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University of South FloridaScholar Commons
Graduate Theses and Dissertations Graduate School
November 2017
Manipulating Electromagnetic waves withenhanced functionalities using Nonlinear andChiral MetamaterialsSinhara Rishi Malinda SilvaUniversity of South Florida, [email protected]
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Scholar Commons CitationSilva, Sinhara Rishi Malinda, "Manipulating Electromagnetic waves with enhanced functionalities using Nonlinear and ChiralMetamaterials" (2017). Graduate Theses and Dissertations.https://scholarcommons.usf.edu/etd/7443
Manipulating Electromagnetic waves with enhanced functionalities using
Nonlinear and Chiral Metamaterials
By
Sinhara Rishi Malinda Silva
A dissertation submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy Department of Applied Physics
College of Arts and Sciences University of South Florida
Major Professor: Jiangfeng Zhou, Ph.D. Myung K Kim, Ph.D. Denis Kariskaj Ph.D Zhimin Shi, Ph.D.
Date of Approval: October 26 2017
Keywords: Bi-stability, Harmonics, Optical Activity, Negative refraction Copyright © 2017, Sinhara Rishi Malinda Silva
DEDICATION
To my loving parents: Quintus Silva and Kamala Silva
ACKNOWLEDGEMENT
First, I would like to express my sincere gratitude to my advisor Dr. Jiangfeng Zhou for the
unwavering support throughout my Ph.D research. His suggestions, patience and knowledge
was invaluable throughout the years. Thank you very much for the guidance and advise given
me to improved my professional and academic skills during my period as a graduate student.
I would like to thank the rest of my thesis committee: Dr. Myung K Kim, Dr. Zhimin Shi, Dr.
Dr. Denis Kariskaj and the chairperson of the committee, Dr. S. Ma (Department of
Chemistry) for their insightful comments and encouragement.
Second, I am thankful to lab colleagues Khagendra Bhattarai, Clayton Fowler and Alexander
Shields for research collaboration, valuable discussions, sharing experiences, and helping me
whenever needed. I would like to thank all the faculty and staff members in the Department
of Physics for their assistance throughout my time at USF.
I would be delighted to thank my family: my parents, my sister, aunt, uncle and cousins for
their constant support, love and inspiration throughout my journey. Finally, I would like to
thank my lovely wife and my bundle of joy, my little son, Mihika.
i
TABLE OF CONTENTS
LIST OF FIGURES .......................................................................................................................... iii ABSTRACT ....................................................................................................................................... vii INTRODUCTION TO METAMATERIALS............................................................................... 1
1.1 Summary ................................................................................................................................ 1
1.2 Electromagnetic properties of materials ........................................................................... 1
1.3 Metamaterials ........................................................................................................................ 4
1.4 Split Ring Resonator (SRR) ................................................................................................ 8
1.5 Nonlinear Metamaterials ................................................................................................... 11
1.6 Chiral Metamaterials .......................................................................................................... 13
1.7 Metamaterials in Terahertz frequency regime. ............................................................... 15
1.8 Terahertz time domain spectroscopy (THz-TDS) ........................................................ 17
1.9 Modeling and simulation of metamaterials ..................................................................... 20
1.10 Retrieval Method used for chiral metamaterials ............................................................ 20
1.11 Sample fabrication .............................................................................................................. 22
TUNABLE BISTABLE NONLINEAR METAMATERIALS ................................................ 24
2.1 Summary .............................................................................................................................. 24
2.2 Design and the experimental set up ................................................................................ 25
2.3 Hysteresis and Bistability................................................................................................... 27
2.4 Using double split ring resonator to control tune the bi-stability. .............................. 31
2.5 Transition time and Switching device ............................................................................. 32
2.6 Nonlinear tunable metamaterials in Terahertz frequency regime. ............................... 36
ENHANCING HARMONIC GENERATION USING NONLINEAR METAMATERIALS ........................................................................................................................ 40
3.1 Summary .............................................................................................................................. 40
3.2 Introduction ........................................................................................................................ 41
ii
3.3 Experimental results .......................................................................................................... 43
3.4 Applying an external reverse bias voltage to the unit cell. ............................................ 48
3.5 Using a back-plate to increase the harmonic power ...................................................... 50
3.6 The arrays of DSRR unit cells .......................................................................................... 51
3.7 A model for the DSRR structure ..................................................................................... 54
CHIRAL METAMATERIALS ....................................................................................................... 56
4.1 Summary .............................................................................................................................. 56
4.2 Introduction ........................................................................................................................ 56
4.3 Experimental Design ......................................................................................................... 58
4.4 Mechanism of Chirality of 2D-HMM ............................................................................. 59
4.5 Equivalent Wave Equation in Bi-Isotropic Medium ..................................................... 62
4.6 Results .................................................................................................................................. 64
MAGNETICALLY TUNABLE TERAHERTZ METAMATERIALUSING POLYMERIC MICROACTUATORS ..................................................................................................................... 75
5.1 Introduction ........................................................................................................................ 75
5.2 Simulation results ............................................................................................................... 77
5.3 Experimental results .......................................................................................................... 81
CONCLUSION AND FUTURE WORK .................................................................................... 83
6.1 Conclusion .......................................................................................................................... 83
6.2 Future work......................................................................................................................... 85
6.2.1 Wave front control by nonlinear metamaterial in terahertz frequency. .............. 85
6.2.2 Second Harmonic generation using phase matching ............................................. 86
6.2.3 Enhanced second harmonic generation in THz frequency regime. .................... 88
REFERENCES ................................................................................................................................. 89 LIST OF PUBLICATIONS .......................................................................................................... 102
iii
LIST OF FIGURES
Figure 1-1 : A schematic showing the classification of materials based on the electric and magnetic properties. ............................................................................................................................ 5 Figure 1-2 : a) Wire structure b) SRR structure c) effective permittivity dependence d) effective permeability dependence .............................................................................................. 6 Figure 1-3 : Realizing negative refractive index ............................................................................... 7 Figure 1-4 : Split Ring Resonator (SRR) as a LC circuit ................................................................ 8 Figure 1-5 : The excitation of the SRR using external (a) electric field (b) magnetic field ....... 9 Figure 1-6 : Induced current modes of the SRR in electric field excitation. ............................. 10 Figure 1-7 : (a) Surface mount varactor diode - SMV1231-079LF (b) Dependence of capacitance of a varactor diode with its reverse bias voltage ...................................................... 12 Figure 1-8 : The electromagnetic spectrum and THz gap is circled in red............................... 15 Figure 1-9 : THz-Time Domain Spectroscopy system ................................................................. 18 Figure 1-10 : THz-TDS experimental set-up ................................................................................ 18 Figure 1-11 : THz-Time Domain reflection Spectroscopy system ............................................. 19 Figure 1-12 : Steps in Photolithography ......................................................................................... 23 Figure 2-1 : Mechanism of the SRR switching device. ................................................................. 24 Figure 2-2 : Experimental set-up ..................................................................................................... 25 Figure 2-3 : (a) Unit cell structure (b) surface current distribution (c) Resonance ................... 26 Figure 2-4 : Resonance frequency dependence for continuously increasing (red) and decreasing (blue) external signal frequency with constant power Ppump=-25dBm ................ 27
