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Tyler Wynkoop
Candidate
Electrical and Computer Engineering
Department
This thesis is approved, and it is acceptable in quality and form for publication:
Approved by the Thesis Committee:
Mark Gilmore, Chairperson
Edl Schamiloglu
Christos Christodoulou
DESIGN AND CHARACTERIZATION OF SPLIT-RING
RESONATOR ARRAYS IN WAVEGUIDE
by
TYLER WYNKOOP
B.S., APPLIED MATHEMATICS,
UNIVERSITY OF NEW MEXICO, 2012
THESIS
Submitted in Partial Fulfillment of the
Requirements for the Degree of
Masters of Science
Electrical Engineering
The University of New Mexico
Albuquerque, New Mexico
May 2015
iii
DEDICATION
Toiling,--rejoicing,--sorrowing,
Onward through life he goes;
Each morning sees some task begin,
Each evening sees it close
Something attempted, something done,
Has earned a night's repose.
-H. W. Longfellow, The Village Blacksmith
To my nephew, Colin Dean, and my niece, Norah McKinley
iv
ACKNOWLEDGMENTS
First and foremost, thank you to my advisor, Dr. Mark Gilmore. When I met you, I was a
wayward physics undergrad, unsure about my future. Now I am a young professional
engineer with a path to a fulfilling career. You provided the framework for this thesis and
constant insight and support.
Thank you especially to Chris Woehrle and Firas Ayoub. This thesis and MTBX
wouldn’t exist were it not for your help, always offered freely and without expectation.
You helped me learn CST, manufactured countless SRR arrays, and provided use of your
equipment to help me take data. I cannot thank you enough.
Thank you to my mentors and friends in the ECE department who loaned your
equipment, expertise, and support, including Dr. Alan Lynn, Dr. Sal Portillo, Dr. Rich
Compeau, Tiffany Hayes, Yue Zhang, Scott Betts, Tem Kuskov, Xuyuan Pan, and
Connor Dolan
Thank you to Rev. Dr. Lin Lilley, Father John Barton, and my church family at St.
Thomas of Canterbury. Your spiritual and emotional support held me up through the
most difficult times, and the countless delicious meals and good company made my time
with you unforgettable.
Thank you to my dear friend, Josh Plank, for more than words can express.
This work was supported by AFOSR MURI Grant FA9550-12-1-0489.
v
DESIGN AND CHARACTERIZATION OF SPLIT-RING
RESONATOR ARRAYS IN WAVEGUIDE
by
TYLER WYNKOOP
B.S., APPLIED MATHEMATICS,
UNIVERSITY OF NEW MEXICO, 2012
M.S., ELECTRICAL ENGINEERING
UNIVERSITY OF NEW MEXICO, 2015
ABSTRACT
Split-ring resonator (SRR) arrays are a foundational component to many metamaterial
based structures, be they high-power microwave sources, lenses, or waveguide systems.
SRR arrays remain poorly understood in some regards, but most notably in time domain
response. To study this behavior, an S-band system was simulated and constructed based
on two filter designs explored in previous work.
New SRRs were designed based on previous work, then scaled to the relevant
frequency band. Extensive simulations characterized the effect that various changes in
geometry had on SRR resonant frequency. Once properly scaled, simulations were
performed to analyze the two filter designs, a band pass and a band stop. Concurrently, a
cold test stand was constructed. Experimental and simulated results demonstrate proof of
concept for both filter designs, shed light on evanescent and double-negative signal
propagation, demonstrate SRR turn-on time, and provide a solid foundation for future
studies.
vi
CONTENTS
LIST OF FIGURES ..................................................................................... ix
Chapter 1:Introduction .................................................................................1
1.1 Motivation ..............................................................................................1
1.2 The History of Metamaterials.................................................................2
Chapter 2: Physics Review .........................................................................10
2.1 The Electromagnetics of Split-Ring Resonators ..................................10
2.1.1 Maxwell’s Equations and Permeability .........................................10
2.1.2 Scaling ............................................................................................12
2.1.3 Enhancing diamagnetic response and the SRR ..............................12
2.2 SRR Loaded Waveguides .....................................................................15
2.2.1 Constitutive Parameters and Propagation ......................................15
2.2.2 Band Stop Filter..............................................................................17
2.2.3 Band Pass Filter ..............................................................................18
Chapter 3: Split-Ring Resonator Design ...................................................20
3.1 Size constraints of the Split-Ring Resonator........................................20
vii
3.2 Inductance, Capacitance, and Calculating Resonant Frequency ..........21
3.3 Controlling SRR Resonance .................................................................24
3.4 Array Design ........................................................................................29
Chapter 4: Experimental Setup .................................................................33
4.1 Test Stand Parameters and Motivation.................................................33
4.2 Test Stand .............................................................................................35
4.3 Split-Ring Resonators in MTBX ..........................................................40
Chapter 5: Results and Data ......................................................................46
5.1 Simulation Results and Data ................................................................46
5.1.1 Basic SRR Results ..........................................................................46
5.1.2 Band Stop System ..........................................................................49
5.1.3 Band Pass System...........................................................................53
5.2 Experimental Results and Data ............................................................56
5.2.1 Band Stop System ..........................................................................56
5.2.2 Band Pass System...........................................................................61
Chapter 6: Conclusions ...............................................................................65
6.1 Discussion.............................................................................................65
viii
6.1.1 Resonant Frequency Comparisons .................................................65
6.1.2 Time Behavior ................................................................................66
6.1.3 Quality Factor and Rise Time ........................................................71
6.1.4 An Attempt at a Physical Explanation ...........................................73
6.2 Summary and Conclusion ....................................................................74
6.3 Future Work..........................................................................................76
References.....................................................................................................79
ix
LIST OF FIGURES
1.2.1
Pendry’s original SRR design (Pendry, et al. 1999) ...............................5
1.2.2
Shelby’s experimental test set up and results. (Shelby, Smith and
Schultz 2001) ..........................................................................................6
1.2.3
Marqués’ system to demonstrate propagation below cutoff (Marqués,
Martel, et al. 2002) ..................................................................................7
1.2.4
Shiffler’s experimental setup (Shiffler, et al. 2013) ...............................8
2.1.1
Asymptotic behavior of μr with respect to frequency for a loop
resonator (Pendry, et al. 1999) .................................................................13
2.1.2
Effect of losses on SRR resonance. (a) 𝜎 = 200 S/m, (b) 𝜎 = 2000
S/m (Pendry, et al. 1999) ........................................................................14
2.1.3
Orientation of SRRs in waveguide systems to interact with TE10
modes ......................................................................................................17
3.2.1
SRR Circuit diagram (Baena, Marqués and Medina 2004) ....................22
3.3.1
Simulated SRR with bounding box. The box shown does not represent
the SRR unit cell. .....................................................................................24
3.3.2
S-parameter graph showing SRR resonance for a given geometry; 5
mm outer radius, .5 mm ring break distance, .3 mm gap between rings,
.6 mm ring width, .017 mm ring thickness, .7874 mm dielectric
thickness, and a dielectric constant of 2.2ε0. Resonance at 2.763 GHz. ..25
3.3.3
S-parameter graph showing SRR resonance of a geometry identical
Fig 3.3.1, with the gap between rings reduced to .15 mm. Resonance at
2.425 GHz. ...............................................................................................25
3.3.4
S-parameter graph showing SRR resonance of a geometry identical
Fig 3.3.1, with the break in each rings reduced to .25 mm. Resonance
at 2.700 GHz. ............................................................................................ 26
x
3.3.5
S-parameter graph showing SRR resonance of a geometry identical
Fig 3.3.1, with the thickness of the rings increased to .0255 mm.
Resonance at 2.740 GHz. .......................................................................... 26
3.3.6
S-parameter graph showing SRR resonance of a geometry identical
Fig 3.3.1, with the outer radius increased to 7.5 mm. Resonance at
1.542 GHz. ................................................................................................ 27
3.3.7
S-parameter graph showing SRR resonance of a geometry identical
Fig 3.3.1, with the dielectric constant increased to 3.3ε0. Resonance at
2.388 GHz. ................................................................................................ 28
3.3.8
S-parameter graph showing SRR resonance of a geometry identical
Fig 3.3.1, with the substrate thickness increased to 1.1811 mm.
Resonance at 2.723 GHz. .......................................................................... 28
3.4.1
Transmission parameter graphs of a signal sent through a 3x2x3 SRR
array housed in WR284 waveguide. SRRs are identical. (Top) No
separation between unit cells in the z-direction. (Bottom) 10 mm
separation between unit cells in the z-direction. ....................................... 31
4.2.1
Basic MTBX schematic ............................................................................ 35
4.2.2
MTBX band stop waveguide system ........................................................ 36
4.2.3
MTBX band pass waveguide system ........................................................ 36
4.2.4
HP8620C sweep oscillator mainframe and HP86290C RF plug-in.......... 37
4.2.5
Mini-Circuits ZYSW-2-50DR diode switch ............................................. 37
4.2.6
Stanford Research Systems DG535 digital delay/pulse generator ........... 38
4.2.7
Fig 4.2.7: Buffer circuit used, SN74LS244 buffer chip, and LM7805C
regulator .................................................................................................... 38
4.2.8
Picosecond Pulse Labs 5828A-107 amplifier (top) and Krytar 209A
detector (bottom) ....................................................................................... 39
4.2.9
Complete MTBX setup ............................................................................. 40
4.3.1
Final SRR unit cell design used in MTBX ............................................... 41
4.3.2
An SRR card for the band stop system ..................................................... 41
xi
4.3.3
SRR cards in band stop system. The WR284 waveguide is outlined in
light grey, and propagation occurs in the z-direction. .............................. 42
4.3.4
An SRR card for the band pass system ..................................................... 42
4.3.5
SRR cards in band pass system. The waveguide is outlined in light
grey, and propagation occurs in the z-direction. ....................................... 43
4.3.6
Completed SRR card for the band stop system ........................................ 44
4.3.7
Completed SRR card for the band pass system. The final ring was cut
and reattached while searching for maximum power transmission in
the SRR pass band. The tape does not appreciably effect the SRR
behavior..................................................................................................... 44
4.3.8
Foam brick for insertion into the MTBX band stop system ..................... 45
4.3.9
Foam brick and SRR cards inserted into the MTBX band pass system ... 45
5.1.1
Final SRR unit cell design used in MTBX ............................................... 46
5.1.2
Simulated S-parameters for the final SRR design used in MTBX.
Reflection (S11) in red, transmission (S21) in blue. Resonance at 2.763
5.1.3
GHz. .......................................................................................................... 47
Internal electric field excitation of the SRRs at 𝜔𝑡 = 𝜋⁄4: (a) Ex, (b)
5.1.4
Ey, (c) Ez. ................................................................................................... 48
Internal magnetic field excitation of the SRRs at 𝜔𝑡 = 𝜋⁄2: (a) Hx, (b)
Hy, (c) Hz. .................................................................................................. 48
5.1.5
Simulated S-parameters for the band stop system. (Top) transmission,
(Bottom) reflection. Resonance at 2.80 GHz. ........................................... 50
5.1.6
Normalized signal vs. time for the stop band system. (Top), input in
red, reflection in blue, (Bottom) input in red, transmission in blue. ......... 51
5.1.7
Field profiles near 40 ns. (Top) Ey viewed from the right, (Bottom) Hx
viewed from the top. ................................................................................. 52
5.1.8
Simulated S-parameters for the band pass system. (Top) transmission,
(Bottom) reflection. Resonance at 2.73 GHz. ........................................... 53
5.1.9
Normalized signal vs. time for the band pass system. (Top), input in
red, reflection in blue, (Bottom) input in red, transmission in blue. ......... 54
xii
5.1.10
Ey field maps showing negative phase velocity in the SRR array
region. Signal is incident from the left and produces weak standing
waves in the left region, backward waves in the center region, and
forward propagating waves in the right region. ........................................ 55
5.2.1
Measured transmission (top) and reflection (bottom) parameters vs.
frequency for the band stop system .......................................................... 57
5.2.2
Transmission (in blue) and reflection (in green) of the band stop
control case (empty waveguide with no SRR array). ............................... 58
5.2.3
S-parameter data showing sample frequencies for time-based
measurements of the band stop system ..................................................... 58
5.2.4
Time response for various frequencies in the SRR loaded band stop
system (in blue) and the empty control case (in green). 2.4 GHz (a),
2.79 GHz (b), and 3.2 GHz (c). 𝑡 = 0 when the trigger generator
pulses......................................................................................................... 59
5.2.5
Time response of the band stop system at SRR resonance ....................... 60
5.2.6
Transmission (top) and reflection (bottom) parameters vs. frequency
for the band pass system ........................................................................... 62
5.2.7
Transmission (in blue) and reflection (in green) of the band pass
