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Advances In Microwave Metamaterials

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Advances In
MICROWAVE
METAMATERIALS
By
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
UMI Number: 3454032
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P a g e | ii
Copyright Notice
© Copyright by James A. Wigle 2011, all rights reserved.
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
by
___________________________________
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 (Ɛ r > 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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James A. Wigle
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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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James A. Wigle
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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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James A. Wigle
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Table of Contents
Copyright Notice ............................................................................................................................................ ii
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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James A. Wigle
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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
14.4.3
lh ...................................................................................................................... 156
14.4.4
lh ...................................................................................................................... 159
..................................................................................................................... 154
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14.5 Right Handed Scenario .................................................................................................................. 161
14.6
H
14.7
H
14.8
H
14.9
H
................................................................................................................................................ 164
rh
rh ................................................................................................................................................. 166
rh
and
H
rh
and
H
14.10
E
14.11
E
14.12
E
14.13
E
rh
Relationship .......................................................................................................... 167
rh
Limit Checks ......................................................................................................... 168
rh ............................................................................................................................................... 169
rh ............................................................................................................................................... 172
rh
and
E
rh
and
E
rh
Relationship ......................................................................................................... 173
rh
Limit Checks........................................................................................................ 173
14.14 Left Handed Scenario .................................................................................................................. 175
14.15
H
14.16
H
14.17
H
14.18
H
14.19
E
lh .............................................................................................................................................. 177
lh ............................................................................................................................................... 180
lh
and
H
lh
and
H
lh
Relationship ......................................................................................................... 181
lh
Limit Checks ....................................................................................................... 181
lh ............................................................................................................................................... 183
14.20
E
14.21
E
14.22
E
lh .............................................................................................................................................. 186
lh
and
E
lh
and
E
lh
Relationship ......................................................................................................... 187
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.
Table 2.
Table 3.
Table 4.
Table 5.
Table 6.
Table 7.
Quadrant II Metamaterial Design Parameters ................................................................ 53
11 GHz Patch Antenna Characteristics ............................................................................ 73
Meta-Patch Directivity vs. Gain Comparison .................................................................. 90
Meta-Patch Gain Test Results ......................................................................................... 98
Meta-Lens Gain Results ................................................................................................ 115
Metamaterial Properties for IR Test.............................................................................. 131
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
T
his thesis is a doctor of philosophy dissertation for the Electrical and Computer
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
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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 manmade 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
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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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Chapter 1.
Introduction
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µ
Quadrant II
Ɛ < 0 and µ > 0
N > 0 or N < 0 ???
Left handed materials. ???
Very few natural materials at specific frequencies
(e.g. metals at optical frequencies).
Opaque to EM radiation (huge attenuation).
No wave propagation.
Quadrant I
Ɛ > 0 and µ > 0
N>0
Right handed materials.
Almost all natural materials.
Large freq. portions transparent to EM radiation.
E&M wave propagation.
Ɛ
Quadrant III
Ɛ < 0 and µ < 0
N<0
Left handed materials.
No natural materials; man-made metamaterials only.
Small freq. portions semi-transparent to EM radiation.
E&M wave propagation with significant attenuation.
Quadrant IV
Ɛ > 0 and µ < 0
N > 0 or N < 0 ???
Left handed materials. ???
Very few natural materials at specific frequencies.
Opaque to EM radiation (huge 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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Chapter 1.
Introduction
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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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James A. Wigle
Chapter 1.
Introduction
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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
B
y far, the most popular postulated use of metamaterials involves cloaking devices to make
things invisible, particularly within the visual portion of the spectrum 2. 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
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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.
Advances in Microwave Metamaterials
James A. Wigle
Chapter 2.
Metamaterial Artifacts & Curiosities
| Page|9
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
|Page|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
|Page|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
|Page|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.
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James A. Wigle
Chapter 2.
Metamaterial Artifacts & Curiosities
|Page|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
|Page|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
F
or a better understanding of mechanisms behind these investigations, this chapter
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
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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
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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 Ɛ r 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
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)√
√
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
|
Figure 4. Fish in n<0 material, Source = Jason Valentine of Berkeley
These
phenomena
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5
are
Line normal to
interface
believed to be the result of „mental
tricks‟, much like that of watching
a fish swimming in water from a
stream bank.
To the shore
observer, the fish appears to swim
in a location where it is not
actually located.
n1
Material1
Material2
n2
Referring to
Figure 5, the fish‟s location is
along the n2 > 0 solid red line
trace, but to the observer it‟s
located along the dashed black „ray
5
ϴ2
ϴ2
Angle of Angle of
refraction refraction
for
for
n2 < 0
n2 > 0
Figure 5. Illustration of Snell's Law with Negative Index of Refraction
Use authorized 29 March 2011 by Sarah Yang, Office of Public Affairs, Berkeley University.
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James A. Wigle
Chapter 3.
Metamaterial Theory
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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.
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James A. Wigle
Chapter 3.
Metamaterial Theory
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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
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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
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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.
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James A. Wigle
Chapter 3.
Ē
Metamaterial Theory
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+++++++++++++++++++++
Metal
--------------------
Low Frequency Effects
Ē
-------------------Metal
+++++++++++++++++++++
Figure 6. Low Frequency E-Field Effects upon Metals
Ē
+++++++++++++++++++++
Metal
--------------------
Plasma Frequency
Effects
Ē
-------------------Metal
+++++++++++++++++++++
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.
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James A. Wigle
Chapter 3.
Metamaterial Theory
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(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.
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James A. Wigle
Chapter 3.
Metamaterial Theory
N>0
N<0
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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
7
Palik, E.D. Handbook of Optical Constants of Solids, 2nd edition. Academic Press ©1998.
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James A. Wigle
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Metamaterial Theory
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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.
⁄
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Free space permeability.
James A. Wigle
Chapter 3.
Metamaterial Theory
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⃑⃑ = Magnetic field vector.
= Current density vector.
= Conductivity of the medium.
The point form of Maxwell‟s free space equations:
⃑⃑
⃑⃑
(
)
⃑
⃑
E
Right
Orientation
Handed
⃑⃑
(
)
⃑
K&S
H
With the original assumptions of the region noted above,
Figure 9. Right Handed Orientation
for both ⃑ and ⃑ , two of Maxwell‟s free space equations become:
and time dependence of
⃑⃑
⃑
⃑
⃑⃑
and
⃑ ⃑⃑⃑
⃑
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⃑
⃑
⃑⃑
James A. Wigle
Chapter 3.
⃑
⃑
Metamaterial Theory
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⃑⃑
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
H1
E1
>0
µ1 > 0
n1 < n2
Et1
K 1 & S1
>0
µ2 > 0
n1 < n2
2
En2
(
2
>
1)
Et2
Figure 10. Wave Orientation for N > 0
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James A. Wigle
Chapter 3.
En1
Metamaterial Theory
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E1
>0
µ1 > 0
n2 < n1
n2 < 0
Et1
H1
K 1 & S1
<0
µ2 < 0
n2 < n1
n2 < 0
2
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
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James A. Wigle
Chapter 3.
Metamaterial Theory
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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.
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James A. Wigle
Chapter 3.
Metamaterial Theory
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3.11 Bulk Plasma Frequency
Einternal
Refer to the Plasma Frequency subsection of the
Metamaterial Theory chapter, for a higher level description
of plasma frequency.
That subsection should assist in
understanding this more detailed subsection. As the reader
goes through this derivation, it will become evident that this
is a resonant effect if either permittivity or permeability is
less than zero
----------------
Bulk
Metal
++++
++++
++++
++++
++++
Eexternal
X direction
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:
⃑
⃑
(
̈) ̂
nd
Equation 2. Newton's 2 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.
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James A. Wigle
Chapter 3.
Metamaterial Theory
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Page|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
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James A. Wigle
Chapter 3.