iv
Figure 2-5 : (a) Resonance frequency dependence on pump signal frequency (b) Resonance frequency dependence on pump signal power. ............................................................................. 28
Figure 2-6 : (a) Resonance frequency dependence with the angle of the E filed (Black: E field along the gap of the structure, Green: E field is perpendicular to the structure) ..................... 30 Figure 2-7 : (a) Double split ring (DSRR) design (b) Resonance frequency of the DSRR. .... 31 Figure 2-8 : (a) Resonance frequency dependence on pump signal frequency for DSRR (b) Resonance frequency dependence on pump signal power. ................................................... 32 Figure 2-9 : Trigger transitions between the two states .............................................................. 33 Figure 2-10 : Transition time dependence with the resistance .................................................... 34 Figure 2-11 : Optimized time between transition states ............................................................. 34 Figure 2-12 : Varactor diode design for the terahertz applications ............................................ 37 Figure 2-13 : Tunability of the resonance frequency in terahertz frequency regime ................ 37 Figure 2-14 : Experimental results of SRR structure using THz-TDS ...................................... 38 Figure 3-1 : Enhancing second harmonic using (a) SRR (b) DSRR ........................................... 42 Figure 3-2 : Experimental setup with Vector Network analyzer and two horn antennas ....... 43 Figure 3-3 : (a) SRR and DSRR structures (b) Transmission of the SRR and DSRR .............. 44 Figure 3-4 : Outer ring frequency (f1) and inner ring frequency (f2) .......................................... 45 Figure 3-5 : Electric field distribution of the DSRR (a) At the outer ring frequency (b) inner ring frequency .................................................................................................................................... 45 Figure 3-6 : (a) comparison of the enhancement of the second harmonic power (b) the dependence of the second harmonic power to the inner ring size. ............................................ 46 Figure 3-7 : Third harmonic signal power comparing SRR and DSRR structures ................... 47 Figure 3-8 : Using an external bias voltage to tune the resonance frequencies to find the maximum second harmonic ............................................................................................................. 48 Figure 3-9 : (a) S21 (b) Second harmonic (c) third harmonic tuning using an external signal power................................................................................................................................................... 49
v
Figure 3-10 : Using a back-plate to increase the second harmonic power ................................ 50 Figure 3-11 : Interference of second harmonic signal from two unit cells of DSRR .............. 51
Figure 3-12 : Second harmonic of an array of DSRR unit cells .................................................. 53 Figure 3-13 : (a) Equivalent circuit of SRR structure (b) Equivalent circuit of DSRR structure .............................................................................................................................................. 54 Figure 4-1 : Design of the 2D-HMM (a) Schematic diagram and (b) photograph of the chiral metamaterial. ...................................................................................................................................... 59 Figure 4-2 : (a) Surface current distribution of 2D-HMM at 3.4 GHz. (b) Schematic drawing
of the current. (c) Calculated polarization vector, P ⃗, and magnetization vector, M ⃗. Here, Px (Py) and Mx (My) are the x(y)-components of the polarization vector and the magnetization vector, respectively. ........................................................................................................................... 61 Figure 4-3 : Induced current paths inside the material ................................................................. 62 Figure 4-4 : Transmission and optical Activity. The amplitude (a) and phase (b) of simulated (solid curves) and measured (symbols) transmission coefficients for LCP (red) and RCP (blue) incident waves along the normal direction. (c) Polarization azimuthal rotation angle θ, and (d) Ellipticity η. Inset in (c) shows simulated (black) and measured (red) polarization states of a transmitted wave at 3.4 GHz. .......................................................................................................... 66
Figure 4-5 : Effective medium parameters. The effective permittivity 𝜖 (a), permeability 𝜇 (b),
chirality (c), and refractive indices, 𝑛 (d), 𝑛 + (e) and 𝑛 − (f) of 2D-HMM calculated from simulated (solid lines) and measured (symbols) transmission and reflection coefficients of circular polarization. The red and blue colors represent the real and imaginary parts of the effective parameters, respectively. ................................................................................................... 67
Figure 4-6 : Figure of merit for 𝒏 −- and absorption for LCP. FOM calculated from simulated (red solid curve) and measured (red circle) data. Absorption (blue solid curve) and normalized absorption (blue dashed curve) calculated from simulation. ....................................................... 71
Figure 4-7 : Wave refraction in effective medium. (a-d) Real part of electric field,𝑬 = 𝒛𝑬𝒛, for a linearly polarized EM wave propagating through an isotropic wedge with a negative (a,c)
and a positive (b,d) refractive index. (e) amplitude and (f) phase of 𝑬 along the propagation path ...................................................................................................................................................... 74 Figure 5-1 : Behavior of the pillars without the magnetic field and with the magnetic field .. 77 Figure 5-2 : Simulating Microactuators .......................................................................................... 78
vi
Figure 5-3 : (a) Magnitude (b) Phase of the transmission signal ................................................. 79 Figure 5-4 : Images of the pillars (Top View), (b) geometrical size (Top View) ...................... 80
Figure 5-5 : Applying the magnetic field to the sample - actual experimental setup................ 80 Figure 5-6: a) THz signal intensity b) Transmission c) Phase change of the THz signal with magnetic field. .................................................................................................................................... 82 Figure 6-1 : Focusing terahertz beam to a focal point ................................................................. 85 Figure 6-2 : Second harmonic generation using phase matching in a crystal ............................ 86 Figure 6-3 : Refractive index matching through satisfying phase matching condition ............ 87 Figure 6-4 : Enhancing second harmonic using U shaped SRRs ................................................ 88
vii
ABSTRACT
Metamaterials are artificial structures, which periodically arranged to exhibit fascinating
electromagnetic properties, not existing in nature. A great deal of research in the field of
metamaterial was conducted in a linear regime, where the electromagnetic responses are
independent of the external electric or magnetic fields. Unfortunately, in linear regime the
desired properties of metamaterials have only been achieved within a narrow bandwidth,
around a fixed frequency. Therefore, nonlinearity is introduced into metamaterials by merging
meta-atoms with well-known nonlinear materials. Nonlinear metamaterials are exploited in
this dissertation to introduce and develop applications in microwave frequency with
broadband responses. The nonlinearity was achieved via embedding varactor diode on to split
ring resonator (SRR) design, which demonstrates tunability in resonance frequency and phase
of the transmission signal. SRR exhibits power and frequency dependent broadband tunability
and it is realized for external electro-magnetic signals. More importantly, the nonlinear SRR
shows bi-stability with distinct transmission levels, where the transition between bi-states is
controlled by the impulses of pump signal and it can be used as a switching device in
microwave regime. In order to increase its functionality in other frequencies, a new design,
double split ring resonator (DSRR) is introduced with two rings, which has two distinct
resonance frequencies. The double split ring resonator also demonstrate similar behavior as
the SRR but it is broadband. Furthermore, by designing the structure such that the inner ring
viii
has a frequency twice as outer ring resonance frequency; we observed the enhancement of
harmonic generation. We exhibit enhancement in second harmonic generation and methods
that can use to increase the harmonic signal power. Arranging the unit cells in an array and
particular orientation further increases the harmonic power. In addition, we show that using
a back plate to create a cavity will help to increase harmonic power. Furthermore, we have
demonstrated that applying an external DC voltage can be used to tune resonance frequency
as well as phase of the signal. Exploring these ideas in THz frequency regime is also important.
So simulation results were obtained with advanced designs to achieve non-linearity in terahertz
frequency regime to realize tunability, hysteresis and bi-stable states.