control case (no SRR array). ..................................................................... 63
5.2.8
S-parameter data showing sample frequencies for time-based
measurements of the band pass system ..................................................... 63
5.2.9
Time response for various frequencies in the SRR loaded band stop
system (in blue) and the empty control case (in green). 2.7 GHz (a),
2.84 GHz (b), and 3.0 GHz (c). Figure (d) shows time response for a
chosen frequency above cutoff. Since the cutoff varies between the
loaded and control systems, the loaded system trace (in blue) is at 3.6
GHz, and the control system trace (in green) is at 4.0 GHz. .................... 64
6.1.1
Transmitted signals vs. time of various configurations, (blue) 3.0
GHz, (green) 4.0 GHz, (red), 2.84 GHz.................................................... 67
xiii
6.1.2
Normalized transmitted signals of various configurations, (green)
offset by 6 ns, (red) offset by -14 ns and multiplied by 2.3 ...................... 68
6.1.3
Natural
log transformations of normalized transmitted signals of
various configurations, (green) offset by -6 ns, (red) offset by -14 ns
and multiplied by 2.3 ................................................................................ 69
6.1.4
Normalized transmitted signals of various configurations, (green)
offset by -6 ns, (red) offset by -26 ns and multiplied by 2.3 to show
form consistency. ...................................................................................... 69
6.1.5
Natural log transformation of SRR pass band signal, offset by -14 ns.
Blue and black linear fits, with calculated time constants. ....................... 70
1
Chapter 1:Introduction
1.1 Motivation
Metamaterial physics is a relatively new field principally concerned with constitutive
parameters that are tunable with the frequency of an input wave. Applications of systems
that make use of these properties are generating significant interest, including in the area
of high-power microwave devices for military and communications technology. Nearterm applications include miniaturized microwave sources and networks, more efficient
sources, new filter designs, new antennas, lenses free from diffraction limitations, and
many more (Shamonina and Solymar 2007).
One fundamental scheme for metamaterial systems is to generate negative magnetic
permeability by using split-ring resonators (Pendry, et al. 1999). When coupled with
other components or effects, double negative and single negative behavior can be
engineered into a device or system. Furthermore, a comprehensive understanding of splitring resonators has the potential to shed light on many other metamaterial physics
questions, leading to more robust and varied applications.
In this work, split-ring resonators have been used to create band pass and band stop
filters. These systems provide a foundation on which to explore split-ring resonator time
domain behavior. Both systems have experimentally demonstrated proof-of-concept and
the first steps toward a physical understanding of temporal behavior have been taken.
However, questions remain about many behaviors observed in the split-rings themselves.
These include mode interaction, calculation of constitutive parameters, impedance
calculation, performance comparisons between similar designs, resonant energy
Chapter 1:Introduction
2
saturation, resonance excitation, time-based phenomena, and others. The experimental
and simulation work presented here builds on the foundation provided by others, and
attempts to address some unanswered questions. Furthermore, it provides a solid basis
from which other questions may be addressed in the future.
1.2 The History of Metamaterials
The historical perspective on metamaterials is rather limited in comparison to other
subfields in applied electromagnetics. Obviously, this is directly due to the newness of
the field. While fields such as plasma physics and antenna design began to take shape in
the late 1800s, the beginning of metamaterial physics came much later. It is widely
considered to begin with the development of the split ring resonator (Shamonina and
Solymar 2007). The relative youth of the field strongly relates to the modern
understanding of electromagnetics. Without the advancements in microwave engineering
and plasma physics, the potential of the split ring resonator, and thus metamaterials as a
whole, would not have been studied.
While the development of true metamaterials did not occur until much later, in 1967,
V. G. Veselago, of the USSR, published the first theoretical work concerned with
negative indices of refraction (translated to English a year later), (Veselago 1968). He
began his exploration with a hypothesis of the effect of simultaneously negative
permeability µ and permittivity ε on the index of refraction of a medium. After
considering various cases, he concluded that such a medium is both allowable by physical
laws and behaves differently than a conventional medium.
Veselago started with Maxwell’s equations and paid special attention to the
appearance of µ and ε. He noted that Maxwell’s equations reduce to
𝜔
𝒌 × 𝑬 = 𝜇𝑯
𝑐
Eq. 1.2.1
Chapter 1:Introduction
3
𝜔
𝜀𝑬
Eq. 1.2.2
𝑐
for a monochromatic plane wave with angular frequency ω, wavenumber k, magnetic
𝒌×𝑯 = −
field amplitude H, and electric field amplitude E, with the free space speed of light c. In
conventional media, these equations show that E, H, and k form a right-handed triplet
and require a positive k. However, when µ and ε are simultaneously negative, these
vectors instead form a left-handed triplet where k is negative. Thus, the ‘handedness’ of
the system determines the sign of k. Such media have hence been referred to as doublenegative media (DNG), left-handed media (LHM), or less commonly, Veselago media.
(Veselago 1968)
It is important to note that the Poynting vector is not affected by permeability or
permittivity, and, as such, remains the same in left-handed media. However, the wave
number k reverses direction. In other words, the direction of energy flow and the
direction of propagation are opposite, as are the phase and group velocities. Veselago
noted that the relationship between the index of refraction and the constitutive parameters
was ambiguous. In the equation
Eq. 1.2.3
𝑛2 = 𝜀𝜇
a simultaneously negative µ and ε would not necessarily affect the sign of n. However, in
DNGs, Maxwell’s equations result in a negative wavenumber, thus phase velocity is
negative, and thus n is necessarily negative (Veselago 1968).
Veselago speculated about how a DNG media might be obtained. He noted the
appearance of negative terms in the permittivity tensor of gyrotropic plasmas. If magnetic
monopoles were discovered, he theorized that a plasma of magnetic and electric charges
would create DNG behavior. Unfortunately, he did not consider the notion of
macroscopic resonators, but by beginning his exploration of DNG media by considering
plasmas, he provided a way forward for future work (Veselago 1968).
Chapter 1:Introduction
4
After Veselago’s treatment of DNGs, the subject was not seriously addressed again
for about 30 years. The principle limitation was that no material existed with a negative
permeability. During that time, thin wire arrays (also known as artificial or metallic
plasmas) became a reliable way of generating a negative permittivity (Marqués, Martin
and Sorolla 2008), and so more of the metamaterial groundwork was laid, but the
problem remained with permeability.
Throughout the 1990’s, Pendry et al. did significant work with ‘metallic plasmas,’
most commonly in the form of thin wire arrays. These arrays demonstrated negative
electric permittivity for specific frequencies given a specific array densities. Pendry then
generalized the principles of his work to create a similar theory for magnetic
permeability. His first work on periodic magnetic structures was published in 1999
(Pendry, et al. 1999).
Pendry began from a different perspective than Veselago. Rather than being
specifically concerned with negative permeability, he attempted to create a system in
which permeability could be chosen; that is, tuned according the frequency of an indecent
wave. Understanding that the magnetic response of individual atoms and molecules was
directly related to the frequency of an impinging wave, he sought to scale this response to
the gigahertz regime. In the introduction of his paper, he proposed the use of macroscopic
engineered ‘unit cells’ as analogs of individual atoms. The size of such a unit cell would
then be determined by the wavelength of the incident wave. Pendry proposed that
effective unit cells could be engineered as long as a characteristic dimension was much
less than the wavelength. The next step was to construct a method to calculate an
effective permeability. He approached the problem using Maxwell’s equations in integral
form. By reasoning that µ could be averaged over the volume of a single unit cell, it
would then be possible to repeat this cell to create a bulk material with an engineered µ.
Effective permeability would then be the average of the absolute permeabilities inside
and outside the structure, weighted by volume (Pendry, et al. 1999).
Chapter 1:Introduction
5
Pendry worked through several ‘unit cell’ designs, demonstrating that by adding a
capacitance to an inductive structure, the system becomes resonant. The magnetic
response is then amplified greatly, either increasing or decreasing it to singularity (in a
lossless system) on either side of the resonance. He ultimately settled on a ring geometry
consisting of nested, concentric ‘c’ shaped planar rings, called split-ring resonators
(SRRs) (Pendry, et al. 1999),
The
Pendry’s
principle
SRRs
drawback
was
of
strong
anisotropy, but his understanding
and development would pave the
way for the creation of many split
ring designs. More fundamentally,
however, his SRR design revealed
the possibility to create media with
negative permeability for specific
Fig 1.2.1: Pendry’s original SRR design
(Pendry, et al. 1999)
frequencies. Media borne out of his
periodic unit cell approach, with tunable µ and ε, are what are now commonly referred to
as ‘metamaterials’ (Pendry, et al. 1999).
Here, a note should be made about the relationship between double-negative media
(DNGs) and metamaterials. Though the community is still in disagreement about a
precise definition of what a metamaterial is, this work will use the definition from the
previous paragraph. In this sense, metamaterials enable a designer to choose constitutive
parameters. The most prevalent use is to create a double-negative system, but it should be
noted that metamaterial may be used to create systems that are double-negative, singlenegative (in either parameter), or conventional.
Following Pendry’s development of the SRR, the field of metamaterials widened
rapidly for the next several years. Two years later, Shelby et al. created the first bulk
Chapter 1:Introduction
6
double negative medium (Shelby, Smith and Schultz 2001). This medium was a
composite of split ring resonators designed to resonate in X-band, and a thin wire array.
The experimental setup was quite simple. A triangular metamaterial slab was placed in
front of a wave with a frequency that matched the SRRs’ resonance. A detector swept
radially on the other side to find the angle of refraction. Comparing the angle of
refraction with a slab of Teflon, Shelby successfully demonstrated that the metamaterial
slab had a negative index of refraction, and thus was double-negative. This experiment
conclusively proved Veselago’s original theory.
Fig 1.2.2: Shelby’s experimental test set up and results (Shelby, Smith and Schultz 2001).
From here, the discussion moves away from metamaterial physics as a whole and
becomes more focused to the vein relevant to this thesis. With the success of the SRR,
many applications of sub-wavelength structures grew, such as resonant nano-particles,
superlenses and perfect lenses, photonic band-gap materials, and many more (Shamonina
and Solymar 2007). The most relevant branches to this work are slow-wave structure
design and resonant-structure based filters.
Chapter 1:Introduction
7
Fig 1.2.3: Marqués’ system to demonstrate propagation below cutoff (Marqués, Martel, et al.
2002)
In 2002, Marquez et al. proposed a system which utilized Pendry’s original SRR
design to create a simple band-pass filter in a waveguide. By loading a waveguide with
SRRs with a resonant frequency below the waveguide cutoff, a pass band would exist
around that resonance. Marquez’s system consisted of WR-137 waveguide adapted down
to a 6 mm by 6 mm square waveguide loaded with SRRs, then returned to WR-137.
SRRs in the system were designed to resonate at approximately 6 GHz, while the cutoff
frequency for the square waveguide section was about 25 GHz. Theoretical modeling and
experiments both showed a pass band to exist near the resonance of the split rings.
However, an in depth exploration of the physics and experimental verification of negative
permeability were not provided (Marqués, Martel, et al. 2002).
In 2005, the system was revisited by Hrabar et al. (Hrabar, Bartolic and Sipus 2005).
The objective was to gain a more comprehensive understanding of the physics, and in
turn develop more applications for the system. Realizing the potential to propagate a
signal though waveguide many times below cutoff frequency, the group explored
whatever minimum size there might be for the backward wave system described by
Marquez.
Hrabar’s work represented a clear step forward in the physical understanding of the
cutoff system. He derived dispersion relations and took anisotropic effects and waveguide
Chapter 1:Introduction
8
modes into consideration. Secondly, he introduced a counterpart system using an abovecutoff waveguide loaded with SRRs. In this system, µ is negative at certain frequencies
and modes due to the SRRs, but ε remains positive. With only one constitutive parameter
less than zero, an incident wave becomes evanescent. Thus frequencies near the SRR
resonance are blocked, while others pass (Hrabar, Bartolic and Sipus 2005).
Hrabar demonstrated that backward propagation occurred in the cutoff system by
comparing the electrical length of loaded below-cutoff guide and empty above-cutoff
guide. He showed that the loaded below-cutoff guide appeared electrically shorter by
comparing input and output phase, and concluded that the phase velocity is negative, thus
the system is DNG. This claim is disputed, however, in that wrapped phase could also
account for the electrical length difference. Thus two solutions are available for the
observed data, very high phase velocity, or negative phase velocity (Hrabar, Bartolic and
Sipus 2005).
As metamaterial research
grew,
the
United
States
government took an interest in
the field. The experiments done
by Hrabar and Marquez verified
that
miniaturization
microwave
systems
of
was
demonstrably achievable, and
military
scientists
were
intrigued by the possibility of
using
similar
ideas
Fig 1.2.4: Shiffler’s experimental setup (Shiffler, et al. 2013)
to
miniaturize high-power microwave sources. Thus in 2013, Shiffler et al. from Kirtland
Air Force Base in Albuquerque, New Mexico published work concerned with the
robustness of Marquez’s original pass band system (Shiffler, et al. 2013).
Chapter 1:Introduction
9
Shiffler reconstructed Marquez’s pass band system identically except for a single
change (Shiffler, et al. 2013). Rather than a variable length of the cutoff guide, a static
system, five unit cells long was used. Next, the pass band was analyzed by progressively
‘damaging’ different cells according to what might occur in a high-power environment. A
short-circuit was created by replacing a cell by a solid copper disk, and an open-circuit
was create by removing the copper altogether. The work proved the repeatability of
