

Since

⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

Where: (no time or position dependence, no position dependence  isotropic medium)

So,

⃑⃑ ⃑⃑ ⃑⃑ 2 ⃑⃑ ⃑⃑ ⃑ 3

⃑⃑ ⃑⃑ ⃑⃑ ⃑ * + ⃑⃑ ⃑⃑ ⃑ ( ) ⃑⃑

⃑⃑ ⃑ ⃑⃑ ⃑⃑ (where ⃑⃑ ⃑⃑ )

Also,

̂ ̂ ̂

⃑⃑ | |

⃑⃑ ̂ 6 7 ̂ [ ] ̂ 6 7

Where:

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⃑⃑ ⃑ ⃑⃑ ⃑⃑

⃑⃑ ̂ ̂ ̂

⃑ ̂ ̂ ̂

̂ ̂ ̂

And,

⃑

So,

[ ] ⃑⃑ ⃑⃑

⃑⃑ , ̂ ̂ ̂-

Thus,

( ⃑⃑ ⃑ )

Therefore:

⃑⃑ ( ) ( ⃑ )

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⃑⃑ ( ) ( ⃑ )

Putting it altogether yields:

⃑⃑

( ⃑⃑ ⃑ ) ( ⃑⃑ ⃑ ) ̂ 0 1

⃑⃑

( ⃑⃑ ⃑ ) ̂ [ ]

( ⃑⃑ ⃑ ) ̂ , -

( ⃑⃑ ⃑ ) ̂ [ ]

( ⃑⃑ ⃑ ) ⃑⃑ { ̂[ ] ̂, - ̂[ ]}

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14.3.2 ⃑⃑ ⃑⃑⃑ rh

̂ ̂ ̂

⃑⃑ ⃑⃑ | |

⃑⃑ ⃑⃑ ̂[ ] ̂, - ̂, -

Recall from previous sub-subsection:

( ⃑⃑ ⃑ )

This gives:

̂ ⃑⃑

( ⃑⃑ ⃑ ) ( ⃑⃑ ⃑ ) ̂ 0 1

̂ ( ⃑⃑ ⃑ ) ⃑⃑ { ̂[ ] ̂, - ̂[ ] }

̂ Now, comparing solutions for ⃑⃑ and ⃑⃑ yields:

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̂ ( ⃑⃑ ) ⃑⃑

̂ ⃑⃑ ( ⃑⃑ )

From the previous sub-subsection:

⃑⃑ ⃑ ⃑⃑ ⃑⃑ (where ⃑⃑ ⃑⃑ )

Therefore:

⃑ ⃑⃑ ⃑ ⃑⃑ ⃑⃑ (where ⃑⃑ ⃑⃑ )

14.3.3 ⃑ rh

Faraday‟s point form:

⃑⃑ ⃑⃑

Since,

⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

Where: (no time or position dependence, no position dependence  isotropic medium)

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So,

⃑⃑ ⃑⃑ ⃑⃑

⃑⃑ ⃑⃑ 2 ⃑⃑ ⃑⃑ ⃑ 3

⃑⃑ ⃑⃑ ⃑⃑ ⃑ * + ⃑⃑ ⃑⃑ ⃑ ( ) ⃑⃑

⃑⃑ ⃑ ⃑⃑ ⃑⃑ (where ⃑⃑ ⃑⃑ )

Also,

̂ ̂ ̂

⃑⃑ | |

⃑⃑ ̂ 6 7 ̂ [ ] ̂ 6 7

Where:

⃑⃑ ⃑ ⃑⃑ ⃑⃑

⃑⃑ ̂ ̂ ̂

⃑ ̂ ̂ ̂

̂ ̂ ̂

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And,

⃑

So,

[ ] ⃑⃑ ⃑⃑

⃑⃑ , ̂ ̂ ̂-

Thus,

( ⃑⃑ ⃑ ) ⃑⃑

Therefore:

⃑⃑ ( ) ( ⃑ )

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⃑⃑ ( ) ( ⃑ )

Putting it altogether yields:

⃑⃑

( ⃑⃑ ⃑ ) ( ⃑⃑ ⃑ ) ̂ 0 1

⃑⃑

( ⃑⃑ ⃑ ) ̂ [ ]

( ⃑⃑ ⃑ ) ̂ , -

( ⃑⃑ ⃑ ) ̂ [ ]

( ⃑⃑ ⃑ ) ⃑⃑ { ̂[ ] ̂, - ̂[ ]}

14.3.4 ⃑⃑ ⃑⃑ rh

̂ ̂ ̂ ⃑ ⃑⃑ | |

⃑ ⃑⃑ ̂[ ] ̂, - ̂, -

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Recall from previous sub-subsection:

( ⃑⃑ ⃑ )

This gives:

⃑ ⃑⃑

( ⃑⃑ ⃑ ) ( ⃑⃑ ⃑ ) ̂ 0 1

⃑ ( ⃑⃑ ⃑ ) ⃑⃑ { ̂[ ] ̂, - ̂[ ] }

⃑ Now, comparing solutions for ⃑⃑ and ⃑⃑ yields:

⃑ ( ⃑⃑ ) ⃑⃑

⃑ ⃑⃑ ( ⃑⃑ )

From the previous sub-subsection:

⃑⃑ ⃑ ⃑⃑ ⃑⃑ (where ⃑⃑ ⃑⃑ )

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Therefore:

⃑ ⃑⃑ ⃑ ⃑⃑ ⃑⃑ (where ⃑⃑ ⃑⃑ )

14.4 ⃑⃑ ⃑⃑⃑ lh and ⃑⃑ ⃑⃑ lh

⃑ ⃑ The same methods will be used to solve for ⃑⃑ and ⃑⃑ , for left handed materials. Again, we begin with Maxwell‟s laws, Ampere‟s and Faraday‟s point, or differential, forms:

⃑⃑ ⃑⃑

( , assumes no surface currents )

⃑⃑ ⃑⃑

Where: ⃑⃑ = Magnetic field intensity vector.

⃑⃑ = Electric flux density vector.

t = Time.

= Electric conduction current density vector.

⃑⃑ = Electric field intensity vector.

⃑⃑ = Magnetic flux density vector.

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A general form of plane wave propagation will be used:

⃑ ⃑⃑ ⃑ ⃑⃑ ⃑⃑ [ ( )] ⃑⃑

Where: = Radian frequency ( = 2πf, where f = frequency).

= Observation vector.