Metamaterial Theory
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nd
Equation 5. Modified Newton's 2 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.
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James A. Wigle
Chapter 3.
Metamaterial Theory
|
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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)
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James A. Wigle
Chapter 3.
Metamaterial Theory
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Page|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, ω p 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.
Єeff
Metamaterial Theory
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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 ≈
1023/cm2.
Thus, ωp is usually
within the ultra violet region of the
„visible‟ spectrum.
It is usually
advantageous
lower
frequency,
to
especially
this
Metal with large n
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
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Page|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 macroscale 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.
⃑
⃑
(
̈) ̂
nd
Equation 6. Newton's 2 Law
Where: ⃑ = Force vector.
m = Average mass of electrons.
⃑ = Acceleration of electrons.
̂ = Unit direction of accelerated electrons.
x = Average position of electrons.
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James A. Wigle
Chapter 3.
Metamaterial Theory
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̈ = 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.
∯ ⃑⃑
Since ⃑⃑
(Gauss‟ Law)
⃑ , in free space, Gauss‟ Law can be modified to:
Advances in Microwave Metamaterials
James A. Wigle
Chapter 3.
Metamaterial Theory
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Page|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.
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James A. Wigle
Chapter 3.
Metamaterial Theory
|
Page|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.
++++++++++++
++++++++++++
++++++++++++
1
++++++++++++
++++++++++++
E
+++
+++++++++
E = E1 + E2
-------. ---
Via Superposition
---------------------------------- - - - - - 2- - - - - ------------
E
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
|
Page|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
|
Page|43
E1
r1
l
r
Observation
Point
2
E2
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
|
Page|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.
Absorptio
n
SPR
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
|
Page|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.
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James A. Wigle
Chapter 3.
Metamaterial Theory
|
Page|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.
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James A. Wigle
Chapter 3.
Metamaterial Theory
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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
T
his chapter illuminates the quadrant II
metamaterial design.
The metamaterial
structure was designed and manufactured using available
resources.
preferred
Although available resources did not afford
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 used 8.
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
|Page|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.
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James A. Wigle
Chapter 4.
Quadrant
II,
10.5 GHz Metamaterial
|Page|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:
2
. /
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
|Page|51
r, or wire radius
a
a, or grid spacing periodicity
Figure 21. Thin wire metamaterial structure
2
. /
. /
. /
2
2
8
2
9
Given that „a‟ cannot be solved explicitly, I solicited help
10
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
10
2
6
79
Mathematician Mr. James King of Denver, Colorado.
Advances in Microwave Metamaterials
James A. Wigle
Chapter 4.
Quadrant
II,
10.5 GHz Metamaterial
|Page|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,
FR-4 material was used11. This material has a εr of 4.34 at
Figure 22. Proto Circuit Board Machine
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
|Page|53
Table 1. Quadrant II Metamaterial Design Parameters
Designed:
Manufactured:
a
(mesh period in mm)
7.20
7.39 ±0.1
r
(wire diameter in mm)
0.60
0.44 ±0.051
Calculated ωp
(plasma freq. in GHz)
10.54
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).
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James A. Wigle
Chapter 4.
Quadrant
II,
10.5 GHz Metamaterial
|Page|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
|Page|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, S 11 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
|Page|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
|Page|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
T
hough most laws of physics are reversible with respect to time, they do not create nor
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
|Page|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 sheets 16 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
Rogers 4003 substrate, on much larger boards.
Lexan
This
should afford much better trace tolerances, known and
lower permittivity at our frequency of interest, larger
Copper „S‟ or inverted „S‟
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
16
FR-4, double sided 0.5 ounce copper, µr = 1.0, εr = 4.34 at 1 GHz.
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
|Page|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
|Page|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
|Page|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
T
hough this may seem intuitive, I never found a single reference to metamaterial reciprocity,
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 nonpropagating 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
|Page|64
Metamaterial
S21
S12
Port 1
Vector
Network
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.
2
Where: R = Range from antenna phase center.
Advances in Microwave Metamaterials
James A. Wigle
Chapter 6.
Metamaterial Reciprocity for Quadrant
II
|Page|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.
Advances in Microwave Metamaterials
James A. Wigle
Chapter 6.
Metamaterial Reciprocity for Quadrant
II
|Page|66
Figure 29. Metamaterial Reciprocity, same polarization
Advances in Microwave Metamaterials
James A. Wigle
Chapter 6.
Metamaterial Reciprocity for Quadrant
II
|Page|67
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
A
n antenna was required to drive and test the wire mesh metamaterial close to, and on either
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
|
Page|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, ε r = 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.
Advances in Microwave Metamaterials
Therefore, to obtain the objective 50 Ω
James A. Wigle
Chapter 7.
Patch
Antenna
|
Page|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
( ⁄)
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.
Advances in Microwave Metamaterials
James A. Wigle
Chapter 7.
Patch
Antenna
|
Page|71
One side of PC board
ϕ
≥3λ
l
ρ
l
≥3 λ
Figure 31. Patch Antenna Schematic
Advances in Microwave Metamaterials
James A. Wigle
Chapter 7.
Patch
Antenna
|
Page|72
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
|
Page|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 S 11
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. S 11
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
Resonant Frequency
Zo
S11
Frequency Bandwidth
Gain
Polarization Isolation
Value
11.01 GHz
50.10 Ω
-53.1 dB
8.56%
5.9 dBi
21.9 dB
Comments
At 11.01 GHz
At 11.01 GHz
At 11.0 GHz
At 11.0 GHz
It should be understood that the antenna actually radiates during the S 11 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
|
Page|74
Figure 36. 11 GHz Patch, S11
Advances in Microwave Metamaterials
James A. Wigle
Chapter 7.
Patch
Antenna
|
Page|75
Figure 37. 11 GHz Patch, Impedance
Advances in Microwave Metamaterials
James A. Wigle
Chapter 7.
Patch
Antenna
|
Page|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].
2
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.
150
160
Patch
-170
170 0
Antenna
-160
-5
140
-140
-130
-15
120
Page|77
-150
-10
130
|
-120
-20
110
-110
-25
100
-100
-30
90
-90
-35
-80
80
-70
70
-60
60
-50
50
-40
40
30
20
10
-10
0
-20
-30
deg
E-Plane
H-Plane
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
|
Page|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
H
euristically, directivity improvement will be shown, when employing the wire mesh
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
Figure 39. Metamaterial with Patch
Directivity
|
Page|80
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
|
Page|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.
between frequencies.
Improved
Directivity
|
For lucidity, Figure 43 graphs only the best three frequencies.
Page|82
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
|
Page|83
1.2
Meta-Antenna, 9.8-10.9 GHz Freq's
1
9.8 GHz
9.9 GHz
0.8
10.0 GHz
10.1 GHz
10.2 GHz
0.6
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
|
Page|84
1.2
Meta-Antenna, 11.0-12.0 GHz Freq's
1
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
Advances in Microwave Metamaterials
James A. Wigle
Chapter 8.
Improved
Directivity
1.2
|
Page|85
1.2
Metamaterial Improvement for 11 GHz Patch
Antenna
Meta-Antenna Radiation Pattern Response
1
1
10.2 GHz
10.8 GHz
0.8
0.8
11.7 GHz
0.6
0.6
0.4
0.4
0.2
-10
MetaAntenna at
10.8 GHz
0.2
0
-30
Patch
Antenna
11.0 GHz
0
deg
10
Figure 43. Normalized Meta-Ant Best 3
30
-40
-20
0 deg
20
40
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
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Page|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
U
sing Sir Pendry‟s thin wire model[33], a metamaterial was created to perform within the
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.