A negative refractive index can be realized in metamaterials consisting of strong magnetic and
electric resonators with responses at the same frequency band. However, high loss and narrow
bandwidth resulting from strong resonances have impeded negative index optical components
and devices from reaching expected functionalities (e.g. perfect lens). Here, we demonstrate
experimentally and numerically that a 2D helical chiral metamaterial exhibits broadband
negative refractive index with extremely low loss. With Drude-like dispersion, its permittivity
leads to zero-index, and broadband chirality further brings the index to negative values for
left-handed circularly polarized light in the entire range below the plasma frequency. Non-
resonant architecture results in very low loss (<2% per layer) and an extremely high Figure-
of-merit (>90).
ix
Tunable THz metamaterials has shown great potential to solve the material challenge due to
the so-called “THz gap”. However, the tunable mechanism of current designs relies on using
semiconductors insertions, which inevitably results in high Ohmic loss, and thereby
significantly degrades the performance of metamaterials. In this work, we demonstrate a novel
tunable mechanism based on polymeric microactuators. Our metamaterials are fabricated on
the surface of patterned pillar array of flexible polymers embedded with magnetic
nanoparticles. The transmission spectrum of the metamaterial can be tuned as the pillars are
mechanically deformed though applied magnetic field. We observed and measured several type
of deformation including bending, twisting and compressing when the applied magnetic field
is polarized along different direction with respected to the axis of the magnetic particles.
Compared to previous semiconductor based tunable mechanism, our structure has shown
much lower loss. We demonstrate using simulations and experimentally that with an external
magnetic field, we can achieve phase modulation using magnetic polymeric micro-actuators.
1
CHAPTER ONE
INTRODUCTION TO METAMATERIALS
1.1 Summary
Study of materials and its properties has been an area of major interest for the scientific
community since the beginning of technological evolution. It is known as material science,
which is highly interdisciplinary between physics, chemistry and engineering, and it directly
effects the advancement of new technology and innovations. Nowadays in materials science,
the main goal is to fully understanding materials that exist in nature and exploits their
properties in order to create new materials with desired properties.
1.2 Electromagnetic properties of materials
Materials gain their properties from its arrangement of electrons in their atomic structure.
Most of the natural materials do not have electric or magnetic behavior since the motion of
electrons is randomized. However, in the presence of electromagnetic waves, most materials
change their electric and magnetic properties according to the direction of electric or magnetic
field of the electromagnetic wave by generating or rearranging the orientation of the electric
or magnetic dipoles in atomic level. With induced dipoles, these materials change path of the
incident electromagnetic waves by altering permittivity and permeability of the material.
2
Interaction of materials with electromagnetic waves is described by Maxwell’s equations.
∇. �̅� = 𝜌
𝜀0
∇. �̅� = 0
∇ x �̅� = −𝜕𝐵
𝜕𝑡
∇ x �̅� = 𝐽 +𝜕𝐸
𝜕𝑡
With an external field, we can obtain polarization (P) and magnetization (M) of a material from
the relationships below,
�̅� = 𝜀0 �̅� + �̅�
�̅� = 𝜇0 �̅� + �̅�
Where, �̅� and 𝐻 are local electric and magnetic fields in the media
�̅� and �̅� are electric and magnetic flux
�̅� and �̅� are polarization and magnetization fields induced by external fields.
In linear electromagnetic materials, polarization �̅� is linearly proportional to external electric
filed �̅� , and magnetization �̅� is linearly proportional to external magnetic field 𝐻 .
3
We can express polarization and magnetization using electric susceptibility (𝛸𝑒 ) and magnetic
susceptibility 𝛸𝑚
�̅� = 𝜀0 𝛸𝑒 �̅�
�̅� = 𝜇0 𝛸𝑚 �̅�
�̅� = 𝜀0 �̅� + 𝜀0 𝛸𝑒 �̅� = 𝜀0 (1 + 𝛸𝑒 )�̅� = 𝜀0 𝜀𝑟 �̅�
�̅� = 𝜇0 �̅� + 𝜇0 𝛸𝑚 �̅� = 𝜇0 (1 + 𝛸𝑚 )𝐻 ̅̅ ̅ = 𝜇0 𝜇𝑟 𝐻 ̅̅ ̅
Where, relative permittivity and relative permeability is defined by,
(1 + 𝛸𝑒 ) = 𝜀𝑟
(1 + 𝛸𝑚 ) = 𝜇𝑟
Materials found in nature have fixed values of permittivity and permeability defined by its
atomic structure. In order to use these materials in different applications most of the time
scientists have to create materials with desired permittivity and permeability values and to alter
those parameters researchers modify materials by changing its atomic structure. In
applications, altering materials in atomic level is not preferred since it is not cost effective and
time consuming. As a solution to adjust these material parameters researches move towards
artificial materials.
4
1.3 Metamaterials
Metamaterials are artificial structures, which periodically arranged to exhibit fascinating
electromagnetic properties, not existing in nature. Veselago in 1968[1] introduced an unusual
phenomena which can be expected in a hypothetical left-handed substance in which the triplet
of field vectors Electric (E), Magnetic (H) and the wave vector (k) form a left-handed system.
Furthermore, he showed that the pointing vector or energy flow of these electromagnetic(EM)
medium is counter directed with respect to wave vector or phase velocity. Hence, these are
called left handed materials (LHM) [2], Smith [3] in 2000 experimentally demonstrated its
fascinating behavior with resonating element structures which are termed meta-atoms as
elucidated by Pendry in 1999[4-6]. These manmade atoms are geometrically engineered to have
both permittivity and permeability to be simultaneously negative and therefore called negative
index materials (NIM). These NIMs demonstrate peculiar behaviors in many regions of the
electromagnetic spectrum. The left handed metamaterials have been implemented to develop
a wide variety of devices with uncommon functions such as negative index of refraction[3, 4],
capability of super-resolution or perfect lensing [5] and EM invisible cloaking[6, 7].
Any material in general can be characterized using microscopic parameters permittivity (ε) and
permeability (µ) as shown in Figure 1-1. The quadrants represents different material regions,
with properties associated with its impedance and refractive index.
5
Figure 1-1 : A schematic showing the classification of materials based on the electric and
magnetic Properties.
Simultaneously negative permittivity and permeability lead to peculiar behaviors in materials.
Materials with negative permittivity are found routinely in nature. All naturally occurring
metals have negative permittivity up to the plasma frequency (ωp). The plasma frequencies are
in the ultraviolet or visible range, and are many orders of magnitude higher than microwave
frequencies. For constructing metamaterials with metallic structures, it is desirable to scale the
value of ε to reasonable value (~ -1) to improve the impedance matching and transmission. It
has been theoretically demonstrated that an array of thin metal wires [8] as shown in Figure 1-
2(a) present a very low plasma frequency in microwave frequency.
The Drude model for dielectric function of a metal given by
6
𝜀(𝜔) = 1 −𝜔𝑝2
𝜔(𝜔 + 𝑖𝛾)
Where ω is the damping frequency and ωp is the plasma frequency given by,
𝜔𝑝 = √𝑛𝑒2
𝑚𝜀0
Where n is the electron density and m is the effective mass of electrons
An array of thin metal wires lowers the plasma frequency due to the diluted effective electron
density and increases its effective mass. The frequency dependence of the effective permittivity
(ε) is illustrated in Figure 1-2(c).
Figure 1-2 : (a) Wire structure (b) SRR structure (c) effective permittivity dependence
(d) effective permeability dependence
a(a)
(d)(c)
(b)
7
It has been demonstrated that the negative permeability can be realized using split ring
resonator structure [2] as shown in Figure 1-2(b). By applying a magnetic field (H)
perpendicular to the structure, it will induce an alternative current in the metallic ring. Near
the resonant frequency (ω), the magnetic response would be strongest and its frequency
dependence of the permeability is shown in Figure 1-2(d). The effective permeability of the
SRR structure is given by the following equation.
𝜇(𝜔) = 1 −𝜔𝑚𝑝2
𝜔2 − 𝜔02 + 𝑖Γ𝜔
The real part of effective permeability due to SRRs has negative value within a narrow
frequency band above the resonant frequency.
However, combining Split ring resonators with continuous wires arrays, we can realize
negative refractive index n for any desired frequency as shown by Figure 1-3.