⃑ = Propagation vector. and, ⃑⃑ ⃑⃑ , ⃑ ⃑⃑ , ⃑ ⃑⃑

The above generic definitions are in keeping with the right handed definition, as well as the left handed definitions. Note that this is the same as for the right handed materials, since ⃑ has a generic direction definition.

⃑ ̂ ̂ ̂

⃑⃑ ̂ ̂ ̂

⃑ ̂ ̂ ̂

̂ ̂ ̂

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Once again, this does not assume any specific component directions for ⃑⃑ and ⃑⃑ , even if it is a left handed material.

14.4.1 ⃑⃑ lh

Since

⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

Where: (no time or position dependence, no position dependence  isotropic medium)

-12 = Permittivity of free space, 8.854 x 10 F/m.

So, using Ampere‟s point form:

⃑⃑ ⃑⃑ ⃑⃑ 2 ⃑⃑ ⃑⃑ ⃑ 3

⃑⃑ ⃑⃑ ⃑⃑ ⃑ * + ⃑⃑ ⃑⃑ ⃑ ( ) ⃑⃑

⃑⃑ ⃑ ⃑⃑ ⃑⃑ (where ⃑⃑ ⃑⃑ )

Note, this is the same for right handed materials.

Now solving another way:

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̂ ̂ ̂

⃑⃑ | |

⃑⃑ ̂ 6 7 ̂ [ ] ̂ 6 7

Where, from before:

⃑⃑ ⃑ ⃑⃑ ⃑⃑

⃑⃑ ̂ ̂ ̂

⃑ ̂ ̂ ̂

̂ ̂ ̂

And,

⃑

So,

[ ] ⃑⃑ ⃑⃑

⃑⃑ , ̂ ̂ ̂-

Thus,

( ⃑⃑ ⃑ )

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Therefore:

⃑⃑ ( ) ( ⃑ )

Putting it altogether yields:

⃑⃑

( ⃑⃑ ⃑ ) ( ⃑⃑ ⃑ ) ̂ 0 1

⃑⃑

( ⃑⃑ ⃑ ) ̂ [ ]

( ⃑⃑ ⃑ ) ̂ , -

( ⃑⃑ ⃑ ) ̂ [ ]

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( ⃑⃑ ⃑ ) ⃑⃑ { ̂[ ] ̂, - ̂[ ]}

Note that ⃑⃑ ⃑⃑

14.4.2 ⃑⃑ ⃑⃑⃑ lh

̂ ̂ ̂ ⃑ ⃑⃑ | |

⃑ ⃑⃑ ̂[ ] ̂, - ̂, -

Recall from previous sub-subsection:

( ⃑⃑ ⃑ )

This gives:

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⃑ ⃑⃑

( ⃑⃑ ⃑ ) ( ⃑⃑ ⃑ ) ̂ 0 1

⃑ ( ⃑⃑ ⃑ ) ⃑⃑ { ̂[ ] ̂, - ̂[ ] }

⃑ ⃑ Note that ⃑⃑ ⃑⃑

⃑ Now, comparing solutions for ⃑⃑ and ⃑⃑ yields:

̂ ( ⃑⃑ ) ⃑⃑

̂ ⃑⃑ ( ⃑⃑ )

As expected, this is the same for right handed materials, since ⃑⃑ ⃑⃑ and ⃑⃑ ⃑⃑

⃑ ⃑ ̂ and ⃑⃑ ⃑⃑ . Since ⃑⃑ ( ⃑⃑ ) , as described previously.

From the previous sub-subsection:

⃑⃑ ⃑ ⃑⃑ ⃑⃑ (where ⃑⃑ ⃑⃑ )

̂ ⃑⃑ ( ⃑⃑ ) ⃑⃑

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Therefore:

⃑ ⃑⃑ ⃑ ⃑⃑ ⃑⃑ (where ⃑⃑ ⃑⃑ )

Note that this is the same for right handed materials.

14.4.3 ⃑ lh

Faraday‟s point form, for left handed materials:

⃑⃑ ⃑⃑

Since,

⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

Where: (no time or position dependence, no position dependence  isotropic medium)

µ = Permeability of free space, 4π x 10-7 H/m.

So,

⃑⃑ ⃑⃑ ⃑⃑

⃑⃑ ⃑⃑ 2 ⃑⃑ ⃑⃑ ⃑ 3

⃑⃑ ⃑⃑ ⃑⃑ ⃑ * + ⃑⃑ ⃑⃑ ⃑ ( ) ⃑⃑

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⃑⃑ ⃑ ⃑⃑ ⃑⃑ (where ⃑⃑ ⃑⃑ )

Note that ⃑⃑ ⃑⃑ and ⃑⃑ ⃑⃑ .

Solving another way:

̂ ̂ ̂

⃑⃑ | |

⃑⃑ ̂ 6 7 ̂ [ ] ̂ 6 7

Where:

⃑⃑ ⃑ ⃑⃑ ⃑⃑

⃑⃑ ̂ ̂ ̂

⃑ ̂ ̂ ̂

̂ ̂ ̂

And,

⃑

So,

[ ] ⃑⃑ ⃑⃑

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⃑⃑ , ̂ ̂ ̂-

Thus,

( ⃑⃑ ⃑ ) ⃑⃑

Therefore:

⃑⃑ ( ) ( ⃑ )

Putting it altogether yields:

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⃑⃑

( ⃑⃑ ⃑ ) ( ⃑⃑ ⃑ ) ̂ 0 1

⃑⃑

( ⃑⃑ ⃑ ) ̂ [ ]

( ⃑⃑ ⃑ ) ̂ , -

( ⃑⃑ ⃑ ) ̂ [ ]

( ⃑⃑ ⃑ ) ⃑⃑ { ̂[ ] ̂, - ̂[ ]}

Note that ⃑⃑ ⃑⃑ .

14.4.4 ⃑⃑ ⃑⃑ lh

̂ ̂ ̂ ⃑ ⃑⃑ | |

⃑ ⃑⃑ ̂[ ] ̂, - ̂, -

Recall from previous sub-subsection:

( ⃑⃑ ⃑ )

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( ⃑⃑ ⃑ )

This gives:

⃑ ⃑⃑

( ⃑⃑ ⃑ ) ( ⃑⃑ ⃑ ) ̂ 0 1

⃑ ( ⃑⃑ ⃑ ) ⃑⃑ { ̂[ ] ̂, - ̂[ ] }

⃑ ⃑ Note that ⃑⃑ ⃑⃑ .