Baseline Setup:
Improved
Gain
C2
Page|88
TP4
TP
Meta-Patch Gain Test
PP1
|
C6
C3
C4
PP2
C4
PP2
Microwave Anechoic Chamber
C6 only used for baseline setup
Loss1
Loss2
SA
C1
Cn
= Cables
PPn = Patch Panels
TPn = Test Points
Lossn = Segment Losses
TP2
C5
TP5
Figure 45. Meta-Patch Gain Baseline Test Schematic
Test Setup:
Meta-Patch Gain Test
PP1
Gt
Gr
C2
C3
Microwave Anechoic Chamber
C1
Cn
= Cables
PPn
= Patch
TPn = Test Points
Panels
SA
C5
TP1
Figure 46. Meta-Patch Gain Test Schematic
Advances in Microwave Metamaterials
James A. Wigle
Chapter 9.
Improved
Gain
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Page|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].
2
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
|
Page|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
Advances in Microwave Metamaterials
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
---
James A. Wigle
Chapter 9.
Improved
Gain
|
Page|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
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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
|
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+
√
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
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1.2
Meta-Patch at 11.45 GHz
1
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
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James A. Wigle
Chapter 9.
Improved
Gain
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11 GHz Patch, E-Plane
-178
178
-174
-170
-166
1661701740
-162
162
-158
158
-154
154
-150
150
-146
146
-5
-142
142
-138
138
-134
134
-10
-130
130
-126
126
-122
-15
122
-118
118
-114
114
-20
-110
110
-106
106
-25
-102
102
-98
98
-30
-94
94
-90
90
-35
-86
86
-82
82
-78
78
-74
74
-70
70
-66
66
-62
62
-57.9
58
-54
54
-50
50
-45.9
46
-42
42
-38
38
-34
34
-30
30
-26
26
-22
22 18
-18
-14
14 10 6
2 -2 -6 -10
deg
Figure 48. Patch Antenna, Normalized dB E-Plane Pattern at 11.45 GHz
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James A. Wigle
Chapter 9.
Improved
Gain
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HP11966E, E-Plane
-178
178
-174-170-166
1661701740
-162
162
-158
158
-154
154
-150
150
-146
146
-142
142
-10
-138
138
-134
134
-130
130
-20
-126
126
-122
122
-118
118
-30
-114
114
-110
110
-40
-106
106
-102
102
-98
-50
98
-94
94
-90
90
-60
-86
86
-82
82
-78
78
-74
74
-70
70
-66
66
-62
62
-58
58
-54
54
-49.9
50
-46
46
-42
42
-38
38
-34
34
-30
30
-26
26
-22
2218.1
-18
-14
14 10 6
2 -2 -6 -10
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
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"E34" Antenna w/Narda 601A Feed, E-Plane
-178
178
-174-170-166
1661701740
-162-158
162
158
-154
154
-150
150
-146
146
-142
142
-10
-138
138
-134
134
-130
130
-20
-126
126
-122
122
-118
118
-30
-114
114
-110
110
-40
-106
106
-102
102
-98
-50
98
-94
94
-90
90
-60
-86
86
-82
82
-78
78
-74
74
-70
70
-66
66
-62
62
-58
58
-54
54
-49.9
50
-45.9
46
-42
42
-38
38
-34
34
-30
30
-26
26
-22
22 18
-14 -18
14 10 6
2 -2 -6 -10
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
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Page|98
Table 4. Meta-Patch Gain Test Results
Antenna
Description
HP11966E, bi-ridge
Meta-Patch
Patch (no metamaterial)
E34, flared waveguide horn
Gain
Gain
(linear)
7.669
2.572
0.927
64.976
(dBi)
8.85
4.10
-0.33
18.13
Expected
Value
(dBi)
10-15
2-10
1-4
12-17
Manufacture
Specifications
(dBi)
≈12.122
N/A
N/A
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 antenna 23. 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].
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.
23
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James A. Wigle
Chapter 9.
Improved
Gain
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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
G
iven material already presented within the Metamaterial Reciprocity for Quadrant II
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
|
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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:
Gt/r
C1
PP1
Gr/t
C2
Microwave
Chamber
Cn
= Cables
PPn = Patch Panels
PP2
C3
Anechoic
SA
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].
2
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James A. Wigle
Chapter 10.
Meta-Antenna Reciprocity
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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.
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James A. Wigle
Chapter 10.
Meta-Antenna Reciprocity
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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
MetaAnt Receive, E-Plane,
Normalized Power (dB)
MetaAnt Transmit, E-Plane,
Normalized Power (dB)
-20
-25
Figure 52. Meta-Antenna Reciprocity, Normalized dB E-Plane
Advances in Microwave Metamaterials
James A. Wigle
Chapter 10.
Meta-Antenna Reciprocity
|
Meta-Antenna Reciprocity, H-Plane
Page|104
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,
Normalized Power (dB)
-30
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
W
hilst contemplating antenna enhancement using metamaterials, I pondered making a
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
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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
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Meta-Lens Pattern Test Setup:
Gt
C1
PP1
Gr
C2
C3
PP2
Microwave Anechoic Chamber
Cn
PPn
= Cables
= Patch Panels
SA
C5
C4
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
|
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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
-90
-84
-78
-72
-66
-60
-54
-48
-42
-36
-30
-24
-18
-12
-6
0
6
12
18
24.1
30
36
42
48
54
60.1
66
72
78
84
90
0
deg
Figure 56. Double Beam Meta-Lens, 10.83 GHz
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James A. Wigle
Chapter 11.
Multifaceted MetaLens
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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
-90
-84
-78
-72
-66
-60
-54
-48
-42
-36
-30
-24
-18
-12
-6
0
6
12
18
24
30
36
42
48
54
60
66
72
78
84
90
0
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.
performing as expected.
Multifaceted MetaLens
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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
Baseline Setup:
PP2
C2
Page|111
TP
TP3
Meta-Lens Gain Test
|
C7
C3
C4
PP1
Microwave Anechoic Chamber
C7 only used for baseline setup
Loss1
Loss2
SA
C1
C5
C6
Cn
= Cables
PPn = Patch Panels
TPn = Test Points
Lossn = Segment Losses
TP5
TP2
Figure 58. Meta-Lens Gain Test, Baseline Test Setup
Test Setup:
Gt
Meta-Lens Gain Test
PP2
Gr
C2
C3
C4
PP1
Microwave Anechoic Chamber
Cn
PPn
C1
= Cables
= Patch Panels
SA
C
C5
TP1
Figure 59. Meta-Lens Gain Test Setup
Advances in Microwave Metamaterials
James A. Wigle
Chapter 11.
Multifaceted MetaLens
(
|
Page|112
)
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.
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].
2
Where: R = Range from antenna phase center.
D = Longest physical dimension of antenna.
λ = Operational wavelength.
11.3.2 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
Advances in Microwave Metamaterials
James A. Wigle
Chapter 11.
Multifaceted MetaLens
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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).
Advances in Microwave Metamaterials
James A. Wigle
Chapter 11.
Multifaceted MetaLens
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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
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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 dB i ; very close indeed.
Table 5. Meta-Lens Gain Results
Antenna
Description
HP11966E
E34
Patch (no lens)
Meta-Lens at +49°
Meta-Lens at -48°
Gain
Gain
(linear)
13.603
37.123
1.200
2.503
1.748
(dBi)
11.34
15.70
0.78
3.99
2.43
Expected
Value
(dBi)
10-15
12-17
1-4
2-10
2-10
Manufacture
Specifications
(dBi)
12.0425
N/A
N/A
N/A
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].
Advances in Microwave Metamaterials
James A. Wigle
Chapter 11.
Multifaceted MetaLens
|
Page|116
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
Meta-Property Test
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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
Again, I thank the great Dr. Gozhenko for
the use of his work. Below describes the geometry of
n1
n2
the test scenario, and provides the critical equations
for the indices of refraction, both for negative and
positive indices.
n1
Δ
Note the solution works toward
measurable parameters, such as α and d, instead of β
δ-
δ+
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
Meta-Property Test
|
Page|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
n1
( )
A
α
|
|
( )
|
|
|
|
|
n2
γ
E
β
|
( )
d
n1
B
C
δ+
( )
Figure 62. n > 0 Ray Trace
Advances in Microwave Metamaterials
James A. Wigle
Chapter 12.