Marquez and Hrabar’s systems and provided the first foundational studies of
metamaterial systems in a high-power environment.
10
Chapter 2: Physics Review
2.1 The Electromagnetics of Split-Ring Resonators
Split ring resonators were originally designed by mimicking the magnetic response of
individual atoms, then scaling to larger wavelengths, and enhancing specific behaviors
(Pendry, et al. 1999). This construction will be followed below, starting with magnetic
response of atoms. Then it will address scaling with frequency, followed by the
enhancement of the magnetic response with results in negative permeability.
2.1.1 Maxwell’s Equations and Permeability
Diamagnetic response is a fundamental reaction of all atoms to an externally applied
magnetic field. A classical analogy may represent electron orbits as closed current loops.
When an external magnetic field is applied to the loop, it experiences a net magnetic flux.
A common interpretation of Faraday’s law is that when such a current loop experiences a
change in magnetic flux B, the loop will source current in an attempt to preserve a zero
(or constant) flux. In turn, currents will be generated in the loop such that the internal
induced field opposes the external applied field (Balanis 1989).
∇×𝑬=−
𝜕𝑩
𝜕𝑡
Eq. 2.1.1
Chapter 2: Physics Review
11
More simply, a voltage is induced in the loop by the changing magnetic field. This
voltage motivates current, which itself generates a magnetic field by Ampere’s Law,
where Jf is the free current, and D is the electric displacement (flux) field.
∇ × 𝑯 = 𝑱𝒇 +
𝜕𝑫
𝜕𝑡
Eq. 2.1.2
Noting the negative sign in eq. 2.1.1, the induced internal field is opposite of the
external applied field, but is typically several orders of magnitude weaker (Marqués,
Martin and Sorolla 2008).
Averaged over many atoms, this effect is known as diamagnetism. In a purely
diamagnetic material, the amplitude of the magnetic field is weaker in the material than
in free space for the same amount of energy. In linear bulk materials, magnetization M is
defined as
𝑴 = 𝜒𝑚 𝑯
Eq. 2.1.3
where H is the applied external magnetic field and χm is the magnetic susceptibility. M is
the magnetic response of the material to an external applied field. While generally
considered macroscopically, in the case of a single atom, it is the internal induced field.
In turn, χm, is ratio of the external applied field to the internal induced field.
𝜒𝑚 =
𝑴
𝑯
Eq. 2.1.4
For a single diamagnetic atom, M should be in the opposite direction as H and much
smaller. That is, χm should be very small and negative.
Now defining relative permeability µr,
𝜇𝑟 = 1 + 𝜒𝑚
Eq. 2.1.5
it is clear that a diamagnetic response should produce a relative permeability of slightly
less than unity. Indeed, this is the behavior observed in purely diamagnetic materials such
Chapter 2: Physics Review
12
as water, copper, and many oxide minerals. By extension, to decrease relative
permeability, the magnitude of the internal induced magnetic field must be increased.
2.1.2 Scaling
Because the physical size of the electron orbits is so small, this construction has assumed
that the magnetic field does not vary spatially across the atomic diameter. Rather than an
atom, instead consider a small conducting loop (that is, 𝜆 ≫ ℓ, where ℓ is the
characteristic dimension). Like a diamagnetic atom it will respond to an external field
with a weak induced internal field in the opposite direction. The superposition of the
fields results in a total field slightly weaker than the applied field. For the local area,
𝜒𝑚 < 0 and 𝜇𝑒𝑓𝑓 < 1 where μeff replaces μr for the macroscopic system.
Now suppose, hypothetically, that the loop was able to motivate enough current such
that when an external magnetic field is applied, the induced response perfectly preserves
zero net flux inside the loop. In other words, the internal induced field is equal to and
opposite of the external field. The superposition of these fields cancels totally, and the
resultant total field amplitude is zero. In such a case, |𝑴| = 𝑯, 𝜒𝑚 = −1, and 𝜇𝑒𝑓𝑓 = 0.
Continuing this logic, suppose the loop was able to reflexively generate current, such
that when an external magnetic field is applied, the induced response is greater than, and
opposite to the external applied field. The superposition of the internal and external fields
results in a total field in the opposite direction to the applied field. In this case, |𝑴| > 𝑯,
𝜒𝑚 < −1 and 𝜇𝑒𝑓𝑓 < 0. Thus negative permeability has been achieved.
2.1.3 Enhancing diamagnetic response and the SRR
The question is then how to create such a system where the internal induced response is
stronger than the external applied field. While the answer is actually quite simple, it
introduces a level of complexity: add capacitance to the loop. In one instance, Hrabar et
al. did this simply by a brute-force approach to demonstrate the principle. They created
Chapter 2: Physics Review
13
metal C-shaped rings and loaded the gaps with chip capacitors (Hrabar, Bartolic and
Sipus 2005). Perhaps more elegantly, a distributed capacitance can be created easily by
Pendry’s original split-ring design. (Pendry, et al. 1999)
As is well understood, any LC circuit is resonant. Physically, energy in the resonator
oscillates between the electric field, stored in the capacitance, to the magnetic field,
stored in the inductance, and vice versa. This rate of oscillation is the resonant frequency.
Suppose now that an incident wave with frequency ω passes over a capacitively loaded
loop with some known resonant frequency ω0. The magnetic field of the loop superposes
with the magnetic field of the wave. If the wave is near the loop’s resonant frequency,
that is 𝜔 ≈ 𝜔0 , the fields will interfere entirely constructively or entirely destructively.
If 𝜔 < 𝜔0 , the fields add constructively,
creating a strong paramagnetic response. If
𝜔𝑖 > 𝜔0 , the fields add destructively, creating
a strong diamagnetic response. Losses cap the
total energy that can be stored in the resonator,
and thus limit the magnitude of the interference
(Pendry, et al. 1999). If losses are sufficiently
low (which is easily achievable with copper),
the diamagnetic response may be strong
Fig 2.1.1: Asymptotic behavior of μr with
respect to frequency for a loop resonator
(Pendry, et al. 1999)
enough to create 𝜇𝑒𝑓𝑓 < 0 as described above.
The effective permeability is then tunable with
the incident frequency. That is,
𝜇𝑒𝑓𝑓 (𝜔) =
𝐹𝜔02
𝜔 2 − 𝜔02 − 𝑗𝜔Γ
Eq. 2.2.6
Chapter 2: Physics Review
14
where F and Γ are constants related to the geometry of the SRR (Pendry, et al. 1999)
(Smith, Vier, et al. 2000).
A few additional notes should be made. Firstly, and most importantly, the direction of
incidence is foundationally important to the understanding of the split-ring resonator.
They are not free with align to an arbitrary incident wave, like an atom. The polarization
of the incident wave must be such that the magnetic component is normal to the ring’s
diameter (Marqués, Martin and Sorolla 2008). In order for SRRs to produce tunable
permeability, the direction of the incident wave and its polarization must be correct. In
this project, these factors are precisely controlled, as will be shown in later sections.
Secondly, as stated above, the incident wavelength must be sufficiently large such that
Fig 2.1.2: Effect of losses on SRR resonance. (a) 𝜎 = 200
S/m, (b) 𝜎 = 2000, S/m (Pendry, et al. 1999)
the incident magnetic field does not appear to vary appreciably across the ring diameter
(Pendry, et al. 1999). While this sizing is arbitrary, typically the ring diameter an order of
magnitude smaller than the wavelength is sufficient to guarantee the desired behavior.
Thirdly, care must be taken to reduce losses where possible. As shown in figure 2.1.2, too
much resistance can destroy the strength of the diamagnetic SRR response, and result in
failure to achieve a negative µeff (Pendry, et al. 1999).
Chapter 2: Physics Review
15
2.2 SRR Loaded Waveguides
This work is concerned with two metamaterial devices: a band stop and a band pass filter.
Both systems are variations on a waveguide loaded with an SRR array. The approach
taken will be from a frequency domain perspective and assumes that the SRRs are
already excited. The excitation of the SRRs and the initial time behavior are as yet poorly
understood and will be addressed later.
2.2.1 Constitutive Parameters and Propagation
For a wave traveling in a lossy bulk medium, its normalized amplitude has the form:
𝜓(𝑧) = 𝑒 −(𝛼+𝑗𝑘)𝑧
Eq. 2.2.1
where α represents losses and in general,
𝑘 = 𝜔√𝜇𝑒𝑓𝑓 𝜀𝑒𝑓𝑓
Eq. 2.2.2
where 𝜇𝑒𝑓𝑓 = 𝜇 and 𝜀𝑒𝑓𝑓 = 𝜀 for a bulk homogenous medium. It is evident that when
either µeff or εeff (but not both) is negative, a j falls out of the radical and results in the
imaginary term becoming real. Thus k becomes an attenuating factor, and the mode
becomes evanescent (Pozar 2012), (Jackson 1999).
When both constitutive parameters are negative, eq. 2.2.1 and 2.2.2 are apparently
unaffected. However, Veselago’s manipulation of Maxwell’s equations shows that:
𝒌×𝑬=
and
𝜔
𝜇𝑯
𝑐
Eq. 2.2.3
Chapter 2: Physics Review
16
𝒌×𝑯=−
𝜔
𝜀𝑬
𝑐
Eq. 2.2.4
These two relations show a circular relationship between k, H, and E. If µ and ε are
both positive, the relationship is right-handed with positive k. If they are both negative,
the relationship is left-handed with negative k. Thus, the ‘handedness’ of k, H, and E
informs our choice of the sign of the radical in eq. 2.2.2. Put simply, if µ and ε are both
negative, then by Maxwell’s equations, k is negative (Veselago 1968). Indeed this is the
case physically (Shelby, Smith and Schultz 2001), (Smith, Schurig and Pendry 2002),
(Valanju, Walser and Valanju 2002).
Thinking of it another way, it was demonstrated above that negative permeability
corresponds to internal magnetic fields that are greater in amplitude and opposite in
direction to an incident field. Analogously, negative permittivity is the same for electric
fields. For a double-negative medium, it would be expected that both the electric and
magnetic field components of a wave would switch direction, resulting in a reversal of
phase velocity, and thus k (Marqués, Martin and Sorolla 2008), (Foteinopoulou,
Economou and Soukoulis 2003).
However, even in a DNG system, the direction of energy flow does not change. The
Poynting vector is defined as,
𝑺 = 𝑬 × 𝑯∗
Eq. 2.2.5
Thus S, E, and H always form a right-handed set (Veselago 1968). The direction of
energy flux, determined by S, is not dependent on the signs of the constitutive
parameters. Furthermore, when µ and ε are negative, S and k are in opposite directions.
That is, phase velocity and group velocity are in opposite directions (Marqués, Martin
and Sorolla 2008).
Chapter 2: Physics Review
17
2.2.2 Band Stop Filter
A section of rectangular waveguide is loaded with an array of split ring resonators
aligned such that their surface plane is parallel to the waveguide’s short walls, as shown
in figure 2.1.2. Since the SRRs resonant frequency can be selected, it is chosen to be
some frequency between the TE10 and the TE20 cutoffs. In this range, only the
fundamental mode propagates in the waveguide. This is also the recommended operating
range for standard commercial WR indexed waveguides.
Fig 2.1.3: Orientation of SRRs in waveguide systems to interact with TE 10 modes
Requiring the TE10 mode is done for several practical reasons, but principally to make
the design physically simple, and thus easier to understand than a more sophisticated
system. For more discussion on the why the TE10 mode regime was chosen, please see
Chapter 4.
For a guided wave, in reference to eq. 2.2.2,
𝜀𝑒𝑓𝑓 = 𝜀0 (1 −
𝜔𝑐2
)
𝜔2
Eq. 2.2.6
Chapter 2: Physics Review
18
where ωc is the cutoff frequency of the waveguide, and determined by its inside physical
dimensions (Jackson 1999). If 𝜔 > 𝜔𝑐 , the wave propagates and the treatment is
otherwise unaffected.
The choice of the TE10 mode guarantees a magnetic field component will be normal
to the surface plane of the SRRs. This allows the wave and the SRRs to interact, and
provided the wave’s frequency is just above the SRR resonance, the wave will experience
negative effective permeability μeff. As discussed above, this results in an exponential
decay in amplitude, thus blocking the signal’s transmission. It is important to note,
however, that the signal is only blocked very close to the SRR resonant frequency in the
small range where 𝜇𝑒𝑓𝑓 < 0. When the signal is not near the SRR resonant frequency, the
structures do not interact with the wave and appear transparent, allowing full propagation.
2.2.3 Band Pass Filter
Consider again equations 2.2.1, 2.2.2, and 2.2.6:
𝜓(𝑧) = 𝑒 −(𝛼+𝑗𝑘)𝑧
Eq. 2.2.1
𝑘 = 𝜔√𝜇𝑒𝑓𝑓 𝜀𝑒𝑓𝑓
Eq. 2.2.2
𝜀𝑒𝑓𝑓
𝜔𝑐2
= 𝜀0 (1 − 2 )
𝜔
Eq. 2.2.6
To construct a band pass filter, a section of waveguide is used such that the incident
signal is below cutoff for the waveguide system, that is, 𝜔 < 𝜔𝑐 . As long as this is the
case, 𝜀𝑒𝑓𝑓 < 0 according to eq. 2.2.6.
As discussed previously, for a small bandwidth above SRR resonance, 𝜇𝑒𝑓𝑓 < 0.
Therefore, if the resonant frequency of the SRRs in this system is below the waveguide
cutoff frequency, that is, 𝜔0 < 𝜔𝑐 , then for that bandwidth, the system is DNG.
Propagation occurs as a backward wave with reversed phase velocity. Group velocity
Chapter 2: Physics Review
19
remains in the same direction as conventional media, however, so the direction of energy
flow remains unchanged.
20
Chapter 3: Split-Ring Resonator Design
Though the basic physical principles of the split-ring resonator have been explored in
chapter 2, designing a functional SRR is another matter. There are many basic geometries
for the SRR including Pendry’s original edge-coupled SRR, the broadside-coupled SRR,
2-loop spirals, 3-loop spirals, helices, and dozens of other designs, both planar and
volumetric (Marqués, Martin and Sorolla 2008) (Marqués, Mesa, et al. 2003). They all
operate on the basic principles already discussed. Each design has drawbacks and
advantages based on how their individual operational characteristics interact.
This work is concerned with the edge-coupled SRR (EC-SRR), which will henceforth
be simply referred to as SRR. This design was the first SRR design developed and is still
considered to be the most robust (Marqués, Martin and Sorolla 2008).
3.1 Size constraints of the Split-Ring Resonator
The principle limiting factor for SRR design is its physical size. The treatment in Chapter
2 assumed some characteristic dimension d to be small compared to the wavelength. This
dimension is now taken to be the outer diameter of the SRR’s largest ring. That is,
Chapter 3: Split-Ring Resonator Design
21
𝜆≫𝑑
Eq. 3.1.1
The magnetic interaction of the SRR requires, through Faraday’s and Ampere’s laws,
that the external magnetic field be changing. It is important to specify that the change
must happen only in time, not space. In other words, the amplitude of the wave at any
given point must be approximately constant across the SRR diameter. This guarantees the
magnetic response is likewise spatially constant.
If this were not the case, a number of consequences could destroy the magnetic
response of the SRR. Firstly, the orientation of the magnetic field component must be
normal to plane of the SRR. This necessitates the electric field component be directed
parallel to the SRR plane. If the SRR is not ‘small’, spatial variation in the electric field
can set up errant voltages on the SRR. These voltages interfere with the flow of current
which create the fundamental magnetic response, reducing or destroying the response
behavior.
Secondly, if the magnetic field varies across the SRR diameter, the magnetic response
will merely be that of the spatial average. This is problematic for several reasons. The
amplitude of the time-varying magnetic response will always be weaker at the maxima of