Figure 1-3 : Realizing negative refractive index
8
1.4 Split Ring Resonator (SRR)
Among basic metamaterial structures, split ring resonator (SRR) is known as the most
common, best characterized [9, 10] metamaterial structure since its discovery. In SRR
structure, when an external electromagnetic waves incident on it, the external electric and
magnetic fields induce an oscillating current within the structure. The oscillating ring current
produces a magnetic dipole moment perpendicular to the structure, which allows us to
manipulate effective permeability (µ𝑒𝑓𝑓) of the medium.
SRR structure can be viewed as a LC resonator as shown in Figure 1-4, where the capacitance
C in a single SRR primarily results from the gap and inductance L determined by its structure.
Figure 1-4 : Split Ring Resonator (SRR) as a LC circuit
The anticipated LC resonant frequency can be estimated using the geometrical parameters[11]
of the SRR, such as width of metal (w), gap of the capacitor (g), thickness of metal (t) and
length of the coil (l). The capacitance, inductance and hence, resonance of the SRR can be
written as the following equations.
9
𝐶 = 𝜀0𝜀𝑐𝑤𝑡𝑔
𝐿 = 𝜇0𝑙2
𝑡
} 𝜔𝐿𝐶 =1
√𝐿𝐶=
𝑐0
𝑙√𝜔𝑔𝜀𝑐
The advantage of using these SRR meta-atoms is that its geometry is scalable to translate its
operability in many other frequency regimes.
The induced ring current in a SRR structure can be excited using external electric field or
magnetic field depending on SRR’s orientation to the incoming electromagnetic fields, which
is known as the excitation configuration.
The Figure 1-5(a) and Figure 1-5(b) shows the excitation of the SRR using external electric
field and magnetic field.
Figure 1-5 : The excitation of the SRR using external (a) electric field (b) magnetic field
When we excite the ring with the electric field, the electric field should be along the gap.
However, the magnetic field excitation happen when the magnetic field is perpendicular to the
�̅�
�̅�
�̅�
�̅�
�̅�
�̅�
10
structure. So that according to the Lenz law there will be an induced current in the ring because
of the incident magnetic field.
The Figure 1-6, shows the induced current modes in a SRR. The LC circuit model gives a
simple description of the resonance of the SRR and the induced current behavior is shown by
the Figure 1-6(a), with current oscillates in a loop, which is known as the LC resonance mode.
The higher order current modes or resonances shown in Figure 1-6(b) and Figure 1-6(c).
Figure1-6(b) current is known as the dipole resonance current.
Figure 1-6 : Induced current modes of the SRR in electric field excitation.
Metamaterials are not the only artificial structures that are used to control electromagnetic
waves. There are other man-made structures such as photonic crystals[12], metallic hole
arrays[13, 14] frequency selective surfaces[15] which are used to manipulate electromagnetic
waves in different frequency regimes. But metamaterials can be distinguished by the following
attributes that are inherent to metamaterials [16].
i) The structure can be described by a set of effective homogeneous electromagnetic
parameters, permittivity (ε) and permeability (µ).
11
ii) The parameters are determined by the collective response of these structures.
iii) Structures are placed periodically.
iv) The ratio of the operating wavelength to lattice constant is of the order of ten or more.
1.5 Nonlinear Metamaterials
Nonlinearity has also influenced the advancement in the field of Metamterials in many ways.
A great deal of research in the field of metamaterial was conducted in a linear regime, where
the electromagnetic responses are independent of the external electric or magnetic fields.
Unfortunately, in linear regime the desired properties of metamaterials have only been
achieved within a narrow bandwidth, around a fixed frequency. Therefore, nonlinearity is
introduced into metamaterials by merging meta-atoms with well-known nonlinear materials.
Among different ways to introduce nonlinearity to the metamaterial, embedding an additional
controllable media[17, 18] or inserting nonlinear elements[10, 19, 20] to the structure has led
to extensive studies on nonlinear metamaterials and demonstrated novel ways to obtain
nonlinear responses in metamaterials. Studies have shown nonlinear metamaterials show
fascinating features such as resonant frequency tuning[21-23], bi-stability[24, 25], harmonic
generation[26-28], parametric amplification[29] , modulation[30] and backward phase
matching[31] .
Although there are many methods used to obtain nonlinearity in microwave frequency regime,
introducing varactor diodes to the structure has been most preferred and more effective in
applications. It has been demonstrated that applying DC bias voltage [32-34], external
12
electromagnetic fields [10, 23, 30] or heat [18, 35] can be used to achieve non-linearity and
tunability of characteristic properties in varactor loaded nonlinear-metamaterials. Varactor
diode has a dependence of its capacitance for its applied reverse bias voltage and its behavior
is shown in Figure 1-7(b). In this dissertation, I have used SMV 1231-79LF, surface mount
diode for nonlinear metamaterial applications as shown in Figure 1-7(a) which has a
capacitance of 2.35pF at zero bias voltage and decreases as reverse bias voltage increases.
Figure 1-7 : (a) Surface mount varactor diode - SMV1231-079LF (b) Dependence of
capacitance of a varactor diode with its reverse bias voltage
Capacitance of the varactor diode is a function of reverse bias voltage as shown in Figure 1-
7(b), which is given by the following relationship
𝐶(𝑉𝐷) = 𝐶0 ( 1 −𝑉𝐷𝑉𝑃)−𝑀
DC rest capacitance 𝐶0 = 2.35𝑝𝐹 , Intrinsic potential 𝑉𝑃 = 1.5𝑉 and M=0.8
𝐼𝐷(𝑉𝐷) = 𝐼0(𝑒𝑉𝐷 𝑉𝑇⁄ − 1)
𝐶 =𝑑𝑄𝐷
𝑑𝑉𝐷
13
𝑉𝐷 = 1
𝐶 ∫ 𝐼 𝑑𝑡 =
𝑡
0
𝑉𝑃 [ 1 − (1 −𝑄
𝐶° 1 − 𝑀
𝑉𝑃)
1(1−𝑀)⁄
]
= 𝑞 + 𝑎𝑞2 + 𝑏𝑞3 + 𝑐𝑞4 +⋯
In the structure we present has an extra capacitance 𝐶𝑔 occurs from the gap in the structure.
Voltage across the capacitance 𝑉𝐶 = 𝑞
𝐶𝑔
𝐼 = 𝑑𝑞
𝑑𝑡
𝐿 𝑑2𝑞
𝑑𝑡2+ 𝑅
𝑑𝑞
𝑑𝑡+
𝑞
𝐶𝑔+ 𝑞 + 𝑎𝑞2 + 𝑏𝑞3 = 𝑉0 cos𝜔𝑡
Where 𝑞 = 𝑄
𝐶0 is the normalized charge and a,b,c are second, third and fourth order Taylor
expansion coeffficients given by;
𝑎 = −𝑀
2𝑉𝑃
In my work, I have used SRR structures where the gap is filled with a surface mount varactor
diode, and the resonance behavior is sensitive to the magnitude of the electric field localized
within the gap. This magnitude is in turn related to the current flow inside the ring generated
by the external magnetic or electric field.
1.6 Chiral Metamaterials
Chiral medium is a medium, which consist of particles that cannot be superimposed by their
mirror images. The main characteristic of the chiral media is that when the electromagnetic
14
wave propagates through the material, there is a cross coupling between the electric field and
magnetic field, which is described using a dimensionless chirality parameter κ.