(
( )
,
6
[
Via Snell‟s law,
( )
)
( )
( )
( )
( )
( )
Via a well-known trigonometric identity,
Meta-Property Test
( )
( )
Page|120
( )
( )
( )-
7
( )
6
( )
( )
7
( )
( )
√
( )
( )
( )
( )
( )
√
( )
√
( )
]
( ) , the solution now becomes
( )
(
)
√
( )
[
/ √
.
( )
( )
.
/
.
/
( )
]
( )
√
√.
( )
]
( )
√
( )
[
( )
√
( )
[
Advances in Microwave Metamaterials
|
)
(
Via a well-known trigonometric identity,
( )
IR
/
( )
]
James A. Wigle
Chapter 12.
IR
Meta-Property Test
√.
[
Page|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 measureable 26. 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
|
|
n2
|
|
( )
|
|
|
E
( )
|
|
d
α
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.
6
[
Via Snell‟s law,
( )
Meta-Property Test
( )
( )
Via a well-known trigonometric identity,
IR
( )
( )
( )
( )
( )
6
( )
Page|122
( )
7
( )
( )
√
( )
7
|
√
( )
√
( )
]
( ) , 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.
12.2.3 Solutions for n, r,
r
IR
Meta-Property Test
|
Page|123
, 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
(
)
( )
6
(
)
( )
√
Note that the displacement,
( )
)
( )
)
{
( )
( )
6
( )
( )
( )
( )
( )
( )
( )
7
( )
( )
( )
7
( )
( )
6
( )
( )
( )
7 }
( )
( )
7
( )
, 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
Meta-Property Test
|
Page|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
(
)
( )
6
(
)
( )
√
( )
)
( )
)
Advances in Microwave Metamaterials
( )
/
( )- √(
{
( )
( )
6
( )
( )
( )
( )
( )
( )
( )
7
( )
( )
( )
7
( )
6
( )
( )
( )
( )
7 }
( )
( )
7
( )
James A. Wigle
Chapter 12.
Note that the displacement,
IR
Meta-Property Test
|
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, 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
provide information leading to the value of the loss tangent, δloss [46, 50].
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
Meta-Property Test
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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.
Test Setup:
IR
Meta-Property Test
Material under test,
which can rotate.
IR Material Properties Test
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Page|127
Carbon loaded Neoprene, or
other absorber such as
carbon paper with known
resistivity
Gt
α
n>0
n=0
PP1
C3
IR camera
TP1
C
n<0
D
Anechoic
Chamber
C2
Cn
= Cables
PPn = Patch Panels
TPn = Test Points
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
Meta-Property Test
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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
Meta-Property Test
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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
Advances in Microwave Metamaterials
James A. Wigle
Chapter 12.
IR
Figure 70. 10.8 GHz Material Displacement w/Meta
Meta-Property Test
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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
Meta-Property Test
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Table 6. Metamaterial Properties for IR Test
Operational
Frequency
(GHz)
17.0
10.8
n
µr
1.4
2.1
1.0
+0.9
or
-0.9
+0.8
or
-0.8
+1.0
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
Line Generator
Vertical
Line
Laser
Figure 72. Optical Test Schematic
Advances in Microwave Metamaterials
James A. Wigle
Chapter 12.
IR
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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.
µWave Meta-Property Test
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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 &
receiver in
far field
Gt
Material under test,
which can rotate.
α
Gr
n>0
n=0
PP1
C2
Anechoic Chamber
Cn
PPn
C1
n<0
D
= Cables
= Patch Panels
C3
PP
C4
Port1
Vector
Network
Analyzer
(S21)
Port2
Figure 73. Microwave Material Properties Test Schematic
Advances in Microwave Metamaterials
James A. Wigle
Chapter 13.
µWave Meta-Property Test
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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
G
iven that electromagnetic material parameters can take on any sign, Dr. Pinchuk and I
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
|
Page|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
1
N1 > 0
N2 > 0
N1 < N2
Known
2
N1 > 0
N2 > 0
N1 > N2
Known
3
N1 > 0
N2 > 0
N1 = N2
Known
4
N1 > 0
N2 < 0
N1 > N2
5
N1 > 0
N2 = 0
N1 > N2
6
N1 = 0
N2 > 0
N1 < N2
7
N1 = 0
N2 < 0
N1 > N2
8
N1 < 0
N2 < 0
N1 < N2
9
N1 < 0
N2 < 0
N1 > N2
10
N1 < 0
N2 < 0
N1 < N2
11
N1 < 0
N2 < 0
N1 = N2
12
N1 < 0
N2 = 0
N1 < N2
13
N1 = 0
N2 = 0
N1 = N2
Recently
Described [31]
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Unknown
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Unknown
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Unknown
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Unknown
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Unknown
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Unknown
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Unknown
Advances in Microwave Metamaterials
⃑
⃑
⃑
⃑
2
Γ
2
Γ
2
Γ
2
Γ
TBD
TBD
TBD
TBD
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TBD
TBD
James A. Wigle
Chapter 14.
14.3 ⃑
⃑⃑⃑ rh and ⃑
Fresnel Coefficient Matrix
|
Page|138
⃑⃑ 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.
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|139
⃑ = 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.
⃑
̂
⃑⃑
̂
⃑
̂
̂
̂
̂
̂
̂
̂
̂
̂
̂
z
Ez
Ho
Eo
k
Ey
Ex
y
X
Figure 74. Vector Orientations
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
14.3.1
Fresnel Coefficient Matrix
|
Page|140
⃑⃑ rh
Since
⃑⃑
⃑⃑
⃑
⃑
(no time or position dependence, no position dependence  isotropic medium)
Where:
So,
⃑⃑
⃑⃑
⃑⃑
⃑⃑
⃑
(where ⃑
⃑
⃑
⃑ ⃑
⃑
⃑ ⃑
⃑ ⃑
2⃑
*
+
⃑
3
⃑ ⃑
(
)
]
̂6
⃑
)
Also,
̂
̂
̂
⃑⃑
||
||
⃑⃑
̂6
7
̂[
7
Where:
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
⃑⃑
|
Page|141
⃑ ⃑
⃑⃑
⃑⃑
̂
⃑
Fresnel Coefficient Matrix
̂
̂
̂
̂
̂
̂
̂
̂
And,
⃑
So,
⃑⃑
⃑⃑
⃑⃑
,
[
̂
]
̂
̂-
Thus,
(⃑ ⃑
)
(⃑ ⃑
)
(⃑ ⃑
)
Therefore:
Advances in Microwave Metamaterials
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|142
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
Putting it altogether yields:
⃑⃑
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
⃑⃑
⃑⃑
{ ̂[
Advances in Microwave Metamaterials
]
̂,
̂
(⃑ ⃑
)
[
]
̂
(⃑ ⃑
),
-
̂
(⃑ ⃑
)
]
[
-
̂[
]}
(⃑ ⃑
)
James A. Wigle
Chapter 14.
14.3.2 ⃑
Fresnel Coefficient Matrix
|
Page|143
⃑⃑⃑ rh
̂
⃑
⃑
|
⃑
⃑
̂[
̂
̂
|
]
-
̂,
̂,
-
Recall from previous sub-subsection:
(⃑ ⃑
)
(⃑ ⃑
)
(⃑ ⃑
)
This gives:
̂
̂
⃑⃑
⃑⃑
{ ̂[
Now, comparing solutions for ̂
Advances in Microwave Metamaterials
]
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
-
̂,
⃑⃑
and
⃑⃑
̂[
]}
(⃑ ⃑
)
yields:
James A. Wigle
Chapter 14.