⃑ Now, comparing solutions for ⃑⃑ and ⃑⃑ yields:

⃑ ( ⃑⃑ ) ⃑⃑

⃑ ⃑⃑ ( ⃑⃑ )

Note that this also holds true for right handed materials. Now, from the previous sub-subsection:

⃑⃑ ⃑ ⃑⃑ ⃑⃑ (where ⃑⃑ ⃑⃑ )

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⃑ ( ⃑⃑ ) ⃑⃑ ⃑⃑

Therefore:

⃑ ⃑⃑ ⃑ ⃑⃑ ⃑⃑ (where ⃑⃑ ⃑⃑ )

Note that this is the same as for right handed materials.

14.5 Right Handed Scenario

Figure 75 below details the right handed scenario for developing the normal Fresnel coefficients, as this will demonstrate the validity of our path to redefining them for left handed materials. Note that we make few assumptions, outside of normal incidence. The permittivity, permeability, and index of refraction can have any sign, as well as be a complex number. Abstraction to oblique angles, is relatively straight forward using trigonometric functions, and is left out of this work for clarity.

Therefore the following subsections will obtain the reflection, and , and transmission,

and , coefficients for E and H. This method will provide the basis for deriving the left handed cases.

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Matrix Scenarios 1 to 3 (all RHM)

n1 n2

Er Є1 Є2

µ1 µ2 Kr Hr

K n < n t 1 2 Ki Ht H n1 > n2 i

n1 = n2 RHM RHM

Figure 75. Right Handed Material Fresnel Coefficients, Scenarios 1-3

n1 and n2 are indices of refraction, for medium 1 and 2, respectively.

√

( )

Via our former model definitions:

⃑⃑ ( ⃑ ) ⃑⃑

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( ⃑⃑ ⃑ ) ⃑⃑

Where: ω = 2πf = Radian frequency.

f = Frequency.

t = Time.

= Observation vector.

⃑ = Propagation vector for incident wave.

⃑ = Propagation vector for reflected wave.

⃑ = Propagation vector for transmitted wave.

⃑⃑ ⃑⃑

⃑ ⃑⃑

⃑⃑ = Incident electric field vector.

⃑⃑ = Reflected electric field vector.

⃑⃑ = Transmitted electric field vector.

⃑⃑ = Incident electric field amplitude vector.

⃑⃑ = Reflected electric field amplitude vector.

⃑⃑ = Transmitted electric field amplitude vector.

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Note that mathematical signs are not modified for direction in the general wave definitions, since

⃑ ⃑ ⃑ each , , and have previously been defined generically. This does assume tangential ⃑⃑ and ⃑⃑ , with normal incidence.

Via well-known boundary conditions:

⃑⃑ ⃑⃑ ⃑⃑ (No normal ⃑⃑ components)

Note that our scenario assumes normal incidence, with tangential components for ⃑⃑ and ⃑⃑ , but oblique incidence requires angle inclusion.

14.6 Hrh

⃑ ⃑

⃑ ⃑ ( )

⃑ ⃑

⃑ Where: = Free space wave propagation vector.

= Free space wavelength.

Given:

⃑ ⃑⃑ ⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

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we then obtain:

⃑ ⃑ ⃑ ⃑⃑ ⃑⃑ ⃑⃑

⃑ ⃑ ⃑ ⃑⃑ [ ⃑⃑ ] ⃑⃑

⃑ ⃑ ⃑ ( ⃑⃑ ) ( ⃑⃑ ) ( ⃑⃑ )

Again:

⃑ ⃑⃑ ⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

So we have:

( ⃑⃑ ) ( ⃑⃑ ) ( ⃑⃑ )

( ⃑⃑ ⃑⃑ ) ⃑⃑

⃑⃑ ⃑⃑ ⃑⃑

Recall ⃑⃑ ⃑⃑ ⃑⃑ , so combining these two equations yields:

⃑⃑ ⃑⃑ [ ⃑⃑ ⃑⃑ ]

⃑⃑ 2 ⃑⃑ ⃑⃑

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⃑⃑ ⃑⃑ 2 ⃑⃑ ⃑⃑ [ ]

2 ⃑⃑ ⃑⃑

⃑⃑ 2

⃑⃑

14.7 Hrh

Recall from the previous subsection:

⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

Combining these two equations yields:

⃑⃑ ⃑⃑ [ ⃑⃑ ⃑⃑ ]

⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ [ ] ⃑⃑

⃑⃑ ⃑⃑ ⃑⃑ [ ]

⃑⃑ [ ] [ ] ⃑⃑

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0 1 ⃑⃑ ⃑⃑

0 1

⃑⃑ ⃑⃑ ( )

⃑⃑

14.8 Hrh and Hrh Relationship

From previous subsections:

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14.9 Hrh and Hrh Limit Checks

Mathematical limit checks will be performed on the solutions recently derived. This is used simply as a „sanity check‟ and will be employed later for the left handed coefficient solutions.

From a previous subsection, we can obtain the following relationship:

0 1

Now I shall take the limits as the various indices of refraction approach a limit.

( )

From a previous subsection, we can obtain the following relationship:

Now I shall take the limits as the various indices of refraction approach a limit.

( )

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 169

( ) 2

The Fresnel reflection coefficients are as expected. As the second material‟s index of refraction becomes a very high value, the material acts more like a short circuit and provides total reflection, with an

inversion ( ). As the two materials become equal in electromagnetic nature, they essentially

become the same material, without interface, and provide no reflection ( ). As the first material‟s index of refraction becomes a very high value, the second material acts more like an open circuit and

provides total reflection, without an inversion ( ).

As well, the Fresnel transmission coefficients are as expected. As the second material‟s index of refraction becomes a very high value, the material acts more like a short circuit and provides total

reflection; thus, transmission is zero ( ). As the two materials become equal in electromagnetic nature, they essentially become the same material, without interface, and provide no reflection with total

transmission ( ). As the first material‟s index of refraction becomes a very high value, the second material acts more like an open circuit and provides total reflection. Even for right handed materials, the

transmission coefficient becomes the value 2, which continues to plague humanity and us ( 2).

14.10 Erh

As derived previously:

⃑ ⃑⃑ ⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

⃑⃑ ( ⃑ ⃑⃑ )

It follows that:

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 170

⃑ ⃑⃑ ( ⃑⃑ )

⃑ has a generic direction for each wave type (i, r, t). As previously indicated, if we use boundary conditions on a normal interface, we obtain:

⃑⃑ ⃑⃑ ⃑⃑

Therefore, we can derive:

⃑ ⃑ ⃑ ( ⃑⃑ ) ( ⃑⃑ ) ( ⃑⃑ )

̂ ̂ ̂ Since :

⃑ ⃑

⃑ ⃑ ( )

⃑ ⃑

⃑ Where: = Free space wave propagation vector.