Conventional metamaterials need to have negative permittivity (ε) and negative permeability
(μ) simultaneously to achieve negative index of refraction. It has been experimentally shown
that by combining artificial structures of negative permittivity with negative permeability will
allow the combined media to have negative index materials. However, Chiral metamaterial is
a new class of metamaterials with a simpler way to achieve negative index and therefore
negative refraction. A Chiral media gives different responses to left circularly polarized (LCP)
light and right circularly polarized (RCP) light due to the asymmetry of the medium. The
refractive index of LCP (𝑛−) and RCP (𝑛+) is given by the following equation in a chiral media
𝑛± = √𝜖𝜇 ± 𝜅
In order to achieve negative refraction for left circularly polarized waves, chiral parameter (κ)
needs to be greater than √εμ according to the above equation. In natural materials such as
quartz and sugar solutions, √εμ is much larger than one and κ is much smaller than one. So
achieving negative refraction is impossible in natural chiral materials. However, using chiral
metamaterials, we can artificially design macroscopic parameters, permittivity and permeability
to realize desired values and force √𝜖𝜇 to become less than zero. The idea of chiral
metamaterial is that, when ε and μ of a chiral medium are small and very close to zero, the
chirality can make the refractive index for one circular polarization to become negative, even
when κ is small.
15
In this dissertation, I present a study of the wave propagation properties in chiral
metamaterials and show that negative index can be realized in chiral metamaterials with a
strong chirality, with neither ε nor μ negative required. We have developed a retrieval
procedure, adopting uniaxial bi-isotropic model to calculate the effective parameters n ±, κ, ε
and μ of the chiral metamaterials. In addition, we demonstrate that the chiral metamaterial will
be introduced. Strong chiral behaviors such as optical activity and circular dichroism.
1.7 Metamaterials in Terahertz frequency regime.
The terahertz frequency spectral band is loosely defined between 0.3THz to 30 THz, which
bridges the worlds of electronics and optics. It is referred to as the terahertz gap as shown in
Figure 1-8.
Figure 1-8 : The electromagnetic spectrum and THz gap is circled in red
Only recently that the terahertz technology has been developed to the point where the
frequencies can be generated, detected, and manipulated routinely.
Amongst many techniques [36-38], terahertz time-domain spectroscopy (THz-TDS) [39]
utilizing a femtosecond laser source is a potential candidate for generation and detection of
THz gap
Frequency (Hz)
Electronics Photonics
16
broadband coherent terahertz radiation. There are many distinctive properties associated with
terahertz frequencies. Its radiation is nonionizing and favorable to applications seeking human
exposure and security screening [40, 41]. And also substance identification[41-43] and
sensors[44] are among other important applications in terahertz technology.
The terahertz gap not only defines a spectral region that has rarely been explored, but also a
region where the electromagnetic responses of most natural materials diminish. Naturally
occurring materials do not exhibit strong magnetic or electric responses between 1 and 3 THz
[45] and conventional magnetic materials tend to have resonances at frequencies below the
gigahertz range, while metals possess resonances at frequencies beyond the mid- infrared
owing to phonon modes. Metamaterials have become versatile tools for terahertz applications
and terahertz- manipulating devices that outperform their predecessors. Moreover, because of
their customizable characteristics in any frequency regime, metamaterials are capable of many
applications, which is not possible with conventional materials.
THz science and technology offers great potential in many important applications including
short-range secure communication, non-destructive detection, identification of chemical
materials, and bio-medical diagnostics. These applications are widely used in spectroscopy,
sensing, human health, and homeland security.
17
1.8 Terahertz time domain spectroscopy (THz-TDS)
Terahertz time-domain spectroscopy (THz-TDS) has demonstrated its effectiveness in
material property characterization, imaging, and sensing since it can directly measure the
electrical field of THz wave. The Fourier transforms of this time domain data gives the
amplitude and phase information of the THz wave pulse, therefore, providing the wideband
spectrum and the real and imaginary parts of the dielectric constant. In addition, it provides
precise measurements of the refractive index and absorption coefficient of samples[46].
A typical THz-TDS system is shown in Figure 1-9. An ultrafast laser beam is split into two
beams. One beam is employed to excite the transmitter and the other one is used at the
receiver. The transmitter is based on photoconductive coplanar transmission lines and a high
DC bias is applied between two transmission lines. Electron-hole pairs are created at one point
between these two lines where it is bombarded by the ultrafast laser pulse. These free carriers
are then accelerated in the static bias field to form a transient photocurrent, and this fast
current radiates electromagnetic waves with the frequency just within the range of a few THz.
The THz radiation is collected by silicon lens that are attached on the back of chip, and then
collimated by a para-boloidal mirror. Through the same structure with a para-boloidal mirror
and silicon lens, the freely propagating THz radiation is collected by the receiver. The probe
laser beam triggers the receiver’s photoconductive switch and creates the current proportional
to the amplitude of THz radiation received by the chip. By changing the path length of the
probe laser beam, the time domain THz radiation pulse can be measured. The
photoconductive switch is just like a sampling gate that opens for just several hundred-
18
femtosecond signals. In this way, it is easy to get a THz pulse of time domain, as well as the
frequency domain information by a Fourier transformation.
Figure 1-9 : THz-Time Domain Spectroscopy system
Figure 1-10 : THz-TDS experimental set up
800nm,
<100fs
Ultrafast laser
Lock-in amplifier
Variable
Delay
Sample
THz Transmitter
THz
Receiver
Current
Pre-amplifier
Optical
Chopper
DC Power
supply 60V
Attenuator
Beam
Splitter Mirrors
THz
Signa
l
2nd Pulse
1st Pulse
19
Lock
ed-i
n
Am
pli
fier
Curr
ent
Am
pli
fier
Opti
cal
Fib
er
Fig
ure
1-1
1 : T
Hz-
Tim
e D
om
ain
ref
lect
ion
Sp
ectr
osc
op
y sy
stem
20
1.9 Modeling and simulation of metamaterials
Before the fabrication of any metamaterial, we model and optimize the structure according to
its application using simulation software called Computer Simulated Technology Microwave
Studio (CST-MW). It is based on Finite Integration Technique (FIT) and in general, purpose
electromagnetic simulations, FIT applied to Cartesian grids in the time domain is
computationally equivalent to the standard Finite Difference Time Domain (FDTD) method.
For high frequency electromagnetic applications, time domain simulations methods are highly
desirable, especially when broadband results are needed. FIT therefore shares FDTD’s
advantageous properties like low memory requirements and efficient time stepping algorithm.
Simulation results of CST can be obtained with scattering parameters(s- parameters) which
can be used to retrieve transmission, reflection and other characteristic parameters such as
permittivity, permeability and index of refraction.
1.10 Retrieval Method used for chiral metamaterials
The effective impedance and refractive indices for RCP (+) and LCP (-) are calculated from
the complex transmission and reflection coefficients, 𝑇 and 𝑅, and are obtained from
simulations and experiments using the following equations:
2
2
(1 )
(1 )
R T Tz
R T T
0
1 1ln 1
1
i zn R
k d T z
21
𝑘0 is the wave vector in free space and 𝑑 is the thickness of the effective bi-isotropic medium.
The branches of the square root and the logarithm function in the formula above have to be
chosen carefully according to the energy conservation principle, i.e. the real part of the
impedance is positive, 𝑅𝑒(𝑧) > 0, and continuity of 𝑛± versus frequency is maintained.
Finally, for other material parameters, we derive
𝑘 = (𝑛+ − 𝑛−)/2,𝜇 = 𝑧(𝑛+ + 𝑛−)/2
𝜖 = (𝑛+ + 𝑛−)/2𝑧.
The experimental measurements were carried out by using an AV3629 network analyzer with
two broadband linearly polarized horn antennas in an anechoic chamber. Using the linear co-
polarization and cross-polarization transmission coefficients and , we can obtain the
transmission coefficients of the circularly polarized waves as follows: xx yxT T iT . The
optical activity is revealed via the polarization azimuthal rotation angle, arg( ) arg( )
2
T T ,
and the circular dichroism of the transmitted waves is characterized by the ellipticity,.