the wave than a properly sized SRR. If the ring is too large, induced currents in the ring
may be created in opposing directions, destroying the magnetic response altogether.
3.2 Inductance, Capacitance, and Calculating Resonant Frequency
The resonant frequency of an SRR is its defining characteristic. The resonant behavior
allows the wave and the SRR to interact, which in turn creates negative effective µ.
Chapter 3: Split-Ring Resonator Design
22
Therefore, the next question to be addressed is how to determine the resonant frequency
of an SRR.
The SRR can be quite neatly reduced to the circuit model shown in fig 3.1.1. The
inductance simply comes from the ring-like nature of the rings. The source of capacitance
is a bit more complex. Firstly, it comes from the break in each ring, Cb. These breaks are
in series with each other and with the inductance. Secondly, a capacitance per unit length,
Cpul exists between the inner and outer rings (Baena, Bonache, et al. 2005).
Fig 3.2.1: SRR Circuit diagram (Baena, Marqués and Medina 2004)
Theoretically, the resonant frequency of the circuit in figure 3.1.1 is quite simple to
calculate,
𝜔0 = 2𝜋𝑓0 =
1
√𝐿𝐶
Eq. 3.2.1
where analytical formulas exist that one may naively assume are good estimates of L and
C. The difficulty in this equation is accurately estimating the value of C. If the resonator
has some thickness h (the thickness of the copper cladding on printed circuit material, for
instance) and the rings have some width, w, then typically ℎ ≪ 𝑤. Consequently, a
parallel plate approximation for either the Cb or Cpul will be very poor as both these
Chapter 3: Split-Ring Resonator Design
23
capacitances are dominated by fringe effects. Furthermore, the fringe effects may not be
symmetric. Since SRRs are generally printed on a dielectric substrate, the dielectric
constant of the substrate will modify the fringe capacitance similar to dielectric loading
of a parallel plate capacitor. However, common manufacturing methods leave the SRR
exposed on one face, and attached to the substrate on the reverse, which loading the
fringe capacitance asymmetrically.
If analytical methods cannot provide accurate calculation of resonant frequency, then
a computational approach must be taken. The principle tool for determining SRR
resonant frequencies used in this work was by utilizing transmission and reflection
parameters. As was explored in Chapter 2, when an incident wave interacts with an SRR
just above the SRR resonance, a negative effective µ is created. If the only effect is
negative µ, then the incident mode will become evanescent in the SRR region, and
significant power will be reflected. If the incident wave is below (or sufficiently above)
SRR resonance, the SRR will appear transparent, and the wave will propagate unaffected.
That is, an SRR loaded region of empty space should appear transparent for all
frequencies except the SRR resonance (or to be more specific, a small bandwidth
marginally above the SRR resonance) where it instead appears opaque.
Using this approach, transmission and reflection through a region can be calculated
simply and efficiently by use of commercially available field solvers, such as CST
Microwave Studio or HFSS. This work used CST exclusively.
Chapter 3: Split-Ring Resonator Design
24
3.3 Controlling SRR Resonance
The ‘control knobs’ on SRR
resonant
frequency
rest
with
changing the L and C values of the
SRR. That is, according to eq.
3.2.1, increasing either value will
lower the resonant frequency,
while reducing either value raises
resonant
frequency.
However,
SRR resonance is still a problem
in optimization. Almost every
Fig 3.3.1: Simulated SRR with bounding box. The box
shown does not represent the SRR unit cell.
adjustment of L or C will have other consequences. This leads to numerous SRR designs
that may have the similar resonant frequencies, but vastly different operational
parameters.
All SRR variations explored below are of the EC-SRR (Pendry’s original circular
design) made from copper and printed on a dielectric substrate. In order to design SRRs
with a chosen resonant frequency, the following simulation set was performed in CST
Microwave Studio. A plane wave (with the H-field oriented in the x-direction) is incident
on the SRR array from the –z direction. The geometry shown represents a single cell of
this array that repeats infinitely in x and y, simulated by using unit-cell boundary
conditions available in newer versions of CST. This boundary type requires the frequency
domain solver and uses a tetrahedral mesh. Simulation convergence is guaranteed
internally by using adaptive mesh refinement and starting with a relatively fine mesh
step.
Chapter 3: Split-Ring Resonator Design
25
Consider capacitance first. There are a number of ways to increase capacitance, and
thus lower the resonant frequency. (An SRR designer will almost always be trying to
lower resonant frequency, and so such an approach is taken here.) In the accompanying
figures for each method, the method parameter has been increased, or reduced, if
applicable, by 50% in comparison to figure 3.3.1, while all others are held constant.
𝑓0 = 2.76 𝐺𝐻𝑧
Fig 3.3.2: S-parameter graph showing SRR resonance for a given geometry; 5 mm outer radius, .5 mm
ring break distance, .3 mm gap between rings, .6 mm ring width, .017 mm ring thickness, .7874 mm
dielectric thickness, and a dielectric constant of 2.2ε0. Resonance at 2.763 GHz.
1. Reduce the space between the inner and outer rings. The lower limit is the
breakdown voltage between the rings or the ability to manufacture small gaps, whichever
is larger. However, the capacitance gain by this method is comparatively small, and is
only suitable as a very fine adjustment for large systems.
𝑓0 = 2.43 𝐺𝐻𝑧
Fig 3.3.3: S-parameter graph showing SRR resonance of a geometry identical Fig 3.3.1, with the gap
between rings reduced to .15 mm. Resonance at 2.425 GHz.
Chapter 3: Split-Ring Resonator Design
26
2. Reduce the distance of the break of a single ring. Physically similar to the first
method in that it has no drawbacks or complexities, but limited in scale.
𝑓0 = 2.70 𝐺𝐻𝑧
Fig 3.3.4: S-parameter graph showing SRR resonance of a geometry identical Fig 3.3.1, with the break in
each rings reduced to .25 mm. Resonance at 2.700 GHz.
3. Increase the thickness of the SRR (via thicker cladding on printed circuit material)
Once again, similar to the first two methods. This has the additional effect of marginally
increasing inductance. While this is an advantage, the effect is almost negligible.
𝑓0 = 2.74 𝐺𝐻𝑧
Fig 3.3.5: S-parameter graph showing SRR resonance of a geometry identical Fig 3.3.1, with the
thickness of the rings increased to .0255 mm. Resonance at 2.740 GHz.
4. Increase the outer diameter of the SRR. This is absolutely the best choice, and has
many benefits. Increasing the outer diameter increases the ring circumference, and thus
directly increases both the capacitance per unit length between the rings as well as the
total inductance of the SRR. Additionally, it provides more geometrical space inside the
Chapter 3: Split-Ring Resonator Design
27
SRR to implement other methods that may otherwise be limited by manufacturing
constraints. This method is a very good ‘course adjustment’ for the resonant frequency.
For microwave systems, this method can move the resonant frequency by gigahertz for
relatively small changes in SRR size. The principal drawback, however, is the
subwavelength requirement on SRRs, which can be quite demanding. This requirement is
relaxed in guided systems, and so SRR loaded waveguides can make use of this method
beyond what a bulk SRR material might.
𝑓0 = 1.54 𝐺𝐻𝑧
Fig 3.3.6: S-parameter graph showing SRR resonance of a geometry identical Fig 3.3.1, with the outer
radius increased to 7.5 mm. Resonance at 1.542 GHz.
5. Increase the dielectric constant of the substrate. This is one of the most effective
ways to lower the resonant frequency, while not affecting other SRR parameters at all.
This method can be used effectively to counteract size limitations. However, the
drawbacks here are extensive in the context of a larger system. Firstly, high-dielectric
substrates are often very lossy, causing problems with the storage of energy and limiting
the magnetic response of the SRR. Secondly, dielectric effects must be considered
carefully in waveguide systems. Dielectric loading shifts cutoff frequencies, and may
cause unwanted modes to propagate, or prevent DNG systems from operating at all.
Lastly, SRR designs are often stepping-stones to more complex DNG or SNG systems,
such as complimentary designs or designs for high-power environments. Dielectric
behavior often does not have an analogue in these systems, and thus using a high
dielectric substrate may not be ideal as a starting point.
Chapter 3: Split-Ring Resonator Design
28
𝑓0 = 2.39 𝐺𝐻𝑧
Fig 3.3.7: S-parameter graph showing SRR resonance of a geometry identical Fig 3.3.1, with the
dielectric constant increased to 3.3ε0. Resonance at 2.388 GHz.
6. Increase the thickness of the dielectric substrate. This method has all the
advantages and disadvantages as method 6, but to a lesser degree. Additional care must
be taken for the larger physical profile.
𝑓0 = 2.72 𝐺𝐻𝑧
Fig 3.3.8: S-parameter graph showing SRR resonance of a geometry identical Fig 3.3.1, with the substrate
thickness increased to 1.1811 mm. Resonance at 2.723 GHz.
Next, and much more simply, is changing the inductance. Increasing the inductance
will lower the resonant frequency. This can only be done in two ways, one of which has
been already mentioned in method 4; increasing the outer diameter of the SRR. The
additional note must be made that since inductance is much more difficult to control, then
this method should be the foundation for beginning a new SRR design. Making the rings
as large as possible for a given incident wavelength will save significant time and effort
in the design process.
Chapter 3: Split-Ring Resonator Design
29
Special attention should be paid to the width of the rings. It is well understood that a
broad, flat bar has less inductance than a single wire with the same cross-sectional area.
Thus to increase inductance, the SRRs should be made narrower. However, decreasing
the area of the rings also decreases the capacitance. A balance must be achieved, and
there is a theoretical equilibrium where changing the ring width in either direction results
in an undesirable increase in resonant frequency. This equilibrium point should be sought
if maximal geometrical efficiency is needed.
Finally, losses are a key aspect to creating a suitable split-ring design (Pendry, et al.
1999). Looking at the SRR as an LC resonant circuit, there is a maximum amount of
energy that can be stored in both the inductance and the capacitance. The more energy
that can be stored in the SRR, the greater the magnetic response. However, this maximum
energy is entirely determined by nonlinear loss effects. Care should be taken to minimize
losses where possible. This includes avoiding using resistive metals or lossy dielectrics as
part of the SRR design. The lower the losses, the more freedom is available in other
aspects, such as array density or unit cell size. Too much loss will stop the SRR structure
from functioning all together. Typically copper and low-loss substrates are sufficient to
create the desired magnetic response.
3.4 Array Design
The magnetic response of SRRs is a phenomenon averaged over a volume. Often inside
an SRR’s diameter, the internal magnetic fields are very high, which leads to a strong
response in µ. However, this µ is only valid inside the SRR’s diameter. Since linear
electromagnetics obeys the law of superposition, the true effective µ is the integral mean
of the permeabilities inside the SRR and the local area around the SRR. Let this be the
definition of a unit cell. That is, the unit cell is the local area around an SRR such that
Chapter 3: Split-Ring Resonator Design
30
when the SRR response creates negative effective permeability, the integrated
permeability across the area is also negative. (The unit cell is also typically constructed to
be cubic for ease of analysis.)
An SRR behaves as a magnetic dipole. Though the SRR itself is planar, the dipole
behavior is not. It is more accurate to consider an SRR’s magnetic response in a sphere
concentric with the SRR with equal radius. Thus an SRR unit cell encompasses this
sphere, rather than simply the SRR.
The discussion so far has assumed an infinite repeating plane of adjacent unit cells
such that the bulk µ across the slab volume was negative (for a specific direction and
polarization of incident waves). It should be noted, however, that unit cells need not be
adjacent, and may have some space between them. It is then necessary to define another
term; array density is the number of unit cells per unit volume. This distinction is
important. To give an example, if two systems have a 3x2x3 array of SRR unit cells but
different array densities, they may behave very differently. See Figure 3.3.8.
It seems self-evident that array density should always be maximum, or in other
words, that neighboring unit cells share a boundary. If not, then the free space between
unit cells might compromise the SRRs’ magnetic response. However, there is a limit to
how close SRRs can be spaced. If the array density is too high, the fields of neighboring
resonators will interfere destructively, reducing or destroying the SRRs’ magnetic
response.
Chapter 3: Split-Ring Resonator Design
31
Fig 3.4.1: Transmission parameter graphs of a signal sent through a 3x2x3 SRR array housed in WR284
waveguide. SRRs are identical. (Top) No separation between unit cells in the z-direction. (Bottom) 10
mm separation between unit cells in the z-direction.
Unfortunately, interference between neighboring rings is not well documented,
though simulations shed some light on the behavior. Consider a unit cell that is
constructed as a cube, whose edge length is just larger than the outer diameter of the
contained SRR. Consider also a bulk medium, such as a prism or lens, constructed from
these cells with maximum array density. Simulations show that when illuminated with an
unguided TEM wave, no interference occurs between the rings. However, when this
medium is placed in a rectangular waveguide and illuminated with a TE10 mode,
interference does occur, but only in the direction of propagation (the z-direction). The
array density may be maximum in the x and y directions and not affect the SRR behavior,
but if the density is too high in the z-direction, interference occurs.
Though not explored thoroughly, a possible explanation might involve the mode
behavior. Since SRRs couple to the magnetic field, and the magnetic field of a TE10 mode
Chapter 3: Split-Ring Resonator Design
32
has a z-component, a case could be made that the array density in the z-direction must
take the mode profile into account. This is an important problem to explore for SRR
loaded waveguides. See the future work section for more details.
When designing an SRR array, it is important to consider its final use. In the band
stop filter explored in this work, the array density need not be high. It only needs to be
high enough to block a signal; a relatively simple task as even a small disruption will
reflect the signal. However, the band pass system requires more sophisticated design.
Array density must be as high as possible while still considering the interference effects
which does not present in TEM systems.
33
Chapter 4: Experimental Setup
4.1 Test Stand Parameters and Motivation
The fundamental interest in this project is to study aspects of SRRs that, so far, have been
not been well addressed. Foremost among these is the time behavior of the rings, or more
directly, their excitation behavior. The experiment has several requirements:

It must have SRRs in a well-controlled environment.

The source of excitation, that is, the incident signal, must be well understood,
particularly in field profile and power.

The frequency of the incident signal must be tunable.

It should allow easy access to the SRRs to allow them to be altered or changed.

It should allow for significant geometrical space to add various diagnostics.

Diagnostics should be able to measure both frequency and time information such as sparameters, output power vs. time, output power vs. frequency, and others.

The SRRs, and thus the entire system, must be large enough to be able to be precisely
manufactured with available equipment.
Previously, two similar systems were proposed and developed that explore basic SRR
physics. These designs are a band pass filter and band stop filter (Marqués, Martel, et al.
2002), (Hrabar, Bartolic and Sipus 2005). Using these systems as a starting point, the
Chapter 4: Experimental Setup
34
above concerns were incorporated to design an experiment. The band stop filter is the
simpler of the two systems. In a length of empty waveguide, a small section is loaded
with an SRR array. The SRR resonant frequency is above the cutoff frequency of the
fundamental mode 𝑓𝑐,𝑇𝐸10 , but below the cutoff frequency of any higher order modes.
That is,
𝑓𝑐,𝑇𝐸10 < 𝑓0 < 𝑓𝑐,𝑇𝐸20
Eq. 4.1.1
where f0 is the SRR resonant frequency and 𝑓𝑐,𝑇𝐸20 is the cutoff frequency for the second
propagating mode. In this system, for the band between 𝑓𝑐,𝑇𝐸10 and 𝑓𝑐,𝑇𝐸20 , the TE10 mode
propagates unhindered except for a small bandwidth near f0 where effective permeability
is negative (Hrabar, Bartolic and Sipus 2005).
The pass band system is composed of three consecutive sections of waveguide. The
first section is empty with some fundamental cutoff frequency fc1. The second section has
a smaller cross-sectional area such that its fundamental cutoff frequency 𝑓𝑐2 > 𝑓𝑐1 . This
smaller sections is loaded with SRRs with resonant frequency f0 between the two cutoff
frequencies, that is,
𝑓𝑐1 < 𝑓0 < 𝑓𝑐2
Eq. 4.1.2
In this system, for the band between fc1 and fc2, the TE10 mode does not propagate
except for a small bandwidth near f0 where the system appears DNG. Otherwise, the
signal is evanescent in the cutoff region (Marqués, Martel, et al. 2002), (Esteban, et al.
2005).
These two systems allow for the first two requirements to be well satisfied. Operating
in the intended frequency regimes demands that only TE10 modes can propagate, as all
other modes are evanescent. As long as the sections of empty waveguide are long
enough, no power from competing modes should reach the SRRs, and certainly not the
system’s output. Furthermore, the TE10 mode has a well understood field profile.
Chapter 4: Experimental Setup
35
To understand frequency behavior, the excitation source must be able to operate
across the entire frequency band of interest, and signal power must be measured on the
output. This provides experimental verification on the precise frequency where SRR
interaction occurs, as well as the boundaries for the band of interest. To understand basic
time behavior, a time comparison must be made between the input signal (or trigger) and
output signal.
4.2 Test Stand
Fig 4.2.1: Basic MTBX schematic
The developed test stand has been named the Metamaterial Time Behavior eXperiment
(MTBX). The core of MTBX is the two SRR loaded waveguide filters described above.
Starting with a target frequency of 3.0 GHz, all parts and designs were constructed for
this frequency regime. The band stop filter was constructed first as it is the simpler
system, shown in the figure below. The band stop system consists of three consecutive 1
foot (30.5 cm) sections of standard bronze WR284 waveguide with inner dimensions
2.840 x 1.340 in (7.214 x 3.404 cm) with brass flanges. Both ends are capped with n-type
coaxial to WR284 adapters. The waveguide supports a pure TE10 mode between 2.08
GHz and 4.15 GHz, which is approximately S-band. For a 3 GHz signal, the free space
Chapter 4: Experimental Setup
36
Fig 4.2.2: MTBX band stop waveguide system
wavelength is 10 cm, and for this sized waveguide, the guided wavelength is about 14
cm. The center section is loaded with an SRR array with a resonant frequency near 3
GHz. The array loading will be discussed in detail in section 4.3. Signal should propagate
uninterrupted for the entire band, except for a small region near the SRR resonance. The
entire system is supported on custom made aluminum clamps. The clamps are lined with
insulating foam to electrically isolate the waveguide from the table, and provide support
for the entire experiment. The modular nature of the setup allows for the center section to
be easily replaced with empty waveguide for benchmarking, and for easy adjustment or
replacement of the SRRs.
The modularity also allows the center section to be reconfigured for the band pass
system. In this setup, the left and right waveguide sections remain the same, but the
center section of WR284 is replaced with a section of WR159. This section is 4.5 inch
(11.43 cm) long and has inside dimensions 1.590 x .795 inch (4.039 x 2.019 cm). Its
fundamental mode cutoff is 3.7 GHz, without any dielectric loading. This section is
mated to the larger WR284 sections with a pair of custom made, brass zero-length
Fig 4.2.3: MTBX band pass waveguide system
Chapter 4: Experimental Setup
37
reducer flanges. These flanges add 1.25 inches (3.175 cm) to the cutoff region.
In this system, the individual SRRs are the same as in the band stop system. However,
the array fills the entire WR159 region. It should be noted that the dielectric backing of
the SRRs, as well as the mechanical support structure for the array, loads the WR159
region with some average 𝜀𝑟 > 𝜀0 , reducing the fundamental cutoff frequency. This is a
significant problem for the filter if it is not considered in the initial design.
For this system, a TE10 mode will propagate in the WR284 sections, but is
evenanescent in the WR159 region below the cutoff frequency (3.7 GHz without
loading). However, near the SRR resonance, a backward wave should appear as a
passband. In previous work, this pass band was experimentally verified, but shown to
have significantly reduced power. The reduced power was reasoned to be caused by
reflection at the boundary between the loaded and unloaded regions. To effectively study
the excitation behavior of the SRR arrays in the structures presented above, an incident
signal of variable frequency must be sent repeatedly with the best possible rise time, and
the response observed. To accomplish this, MTBX uses a CW source and a fast switch.
The source is an HP8620C sweep oscillator mainframe with an HP86290C RF plug-in.
The plug-in is capable of operating in several frequency bands between 2.0 and 18.6
GHz. The configuration used in this setup is 2.0 to 6.2 GHz, with variable power up to
~10 mW.
Fig 4.2.4: HP8620C sweep oscillator
mainframe and HP86290C RF plug-in
Fig 4.2.5: Mini-Circuits
ZYSW-2-50DR diode switch
Chapter 4: Experimental Setup
38
To gate the signal, a Mini-Circuits ZYSW-2-50DR PIN diode switch is used. It is
rated for a nominal rise time of 6 ns when triggered by a TTL input and to keep rise times
to a minimum, is mounted directly to the WR284 waveguide. The switch has an insertion
loss of -2 dB for the relevant band and an operating range from DC to 5 GHz.
Finally, the switch is triggered by a Stanford Research Systems DG535 digital
delay/pulse generator, set to generate a TTL square wave. Time resolved diagnostics,
such as a fast oscilloscope, are also triggered off this device. Buffer circuitry is also
placed in line between the trigger generator and the switch to protect and power the
switch and help reduce losses. To summarize, the input side of MTBX is as follows: The
pulse generator and the CW source oscillator connect in parallel to the PIN diode switch.
The switch, in turn, feeds directly into the waveguide system.
Fig 4.2.6: Stanford Research Systems DG535 digital
delay/pulse generator
Fig 4.2.7: Buffer circuit used,
SN74LS244 buffer chip, and LM7805C
regulator
To understand the output signal, several instruments are used on the output side of the
waveguide. Basic frequency vs. power measurements are made with an HP8566B
spectrum analyzer. This is also used to accurately measure the output frequency of the
source if needed. For more sophisticated S-parameter measurements, an HP8510C
Network analyzer with an HP8517B S-parameter test set is used.
Chapter 4: Experimental Setup
39
Time measurements, on the other hand, require
several additional components. Firstly, a detector
converts the signal power to a voltage. MTBX uses
a Krytar 209A zero bias Schottky detector mounted
directly to the waveguide output. The detector has
a small 3 pF output capacitance that affects the
system rise time and has a negative output polarity.
In the beginning of the experiment, the detector
was routed directly into an oscilloscope for timebased measurements. However, the available scope
could not resolve signal forms much below -10
dBm. A Picosecond Pulse Labs 5828A-107 ultrabroadband amplifier was connected inline between
the detector and the oscilloscope. The amplifier
requires a 12 V supply, but provides +10 dB gain
(up to +12 dBm) with an input impedance of 50 Ω.
It also switches the output polarity of the detector.
Fig 4.2.8: Picosecond Pulse Labs
5828A-107 amplifier (top) and Krytar
209A detector (bottom)
The rise time of the amplifier is rated at 15 ps, and thus does not contribute significantly
to the system’s overall rise time.
Ultimately, the output terminates with an oscilloscope. The oscilloscope must be able
to resolve time behavior in the nanosecond regime, and so must have a sampling rate
upwards of 1 GS/s. The scope used in MTBX is a LeCroy Wavemaster 8300A, rated to 3
GHz and 10 GS/s (in 4-channel mode), configured for a 50Ω input termination to reduce
rise times.
It should be noted that care was taken to reduce rise times from external components
as much as possible. Electronics were chosen with low rise times, and low loss, high
Chapter 4: Experimental Setup
Buffer Circuitry
and Switch Power
40
Trigger Generator
Amplifier Power
Scope
Source
Switch
Spectrum
Analyzer
Amplifier
SRR Loaded Waveguide
Detector
Fig 4.2.9: Complete MTBX setup
frequency coaxial SMA cables were used everywhere possible. Care was also taken to
shorten cable lengths to reduce cable capacitance. The time constant for the system is
𝜏 = 𝐶𝑜𝑢𝑡 𝑅𝑙𝑜𝑎𝑑
Eq. 4.2.1
where 𝐶𝑜𝑢𝑡 = 𝐶𝑑𝑒𝑡 + 𝐶𝑐𝑎𝑏𝑙𝑒 + 𝐶𝑎𝑚𝑝 , the capacitances of the detector, cables, and
amplifier respectively. A static addition must also be made to rise time to allow for the
gate switch to open. In all, empty waveguide tests showed a rise time of 20 ns.
4.3 Split-Ring Resonators in MTBX
The split ring resonators went through numerous development stages before a final
design was developed for MTBX. The final design, shown in figure 4.3.1, is the base
SRR design described in chapter 3. This design is printed on Rogers RT5880 dielectric
substrate with εr = 2.2. The material is .031 inches (.7874 mm) thick with an H1/H1
Chapter 4: Experimental Setup
41
cladding (.017 mm nominal thickness soft copper). This SRR unit cell design is used for
both the band pass and the band stop systems. The design dimensions are as follows:

5.0 mm large ring outer radius

4.1 mm small ring outer radius

.6 mm ring width

.3 mm gap between rings

.5 mm break in each ring

12 mm3 cubic unit cell
To create arrays, coplanar SRRs were
constructed together on ‘cards,’ and the cards are
Fig 4.3.1: Final SRR unit cell design
used in MTBX
repeated to fill the waveguide volume. Because of the orientation of the SRR with respect
to the TE10 field profile, coplanar SRRs are oriented on yz-planes, where z is the
direction of propagation. Thus the cards are repeated in the x-direction. Both waveguide
systems use 3 cards across their volume, though the array density and card configuration
is very different.
Fig 4.3.2: An SRR card for the band stop system
Chapter 4: Experimental Setup
42
Fig 4.3.3: SRR cards in band stop system. The WR284 waveguide is
outlined in light grey, and propagation occurs in the z-direction.
Array density in the band stop system was designed to be low. As shown in fig. 3.3.8,
if the array density is too high, the SRRs begin to destructively interfere. Beyond this,
however, it is relatively easy to block a signal with SRRs. A region of negative effective
permeability need not saturate the waveguide, but only provide enough inconsistency to
disrupt the mode (though how much ‘inconsistency’ in required isn’t well known).
Secondly, the array only needs to be long enough to ensure the signal is blocked, and no
longer. As shown in figure 4.3.3, the MTBX band stop system uses a 3x2x3 array to
accomplish this task.
Propagating a signal, on the other hand, is much more difficult. Accomplishing this
requires much more precise array density. An SRR card for the band pass system is
presented in figure 4.3.4. The array density in this system is maximum. While it has been
Fig 4.3.4: An SRR card for the band pass system
Chapter 4: Experimental Setup
43
shown that this is not necessarily desirable, time constraints on the project did not allow
for proper exploration of array density in backward wave pass bands. Instead, the
approximate array density of previous experiments was preserved to demonstrate proof of
concept. As such, while a pass band below cutoff is expected, a significant loss of power
is also expected. Propagating a signal through an SRR array also requires the SRR array
to be continuous for the entirety of the relevant region. Put more simply, while the band
stop only requires a few SRRs to stop a signal, the band pass requires SRRs down the
entire length of the device. The array used in the MTBX band pass system is 14x1x3 and
allows one cell in each card to partially hang out the end of the cutoff section.
Fig 4.3.5: SRR cards in band pass system. The waveguide is outlined in
light grey, and propagation occurs in the z-direction.
Manufacturing of the SRR cards started with 11 x 9 inch sheets of Rogers RT5880
copper clad dielectric substrate with the specifications given above. One surface was
machined with a CNC mill which cut out the SRR pattern directly fed from CST
simulations. Once milled, the cards were cut free with a razor and cleaned. Touch up was
done with under a low-power microscope with a razor to clear any unwanted copper or
debris from the surface, then given a rinse in methanol to remove fingerprints and
machine oil. Once clean, the copper was coated in ink, then acid-resistant tape, then
soaked in muriatic acid. The acid bath cleared all copper from the back face and removed
any remnant copper left from milling errors. Finally, the cards were washed in methanol
Chapter 4: Experimental Setup
44
Fig 4.3.6: Completed SRR card for the band stop system
again to remove the ink. The whole process took about a day for any given card design,
though multiple cards could be manufactured simultaneously. The final cards for each
design are shown in figure 4.3.6 and 4.3.7.
The mechanical structure that supports the SRR cards is of significant importance.
Experiments prior to MTBX notched the top and bottom walls of the waveguide to
support the cards. In an attempt to avoid disrupting surface currents, MTBX instead used
a low-loss, low dielectric foam to support the cards. The foam is Eccostock PP-2 with a
dielectric constant of 1.03 from 60 Hz – 10 GHz. It was fashioned into waveguide sized
bricks, then cut to fit the SRR cards.
Fig 4.3.7: Completed SRR card for the band pass system. The final ring was
cut and reattached while searching for maximum power transmission in the
SRR pass band. The tape does not appreciably effect the SRR behavior.
Chapter 4: Experimental Setup
45
Inserted SRR cards
Fig 4.3.8: Foam brick for insertion into the MTBX band stop system
Fig 4.3.8: Foam brick and SRR cards inserted into the MTBX band pass system
46
Chapter 5: Results and Data
5.1 Simulation Results and Data
5.1.1 Basic SRR Results
The SRR design geometry used in MTBX
and the corresponding simulations was
presented in chapter 4.3. It was the final
SRR design, after several iterations, so that
it could be used effectively in both the
band pass and band stop systems. It proved
successful in this goal.
The original target frequency for SRR
Fig 5.1.1: Final SRR unit cell design used in
MTBX
resonance was to be 3.0 GHz, but as the experiment developed, problems with dielectric
loading of the band pass system made lowering resonant frequency desirable as long is at
it remained in S-band. With the latest SRR iteration, a target of 2.8 GHz or just below
was sought in the physical experiment. Since initial experiments had shown an upward
shift in SRR resonant frequency of about 50 MHz compared to the planar array
simulations, a further adjustment was made. Figure 5.1.2 shows the resonance for an
infinite planar array of these SRRs at 2.763 GHz.
Chapter 5: Results and Data
47
Fig 5.1.2: Simulated S-parameters for the final SRR design used in MTBX.
Reflection (S11) in red, transmission (S21) in blue. Resonance at 2.763 GHz.
The unit cell boundary conditions which allow for the simulation of the infinite array
are frequency domain exclusive. As such, no time-based behavior or excitation could be
simulated. However, field profiles for the fully resonating structure could be observed.
Figure 5.1.3 gives the E-field behavior for each component, and figure 5.1.4 does the
same for the H-field. Note that the incident wave merely excites the SRRs, it does not
affect the internal field behavior. These field profiles remain unchanged regardless of the
polarization or mode of the incident wave, as long as the SRRs are excited in the first
place (some polarizations do not excite the SRRs, for example). As such, these field
profiles are valid for any system using these SRRs.
The color maps for electric field give a clear indication of separation of charge. The
fields in both the x-direction (a) and z-direction (c) show a cumulative zero field across
the unit cell. However, the y-direction (b) shows a net field with a clear dipole behavior
between the inner and outer rings, oriented vertically expected. Similarly, when the
whole of figure 5.1.4 is taken, solenoidal behavior can clearly be seen. (Consider red
regions to represent positive field vectors with respect to the figure’s axis and blue
regions to be negative. Trace the vectors through each figure.) The resonator acts as a
clear magnetic dipole at a specific phase of excitation.
Chapter 5: Results and Data
48
(a)
(b)
Fig 5.1.3: Internal electric field excitation
of the SRRs at 𝜔𝑡 = 𝜋⁄4: (a) Ex, (b) Ey,
(c) Ez.
(c)
(a)
(b)
Fig 5.1.4: Internal magnetic field excitation
of the SRRs at 𝜔𝑡 = 𝜋⁄2: (a) Hx, (b) Hy, (c)
H z.
(c)
Chapter 5: Results and Data
49
The SRRs behavior as an LC resonator is also evident. Figure 5.1.3 was captured at
maximum E-field amplitude, at 𝜔𝑡 = 𝜋⁄4 where figure 5.1.4 was captured at maximum
H-field amplitude, at 𝜔𝑡 = 𝜋⁄2. The creation of negative effective µ demands the wave
interact strongly with the SRR’s internal magnetic field, which exist at integer multiples
of 𝜋⁄2 of the SRR resonance. Thus, it can be deduced that for polarizations where the E
and H-field are in phase, the electric dipole shown in figure 5.1.3 (b) does not interact
with the incident wave. It will always occur at zero-crossings of the external field
amplitude.
A TE10 mode in waveguide has an Ey component, and an Hx and Hz component, thus
special attention should be made to figure 5.1.3 (b) and figure 5.1.4 (a) and (c). These
internal fields will sum with the incident wave when the SRR is fully excited.
5.1.2 Band Stop System
Understanding the time-behavior of SRRs proved to be a significant challenge
computationally. The initial scheme was to use frequency domain simulations to calculate
S-parameters as is shown in figure 5.1.2, but for the band pass and the band stop systems.
Once completed, the solver would be switched to time-domain and a monochromatic
signal with frequency at SRR resonance would be fed to the input port. However,
frequency and time domains used different mesh styles. Frequency domain defines its
mesh around objects in the system, while time domain simply lays a three dimensional
grid through the geometry. The differences in meshing created a problem with agreement
between the solvers, apparently due to convergence problems. In order to use a
monochromatic signal to study resonance, the exact resonant frequency must be known,
but the frequency and time domain solvers must also show the same behavior. A number
of simulations revealed the two solvers to be finding the SRR resonance at different
frequencies, due to the difference in meshing. Without a convergence study, it would be
impossible to get the solvers to agree, and thus nearly impossible to find the right
monochromatic signal, compounded further by the narrow bandwidth of SRR behavior.
Chapter 5: Results and Data
50
Unfortunately, the complexity of the system put simulation time for each run upwards of
several days, weeks in some cases. This made a convergence study impossible in the
allotted time for the filter systems. (The short run times for the infinite planar array
shown previously did allow for convergence to be demonstrated, however.)
Furthermore, the differences in meshing produce very different results at large mesh
sizes. Frequency domain mesh will always show a resonance, if there is one, though its
frequency may be very inaccurate with poor meshing. Time domain meshing, on the
other hand, may not show resonant behavior at all, with poor meshing. Frequency domain
is then the more reliable of the two for S-parameter calculation, but this discrepancy
makes crossing between the domains very difficult.
Fortunately, an alternate method was devised that would find the correct
Fig 5.1.5: Simulated S-parameters for the band stop system.
(Top) transmission, (Bottom) reflection. Resonance at 2.80 GHz.
Chapter 5: Results and Data
51
monochromatic signal, though at the expense of accuracy. Rather than using frequency
domain simulations to find S-parameters, time domain was used. Based on these results, a
monochromatic signal was selected, but the mesh was preserved from the previous
simulation. The method guarantees finding resonant frequency with the monochromatic
signal, but convergence cannot be guaranteed. That said, a very fine mesh was used, and
refined further in the SRR region. Results may or may not be fully converged, but they
closely match the well converged frequency domain results of the infinite planar array.
Figure 5.1.5 shows the S-parameters for the band stop system. The array in this
system is 3x2x3, and shows clear differences to the infinite planar array. Firstly, the
depth of the resonant valley is much greater, dropping below -80 dB. Secondly, the
bandwidth is much greater. It is interesting to note the triple spike formation at lowest
points. While it is indistinguishable from background noise at this level in practice, it was
Fig 5.1.6: Normalized signal vs. time for the stop band system. (Top), input
in red, reflection in blue, (Bottom) input in red, transmission in blue.
Chapter 5: Results and Data
52
a formation that was repeatedly observed through many simulations. Unit cell simulations
revealed it to be related to SRR array elements in the z-direction. As the number of cells
increases in the z-direction, the number of peaks increase, but also grow closer together.
The trend indicated that the peaks would converge at an arbitrarily large number of cells.
However, because the power is so low at that point, 3 unit cells in the z-direction is
adequate to produce the desired result.
From this information, a monochromatic signal of 2.80 GHz was selected, and the
simulation was run again, preserving the meshing. The input signal was 50 ns long, with
a rise time of 2 ns. The input signal, reflection, and transmission vs. time are shown in
figure 5.1.6. The time signals show that some signal initially gets through the array to the
output port, but as the SRR array begins to resonate, the reflected signal increases. In the
region near 25 ns, it can also be seen that the reflected signal surpasses the input signal in
amplitude, suggesting that energy is stored in the SRR array, then released as the system
reaches an equilibrium.
Fig 5.1.7: Field profiles near 40 ns. (Top) E y viewed from the
right, (Bottom) Hx viewed from the top.
Chapter 5: Results and Data
53
Planar field profiles as a function of time were also taken to study the SRR excitation.
The SRRs initially appear transparent, but as their internal fields increase, the array
begins to reflect the signal. E and H-field maximums near 40 ns (with a 𝜋⁄2 phase shift
between them, to show field maximums) are shown in figure 5.1.7 The input signal is
incident from the left, and the color map is linear. The figures show strong standing wave
behavior in the region on the left, SRR excitation in the center, and no visible signal in
the region on the right.
5.1.3 Band Pass System
The band pass system follows the same methodology, including an identical meshing
scheme. Due to the increased complexity and number of array elements of the system,
simulation times were significantly longer. Figure 5.1.8 shows the S-parameters of the
Fig 5.1.8: Simulated S-parameters for the band pass system.
(Top) transmission, (Bottom) reflection. Resonance at 2.73 GHz.
Chapter 5: Results and Data
54
band pass system. The most prominent difference compared to the band stop system is a
downward shift in resonant frequency. One might reason this is due to meshing issues,
but the mesh setup for band pass system is identical to the band stop system. The only
significant difference is the increased number of unit cells which each require regions of
finer meshing than the background. The resonant frequency here is 2.73 GHz, very close
to the infinite planar array.
2.73 GHz was selected for the monochromatic signal, and as with the band stop
system, the input signal has a rise time of 2 ns and a length of 50 ns. Initially, the signal is
almost completely reflected, but as SRRs begin to resonate, more and more signal is
transmitted. It takes ~35 ns to reach maximum transmission, far more than the
corresponding reflection in the band stop system. It might be reasoned that this is due to
the length of the array, and having to excite each consecutive unit cell in order. Field
profiles show this to indeed be the case.
Fig 5.1.9: Normalized signal vs. time for the band pass system. (Top), input
in red, reflection in blue, (Bottom) input in red, transmission in blue.
Chapter 5: Results and Data
55
In simulation, for the sake of run time, the SRRs and waveguide are PECs and the
dielectric backing is lossless. This may affect the rise time of the transmitted signal. As
was discussed in Chapter 3, losses limit the magnitude of the magnetic response of the
SRR, and thus the fill time for the resonators is long. In turn, the rise time for transmitted
signal may be long. On the other hand, maximum excitation is not needed to transmit a
signal, only enough to generate negative effective permeability. However, the increasing
excitation increased the magnitude of effective permeability, resulting in impedance that
changes as a function of time. Regardless of the excitation needed to propagate the signal,
significant energy is stored in the SRR array compared to the field amplitude, and can
result in artificially inflated transmitted or reflected signal as it is released or absorbed.
This effect can clearly be seen in figure 5.1.6, but also more subtly here in the reflected
signal of figure 5.1.9.
Fig 5.1.10: Ey field maps showing negative phase velocity in the SRR array region.
Signal is incident from the left and produces weak standing waves in the left region,
backward waves in the center region, and forward propagating waves in the right region.
Chapter 5: Results and Data
56
It is evident that full signal transmission was not achieved. This may be due to the
difference in impedance between the loaded and unloaded sections of waveguide. The
transmitted signal amplitude at its max is 55% of the input signal, or roughly -2.6 dB.
The field profiles for the monochromatic signal were mapped as well. As predicted by
theory, the signal should experience DNG behavior when the SRRs are fully resonating.
This manifests itself as backward phase velocity. The simulation shows that this is the
case. Field animations showed phase fronts advance in reverse in the loaded region
compared to the incident and transmitted signal. This is shown in figure 5.1.10 which
shows Ey field profiles in series when the system is at maximum transmission. From the
top down, each step is 40 ps apart. In the region on the left, the incident signal arrives to
the SRR array, and is partially reflected, showing weak standing wave behavior. In the
center region, phase can be seen to advance to the left. In the region to the right, signal
can be seen emerging from the SRR array and propagating to the right. Though
experimental measurements to confirm DNG behavior in the loaded section have not yet
been obtained, these simulations show DNG behavior very clearly.
5.2 Experimental Results and Data
5.2.1 Band Stop System
The stop band system was the first constructed, and immediately demonstrated a
significant loss in power near the resonant frequency predicted by simulations. Initial
experiments involved sweeping the input frequency and mapping at what frequencies the
oscilloscope registered a voltage from the detector. These preliminary frequency sweeps
gave an indication of the transmission parameter S21 of the system, which would be later
be measured more thoroughly with a network analyzer, but more importantly demonstrate
the functionality of the filter.
For comparison, the system was also measured without the SRR array or foam
support block. This will be referred to as the control case. The S-parameters of the
control case is provided in figure 5.2.2. In the above figures, at approximately 2.1 GHz,
Chapter 5: Results and Data
57
Fig 5.2.1: Measured transmission (top) and reflection
(bottom) parameters vs. frequency for the band
stop.system
reflected and transmitted power reverse, showing near total power reflection. This
corresponds to the fundamental cutoff frequency of WR284 waveguide, analytically
calculated to be 2.08 GHz.
Naturally, the features of most interest are the valley in transmission and
corresponding peak in reflection at 2.79 GHz. However, comparison to the empty
waveguide S-parameters shows non-transparent behavior in the 3.2-3.8 GHz region. This
manifested on the oscilloscope as a very small dip in transmitted power.
Chapter 5: Results and Data
58
Fig 5.2.2: Transmission (in blue) and reflection (in green) of the band stop
control case (empty waveguide with no SRR array).
The S-parameter measurements provided a good sense of the frequency behavior of
the system. Using this information, the time response of various frequencies of interest
near the SRR resonance was measured. Three frequencies were selected; the array
resonance itself, as well as a point on either side to make comparisons between frequency
regions of array excitation and non-excitation, and to compare time and signal response
to the control system.
Fig 5.2.3: S-parameter data showing sample frequencies for time-based measurements
of the band stop system
Chapter 5: Results and Data
59
(b)
(a)
(c)
Fig 5.2.4: Time response for various frequencies in the SRR loaded band stop system (in
blue) and the empty control case (in green). 2.4 GHz (a), 2.79 GHz (b), and 3.2 GHz (c).
𝑡 = 0 when the trigger generator pulses.
It should be noted that the LeCroy Wavemaster scope used to capture data is linear
with a lower limit around .5 mV. With respect to the source, the scope can register down
to about -12 dB. With the amplifier added, the scope can register down to -35 dB. In fact,
without the amplifier, the small signal present in figure 5.2.4 (b) is not distinguishable
from background noise.
Furthermore, as mentioned in Chapter 4, rise time is dominated by external
components of the system, such as cables and the switch. Rise times shown in the figures
should not be taken to be the signal rise time, but rather the maximum rise time
Chapter 5: Results and Data
60
achievable with the configuration built. To properly interpret the data, comparison to the
control case is needed. For all time-domain figures presented (figures 5.2.4 and 5.2.9),
the scope was triggered from the function generator that simultaneously triggered the
switch. Time zero is when the scope received the trigger signal. The ~25 ns delay is the
time difference between the trigger-to-scope and the trigger-to-system-to-scope. Group
velocity through the waveguide at 2.79 GHz is approximately 2 x 108 m/s, resulting in
about a 5 ns propagation time through the waveguide alone. Thus it is evident that
propagation time through MTBX is dominated by external components.
The signals in figure 5.2.4 demonstrate several key aspects of the band stop system.
(a) shows that below the SRR resonance, the array elements appear completely
transparent to the input signal as there is no difference between empty waveguide and
Fig 5.2.5: Time response of the band stop system at SRR resonance
SRR loaded guide. (c) demonstrates the same behavior as (a) above the SRR resonance
with the small addition that the array is no longer completely transparent. (b) shows the
resonant behavior resolved in time. In this figure, a small bump in power can be seen on
the front end of the pulse, but power quickly fades away, corresponding to a finite
amount of time needed for the SRRs to begin operating at resonance. A detailed image of