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 171

= Free space wavelength.

Thus:

⃑ ⃑ ⃑ ⃑⃑ [ ⃑⃑ ] ⃑⃑

⃑ ⃑ ⃑ ( ⃑⃑ ) ( ⃑⃑ ) ( ⃑⃑ )

⃑ ⃑ ( ⃑⃑ ⃑⃑ ) ( ⃑⃑ )

⃑⃑ ⃑⃑ ⃑⃑

As previously indicated, if we use well known boundary conditions on a normal interface, we obtain:

⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

Combining the last two lines yields:

⃑⃑ ⃑⃑ ( ⃑⃑ ⃑⃑ ) ⃑⃑ ⃑⃑

⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 172

[ ] ⃑⃑ [ ] ⃑⃑

0 1 ⃑⃑ ⃑⃑

0 1

⃑⃑ ⃑⃑

⃑⃑

14.11 Erh

Now use ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ , along with ⃑⃑ ⃑⃑ ⃑⃑ found in the previous

subsection. This all yields:

⃑⃑ [ ⃑⃑ ⃑⃑ ] ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ 2 ⃑⃑ ⃑⃑

⃑⃑ ⃑⃑ [ ] ⃑⃑ 2 ⃑⃑

2 ⃑⃑ ⃑⃑

2 ⃑⃑ ⃑⃑ ( )

⃑⃑ 2

⃑⃑

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 173

14.12 Erh and Erh Relationship

From previous subsections:

14.13 Erh and Erh Limit Checks

Mathematical limit checks will be performed on the solutions recently derived. This is used simply as a „sanity check‟ and will be employed later for the left handed coefficient solutions.

From a previous subsection, we can obtain the following relationship:

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 174

0 1

Now I shall take the limits as the various indices of refraction approach a limit.

( )

From a previous subsection, we can obtain the following relationship:

Now I shall take the limits as the various indices of refraction approach a limit.

( )

( ) 2

inversion ( ). As the two materials become equal in electromagnetic nature, they essentially

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 175

provides total reflection, without an inversion ( ).

transmission coefficient becomes the value 2, which continues to plague humanity and us ( 2).

14.14 Left Handed Scenario

Figure 76 below details the right handed into left handed scenario for developing the new Fresnel coefficients (matrix scenario #4). Note that we make few assumptions, outside of normal incidence. The permittivity, permeability, and index of refraction can have any sign, as well as be a complex number.

Abstraction to oblique angles, is relatively straight forward using trigonometric functions, and is left out of this work for clarity.

Therefore, the following subsections will obtain the reflection, and , and transmission,

and , coefficients for E and H.

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 176

Matrix Scenario 4 (RHM into LHM)

n1 n2

Er Є1 Є2

µ1 µ2 Kr Hr

Kt n < n H 1 2 Ki t H n1 < 0 i

0 < n2 RHM LHM

Figure 76. Left Handed Material Fresnel Coefficients, Scenario 4

n1 and n2 are indices of refraction, for medium 1 and 2, respectively.

√

( )

Via our former model definitions, for right and left handed materials:

⃑⃑ ( ⃑ ) ⃑⃑

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 177

( ⃑⃑ ⃑ ) ⃑⃑

Note that mathematical signs are not modified for direction in the general wave definitions, since

⃑ ⃑ ⃑ each , , and have previously been defined generically. This does assume tangential ⃑⃑ and ⃑⃑ , with normal incidence.

Via well-known boundary conditions:

⃑⃑ ⃑⃑ ⃑⃑ (No normal ⃑⃑ components)

Note that our scenario assumes normal incidence, with tangential components for ⃑⃑ and ⃑⃑ , but oblique incidence requires angle inclusion.

14.15 Hlh

⃑ ⃑

⃑ ⃑ ( )

⃑ Where: = Free space wave propagation vector.

= Free space wavelength.

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 178

I feel it is worth the extra space here to describe my definitions, since these caused some consternation. Given the scenario illustration (see Figure 76), I have chosen the above definitions.

Note that n1 and n2 keep their mathematical signs, so n1 and n2 can independently be positive or

⃑ negative. As in the right handed material (RHM) case, the sign before the vector indicates the direction, + to the right and – for the opposite direction to the left. We have negatives values for the reflected and left handed material (LHM) cases. We only have the + or – cases, since this is simplified using a normal interface.

⃑ These definitions appear to work for the propagation vectors, , but further investigation is required of the power vectors, ⃑ . I have investigated variations using absolute values, but these do not appear to function properly for all scenarios of ± nx.

Given generic direction definitions, the following still hold true:

⃑ ⃑⃑ ⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

we then obtain:

⃑ ⃑ ⃑ ⃑⃑ ⃑⃑ ⃑⃑

⃑ ⃑ ⃑ ⃑⃑ [ ⃑⃑ ] ⃑⃑

⃑ ⃑ ⃑ ( ⃑⃑ ) ( ⃑⃑ ) ( ⃑⃑ )

Again:

⃑ ⃑⃑ ⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 179

So we have:

( ⃑⃑ ) ( ⃑⃑ ) ( ⃑⃑ )

( ⃑⃑ ⃑⃑ ) ⃑⃑

⃑⃑ ⃑⃑ ⃑⃑

Recall ⃑⃑ ⃑⃑ ⃑⃑ , so combining these two equations yields:

⃑⃑ ⃑⃑ [ ⃑⃑ ⃑⃑ ]

⃑⃑ 2 ⃑⃑ ⃑⃑

⃑⃑ ⃑⃑ 2 ⃑⃑ ⃑⃑ [ ]

2 ⃑⃑ ⃑⃑

2 ⃑⃑ ⃑⃑ ( )

⃑⃑ 2

⃑⃑

Note the sign difference from the RHM definition.

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 180

14.16 Hlh

Recall from the previous subsection:

⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

Combining these two equations yields:

⃑⃑ ⃑⃑ [ ⃑⃑ ⃑⃑ ]

⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ [ ] ⃑⃑

⃑⃑ ⃑⃑ ⃑⃑ [ ]

⃑⃑ [ ] [ ] ⃑⃑

0 1 ⃑⃑ ⃑⃑

0 1

⃑⃑ ⃑⃑ ( )

⃑⃑

Note the sign difference from the RHM definition.