(̂
⃑⃑
|
Page|144
⃑⃑
)
(̂
⃑⃑
Fresnel Coefficient Matrix
⃑⃑
)
From the previous sub-subsection:
⃑⃑
⃑
(where ⃑
⃑ ⃑
⃑
)
Therefore:
⃑
⃑⃑
14.3.3
⃑
(where ⃑
⃑
⃑ ⃑
)
⃑ rh
Faraday‟s point form:
⃑
⃑
Since,
⃑
Where:
⃑⃑
⃑
⃑⃑
⃑
⃑⃑
(no time or position dependence, no position dependence  isotropic medium)
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|145
So,
⃑
⃑
⃑⃑
⃑⃑
⃑
⃑
⃑
⃑⃑
2⃑⃑
⃑ ⃑
⃑⃑
⃑⃑
⃑ ⃑
̂
̂
(where ⃑⃑
*
+
⃑ ⃑
3
⃑ ⃑
⃑⃑
(
⃑⃑
)
)
Also,
̂
⃑
||
||
⃑
̂6
7
⃑
⃑ ⃑
̂[
]
̂6
7
Where:
⃑
⃑
̂
⃑
̂
̂
Advances in Microwave Metamaterials
̂
̂
̂
̂
̂
̂
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|146
And,
⃑
So,
⃑
⃑
⃑
,
[
̂
]
̂
̂-
Thus,
⃑
(⃑ ⃑
)
⃑
(⃑ ⃑
)
⃑
(⃑ ⃑
)
Therefore:
Advances in Microwave Metamaterials
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
James A. Wigle
Chapter 14.
(
Fresnel Coefficient Matrix
|
Page|147
(⃑ ⃑
)
)
Putting it altogether yields:
⃑
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
⃑
⃑
{ ̂[
14.3.4 ⃑
]
̂
(⃑ ⃑
)
[
]
̂
(⃑ ⃑
),
-
̂
(⃑ ⃑
)
]
[
-
̂,
̂[
]}
(⃑ ⃑
)
⃑ rh
̂
⃑
⃑
|
⃑
⃑
̂[
Advances in Microwave Metamaterials
̂
̂
|
]
̂,
-
̂,
-
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|148
Recall from previous sub-subsection:
(⃑ ⃑
)
(⃑ ⃑
)
(⃑ ⃑
)
This gives:
⃑
⃑
⃑
⃑
{ ̂[
]
Now, comparing solutions for ⃑
(⃑
⃑
(⃑ ⃑
)
(⃑ ⃑
)
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
-
̂,
⃑
⃑
̂0
⃑
and
̂[
]}
1
(⃑ ⃑
)
yields:
⃑
)
(⃑
⃑
)
From the previous sub-subsection:
⃑
Advances in Microwave Metamaterials
⃑⃑
(where ⃑⃑
⃑⃑
⃑ ⃑
)
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|149
Therefore:
⃑
⃑
14.4 ⃑
⃑⃑
(where ⃑⃑
⃑⃑⃑ 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.
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|150
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.
⃑
̂
̂
̂
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.
⃑
̂
⃑⃑
̂
⃑
̂
̂
̂
̂
Advances in Microwave Metamaterials
̂
̂
̂
̂
̂
̂
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Once again, this does not assume any specific component directions for ⃑ and ⃑⃑
Page|151
, even if it is a
left handed material.
⃑⃑ lh
14.4.1
Since
⃑⃑
⃑⃑
⃑
⃑
(no time or position dependence, no position dependence  isotropic medium)
Where:
= Permittivity of free space, 8.854 x 10 -12 F/m.
So, using Ampere‟s point form:
⃑⃑
⃑⃑
⃑⃑
⃑
(where ⃑
⃑⃑
⃑
⃑
⃑ ⃑
⃑
⃑ ⃑
⃑ ⃑
2⃑
*
+
⃑
⃑ ⃑
(
3
)
⃑
)
Note, this is the same for right handed materials.
Now solving another way:
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
̂
Fresnel Coefficient Matrix
̂
|
Page|152
̂
⃑⃑
||
||
⃑⃑
̂6
7
̂[
]
̂6
7
Where, from before:
⃑⃑
⃑ ⃑
⃑⃑
⃑⃑
̂
⃑
̂
̂
̂
̂
̂
̂
̂
̂
And,
⃑
So,
⃑⃑
⃑⃑
⃑⃑
,
[
̂
]
̂
̂-
Thus,
Advances in Microwave Metamaterials
(⃑ ⃑
)
(⃑ ⃑
)
(⃑ ⃑
)
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|153
Therefore:
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
Putting it altogether yields:
⃑⃑
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
⃑⃑
Advances in Microwave Metamaterials
̂
(⃑ ⃑
)
[
]
̂
(⃑ ⃑
),
-
̂
(⃑ ⃑
)
]
[
James A. Wigle
Chapter 14.
⃑⃑
{ ̂[
Note that
14.4.2 ⃑
]
⃑⃑
Fresnel Coefficient Matrix
-
̂,
̂[
]}
|
Page|154
(⃑ ⃑
)
⃑⃑
⃑⃑⃑ lh
̂
⃑
⃑⃑
|
⃑
⃑⃑
̂[
̂
̂
|
]
̂,
-
̂,
-
Recall from previous sub-subsection:
(⃑ ⃑
)
(⃑ ⃑
)
(⃑ ⃑
)
This gives:
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
⃑
⃑
⃑⃑
]
⃑
⃑⃑
(̂
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
]}
(⃑ ⃑
-
̂,
⃑⃑
⃑⃑
and
⃑⃑ )
̂[
yields:
⃑⃑ )
As expected, this is the same for right handed materials, since ⃑⃑
⃑
⃑⃑
)
⃑⃑
(̂
⃑⃑
⃑⃑
Page|155
⃑⃑
Now, comparing solutions for ⃑
and ⃑
|
⃑⃑
{ ̂[
Note that ⃑
Fresnel Coefficient Matrix
. Since
(̂
⃑⃑
⃑⃑
⃑⃑
and
⃑⃑
⃑⃑
) , as described previously.
From the previous sub-subsection:
⃑⃑
⃑⃑
Advances in Microwave Metamaterials
⃑
(̂
⃑⃑ )
(where ⃑
⃑
⃑ ⃑
)
⃑
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|156
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,
⃑
⃑
⃑⃑
⃑⃑
⃑
⃑
Advances in Microwave Metamaterials
⃑⃑
2⃑⃑
⃑ ⃑
*
+
⃑ ⃑
3
⃑⃑
⃑ ⃑
(
)
⃑⃑
James A. Wigle
Chapter 14.
⃑
Note that
⃑⃑
⃑
(where ⃑⃑
⃑
⃑ ⃑
⃑⃑
and ⃑⃑
Fresnel Coefficient Matrix
|
Page|157
)
⃑⃑
.
Solving another way:
̂
̂
̂
⃑
||
||
⃑
̂6
7
̂[
]
̂6
7
Where:
⃑
⃑ ⃑
⃑
⃑
̂
⃑
̂
̂
̂
̂
̂
̂
̂
̂
And,
⃑
So,
⃑
Advances in Microwave Metamaterials
⃑
[
]
James A. Wigle
Chapter 14.
⃑
,
̂
Fresnel Coefficient Matrix
̂
|
Page|158
̂-
Thus,
⃑
(⃑ ⃑
)
⃑
(⃑ ⃑
)
⃑
(⃑ ⃑
)
Therefore:
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
(
)
(⃑ ⃑
)
Putting it altogether yields:
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|159
⃑
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
⃑
⃑
{ ̂[
Note that
14.4.4 ⃑
]
⃑
⃑
̂
(⃑ ⃑
)
[
]
̂
(⃑ ⃑
),
-
̂
(⃑ ⃑
)
]
[
-
̂,
̂[
]}
(⃑ ⃑
)
.