this response is given in figure 5.2.5.
Chapter 5: Results and Data
61
The time signal of the control case shows a rise time of about 20 ns, where the time
signal of the stop band shows 35 ns time to reach full function. The last 15 ns of the
signal shows very clear exponential decay. This may be related to the resonant fill time
and quality factor, which will be discussed in chapter 6.
5.2.2 Band Pass System
Preliminary data from the band pass system was not initially encouraging. When the
band pass system was first constructed, the amplifier was not in use, and the pass band
was not detectible through noise. Furthermore, the shift in cutoff frequency due to the
dielectric cards and support structure was much greater than anticipated at more than 500
MHz. However, spectrum analysis revealed a power spike at 2.84 GHz at about -15 dB,
clear on a spectrum analyzer, but invisible on the oscilloscope. The spike was distinct,
but existed in the evanescent region just below cutoff, a frequency region that is not
optimal.
Extensive time was spent making small adjustments to the array position to try to
improve transmission power in the pass band. Eventually, significant power gain was
observed by reversing every other card in the array. The only difference created by
reversing the card is that the outer ring cap now trails the incident wave fronts, instead of
leading it. The reason for this power increase is likely due to a better impedance match
between the empty and loaded sections, though why such a simple change would result in
better matching is unknown. Furthermore, the transmitted power was highest when the
cards alternated in direction, giving some indication that coupling between neighboring
cards is a factor. The data shown in figure 5.2.6 is of the best case observed, with the first
and third cards inserted in reverse.
As in the band stop system, a control was used with no SRR array or foam brick. The
data is shown in figure 5.2.7 is of the empty system. Note the difference in fundamental
cutoff frequencies. The empty system shows a cutoff of approximately 3.8 GHz which
Chapter 5: Results and Data
62
nicely matches the analytical calculation yielding 3.78 GHz. However in the loaded
system, this shifts to below 3.3 GHz, more than 500 MHz less. For some cutoff frequency
fc,
𝑓𝑐 =
2𝜋𝑘𝑐
√𝜇𝑟 𝜀𝑟
Eq. 5.2.1
It is clear from this equation that fc is inversely proportional to the root of εr. Thus it takes
very little change in permittivity to drastically reduce cutoff frequency. For example, if
𝜀𝑟 = 2 for a section of waveguide, fc is reduced by almost 30%.
Fig 5.2.6: Measured transmission (top) and reflection (bottom) parameters vs.
frequency for the band pass system
Chapter 5: Results and Data
63
Fig 5.2.7: Measured transmission (in blue) and reflection (in green) of the band pass
control case (no SRR array).
This dependence was accounted for due to this problem appearing in previous
generations of SRR designs. Analytical calculations accounted for the dielectric of the
substrate and the supporting foam, and put the expected loaded cutoff frequency a just
under 3.7 GHz. Permittivity must have been picked up somewhere in the construction,
likely in the glue used to form the foam into bricks.
Like with the band stop system, the time response of frequencies of interest were
measured. The frequencies are marked in figure 5.2.8. Frequencies of interest here are
below, above, and at the SRR resonance. A fourth measurement was taken in the region
above the fundamental cutoff of the loaded WR159 waveguide.
Fig 5.2.8: S-parameter data showing sample frequencies for time-based measurements
of the band pass system
Chapter 5: Results and Data
64
Fig 5.2.7: Transmission (in blue) and reflection (in green) of the band stop
control system (no SRR array)
(a)
(b)
(c)
(d)
Fig 5.2.9: Time response for various frequencies in the SRR loaded band stop system (in
blue) and the empty control case (in green). 2.7 GHz (a), 2.84 GHz (b), and 3.0 GHz (c).
Figure (d) shows time response for a chosen frequency above cutoff. Since the cutoff varies
between the loaded and control systems, the loaded system trace (in blue) is at 3.6 GHz, and
the control system trace (in green) is at 4.0 GHz.
In figure 5.2.9, both traces in (a) and (c) appear as expected, no signal propagates
below the cutoff frequency of the WR159 waveguide. However, trace (b) shows some
power being propagated below cutoff at the SRR resonance as predicted by the Sparameters. Furthermore, in comparison to propagation above cutoff in (d), there is an
approximately 10 ns delay. Furthermore, the rise time is clearly slower in the SRR pass
band.
65
Chapter 6: Conclusions
6.1 Discussion
Before a meaningful comparison between simulation data and experimental data can be
given, a few differences must be noted. As mentioned previously, the complexity of the
system pushed simulation run time to several days. Because of this, losses were removed
from the system. Metal components were replaced with PEC and dielectrics replaced
with lossless variants. Likewise, the foam support bricks were not included in the
simulation. The foam itself is rated for 𝜀𝑟 = 1.03, and beyond this, only the glue within
the brick might affect dielectric strength. As such, the bricks were disregarded.
The final difference involves the adjustments to the band pass system in an attempt to
maximize transmission power. The orientation of the two outer cards are reversed in the
experiment, which clearly increased transmission power, but for unclear reasons.
Adequate time was not available to test this configuration in simulation.
6.1.1 Resonant Frequency Comparisons
There are five systems to consider; the band pass configuration in simulation and in
experiment, the band stop configuration in simulation and in experiment, and the
simulated infinite planar array. Note that the SRRs in every system are the same.
Chapter 6: Conclusions
66
The resonant frequencies of each case are:
Band Stop, Simulation
2.80 GHz
Band Stop, Experimental
2.79 GHz
Band Pass, Simulation
2.73 GHz
Band Pass, Experimental
2.84 GHz
Infinite Planar Array
2.76 GHz
The most significant difference among these is between the simulated and
experimental values for the band pass system at 110 MHz, representing both the
minimum and maximum values. However, it was observed that moving the SRR array or
foam block even a small distance could change this value by several tens of megahertz. It
stands to reason that the different boundary conditions between the system may affect the
SRR array’s resonance. Some indication of that may be represented by the difference in
values between the simulated data because of the similarity of the simulation schemes.
Further deviation in resonance may have resulted from the capacitance added to the SRRs
in the experiment by the foam support brick and/or the glue used to hold the brick
together. Whatever effect the boundary conditions have on SRR resonance, it cannot be
understood by this data. The mechanical precision of the experiment is not high enough
to address the question.
Conversely, the consistency between these results is a clear reinforcement of SRR
resonant behavior. Resonant behavior was required to create specific attributes, such as
the pass band and the stop band, which varied from system to system. Each system
performed as expected or better.
6.1.2 Time Behavior
Chapter 6: Conclusions
67
The different systems provide different information about various aspects of excitation
behavior. First, consider the band-pass system. Figures 6.1.1-6.1.5 depict the time
responses in of the band pass system under various conditions. Characterize the time
response of the switch and detector/amp, the switch was attached directly to the detector,
and is represented by the blue trace in these figures, and though it was measured at 3.0
GHz, it accurately represents the entire recommended WR284 band (2.6 to 3.95 GHz) As
a benchmark, the data for the control case in figure 5.2.9 (d) was plotted in green. Again,
this trace is the time response of the band pass configuration with no SRR array, only
empty waveguide. The signal frequency is above cutoff for the smaller WR159
waveguide. Finally, the red trace is the time response of the pass band due to SRR
resonant behavior, also shown in in figure 5.2.9 (b).
Fig 6.1.1: Transmitted signals vs. time of
various configurations, (blue) 3.0 GHz,
(green) 4.0 GHz, (red), 2.84 GHz.
In figure 6.1.1, the delay between the blue and green traces represents the propagation
time for the signal to propagate down the waveguide. The final amplitude difference is a
direct result of the reflections at the step between the WR284 and WR159 waveguide
sections, and is not present when compared to an empty section of WR284 of the same
length. The blue trace shows clear linear behavior, likely due to the turn-on time of the
Chapter 6: Conclusions
68
switch itself. The green trace is similarly linear, except for an exponential roll-off as it
reaches maximum amplitude. This roll-off may be due to dispersion or the probe
excitation of the waveguide itself. The system described by the green trace contains no
SRRs and so this roll-off cannot be an effect of the SRR array. Finally, the red trace,
which shows functioning SRR arrays, shows both an exponential rise and roll-off. The
exponential rise is directly due to the SRRs.
Fig 6.1.2: Normalized transmitted signals of
various configurations, (green) offset by -6 ns,
(red) offset by -14 ns and multiplied by 2.3
To give further indication of rise times and forms, figure 6.1.2 shows the same traces
in the previous figure, but with the red trace normalized in amplitude. The traces are also
time shifted, such that each rise time begins at 20 ns. The blue trace is not shifted, the
green is shifted -6 ns, and the red is shifted -14 ns. The initial linear rise on the green and
blue trace is evident, while the red is clearly nonlinear.
Chapter 6: Conclusions
69
Fig 6.1.3: Natural log transformations of
normalized transmitted signals of various
configurations, (green) offset by- 6 ns, (red) offset
by -14 ns and multiplied by 2.3
To verify the forms of the signals, they were transformed by their natural logs, shown
in figure 6.1.3. For such a transformation, linear functions should appear exponential, and
vice versa. This is indeed what appears. The without-waveguide and empty-waveguide
traces appear exponential, except near the top of the waveform, where the emptywaveguide trace becomes linear. The SRR pass band trace, however, shows at least three
linear slopes before reaching maximum amplitude. (Note that the offset in the red trace
Fig 6.1.4: Normalized transmitted signals of
various configurations, (green) offset by -6 ns,
(red) offset by -26 ns and multiplied by 2.3 to
show form consistency.
Chapter 6: Conclusions
70
Fig 6.1.5: Natural log transformation of SRR pass
band signal, offset by -14 ns. Blue and black linear
fits, with calculated time constants.
between 0-20 ns is due to the normalization factor). The third is explained easily by the
waveguide roll-off effect. Figure 6.1.4 shows how closely the SRR pass band and emptywaveguide forms match at later times, approaching maximum.
The first two, however are results of the SRR array and are plotted in figure 6.1.5.
The slope m of these lines are inversely proportional to the exponential time constants of
the phenomena, that is,
𝜏=
1
𝑚
Eq. 6.1.1
and yield time constants of 4.3 ns (blue) and 8.0 ns (black).
Both rise time and signal delay time may shed light on the transmission behavior and
are calculated from the experiment as follows:
Signal Delay
Rise Time
Without Waveguide
20 ns
15 ns
Empty Waveguide
26 ns
24 ns
SRR Pass Band
34 ns
40 ns
Chapter 6: Conclusions
71
6.1.3 Quality Factor and Rise Time
The experimental data of both the band stop (figure 5.2.4) and band pass (figure 5.2.9)
filters shows clear time behavior that can only be attributed to SRR excitation time. In the
band stop system, 35 ns were required for the SRRs to resonate strongly enough to stop
all signal. While the RC rise time of the system is 24 ns, it does not affect the presence or
absence of signal, only its shape. Some time may be attributed to the rise time of the
switch, but even so, the 35 ns signal measured experimentally is greater than the signal
time predicted by the simulations (figure 5.1.6) at 18 ns. Although time did not allow,
using an array only one cell deep may provide clear information on excitation time for a
single resonator.
Conversely, the rise time of the pass band was shown to be significantly faster than
simulation indicates. Rise time in the simulation is about 35 ns (figure 5.1.9) where the
delay time between below- and above-cutoff signals in the pass band is only 10 ns, with
the additional note that the experimental rise time also appears to be a few ns slower than
the control case (figure 5.2.9).
The field maps of the two systems showed that for the band stop system, only the first
resonator group in the array appears to be saturated. In the band pass system, however,
every resonator appears to be saturated. The reason for this is self-evident, but it clearly
effects the rise times of each system.
Understanding excitation behavior is further convoluted by the impedance difference
between the loaded sections and unloaded sections of waveguide. Full power will not be
coupled into the SRR array of any signal unless the sections are matched, a prospect that
has not yet been addressed. Furthermore, if the effective permeability is a function of
time, as suggested in the previous paragraph, impedance is also a function of time. Thus
effects of impedance mismatch and resonant fill time cannot easily be deconvolved.
Chapter 6: Conclusions
72
As might be expected, the resonant behavior of the SRR has some quality factor Q
which corresponds to both bandwidth and fill time as follows,
𝑄=
𝜔0
∆𝜔
Eq. 6.1.2
𝑄=
2𝜏
𝜔0
Eq. 6.1.3
where ω0 is the resonant frequency of the SRRs, Δω is the -3 dB bandwidth, and τ is the
time constant for the rise time. Data for the experimental systems both showed similar
results, yielding an estimate of ∆𝜔 ≈ 40 𝑀𝐻𝑧. The Q for both experimental systems was
calculated to be about 70, resulting in a time constant τ of 45 ns. The clearest
interpretation of this information comes from comparisons made with the band stop
system. In the experiment, signal was received by the detector for a total time of 35 ns
(figure 5.2.5), this represents the total time needed for the SRR array to fully interact with
the incident signal. Simulations show a rise time on the reflected signal of 18 ns (figure
5.1.6), which represents the same behavior. However, from equation 6.1.2, τ was
calculated to be 45 ns. Furthermore, the definition of τ states that for an exponential rise
time, τ is the time needed to rise 63.2% of the final signal amplitude. This indicates a
minimum total rise time of 70 ns. This is significantly longer than the total rise time of
both the simulation and experiment.
However, there a number of additional considerations. Quality factor is the ratio of
power dissipated to power stored in a resonator, and thus is heavily effected by losses.
The simulations did not include losses and are likely to have an inflated Q. Furthermore,
there is also an assumption that the SRRs must be fully resonating to propagate a signal.
As shown in Chapter 2, the SRRs merely need to have strong enough internal fields to
create negative effective permeability, where Q only gives information about maximum
fields. Considering this and the rise time information previously discussed, it is likely that
Q is indeed a factor in SRR excitation, but by itself is insufficient to provide a complete
physical picture. Furthermore, since at least two exponential curves were noted in
experimental data (figure 6.1.5), a single Q dependence is less likely.
Chapter 6: Conclusions
73
6.1.4 An Attempt at a Physical Explanation
The stated goal of this work is to provide a foundation on which to explore time behavior
thoroughly. With this in mind, the first attempts at explaining SRR excitation may be
taken. It should be noted that this section is conjecture, and though it is supported by
experiments and simulation, it cannot yet be asserted as fact.
When the first half-cycle of a wave passes over an SRR near resonant frequency, the
SRR responds diamagnetically. The initial SRR response is only that of a simple closed
current loop, but this diamagnetic current separates charge in the capacitive regions of the
SRR, and stores that diamagnetic energy as a local electric field. On the second halfcycle, the loop responds diamagnetically again, but this time the current is reinforced by
the energy being released by the capacitance that was absorbed in the first half cycle. At
the end of the second half-cycle, the total energy is once again absorbed by the
capacitance. In the third half-cycle, the diamagnetic response is reinforced further by the
energy released by the capacitance, and so on.
Thus the magnetic response of an SRR at any given half cycle is then the sum total of
the diamagnetic responses of all previous half-cycles, modulated by losses. Therefore, the
maximum energy absorbed by an SRR is capped by the maximum energy that can be
stored in its capacitive regions, itself determined by electrical losses. This is when the
SRR reaches saturation, and this saturation time is definitionally predicted by quality
factor.
However, as stated in chapter 2, magnetic permeability becomes negative when the
induced internal field exceeds the external applied field. This point may be far earlier (in
time) than when the SRR reaches saturation. This is indicated clearly in the band stop
system. The rise time is much faster than what is predicted by quality factor, and
according to simulation, the SRR array continues to absorb energy even after signal
transmission disappears.
Chapter 6: Conclusions
74
This notion is also necessary to understand propagating systems, such as the band
pass filter. In order to pass the signal to the next element in the array, an SRR must have
enough energy to achieve negative μ. However, the first SRR may not be saturated and
will continue to absorb energy from the incident signal. Though the incident signal will
propagate passed the first element, it’s amplitude will be reduced. Not until the first
element reaches full saturation will the signal be able to propagate to the next element
with maximum field amplitude. This is supported by the simulation data, where at long
times, the entire band pass array appears to be resonating, but does not pass signal to the
output.
Thus two rise times appear: first, for an individual element to produce negative
permeability; second, for the an individual element to reach saturation. It is reasonable to
assume these rise times will be convolved together, and it may be very difficult to
separate them. The experimental data seems to indicate two rise times correlated to the
SRR array. Furthermore, this understanding also suggests several possible experiments,
such as reducing or increasing losses to bring these two rise times together or further
apart.
6.2 Summary and Conclusion
A basic test stand has been developed and constructed to provide basic understanding of
split-ring resonator array excitation, both experimentally and computationally. A
geometric understanding of SRR resonance was gained, and used to develop an SRR unit
cell with a specific resonant frequency. This unit cell was then synthesized into various
arrays designed to propagate or bock a given signal.
In simulation, frequency domain solvers showed an asymptotic drop in transmission
power at a given frequency and its harmonics for these SRRs, both as an individual unit
cell, and as part of an array in propagating waveguide. Reflected power measured in these
Chapter 6: Conclusions
75
simulations demonstrated that the SRR array was reflecting power, rather than absorbing
it, which is indicative of an evanescent mode. Furthermore, an evanescent mode is
indicative of negative constitutive parameters, and in this case, negative permeability.
The system modeled in simulation was then constructed. Measured data was consistent
with simulations.
The SRRs developed for the band stop system and well simulated in CST were used to
create a band pass system, which theoretically made use of DNG properties. Simulations
for this system proved to be challenging and insufficient knowledge of the CST software
lead to many inconclusive results. Frequency and time domain simulations indicated
resonant behavior in the expected frequency region, but poorly converged results, long
simulation times, and the narrow bandwidth of SRR resonant behavior made finding the
pass band very difficult. The band pass system was constructed alongside simulation
work and quickly demonstrated the existence of a pass band in the expected region.
Though significant attenuation was expected, the MTBX band pass system presented
only about -5 dB attenuation. This allowed linear, time-resolved diagnostics, such as
oscilloscopes to be utilized effectively. High quality time simulations were ultimately
able to resolve the resonant behavior, and very clearly showed backward advancing phase
fronts through the cutoff region, conclusively demonstrating DNG behavior.
The ultimate goal of this work is to provide a foundation for understanding time response
and excitation behavior of SRR arrays. The simulation work provides a solid foundation
by showing time-resolved excitation signals and field profiles through both systems.
Furthermore, the MTBX setup conclusively showed proof of concept for the band pass
and band stop configurations for the in-house designed SRR arrays. Time-resolved
diagnostics show a clear time delay in both filter configurations, indicating SRR
excitation behavior. Calculation of quality factor did not yield sufficient information to
draw a precise conclusion about rise time with respect to fill time. Q is indeed a factor,
but how significant is not yet clear.
Chapter 6: Conclusions
76
6.3 Future Work
The interplay between experimental results and simulation data has yielded a surprising
number of questions that must be answered on the way to making efficient SRR based
devices. While the band pass and band stop systems have been successfully
demonstrated, a great deal of work must be done even to optimize even these relatively
simple geometries. Furthermore, the large task remains of applying the understanding
gained from MTBX to take steps toward SRR based high-power devices.