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 181

14.17 Hlh and Hlh Relationship

From previous subsections:

Note that this form is the same as for RHM (encouraging).

14.18 Hlh and Hlh Limit Checks

Mathematical limit checks will be performed on the solutions recently derived. This is used simply as a „sanity check‟.

From a previous subsection, we can obtain the following relationship:

0 1

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 182

Now I shall take the limit values as the various indices of refraction approach a limit.

( )

From a previous subsection, we can obtain the following relationship:

Now I shall take the limits as the various indices of refraction approach a limit.

( )

( ) 2

The reader must remember that n2 < 0 and n1 > 0 for this scenario. This changes the limits significantly, and produces results identical to those of the right handed material scenarios (matrix scenarios 1 to 3).

The Fresnel reflection coefficients are as expected. As the second material‟s index of refraction becomes a very high absolute value, the material acts more like a short circuit and provides total reflection,

with an inversion ( ). As the two materials become equal in electromagnetic nature, but opposite

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 183

in sign, they essentially become the same material, without interface, and provide no reflection ( ), just like the right handed case with both having the same sign. This is Sir Pendry‟s “Perfect lens” case. As the first material‟s index of refraction becomes a very high value, the second material acts more like an

open circuit and provides total reflection, without an inversion ( ).

As well, the Fresnel transmission coefficients are as expected, and the same as the right handed cases. As the second material‟s index of refraction becomes a very high absolute value, the material acts

more like a short circuit and provides total reflection; thus, transmission is zero ( ). As the two materials become equal in electromagnetic nature, but opposite in sign, they essentially become the same

material, without interface, and provide no reflection with total transmission ( ), just like in the right handed material case with both having the same sign (perfect lens). As the first material‟s index of refraction becomes a very high value, the second material acts more like an open circuit and provides total reflection. Even for right handed materials, the transmission coefficient becomes the value 2, which

continues to plague humanity and us ( 2).

14.19 Elh

As derived previously:

⃑ ⃑⃑ ⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

⃑⃑ ( ⃑ ⃑⃑ )

It follows that:

⃑ ⃑⃑ ( ⃑⃑ )

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 184

⃑ ⃑⃑ ( ⃑⃑ )

⃑ has a generic direction for each wave type (i, r, t). As previously indicated, if we use boundary conditions on a normal interface, we obtain:

⃑⃑ ⃑⃑ ⃑⃑

Therefore, we can derive:

⃑ ⃑ ⃑ ( ⃑⃑ ) ( ⃑⃑ ) ( ⃑⃑ )

̂ ̂ ̂ Since :

⃑ ⃑

⃑ ⃑ ( )

⃑ Where: = Free space wave propagation vector.

= Free space wavelength.

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 185

Again, it is worth the extra space here to describe my definitions. Given the scenario illustration (see Figure 76), I chose the above definitions. Note that n1 and n2 keep their mathematical signs, so n1 and n2 can independently be positive or negative. As in the right handed material (RHM)

⃑ case, the sign before the vector indicates the direction, + to the right and – for the opposite direction to the left. We have negatives values for the reflected and left handed material (LHM) cases. We only have the + or – cases, since this is simplified using a normal interface.

Given the former equations:

⃑ ⃑ ⃑ ⃑⃑ [ ⃑⃑ ] ⃑⃑

⃑ ⃑ ⃑ ( ⃑⃑ ) ( ⃑⃑ ) ( ⃑⃑ )

⃑ ⃑ ( ⃑⃑ ⃑⃑ ) ( ⃑⃑ )

⃑⃑ ⃑⃑ ⃑⃑

As previously indicated, if we use well known boundary conditions on a normal interface, we obtain:

⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 186

Combining the last two lines yields:

⃑⃑ ⃑⃑ ( ⃑⃑ ⃑⃑ ) ⃑⃑ ⃑⃑

⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑

[ ] ⃑⃑ [ ] ⃑⃑

0 1 ⃑⃑ ⃑⃑

0 1

⃑⃑ ⃑⃑

⃑⃑

Note the sign difference from the RHM definition.

14.20 Elh

Now use ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ , along with ⃑⃑ ⃑⃑ ⃑⃑ found in the

previous subsection. This all yields:

⃑⃑ [ ⃑⃑ ⃑⃑ ] ⃑⃑ ⃑⃑ ⃑⃑ ⃑⃑ 2 ⃑⃑ ⃑⃑

⃑⃑ ⃑⃑ [ ] ⃑⃑ 2 ⃑⃑

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 187

2 ⃑⃑ ⃑⃑

2 ⃑⃑ ⃑⃑ ( )

2 ⃑⃑ ⃑⃑

⃑⃑ 2

⃑⃑

Note the sign difference from the RHM definition.

14.21 Elh and Elh Relationship

From previous subsections:

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 1 88

Note that this form is the same as for RHM (encouraging).

14.22 Elh and Elh Limit Checks

Mathematical limit checks will be performed on the solutions recently derived. This is used simply as a „sanity check‟ and will be employed later for the left handed coefficient solutions.

From a previous subsection, we can obtain the following relationship:

0 1

Now I shall take the limit values as the various indices of refraction approach a limit.

( )

From a previous subsection, we can obtain the following relationship:

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 189

Now I shall take the limits as the various indices of refraction approach a limit.

( )

( ) 2

with an inversion ( ). As the two materials become equal in electromagnetic nature, but opposite

in sign, they essentially become the same material, without interface, and provide no reflection ( ), just like the right handed case with both having the same sign (perfect lens). As the first material‟s index of refraction becomes a very high value, the second material acts more like an open circuit and provides total

reflection, without an inversion ( ).

material, without interface, and provide no reflection with total transmission ( ), just like in the right

Advances in Microwave Metamaterials James A. Wigle Chapter 14. Fresnel Coefficient Matrix | P a g e | 190

handed material case with both having the same sign (perfect lens). As the first material‟s index of refraction becomes a very high value, the second material acts more like an open circuit and provides total reflection. Even for right handed materials, the transmission coefficient becomes the value 2, which

continues to plague humanity and us ( 2).

Advances in Microwave Metamaterials James A. Wigle

Chapter 15. Summary

This work illustrated many research efforts that have transpired over numerous years at the

University of Colorado at Colorado Springs. A number of efforts are new to the industry, some simply expand existing new research, and others have yet to be entirely completed.