2 2
2 2
1arctan
2
T T
T T
xxT yxT
22
1.11 Sample fabrication
The applications of metamaterials has a wide range in electromagnetic spectrum. It has been
shown in many applications that we can demonstrate an application in one band of frequencies
in EM spectrum and then transfer the application to any other band of frequencies by simply
changing the unit cell size of the structure according to the wavelength of the operation
frequency.
In this dissertation, part of my research work is in microwave frequency regime and the other
part is in terahertz frequency regime. Sample fabrication in microwave frequency (~1GHz) is
carried out using Photo fabrication method, which is commonly known as PCB making. In
Terahertz frequency (~1THz) regime the structure size is in the order micrometers and we
can fabricate the samples using photolithography techniques. Basic steps that I used to
fabricate my samples for the THz applications are listed below and shown in Figure 1-12.
1. Clean the substrate: before depositing any metal layers, first we need to clean the
substrate with Acetone and then remove acetone with Methanol and finally use DI
water to clean Methanol.
2. Deposit Au/Al: used electron beam evaporation to deposit gold (Au) or Aluminum
(Al) layer of 200μm. When depositing an Au layer we need to use an adhesion layer of
Chromium (Cr) between the substrate and Au.
3. Apply Photoresist: A thin layer of Photoresist is applied on to the Au/Al using a spin
coater. The rotational speed of the spin coater with the photoresist is given in the
23
MSDS of the photoresist. There are two types of photoresist available. They are called
positive photoresist and negative photoresist.
4. Exposure to UV: To transfer the pattern from the mask on to the substrate, I used
Karl Suss Mask Aligner. By varying the photoresist thickness and exposure time
(according to the UV dose of the instrument), we can transfer the pattern with a good
accuracy.
5. Develop: When positive photoresist is exposed to UV light, a developing solution can
be used to wash it away.
6. Etch: An etching solution is used to etch the developed areas.
Figure 1-12 : Steps in Photolithography
Develop
Prepare Wafer(Si) Aceton + Methanol +DI water
Deposit Al or Au (200nm)
Apply Photoresist (AZ 1512) +Prebake
Expose
Mask
Etch
UV light
24
CHAPTER TWO
TUNABLE BISTABLE NONLINEAR METAMATERIALS
2.1 Summary
We demonstrate a nonlinear metamaterial in microwave frequency regime with hysteresis
effect and bi-stable states, which can be utilized as a remotely controllable micro second
switching device. A varactor loaded split-ring resonator (SRR) design which exhibits power
and frequency dependent broadband tunability of the resonance frequency for an external
control signal is used. More importantly, the SRR shows bi-stability with distinct transmission
levels. The transition between bi-states is controlled by impulses of an external pump signal.
Furthermore, we experimentally demonstrate that transition rate can be further reduces to the
order of microseconds.
Figure 2-1 : Mechanism of the SRR switching device.
25
2.2 Design and the experimental set up
A PNA-X Network Analyzer is used to measure the S-parameters of the SRR sample while a
N5181A MXG Analog Signal Generator is used as the pump signal as shown in Figure 2-2.
The experiments were performed with both the pump and the probe signals in such a way as
to have the electric field polarization parallel to the vacant gap of the SRR structure.
Figure 2-2 : Experimental set-up
Unit cell geometry of the metamaterial is shown in Figure 2-3(a) with dimensions of the SRR
structure as H=40mm, h=20mm, G=0.5mm, W=1mm, where (D) is a Hyper-abrupt varactor
diode. The resonator structures are fabricated using copper on a FR4 substrate of thickness
0.2mm. The Figure 2-3(b) shows the surface current induced on the SRR structure at the
resonance frequency 760MHz. The surface current density obtained by finite element
simulations using CST.
Signal
generator Vector Network
Analyzer
Transmitter
Dipole
Antenna
Sample
Pump Signal
Horn Antenna
Receiver
Horn Antenna
Use LabView
for Automation
26
Figure 2-3(c) shows the LC resonance for the SRR structure, which occurs at 760MHz. the
resonance frequency, depends on the geometry of the SRR structure.
Figure 2-3 : (a) Unit cell structure (b) surface current distribution (c) Resonance
Figure 2-1(a) illustrates the SRR being controlled by external signals and being used as a
switchable device in the microwave frequency regime. The SRR structure was exposed to an
external signal for both increasing and decreasing pump frequency with a constant power as
well as both increasing and decreasing pump power at constant frequencies. The behavior of
the SRR’s transmission was observed using the PNA-X Network Analyzer.
(a)
(b)
(c)
27
2.3 Hysteresis and Bistability
Using the PNA-X Network Analyzer, S-parameters of the sample are measured under varying
conditions. In order to show bi-stability as well as a hysteresis effect, measurements of the
SRR under increasing pump frequency and decreasing pump frequency were compared as
shown in Figure 2-4.
Figure 2-4 : Resonance frequency dependence for continuously increasing (red) and
decreasing (blue) external signal frequency with constant power Ppump=-25dBm
The Varactor loaded SRR structure displays a dynamic tuning range of 60MHz for an external
pump signal of a constant power of -25dBm. As shown in Figure 2-4(a), the width of the
hysteresis loop is more than 20MHz for that particular intensity. Figures 2-4(b), 2-4(c), and
2-4(d) display the transmission behavior for the corresponding regions indicated in Figure 2-
28
4(a). In Regions 1 and 3, the frequency dependent transmitted intensity for decreasing and
increasing pump frequencies are similar. However, in region 2, it follows two different paths
demonstrating bi-stability of the metamaterial structure. We observed the sudden jump
between two resonance frequencies when the external pump frequency coincides with the
resonance frequency of the structure.
The hysteresis-type behavior and bi-stability[47] of the resonance frequency can be seen not
only for the external signal frequency as shown in Figure 2.5(a), but also for external signal
intensities with a constant frequency as shown in Figure 2.5(b).
Figure 2-5 : (a) Resonance frequency dependence on pump signal frequency (b) Resonance
frequency dependence on pump signal power.
For very low pump powers and frequencies, hysteresis behavior vanishes and we are no longer
able to observe the jump between two states. In addition, the width of the hysteresis loop
700 725 750 775 800 825 850 875 900720
740
760
780
800
820
Pump Frequency (MHz)
Reso
nan
ce F
req
uen
cy(M
Hz)
-40 -35 -30 -25 -20 -15 -10 -5 0 5
740
760
780
800
820 740 MHz
750 MHz
755 MHz
760 MHz
770 MHz
800 MHz
Pump Power (dBm)
Resonance F
requency (
MH
z)
(a) (b)
29
widens and narrows with the increase and decrease of both the external pump signal power
and frequency indicating an increase in nonlinearity. Furthermore, the direction of the sudden
jump in the resonance frequency is different for varying power and varying frequency. When
increasing pump power, the resonance frequency jumps from a lower frequency to a higher
frequency while the opposite is true for increasing the pump frequency. We found success
when modeling our structure as an RLC circuit. With the increase of the pump power, the
voltage across the diode increases, which results in a decrease in the total capacitance of the
structure and an increase in the resonance frequency. However, sudden transition of the
resonance frequency for external frequency dependence, behaves opposite to external signal
intensity dependence. Furthermore, in order to observe the hysteresis effect external pump
should be applied continuously as shown in Figure 2-6(b) otherwise jump is observed but not
hysteresis effect. Polarization angle of the E filed with respect to the gap of the SRR has an
impact on the behavior of the structure as shown in Figure 2-6(a). When the E filed is parallel
to the gap of the structure we observe the bi-stable states but with the change of the angle for
the same power level the magnitude of the e filed component along the gap reduces and shrink
the hysteresis gap size.