⃑ lh
̂
⃑
⃑
|
⃑
⃑
̂[
̂
̂
|
]
̂,
-
̂,
-
Recall from previous sub-subsection:
(⃑ ⃑
Advances in Microwave Metamaterials
)
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
(⃑ ⃑
)
(⃑ ⃑
)
|
Page|160
This gives:
⃑
⃑
⃑
Note that ⃑
⃑
{ ̂[
⃑
]
⃑
⃑
̂0
(⃑ ⃑
)
(⃑ ⃑
)
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
̂0
(⃑ ⃑
)
(⃑ ⃑
)
1
-
̂,
̂[
]}
1
(⃑ ⃑
)
.
Now, comparing solutions for ⃑
(⃑
⃑
⃑ )
⃑
⃑
and
yields:
⃑
(⃑
⃑ )
Note that this also holds true for right handed materials. Now, from the previous sub-subsection:
⃑
Advances in Microwave Metamaterials
⃑⃑
(where ⃑⃑
⃑⃑
⃑ ⃑
)
James A. Wigle
Chapter 14.
(⃑
⃑ )
Fresnel Coefficient Matrix
⃑
|
Page|161
⃑⃑
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
, and transmission,
, coefficients for E and H. This method will provide the basis for deriving the left handed cases.
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|162
Matrix Scenarios 1 to 3 (all RHM)
n1
Є1
µ1
Er
Kr
n2
Є2
µ2
Hr
Et
Ei
Kt
n1 < n2
n1 > n2
n1 = n2
Ki
Ht
Hi
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:
⃑
Advances in Microwave Metamaterials
(⃑ ⃑
)
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
⃑
(⃑ ⃑
)
⃑
(⃑ ⃑
)
|
Page|163
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.
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|164
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)
(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.
2
= Free space wavelength.
Given:
⃑⃑
Advances in Microwave Metamaterials
⃑⃑
⃑⃑
⃑
⃑
⃑⃑
⃑⃑
⃑⃑
⃑ ⃑
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
⃑
⃑
|
Page|165
we then obtain:
⃑
⃑
⃑
⃑
⃑
(⃑
⃑
⃑ ]
⃑
(⃑
⃑ )
(⃑
[
⃑ )
⃑
⃑
⃑ )
Again:
⃑
⃑
⃑⃑
⃑⃑
⃑⃑
(
⃑⃑ )
⃑ ⃑
So we have:
⃑⃑ )
(
(⃑⃑
⃑⃑
⃑⃑
⃑⃑
⃑⃑ )
⃑⃑
⃑⃑
Recall ⃑⃑
(
⃑⃑ )
⃑⃑
⃑⃑
⃑⃑
⃑⃑
, so combining these two equations yields:
⃑⃑
⃑⃑
⃑⃑
2⃑⃑
Advances in Microwave Metamaterials
[⃑⃑
⃑⃑ ]
⃑⃑
James A. Wigle
Chapter 14.
⃑⃑
⃑⃑
2⃑⃑
2
⃑⃑
2
⃑⃑
⃑⃑
Fresnel Coefficient Matrix
⃑⃑ [
|
Page|166
]
⃑⃑
⃑⃑
2
⃑⃑
14.7
Hrh
Recall from the previous subsection:
⃑⃑
⃑⃑
⃑⃑
⃑⃑
⃑⃑
⃑⃑
Combining these two equations yields:
⃑⃑
⃑⃑
[⃑⃑
⃑⃑
⃑⃑
⃑⃑
⃑⃑
⃑⃑ [
Advances in Microwave Metamaterials
⃑⃑
⃑⃑ [
]
[
⃑⃑ ]
⃑⃑
⃑⃑ [
]
⃑⃑
]
] ⃑⃑
James A. Wigle
Chapter 14.
⃑⃑
0
1
0
1
⃑⃑
Fresnel Coefficient Matrix
|
Page|167
⃑⃑
⃑⃑
(
)
⃑⃑
⃑⃑
⃑⃑
⃑⃑
14.8
Hrh
and
Hrh
Relationship
From previous subsections:
2
2
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
14.9
Hrh
and
Hrh
Fresnel Coefficient Matrix
|
Page|168
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
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:
2
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
|
Page|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 (
14.10
2).
Erh
As derived previously:
⃑
⃑
⃑⃑
⃑⃑
(⃑
⃑⃑
⃑
⃑⃑
⃑ ⃑
)
It follows that:
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
⃑⃑
(⃑
⃑ )
⃑⃑
(⃑
⃑ )
⃑⃑
(⃑
⃑ )
|
Page|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
|
Page|171
2
= 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.
]⃑
[
⃑
1
0
1
|
Page|172
]⃑
[
0
⃑
Fresnel Coefficient Matrix
⃑
⃑
⃑
⃑
14.11
Erh
Now use ⃑
⃑
⃑
⃑
⃑
⃑ , along with ⃑
⃑
⃑
2⃑
⃑
found in the previous
subsection. This all yields:
⃑
[⃑
⃑ ]
⃑
⃑
2
⃑
⃑
2
⃑
⃑
⃑
[
⃑
]⃑
⃑
⃑
⃑
2⃑
⃑
⃑
(
2
(
)
)
2
⃑
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
14.12
Erh
and
Erh
Fresnel Coefficient Matrix
|
Page|173
Relationship
From previous subsections:
2
2
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.
0
1
0
1
Fresnel Coefficient Matrix
|
Page|174
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:
2
Now I shall take the limits as the various indices of refraction approach a limit.
(
)
(
)
(
)
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
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
become the same material, without interface, and provide no reflection (
|
Page|175
). 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).
14.14 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
, and transmission,
, coefficients for E and H.
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|176
Matrix Scenario 4 (RHM into LHM)
n1
Є1
µ1
Er
Kr
n2
Є2
µ2
Hr
Et
Ei
Kt
n1 < n2
n1 < 0
0 < n2
Ht
Ki
Hi
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
⃑
(⃑ ⃑
)
⃑
(⃑ ⃑
)
|
Page|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)
(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.
2
= Free space wavelength.
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|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:
⃑
⃑
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⃑⃑
⃑⃑
⃑⃑
⃑ ⃑
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|179
So we have:
⃑⃑ )
(
(⃑⃑
⃑⃑
⃑⃑ )
⃑⃑
⃑⃑
⃑⃑ )
(
⃑⃑
⃑⃑
⃑⃑
⃑⃑
Recall ⃑⃑
⃑⃑ )
(
⃑⃑
⃑⃑
, so combining these two equations yields:
⃑⃑
⃑⃑
⃑⃑
2⃑⃑
[⃑⃑
⃑⃑
⃑⃑
⃑⃑
2⃑⃑
2
⃑⃑
2
⃑⃑
⃑⃑
⃑⃑ ]
⃑⃑ [
]
⃑⃑
⃑⃑
(
)
2
⃑⃑
Note the sign difference from the RHM definition.
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
14.16
Fresnel Coefficient Matrix
|
Page|180
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.
14.17
Hlh
and
Hlh
Fresnel Coefficient Matrix
|
Page|181
Relationship
From previous subsections:
2
2
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:
Advances in Microwave Metamaterials
0
1
0
1
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|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:
2
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
|
Page|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
2).
continues to plague humanity and us (
14.19
Elh
As derived previously:
⃑
⃑
⃑⃑
⃑⃑
⃑⃑
(⃑
⃑ )
⃑⃑
(⃑
⃑ )
⃑⃑
(⃑
⃑ )
⃑⃑
⃑ ⃑
It follows that:
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
(⃑
⃑⃑
Fresnel Coefficient Matrix
|
Page|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.
2
= Free space wavelength.
Advances in Microwave Metamaterials
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|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.