Hopefully the reader now has a better understanding of metamaterials, and the fundamental physics behind how they achieve designed electromagnetic material properties. This research went beyond modeling and theoretical postulations and manufactured a metamaterial within quadrants I and II. With this metamaterial, improved patch antenna directivity and gain were demonstrated. Taking this concept a bit further, I demonstrated the utility of metamaterials as an antenna lens, proving unique multifaceted lens designs are possible.

Also demonstrated through empirical testing, not just in theory, metamaterial and meta-antenna

Lorentz reciprocity continues to hold true, which otherwise would have significant consequence within the engineering industry. The two new material property tests, along with the quadrant III metamaterial design show promise to open new metamaterial understanding as this research continues. I intend to use these test to demonstrate the transition from quadrant I to quadrant II. I also demonstrated that gain and directivity are not necessarily coincident in regards to frequency.

Fresnel coefficients were also shown to no longer be valid for cases involving a negative index of refraction. This work identified the missing components in what is called the Fresnel Coefficient Matrix in chapter 14. Going a bit further with this concept, an attempt at deriving one case of the new Fresnel coefficients was illustrated.

So many interesting things to do, with so little time...

Chapter 16. Future Investigations

he author has a plethora of questions regarding EM characteristics when propagating T through metamaterials. Some of these questions involve what happens to the following parameters: Total internal reflection, Brewster‟s angle, transmission coefficient, reflection coefficient, dispersion, critical angle, etc.

Dispersion is especially of interest to try and determine, theoretically, since no metamaterial with the appropriate size and bandwidth exists. What happens to light if it were to pass through material with a negative index of refraction? Would this produce a reversed rainbow color scheme?

I have been informed that polarization does not change as an electromagnetic wave passes through a material with a negative index of refraction. I would like to better understand why this is so, when we know the E-field‟s normal component changes its direction.

I would like to investigate manufacturing a flexible circuit board pattern of the SSRR, in order to uniformly wrap it around objects. This opens up lots of possibilities for three dimensional meta-lens applications, especially in regards to antenna patterns, as well as cloaking.

Our physics team is anxious to see if the National Science Foundation proposal is funded. This work should provide unique insight to metamaterial and electromagnetic wave interactions, within the metamaterial itself.

Using flexible circuit boards to create designable, 360º meta-lenses or antenna enhancements is on the to-do list. Chapter 16. Future Investigations | P a g e | 193

Experimentation using the „infrared or microwave experiment to determine material properties‟ is of interest to see if a determination of the plasma frequency can be made, using the transition from opaque to translucent medium information.

Since gain and directivity appear not to be coincident in frequency, I would like to perform more experiments to validate this, in a more „epidemiological‟ manner. It would be interesting to see if any conclusions can routinely be drawn from empirical data. This may help in future metamaterial designs for the community.

Lastly, both derivations for the bulk metal plasma frequency and surface plasma resonance (SPR) begin with electron motion inside metal. I would like to understand why these proofs use Ɛo and not Ɛr ∙Ɛo .

Advances in Microwave Metamaterials James A. Wigle

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46. Norgard J., Metzger D., and Sega R. Analysis of Probe Measurement accuracies and Scattering

47. Veselago, V. G. Negative Refraction as a Source of Some Pedagogical Problems. Acta Physica

Polonica A, Volume 112 (2007), Number 5, from Proceedings of the International School and

Conference on Optics and Optical Materials, ISCOM07, Belgrade, Serbia, September 3-7, 2007.

49. Ulaby F., Michielssen E., and Ravaioli U. Fundamentals of Applied Electromagnetics, 6th Edition.

50. Norgard J. and Sega R. Microwave Fields Determined from Thermal Patterns. SPIE Milestone

Advances in Microwave Metamaterials James A. Wigle

Appendix A – Peer Reviewed Paper

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Advances in Microwave Metamaterials James A. Wigle

Appendix B – Metamaterial Calculator

#!/apps/bin/perl -W # -W turns on ALL warnings for easier code debugging.

################################################################################

# #

# Program = XXXX.pl, Version Day Month 2004 #

# Programmer = Jim Wigle #

# Language = Perl #

# Date = 8 Dec 2009 #

# #

# Equipment Needed: None #

# #

# Other S/W Needed: None #

# #

# This program.... #

# #

################################################################################

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### Enables / disables print statements used for debugging (0=Off, 1=On)

$DEBUG = 0; # Debug switch (0=Off, 1=On)

### Initialize the following GLOBAL program parameters.

$THIS_PROGRAMS_NAME = substr($0,-19,19); # Globally holds name of this Perl program for printing errors to user.

#####$NUMBER_OF_INPUTS= 3; # Globally holds number of command line input arguments required by this program.

### Initialize the following GLOBAL design limits.

### Declare all subroutines here, so they can be used anywhere in this program.

# sub PrintIntro; # Declare subroutine to print information concerning this program. sub PromptForInputs; # Declare subroutine to prompt user for desired coil design parameters. sub GetInputsAndErrorTrap; # Declare subroutine to get calling program's parameters & check for errors. sub Calculations; # Declare subroutine to perform the calculations for the plasmon frequency.

#####sub XXXX; # Declare subroutine to perform the actual coil design calculations. sub PrintResults; # Declare subroutine that will print the coil design results.

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#####sub PrintUsageMessage; # Declare subroutine that prints a message on how to use this program. This is used when input errors are found. sub ExitProgram; # Declare subroutine that exits this program gracefully.

### ###

# Main routine is below. #

### ###

### Initialize & declare variables used in the main routine below.

&PrintIntro(); # Call subroutine to print an informative message to user.

### .....

&Calculations; # Perform calculations.

#####&PrintUsageMessage; # Print the usage information.

&PrintResults(@CoilData); # Call subroutine to print the design results.

&ExitProgram(); # Call subroutine to exit this program gracefully.

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############################################################

################################################################################

# #

# Subroutine = PrintIntro #

# #

# Input argument(s): #

# None #

# #

# Output argument(s): #

# None #

# #

# This subroutine simply prints out an informative message about this #

# program. #

# #

################################################################################ sub PrintIntro

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{ print "\n********************\n" if $DEBUG; # Print only if debugging. print "* Just entered PrintIntro subroutine.\n\n" if $DEBUG; # Print only if debugging.