Since the external electric field is applied across the gap of the SRR structure, Figure 2-6(a)
demonstrates that when the SRR is not excited neither electric or magnetic fields we do not
see any response, but as we increase the angle we see the hysteresis effect and maximum
tunability when electric field aligns with the gap of the SRR. When the electric field is at 90
degrees with respect to the gap of the SRR we do not see any tunability or hysteresis effect,
30
but with the increament of the angle there wil be a component of the electric field that appears
along the gap and that component can excite the ring and give a electric response. As shown
in Figure 2-6(a) we see the maximum hysteresis and tunability for the zero angle since the
applied electric field is aligned with the gap at that configuration. So we can conclude that the
behavior of the structure is not angle independent if we excite the structure using an electric
field. Figure 2-6(b) shows that the hysteresis effect can be obnly obtained when the external
pump signal frequcny is changed without any distruptions. When the applied signal is switched
on and off periodically, the hysteresis effect disserpears and only the tunability and sudden
jump of the resonance frequency occurs. So it is important that in order to make a switching
device we need to continuosly change it’s the external pump signal using pulses.
Figure 2-6 : (a) Resonance frequency dependence with the angle of the E filed (Black: E field
along the gap of the structure, Green: E field is perpendicular to the structure)
7 0 0 7 2 5 7 5 0 7 7 5 8 0 0 8 2 5 8 5 07 3 0
7 4 0
7 5 0
7 6 0
7 7 0
7 8 0
7 9 0
0
3 0
6 0
7 5
9 0
P u m p F re q u e n c y (M H z )
Re
so
na
nc
e F
req
ue
nc
y (
MH
z)
(a) (b)
31
2.4 Using double split ring resonator to control tune the bi-stability.
Previously we demonstrated that a switching device can be implemented using a split ring
resonator. Since a SRR has resonance frequency which defined by its geometry, the switching
device is limited to limited frequencies depending on external pump frequencies and power
which give hysteresis effects. In order to increase its functionality, we designed a new structure
that has two distinct resonance frequencies defined by two rings in the structure. It’s called
the double split ring resonator (DSRR) which shows in Figure 2-7(a) and a varactor diode is
loaded in the common gap. Since it has two rings, it gives two resonance frequencies as shown
in Figure 2-7(b) with an inner ring frequency at 1660MHz and outer ring frequency at
730MHz.
Figure 2-7 : (a) Double split ring (DSRR) design (b) Resonance frequency of the DSRR.
(a) (b)
32
Similar to SRR structure, DSRR can also be tuned by an external signal. Tunability and bi-
stability depends on external pump frequency and power as shown is Figure 2-8(a) and Figure
2-8 (b).
Figure 2-8 : (a) Resonance frequency dependence on pump signal frequency for DSRR (b)
Resonance frequency dependence on pump signal power.
2.5 Transition time and Switching device
The SRR structure is improved by inserting a 1MΩ resistor in parallel to the nonlinear varactor
diode. This modification of the structure decreases the transition time between the states and
improves its applicability, but minimum step size of the time was limited by the delay time of
the instruments used in the experiment. Furthermore, by using a pulse train of appropriate
powers, the hysteresis in Figure 2-8(d) was observed which clearly indicated two states in the
transmission.
fcs = 1650MHz
fcs = 1700MHz
fcs = 1720MHz
Pcs
(a) (b)
33
Figure 2-9 : Trigger transitions between the two states
To explore the transition of the sample under a greater resolution, different measurement
techniques must be utilized. In order to capture the transition, we used the double split ring
resonator (DSRR) that could be pumped at the resonant frequency (1200MHz) of inner ring
and probed at the resonance frequency of outer ring. The power of the pump signal is about
0dBm greater in magnitude than the signal used to probe. Figure 2-10(b) inset graph shows
the double split ring design which was used to capture the transition times in microsecond
speed.
34
Figure 2-10 : Transition time dependence with the resistance
Figure 2-11 : Optimized time between transition states
0 1 2 3 4 5 6-128
-127
-126
-125
-124
-123
-122
-121
-120
-119
-118
-117
Time (s)
Pow
er
(dB
m)
1M
330K
100K
0 2 4 6 8 10 12 14 16 18 20-140
-135
-130
-125
-120
-115
Time (s)
Pow
er
(dB
m)
Probe signals 100k
854
854D
0 2 4 6 8 10 12-126
-124
-122
-120
-118
-116
-114
-112
Time (s)
Pow
er
(dB
m)
1M
100K
330K
(a)
(b) (c)
35
As shown in Figure 2-10(b), by inserting a resistor in parallel to the varactor diode of the
structure we can decrease the transition time by allowing a path to discharge the accumulated
charge across the diode. We experimentally demonstrate that by choosing a particular resistor
value jump up transition and jump down transition time are different and the transition time
depends on the resistor value (Figure 2-10(b) and Figure 2-10(c)). In order to make a switching
device with a switching speed in microseconds we need to choose a correct resistor that
optimizes the both up and down transition times as shown in Figure 2-11. The graph shows
the transition of states for the resistor of 100kΩ at the frequency 854MHz.
Currently, metamaterials with frequency tunability are particular of interest due to the
flexibility of frequency control and has become an essential part of metamaterial devices. It
makes material devices more versatile, adaptive to varying external disturbances and changes
its effective parameters accordingly. Nonlinear metamaterials provide the basis for
manipulating and controlling of the characteristics of a media. Therefore, the functionality
demonstrated above will benefit to better understanding of the nonlinear behavior of the
metamaterials.
We have demonstrated the capability of varactor loaded nonlinear metamaterials in tuning
transmission properties, hysteresis and switching effects in microwave frequency regime. We
can extend this switching effect into the THz frequency regime.
36
2.6 Nonlinear tunable metamaterials in Terahertz frequency regime.
THz technology is extremely challenging due to the lack of suitable materials which are
necessary for the construction of THz devices. The THz NL-MMs with enhanced nonlinear
properties e.g bi-stability and tunability, have the potential to make significant contribution to
solve this material challenge for applications like THz communications, sensing and wave
modulation.
Although in microwave frequency regime, we are able to use surface mount varactor diodes
as the nonlinear elements embedded in SRRs. However, in order to achieve nonlinear
properties at higher wavelengths, a simple scaling of these structures to higher frequencies will
not suffice. First, the response of most metals changes from conductor-like to dielectric-like
at frequencies above a few terahertz, with absorption increasing significantly as the fields are
able to penetrate further into the metal. Second, packaged semiconductor components are not
readily available at frequencies above 100 GHz[48]. So in terahertz frequency regime we have
to fabricate varactor diodes on to the gap of the SRR structure as shown in Figure 2-12 to
achieve nonlinear properties. Doping GaAs in different concentrations in many layers as
shown in the table will assure that there will be capacitance change for applied bias voltage
across the varactor diode.
In this project, tunability of the resonance frequency of the nonlinear metamaterial will be
achieved experimentally by applying a DC bias voltage across its fabricated varactor diode.
37
Figure 2-12 : Varactor diode design for the terahertz applications
Co-planar SRRs as shown in Figure 2-13(a) is designed such that bias voltage can be applied
to every element at the same time to get the tunability of the resonance frequency, similar to
work done for the microwave frequency regime using varactor loaded SRRs.
Figure 2-13 : Tunability of the resonance frequency in terahertz frequency regime
0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.2
0.4
0.6
0.8
1
Frequency (THz)
T(L
inear)
[ 0]
[ 0.01]
[ 0.02]
[ 0.04]
[ 0.08]
[ 0.16]
0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.2
0.4
0.6
0.8
1
Frequency (THz)
R(L
inear)
[ 0]
[ 0.01]
[ 0.02]
[ 0.04]
[ 0.08]
[ 0.16]
(a)
(b) (c)
38
Results as shown in Figure 2-13 were obatined using CST simulations and by changing the
depletion region which can be done experimentally by applying a voltage across the diode.