For MTBX itself, a number of specific questions remain to be explored. Most
prominently is experimental verification of negative phase velocity in the band pass
system. Time based simulations have verifies negative propagating phase fronts in
this region, but experimental confirmation has yet to be achieved. To do this, phase
probes are to be positioned in the empty regions before and after the filter, as well as
inside the filter if possible and required.

Impedance measurements of both systems must be taken. Understanding the
impedance of a given SRR array is critical for many designs and functions, and will
be an important step in toward matching an SRR based DNG system to a
conventional system. Further complicating this task is the fact that SRRs behave
highly nonlinearly as a function of both frequency and time. To accomplish
impedance measurements, VSWR measurements via a slot-line or probes will provide
input impedance vs. frequency data.

Improving transmission power of the band pass system is of interest. So far, SRR
backward wave transmission lines have shown significant insertion loss due to
impedance mismatch between conventional and DNG regions of the line. This work
has shown significantly improved power transmission compared to previous devices,
but reflected power is still a problem. Suggestions to improve this involve varying the
locations of the first and last SRR array elements or tapering the WR284-WR159
interface.
Chapter 6: Conclusions

77
Array density considerations must be explored. The array density may have
significant impact on impedance, bandwidth, and excitation behavior for the entire
array in both system. Array density simulations for the band stop system were
presented in Chapter 3, opening many questions not previously considered.
Simulations with varying array density should be conducted in the band pass system,
and experiments for both systems should be conducted. This can be achieved easily
by manufacturing new cards with larger or smaller array densities and measuring sparameters and time response.

Raising cutoff frequencies of the band pass system should be explored. Most
importantly, the data acquired in this work shows the pass band uncomfortably close
to the WR159 fundamental cutoff frequency. To rectify this, and provide more insight
into questions about theoretical minimum transmission line size, the physical
dimensions of the WR159 section should be further reduced to WR137 or WR112
while keeping the SRR resonance the same. This would elevate the waveguide cutoff
and move the backward pass band out of the evanescent region.

Mode studies should be conducted. One fundamental interest in SRR arrays is the
ability to filter specific modes. The SRR arrays used in MTBX are designed
specifically to couple to a TE10 mode. Higher order mode coupling should be studied
specifically in regard to TE vs. TM modes. MTBX could be easily configured to
study this by designing SRRs to operate in a frequency regime that propagates TE and
TM modes simultaneously in WR284 waveguide.

Continuing work with simulations should be performed. As highly resonant
structures, many field solvers, such as CST, have problems resolving field behavior.
Small mesh sizes, resonant solver modes, high degree of accuracy, and simulation
convergence require simulation times upwards of a week, even on very powerful
computers. While the simulation data provided here gives a good foundation, many
desired simulations were prohibited simply by lack of time and/or computing power.

Further work on time-based excitation is required. This was the focus of this project
to begin with. The foundation is now built, with the physical experiment constructed
and months of simulation finished. However, a robust understanding of basic SRR
Chapter 6: Conclusions
78
excitation behavior in both frequency and time domains is still a long ways out. A
simple, but effective way to the study excitation of a single SRR would be by using
an array only one element deep in the z-direction in the band stop system.
79
References
Baena, J. D., et al. "Equivalent-Circuit Models for Split-Ring Resonators." IEEE
Transactions on Microwave Theory and Techniques 53, no. 4 (2005): 1451-1461.
Baena, J. D., R. Marqués, and F. Medina. "Artificial Magnetic Metamaterial Design by
Using Spiral Resonators." Physical Review B 69 (2004): 014402.
Balanis, Constantine A. Advanced Engineering Electromagnetics. John Wiley & Sons,
Inc., 1989.
Esteban, J., C. Camacho-Peñalosa, J. E. Page, T. M. Martín-Guerrero, and and E.
Márquez-Segura. "Simulation of Negative Permittivity and Negative Permeability
by Means of Evanescent Waveguide Modes—Theory and Experiment." IEEE
Transactions on Microwave Theory and Techniques 53, no. 4 (2005): 1506-1514.
Foteinopoulou, S., E. N. Economou, and and C.M. Soukoulis. "Refraction in Media with
a Negative Refractive Index." Physical Review Letters 90, no. 10 (2003): 107402.
Hrabar, S., J. Bartolic, and and Z. Sipus. "Waveguide Miniaturization Using Uniaxial
Negative Permeability Metamaterial." IEEE Transactions on Antennas and
Propagation 53, no. 1 (2005): 110-119.
Jackson, John David. Classical Electrodynamics, 3rd ed. New York, NY: John Wiley &
Sons, Inc., 1999.
Marqués, R., F. Mesa, J. Martel, and and F. Medina. "Comparative Analysis of Edge- and
Broadside-Coupled Split Ring Resonators for Metamaterial Design—Theory and
Experiments." IEEE Transactions on Antennas and Propagation 50, no. 10
(2003): 2572-2581.
References
80
Marqués, R., J. Martel, F. Mesa, and and F. Medina. "Left-Handed-Media Simulation and
Transmission of EM Waves in Subwavelength Split-Ring-Resonator-Loaded
Metallic Waveguides." Physical Review Letters 89, no. 18 (2002): 183901.
Marqués, Ricardo, Ferran Martin, and Mario Sorolla. Metamaterials with Netative
Parameters. John Wiley & Sons, Inc., 2008.
Pendry, J. B., A. J. Holden, D. J. Robbins, and and W. J. Stewart. "Magnetism from
Conductors and Enhanced Nonlinear Phenomena." IEEE Transactions on
Microwave Theory and Techniques 47, no. 11 (1999): 2075-2084.
Pozar, David M. Microwave Engineering, 4th edition. John Wiley & Sons, Inc., 2012.
Shamonina, E., and and L. Solymar. "Metamaterials: How the subject started."
Metamaterials, no. 1 (2007): 12-18.
Shelby, R. A., D. R. Smith, and and S. Schultz. "Experimental Verification of a Negative
Index of Refraction." Science 292 (2001): 77-79.
Shiffler, D., R. Seviour, E. Luchinskaya, E. Stranford, W. Tang, and and D. French.
"Study of Split-Ring Resonators as a Metamaterial for High-Power Microwave
Power Transmission and the Role of Defects." IEEE Transactions on Plasma
Science 41, no. 6 (2013): 1679-1685.
Smith, D. R., D. C. Vier, N. Kroll, and S. Schultz. "Direct Calculation of Permeability
and Permittivity for a Left-handed Metamaterial." Applied Physics Letters 77, no.
14 (2000): 2246-2248.
Smith, D. R., D. Schurig, and and J. B. Pendry. "Negative Refraction of Modulated
Electromagnetic Waves." Applied Physics Letters 81, no. 15 (2002): 2713-2715.
Valanju, P.M., R. M. Walser, and and A. P. Valanju. "Wave Refraction in NegativeIndex Media: Always Positive and Very Inhomogeneous." Physical Review
Letters 88, no. 18 (2002): 187401.
Veselago, V. G. "The Electrodynamics of Substances with Simultaneously Negative
Values of Espilon and Mu." USPEKHI 10, no. 4 (1968): 509-514.