### Print the message to screen.

# print "\n\n\n"; print " This program will calculate the plasmon radian frequency, given the\n"; print "thin wire mesh model parameters.\n"; print "\n"; print "\n"; print "\n\n\n";

print "\n\n* Leaving PrintIntro subroutine.\n" if $DEBUG; # Print only if debugging. print "********************\n" if $DEBUG; # Print only if debugging.

return (1); # Return to the calling program, indicting error free.

}

################################################################################

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# #

# Subroutine = PromptForInputs #

# #

# Input argument(s): #

# None #

# #

# Output argument(s): #

# $xxxx = ...... #

# #

# This subroutine is designed to ..... #

# #

################################################################################ sub PromptForInputs

{ print "\n********************\n" if $DEBUG; # Print only if debugging. print "* Just entered PromptForInputs subroutine.\n\n" if $DEBUG; # Print only if debugging.

### Initialize parameters and other variables used within this subroutine

### ....

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if ($DEBUG eq 1) # Print only if debugging.

{

print "\n";

print " First parameter = .\n";

print " Second parameter = .\n";

print " Third parameter = .\n";

print "\n";

}

print "\n\n* Leaving PromptForInputs subroutine.\n" if $DEBUG; # Print only if debugging. print "********************\n" if $DEBUG; # Print only if debugging.

return (); # Return to the calling program with coil design input parameters.

}

################################################################################

# #

# Subroutine = GetInputsAndErrorTrap #

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# #

# Input argument(s): #

# @_ = This Perl specific array variable contains all the information #

# passed to this subroutine. It contains only the information #

# entered by the user, which consists of three things: The #

# desired inductance, in uHenrys; the AWG wire gauge (or the #

# actual wire diameter, in inches) to be used; and the desired #

# inner coil diameter, in inches. #

# #

# Output argument(s): #

# $xxxx = ...... #

# #

# This subroutine is designed to ..... #

# #

################################################################################ sub GetInputsAndErrorTrap

{ print "\n********************\n" if $DEBUG; # Print only if debugging. print "* Just entered GetInputsAndErrorTrap subroutine.\n\n" if $DEBUG; # Print only if debugging.

### Initialize passed parameters and other variables used within this subroutine

Advances in Microwave Metamaterials James A. Wigle A p p e n d i x B – Metamaterial Calculator | P a g e | 222 if ($DEBUG eq 1) # Print only if debugging.

{

print "\n";

print " First parameter = .\n";

print " Second parameter = .\n";

print " Third parameter = .\n";

print "\n";

}

print "\n\n* Leaving GetInputsAndErrorTrap subroutine.\n" if $DEBUG; # Print only if debugging. print "********************\n" if $DEBUG;

# Print only if debugging.

return (); # Return to the calling program with coil design input parameters.

}

################################################################################

# #

# Subroutine = Calculations #

# #

# Input argument(s): #

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# @_ = This Perl specific array variable contains all the information #

# passed to this subroutine. It contains only the information #

# entered by the user, which consists of three things: The #

# desired inductance, in uHenrys; the AWG wire gauge (or the #

# actual wire diameter, in inches) to be used; and the desired #

# inner coil diameter, in inches. #

# #

# Output argument(s): #

# $xxxx = ...... #

# #

# This subroutine is designed to ..... #

# #

################################################################################ sub Calculations

{ print "\n********************\n" if $DEBUG; # Print only if debugging. print "* Just entered Calculations subroutine.\n\n" if $DEBUG; # Print only if debugging.

### Initialize passed parameters and other variables used within this subroutine

if ($DEBUG eq 1) # Print only if debugging.

Advances in Microwave Metamaterials James A. Wigle A p p e n d i x B – Metamaterial Calculator | P a g e | 224

{

print "\n";

print " First parameter = .\n";

print " Second parameter = .\n";

print " Third parameter = .\n";

print "\n";

}

### Initialize parameters and other variables used within this subroutine

# my $n = 1.806*10**29; # Electron density of metal used in metamaterial mesh (aluminum =

1.806*10**29 electrons/m**3). my $Pi = 3.14159; # The constant factor of Pi. my $Uo = 7*$Pi*10**-7; # Free space permeability (Henries/m). my $e = 1.60*10**-19; # Charge of an electron (Coulombs). my $c = 2.998*10**8; # Speed of light in free space (m/s). my $Eo = 8.85*10**-12; # Free space permittivity (Farads/m).

$MeshPeriodicity = 0.0072; # Periodicity of metamaterial wire mesh, in m.

$WireRadius = .0006; # Radius of wires used in metamaterial mesh, in m.

$Neff = ($n*$Pi*($WireRadius**2)) / ($MeshPeriodicity**2);

# Neff = the effective electron density (electrons/m**3).

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$Meff = ($Uo*$e**2*$Pi*$WireRadius**2*$n*(log($MeshPeriodicity/$WireRadius))) / (2*$Pi);

# Meff = the effective electron mass (kg).

$PenatrationDepth = 1000*$MeshPeriodicity * ( (2*log($MeshPeriodicity/$WireRadius)) / (3*$Pi) )**0.5;

# Penetration depth in mm.

$Wp1 = ( (2*$Pi*$c**2) / ($MeshPeriodicity**2*log($MeshPeriodicity/$WireRadius)) )**0.5;

# Wp = the plasmon radian frequency (radians/sec).

$Wp2 = ( ($Neff*$e**2) / ($Eo*$Meff) )**0.5;

# Wp = the plasmon radian frequency (radians/sec).

$F1 = ( (2*$Pi*$c**2) / ($MeshPeriodicity**2*log($MeshPeriodicity/$WireRadius)) )**0.5 /

(2*$Pi*10**9); # F = the plasmon frequency (GHz).

$F2 = ( ($Neff*$e**2) / ($Eo*$Meff) )**0.5 / (2*$Pi*10**9);

# F = the plasmon frequency (GHz).

print "\n\n* Leaving Calculations subroutine.\n" if $DEBUG; # Print only if debugging. print "********************\n" if $DEBUG;

# Print only if debugging.

return (); # Return to the calling program with coil design input parameters.

}

################################################################################

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# #

# Subroutine = PrintResults #

# #

# Input argument(s): #

# @_ = This Perl specific array variable contains all the information #

# passed to this subroutine. It contains the calculated air #

# core inductor design parameters, calculated in the CoilCalcs #

# subroutine. #

# #

# Output argument(s): #

# None #

# #

# This subroutine will perform a few remaining calculations and then #

# print the air core inductor design results. #

# #

################################################################################ sub PrintResults

{ print "\n********************\n" if $DEBUG; # Print only if debugging. print "* Just entered PrintResults subroutine.\n\n" if $DEBUG; # Print only if debugging.