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 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
|
Page|186
Combining the last two lines yields:
⃑
⃑
(⃑
⃑
⃑
[
]⃑
⃑
⃑
1
0
1
⃑
⃑
⃑
]⃑
[
0
⃑
⃑ )
⃑
⃑
⃑
⃑
Note the sign difference from the RHM definition.
14.20
Elh
Now use ⃑
⃑
⃑
⃑
⃑
⃑ , along with ⃑
⃑
⃑
found in the
previous subsection. This all yields:
⃑
⃑
Advances in Microwave Metamaterials
[⃑
⃑ ]
⃑
⃑
[
⃑
]⃑
⃑
⃑
2⃑
⃑
2⃑
James A. Wigle
Chapter 14.
2
⃑
2
⃑
⃑
Fresnel Coefficient Matrix
Page|187
⃑
⃑
(
)
2
⃑
⃑
|
2
⃑
Note the sign difference from the RHM definition.
14.21
Elh
and
Elh
Relationship
From previous subsections:
2
2
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James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|188
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
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
|
Page|189
2
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
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 (
).
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 (
Advances in Microwave Metamaterials
), just like in the right
James A. Wigle
Chapter 14.
Fresnel Coefficient Matrix
|
Page|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 (
Advances in Microwave Metamaterials
2).
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
T
he author has a plethora of questions regarding EM characteristics when propagating
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
|
Page|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
Reference List
1.
Bukva, Erica. Navy SBIR 2007.3 – Topic N07-184, Metamaterial-Based Electrically Small
Antenna. 20 August 2007, http://www.navysibr.com/n07_3/n073-184.htm.
2.
Johnson, R. Collin. ‘Metamaterial’ holds promise for antennas, optics. EE Times, 30 April 2001,
http://www.eetimes.com/story/OEG20010430S0110.
3.
Duke
University.
Electromagnetic
Metamaterials.
http://people.ee.duke.edu/~drsmith/about_metamaterials.html.
4.
Author
unknown.
Metamaterial.
Wikipedia,
last
modified
22
January
2009,
http://en.wikipedia.org/wiki/Metamaterial.
5.
Author unknown. New metamaterials that bend light backwards bring invisibility cloaks 1 step
closer.
e!
Science
News,
11
August
2008,
http://esciencenews.com/articles/2008/08/11/new.metamaterials.bend.light.backwards.bring….
6.
Savage, Neil. Metamaterials Breakthrough Brings Invisibility Closer. IEEE Spectrum Online for
Tech Insiders, August 2008, http://www.spectrum.ieee.org/aug08/6564.
7.
Smith David R. Negative Index Metamaterials. University of California at San Diego, 30 January
2009, http://physics.ucsd.edu/~drs/left_home.htm.
8.
Author unknown.
Conform
to
Meta-materials Mimic Ice and Illuminate Why Water-ice Doesn’t Fully
Third
Law
of
Thermodynamics.
Science
Daily,
8
August
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http://www.sciencedaily.com/releases/2008/08/080807144311.htm.
9.
Lovat Giampiero, Burghignoli Paolo, Capolino Filippo, Jackson David R., Wilton Donald R. A
Comparative Study of Conventional and Metamaterial-Based Planar Leaky-Wave Antennas for
Directive Radiation at Broadside.
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Study on High Gain Circular Waveguide Array Antenna with
Metamaterial Structure. Progress in Electromagnetics Research, PIER 60, 207-219, 2006.
11. Khodja Mohamed R. and Marengo Edwin A. Comparative Study of Radiation Enhancement Due
to Metamaterials. Department of Electrical and Computer Engineering, Northeastern University,
Boston Massachusetts.
12. Stollberg, Robert and Hill, Faith Fitch. Physics Fundamentals and Frontiers. Houghton Mifflin
Company ©1975. Pages 426 to 453.
13. Marion, Jerry B., Hornyak William F. Physics for Science and Engineering, Part 2. Saundedrs
College Publishing ©1982. Pages 1066 to 1088.
14. Depine R. A. and Lakhtakia A.
A New Condition to Identify Isotropic Dielectric-Magnetic
Materials Displaying Negative Phase Velocity.
Microwave and Optical Technology Letters,
Volume 41, pages 315-316, 2004. http://dx.doi.org/10.1002/mop.20127.
15. Lindell I. V., Tretyakov S. A., Nikoskinen K. I., Ilvonen S. BW Media – Media with Negative
Parameters, Capable of Supporting Backward Waves. Microwave Optical Technology Letters,
Volume 31, Number 2, pages 129-133, 2001.
16. Padilla Willie J. et. al. Solid-State Metamaterial Device Tames Terahertz Frequency. Science
Daily, 3 March 2009, http://www.sciencedaily.com/releases/2009/02/090224133152.htm.
17. Hapgood, Fred.
How to Make Anything; From Invisibility Cloaks to Optical Computers,
Engineers are turning the Laws of Physics Upside Down. Discovery Magazine. April 2009.
Pages 32-39.
18. Pinchuk, Tolya. University of Colorado at Colorado Springs Metamaterials Lecture Series. JanMay 2009.
19. Edminister, Joseph A. Electromagnetics, 2nd Edition. Schaum‟s Outlines. McGraw-Hill, ©1993.
20. Edminister, Joseph A. and Smith William T.
Electromagnetics.
Schaum‟s Easy Outlines.
McGraw-Hill, ©2003.
21. Hayt, William H. Jr. Engineering Electromagnetics, 5th Edition. McGraw-Hill, ©1989. Pages,
138-143 and 145.
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22. Neaman, Donald A. Semiconductor Physics and Devices, Basic Principles, 3 rd Edition. McGrawHill, ©2003. Page 74.
23. University of Maryland. Clark School Research Team Introduces World’s First Invisibility Cloak.
University of Maryland School of Electrical Engineering‟s Alumni Periodical, spring 2008. Also,
http://www.isr.umd.edu/news/news_stroy.php?id=2545 .
24. Author unknown. UCSD Physicists Develop New Class of Composite Materials with “Reversed”
Physical
Properties
Never
Before
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Daily,
March
2000.
http://www.sciencedaily.com/releases/2000/03/000322091426.htm.
25. Smith David R., Schurig David, Rosenbluth Marshall, Schultz Sheldon, Ramakrishna Anantha S.,
Pendry John B. Limitations on Sub-Diffraction Imaging with a Negative Refractive Index Slab.
23 August 2001.
26. Hao, Yang and Mittra Raj. FDTD Modeling of Metamaterials, Theory and Applications. Artech
House, Inc. © 2009. Page 3.
27. Stefan Enoch, Gèrard Tayeb, Pierre Sabouroux, Nicolas Guèrin, and Patrick Vincent.
A
Metamaterial for Directive Emission. Physical Review Letters. Volume 89, Number 21. 18
November 2002.
28. Pendry, J. B. Negative Refraction Makes a Perfect Lens. Physical Review Letters. Volume 85,
Number 18. 30 October 2000.
29. Smith D. R., Padilla Willie J., Vier D. C., Nemat-Nasser S. C., Schultz S. Composite Medium
with Simultaneously Negative Permeability and Permittivity. Physical Review Letters. Volume
84, Number 18. 1 May 2000.
30. Marqués Ricardo, Martín Ferran, and Sorolla Mario. Metamaterials with Negative Parameters;
Theory, Design, and Microwave Applications. John Wiley & Sons, Inc. © 2008.
31. Ramakrishna Anantha S., Grzegorczyk Tomasz M.
Physics and Applications of Negative
Refractive Index Materials. CRC Press, Taylor & Francis Group. © 2009.
32. Veselago Victor G. The Electrodynamics of Substances with Simultaneously negative Values of ε
and
µ.
Soviet
Physics
Uspekhi
10
(4):
pages
509-514,
doi:
10.1070/PU1968v010n04ABEH003699, 1967 Russian text translated in 1968.