### Initialize variables used within this subroutine

Advances in Microwave Metamaterials James A. Wigle A p p e n d i x B – Metamaterial Calculator | P a g e | 227 if ($DEBUG eq 1) # Print only if debugging.

{

print "\n";

}

### Print the metamaterial design results to screen.

# printf ("\n\n\n"); printf ("*****************************************************\n"); printf ("* The results of this left-handed material design:\n"); printf ("*\n"); printf ("* Mesh periodicity = %.4f mm\n", $MeshPeriodicity); printf ("* Wire radius = %.4f mm\n", $WireRadius); printf ("* Effective electron density = %.4f electrons/m**3\n", $Neff); printf ("* Effective electron mass = %.4f kg\n", $Meff); printf ("* Penetration depth = %.4f mm\n", $PenatrationDepth); print "\n"; printf ("* Plasmon frequency 1st model = $Wp1 (radians/sec), or $F1 (GHz).\n"); printf ("* Plasmon frequency 1st model = $Wp2 (radians/sec), or $F2 (GHz).\n"); printf ("*****************************************************\n"); printf ("\n");

Advances in Microwave Metamaterials James A. Wigle A p p e n d i x B – Metamaterial Calculator | P a g e | 228 print "\n\n* Leaving PrintResults subroutine.\n" if $DEBUG; # Print only if debugging. print "********************\n" if $DEBUG; # Print only if debugging.

return (1); # Return to calling routine, indicating error free.

}

################################################################################

# #

# Subroutine = PrintUsageMessage #

# #

# Input argument(s): #

# None #

# #

# Output argument(s): #

# None #

# #

# This subroutine simply prints out a usage message. It is used mostly #

# when some sort of error has occurred in the input parameters. #

# #

################################################################################ sub PrintUsageMessage

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{ print "\n********************\n" if $DEBUG; # Print only if debugging. print "* Just entered PrintUsageMessage subroutine.\n\n" if $DEBUG; # Print only if debugging.

### Print a message to the screen.

# print "\n"; print "********************************************************\n"; print "* Usage: Simply enter $THIS_PROGRAMS_NAME\n"; print "* at the command prompt.\n"; print "*\n*\n"; print "* -- OR --\n"; print "*\n*\n"; print "* Command Line Usage:\n"; print "*\n"; print "* $THIS_PROGRAMS_NAME \n"; print "* \n"; print "* \n"; print "*\n"; print "* Command Line Example:\n"; print "*\n"; print "* $THIS_PROGRAMS_NAME 28 .008 .375\n"; print "********************************************************\n"; print "\n";

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print "\n\n* Leaving PrintUsageMessage subroutine.\n" if $DEBUG; # Print only if debugging. print "********************\n" if $DEBUG;

# Print only if debugging.

return (1); # Return to the calling program, indicting error free.

}

################################################################################

# #

# Subroutine = ExitProgram #

# #

# Input argument(s): #

# None #

# #

# Output argument(s): #

# None #

# #

# This subroutine is designed to exit this program with Unix error #

# code = 0, indicating that no errors occurred during program execution. #

# #

################################################################################ sub ExitProgram

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{ print "\n********************\n" if $DEBUG; # Print only if debugging. print "* Just entered ExitProgram subroutine.\n\n" if $DEBUG; # Print only if debugging.

if ($DEBUG eq 1)

{

print "\n\n\n";

print "The main program has terminated without errors.\n";

print "\n\n\n";

}

print "\n\n* Leaving ExitProgram subroutine.\n" if $DEBUG; # Print only if debugging. print "********************\n" if $DEBUG; # Print only if debugging.

### Exit the program

# exit (0); # Exit program sending Unix code zero (all OK - error free).

}

Advances in Microwave Metamaterials James A. Wigle

Index

A anechoic chamber, 63, 72, 73, 76, 87, 89, 101, 106, 112, 128

Berkeley, 8, 10, 19

Bulk Plasma Frequency, 32, 69

Cherenkov, 8, 13

Directivity, 76, 79, 80, 81, 87, 90

Doppler, 8, 14

Drude, 6, 10, 12, 13, 20, 21, 49, 50

Duke, 10, 11, 194

Evanescent waves, 9

FR-4, 48, 52, 59, 70, 77 I n d e x | P a g e | 233

Fresnel Coefficient Matrix, 136, 137, 191

Friis, 91, 113

Gauss, 35, 39

Gozhenko, vi, 117, 118, 119, 137

HFSS, 54, 60

Improved Directivity, 53, 79 improved gain, iv, 53, 78, 87, 116 invisibility cloak, 3, 12, 49

Lezec, 13, 14

Lorentz, iv, 14, 63, 64, 65, 67, 100, 104, 191

Maxwell, 28, 136, 138, 149 meta-antenna, iv, 99, 100, 101, 102, 104, 191 meta-lens, iv, 1, 56, 68, 78, 91, 98, 105, 106, 107, 109, 112, 114, 115, 116, 192

Metamaterial Calculator, 214

Nanometa, 12, 13, 23

NIST, 13, 73

Advances in Microwave Metamaterials James A. Wigle I n d e x | P a g e | 234

Patch Antenna, 52, 68, 72, 77, 95, 116

Pendry, 2, 9, 10, 21, 26, 48, 50, 87, 183, 196, 197

Perfect Lens, 9, 196

Pinchuk, iii, vi, 1, 10, 136, 137, 195

Reciprocity, 63, 65, 66, 67, 100, 101, 103, 104 reflection coefficient, 9, 192

Smith, 3, 9, 10, 12, 21, 22, 117, 194, 195, 196

Snell’s law, 5, 15, 120, 122, 126, 135

Song, iii, vi, 1

SSRR, 58, 59, 60, 62, 85, 132, 135, 192

Stutzman, 64, 76, 89, 101, 112, 197

Surface Plasma Resonance, 38, 44

T transmission coefficient, 9, 136, 169, 175, 183, 190, 192

TWT, 7, 8

University of Colorado, 1, iv, vi, 1, 55, 61, 191, 195, 198

University of Maryland, 1, 7, 12, 196

Advances in Microwave Metamaterials James A. Wigle I n d e x | P a g e | 235

Vedral, vi, 54, 55, 60, 61

Veselago, 2, 8, 13, 17, 136, 137, 196, 198

Wigle, 1, ii, iii, 197, 214

Advances in Microwave Metamaterials James A. Wigle