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33. Pendry J. B., Holden A. J., Robbins D. J., and Stewart W. J. Low Frequency Plasmons in ThinWire Structures. Journal of Physics: Condensed Matter 10 (1998), 4785-4809.
34. Schurig D. et. al. Metamaterial Electromagnetic Cloak at Microwave Frequencies. Science 314
(5801): 977, doi: 10.1126/science.1133628.
35. Antipov Sergey P. and Spentzouris Linda K.
Application to Accelerator Physics.
Left-Handed Metamaterials Studies and Their
2005 Proceedings of Particle Accelerator Conference,
Knoxville Tennessee. © 2005 IEEE.
36. Purdue
University.
New
‘broadband’
cloaking
technology
simple
to
manufacture.
http://news.uns.purdue.edu/x/2009a/090520ShalaevCloaking.html.
37. Chen H. S., Ran L. X., Huangfu J. T., Zhang X. M., and Chen K. S. Magnetic Properties of SShaped Split-Ring Resonators. Progress in Electromagnetics Research, PIER 51, pages 231-247,
2005.
38. Balanis, Constantine A. Antenna Theory, Analysis and Design, 3rd edition. John Wiley & Sons,
Inc., © 2005.
39. Stutzman, Warren L. and Thiele, Gary A. Antenna Theory and Design. John Wiley & Sons, Inc.,
© 1981.
40. Corey L., Hopkins G., Kemp J., Mitchell B., and Georgia Institute of Technology. Basic Antenna
Concepts. © 2009 by the authors of their respective sections and Georgia Institute of Technology.
41. Joy, Ed.
Far-Field, Anechoic Chamber, Compact and Near-Field Antenna Measurement
Techniques. © 2007, Ed Joy and Georgia Institute of Technology.
42. Kemp J., Hopkins G., Tripp V., Leatherwood D., Brown G., and Georgia Institute of Technology.
Modeling and Simulation of Antennas. © 2008 by the authors of their respective sections and
Georgia Institute of Technology.
43. Wigle, James. Microwave Engineering Laboratory Manual, 1st edition. © 2010 J. Wigle.
44. Daher J., Santamaria J., and Georgia Institute of Technology. EMC/EMI for Engineers and
Engineering Managers. © 2011 Georgia Institute of Technology.
45. Sebold, John S. Introduction to RF Propagation. John Wiley & Sons, Inc., © 2005.
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46. Norgard J., Metzger D., and Sega R. Analysis of Probe Measurement accuracies and Scattering
Effects. University of Colorado at Colorado Springs, © 1991.
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.
48. Inan, Umran S. and Inan, Aziz S. Electromagnetic Waves. Prentice Hall, © 2000.
49. Ulaby F., Michielssen E., and Ravaioli U. Fundamentals of Applied Electromagnetics, 6th Edition.
Prentice Hall. © 2010.
50. Norgard J. and Sega R. Microwave Fields Determined from Thermal Patterns. SPIE Milestone
Series, Volume MS 116. SPIE Optical Engineering Press, © 1995.
Advances in Microwave Metamaterials
James A. Wigle
Appendix A – Peer Reviewed Paper
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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....
#
#
################################################################################
Appendix B –Metamaterial Calculator
|
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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.
Advances in Microwave Metamaterials
James A. Wigle
Appendix B –Metamaterial Calculator
#####sub PrintUsageMessage;
|
Page|216
# 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;
#####&PrintUsageMessage;
# Perform calculations.
# 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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James A. Wigle
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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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James A. Wigle
Appendix B –Metamaterial Calculator
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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 "********************\n" if $DEBUG;
# Print only if debugging.
# Print only if
debugging.
return (1);
# Return to the calling program, indicting error free.
}
################################################################################
Advances in Microwave Metamaterials
James A. Wigle
Appendix B –Metamaterial Calculator
#
|
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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 $DEBUG;
# Print only if debugging.
### Initialize parameters and other variables used within this subroutine
#
### ....
#
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James A. Wigle
Appendix B –Metamaterial Calculator
if ($DEBUG eq 1)
|
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# 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 "********************\n" if $DEBUG;
return ();
# Print only if debugging.
# Print only if debugging.
# Return to the calling program with coil design input parameters.
}
################################################################################
#
#
#
Subroutine = GetInputsAndErrorTrap
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#
James A. Wigle
Appendix B –Metamaterial Calculator
#
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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 $DEBUG;
# Print only if debugging.
### Initialize passed parameters and other variables used within this subroutine
#
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James A. Wigle
Appendix B –Metamaterial Calculator
if ($DEBUG eq 1)
|
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# 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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#
#
#
James A. Wigle
Appendix B –Metamaterial Calculator
#
@_ = This Perl specific array variable contains all the information
|
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#
#
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)
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# Print only if debugging.
James A. Wigle
Appendix B –Metamaterial Calculator
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{
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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Appendix B –Metamaterial Calculator
#
|
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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 $DEBUG;
# Print only if debugging.
### Initialize variables used within this subroutine
#
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James A. Wigle
Appendix B –Metamaterial Calculator
if ($DEBUG eq 1)
|
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# Print only if debugging.
{
print "\n";
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
printf ("* Wire radius
= %.4f mm\n", $MeshPeriodicity);
= %.4f mm\n", $WireRadius);
printf ("* Effective electron density = %.4f electrons/m**3\n", $Neff);
printf ("* Effective electron mass
printf ("* Penetration depth
= %.4f kg\n", $Meff);
= %.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
Appendix B –Metamaterial Calculator
print "\n\n*
Leaving PrintResults subroutine.\n" if $DEBUG;
|
Page|228
# 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
Advances in Microwave Metamaterials
James A. Wigle
Appendix B –Metamaterial Calculator
|
Page|229
{
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 <Inductance in microHenrys>\n";
print "* <AWG wire gauge or actual wire diameter in inches>\n";
print "* <Desired inner coil diameter in inches>\n";
print "*\n";
print "*
Command Line Example:\n";
print "*\n";
print "* $THIS_PROGRAMS_NAME 28 .008 .375\n";
print "********************************************************\n";
print "\n";
Advances in Microwave Metamaterials
James A. Wigle
Appendix B –Metamaterial Calculator
print "\n\n*
Leaving PrintUsageMessage subroutine.\n" if $DEBUG;
|
Page|230
# 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
Advances in Microwave Metamaterials
James A. Wigle
Appendix B –Metamaterial Calculator
|
Page|231
{
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 "********************\n" if $DEBUG;
# Print only if debugging.
# 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
B
Berkeley, 8, 10, 19
Bulk Plasma Frequency, 32, 69
C
Cherenkov, 8, 13
D
Directivity, 76, 79, 80, 81, 87, 90
Doppler, 8, 14
Drude, 6, 10, 12, 13, 20, 21, 49, 50
Duke, 10, 11, 194
E
Evanescent waves, 9
F
FR-4, 48, 52, 59, 70, 77
Index
|
Page|233
Fresnel Coefficient Matrix, 136, 137, 191
Friis, 91, 113
G
Gauss, 35, 39
Gozhenko, vi, 117, 118, 119, 137
H
HFSS, 54, 60
I
Improved Directivity, 53, 79
improved gain, iv, 53, 78, 87, 116
invisibility cloak, 3, 12, 49
L
Lezec, 13, 14
Lorentz, iv, 14, 63, 64, 65, 67, 100, 104, 191
M
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
N
Nanometa, 12, 13, 23
NIST, 13, 73
Advances in Microwave Metamaterials
James A. Wigle
Index
|
Page|234
P
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
R
Reciprocity, 63, 65, 66, 67, 100, 101, 103, 104
reflection coefficient, 9, 192
S
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
U
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
Index
|
Page|235
V
Vedral, vi, 54, 55, 60, 61
Veselago, 2, 8, 13, 17, 136, 137, 196, 198
W
Wigle, 1, ii, iii, 197, 214
Advances in Microwave Metamaterials
James A. Wigle
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