Transmission and reflection of the THz wave are shown in Figure 2-13(b) and (c) and it
demonstrates that we can obtain near a 40% tunability by changing the depletion thickness of
the varactor diode.
Here our goal is to use a bias voltage to tune the resonance frequency and observe bistability
and switching effects in THz frequency regime.
Figure 2-14 : Experimental results of SRR structure using THz-TDS
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (THz)
Tra
nsm
issio
n
Experiment
Simulation
(a) (b)
(c) (d)
0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
X: 0.8788
Y: 0.4128
Frequency (THz)
Tran
smis
sion
Experiment
Simulation
39
Preliminary experimental results were obtained without varactor diode structure as shown in
Figure 2-14. SEM images of the fabricated SRR are shown in Figure 2-14(a), Figure 2-14(b)
and Figure. 2-14(c), d illustrate the experimental and simulated results. Experiments were
carried out using THz-TDS and we observed that experimental results agree with the
smulation results. Next step is to fabricate the varactor diode in the gap of the SRR structure
and do the measurements to get the tunability of the properties of the metamaterial and
demonstrate switchable effects in the terahertz frequency regime.
40
CHAPTER THREE
ENHANCING HARMONIC GENERATION USING
NONLINEAR METAMATERIALS
3.1 Summary
In this chapter, we demonstrate a nonlinear metamaterial which can enhance higher order
harmonics in microwave frequency regime. Nonlinearity in the structure is introduced by
adding a varactor diode in the common slit of the double split ring resonator (DSRR) design.
By engineering the structure such that the inner ring resonance frequency of the DSRR is twice
as the outer ring resonance frequency, we have demonstrated that the second harmonic of the
outer ring can be enhanced significantly. By comparing with a single ring (SRR) unit cell
structure, DSRR has an enhancement factor of 70. Furthermore, we elucidate that the second
harmonic signals generated by two identical double split rings interfere and we can use its
constructive interference positions to construct an array of DSRRs that gives a maximum
second harmonic signal with phase matching condition. When the phase matching condition
occurs the enhancement factor of an array is more than the contribution of individual unit
cells which proves that the enhancement is due to not only constructive interference of the
second harmonic generated by individual unit cells but also due to cavity mechanism occurred
in the array structure.
41
3.2 Introduction
Recent studies[49] have shown that nonlinear resonating structures such as SRRs can be
advantageous to use in nonlinear applications since it has been observed that Split ring
resonator produces a huge electric field at its gap at the resonance even it is exposed to an low
incident electromagtic waves. In the study, we used double split ring resonator(DSRR)
structure which has two distinct resonance frequencies.
In order to increase the nonlinear polarization generated by a non linear medium we can either
increase suseptibility χ(n) or increase the applied electric field in the medium which explain by
the nonlinear polarization equation.
Wave equation for Propagation of Electromagnetic waves in a nonlinear medium
∇2𝜖 − 1
𝑐2 𝜕2𝜖
𝜕𝑡2 = 𝜇0
𝜕2𝑃
𝜕𝑡2
Polarization density
𝑃 = 𝜀0(𝜒(1)𝜖 + 𝜒(2)𝜖2 + 𝜒(3)𝜖3 +⋯)
𝑃 = 𝜀0𝜒(1)𝜖 + 𝑃𝑁𝐿
Induced polarization is given by
𝑃𝑁𝐿 = 𝜀0(𝜒(2)𝜖2 + 𝜒(3)𝜖3 +⋯)
Where 𝜖 is the applied external electric field.
42
The efficiency of the nonlinear processes is often limited by a weak nonlinear response of
natural materials [50] and hence nonlinear polarization is typically very weak. The nonlinear
susceptibilities 𝜒(i) for natural nonlinear materials (e.g. nonlinear crystals) used in photonics
applications are usually extremely small: for example 𝜒(2) ~10-12 m/V and 𝜒 (3) ~ 10-24 m2/V2.
Therefore, the nonlinear effects are typically very weak, and only observed in very strong EM
fields (with E ~ 108 V/m, comparable to interatomic electric fields) such as those created by
pulsed lasers. Recent studies suggested that the resonance nature of a metamaterial
electromagnetic response could be utilized to engineer the composite medium with the
enhanced nonlinear properties in a desired frequency band.
Figure 3-1 : Enhancing second harmonic using (a) SRR (b) DSRR
Figure 3-1 indicates the general idea of this chapter, which shows that double split ring
resonator, can be used to enhance the second harmonic signal power.
(a) (b)
43
3.3 Experimental results
Experimentally, second harmonic was measured using a vector network analyzer and the
experimental setup is shown in Figure 3-2. Two horn antennas were used as transmitter and
the receiver. Vector network analyzer transmit a band of frequencies and Nonlinear DSRR
will have a response according to its geometry. Since the maximum current density occurs at
the resonance, we will see the harmonics as the multiples of the fundamental resonance
frequency of the structure. The receiver will get all those frequencies and we can measure
power of the second harmonic, third harmonic etc. using vector network analyzer.
Figure 3-2 : Experimental setup with Vector Network analyzer and two horn antennas
Figure 3-3(a) shows the split ring resonator (SRR) and double split ring resonator (DSRR)
designs which we used in this chapter. Figure 3-3(b) shows the transmission through SRR and
DSRR where SRR has a fundamental resonance frequency at 𝑓1 = 822MHz and DSRR has
two distinct peaks with fundamental resonance frequencies at 𝑓1 = 825MHz and
𝑓2 = 1670MHz corresponding to the two rings. The fundamental resonance frequencies
depends the ring structure and it can be changed by modifying its geometry. The current
Vector Network
Analyzer
Sample Horn Antenna
Transmitter Horn Antenna
Receiver
44
density of the DSRR unit cell at the resonance frequencies are shown near each resonance
frequency, which shows a LC resonance mode for the fundamental frequencies.
Figure 3-3 : (a) SRR and DSRR structures (b) Transmission of the SRR and DSRR
In order to enhance the nonlinearity of the design we design the fundamental frequency of the
inner ring as close to the outer ring frequency. Figure 3-4 shows the relationship between the
inner ring frequency and outer ring frequency by plotting outer ring frequency with half of the
inner ring frequency.
(a) (b)
0.8 1.0 1.2 1.4 1.6 1.8
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
Frequency (GHz)
Tra
nsm
issio
n
SRR
DSRR
DSRR
SRR
𝒇𝟐
𝒇𝟏
45
Figure 3-4 : Outer ring frequency (f1) and inner ring frequency (f2)
At the resonance, the surface current of each ring is maximized. Since the gap of the resonator
acts as a capacitor the induced charge accumilates near the gap and produces an electric feld
near the gap. Figure 3-5 shows the electric field distributions at the gaps for inner ring
resonance frequency and outer ring resonance frequency. We can observe clearly that the
electric field is maximized at the gaps as well as near the varcator diode, the nonlinear element
of the structure which contributes to the enhancement of the harmonics.
Figure 3-5 : Electric field distribution of the DSRR (a) At the outer ring frequency (b) inner
ring frequency
𝒇𝟏
0.750 0.775 0.800 0.825 0.850 0.875 0.9000.5
0.6
0.7
0.8
0.9
1.0
1.1
Frequency(GHz)
Tra
nsm
issio
n
𝒇𝟐 𝟐⁄
(a) (b)
46
In order to maximize the second harmonic and hence maximize other higher order harmonics,
DSRR structure was fabricated such that outer ring size is 40mm x 40mm and inner ring size
is 20mm x 20mm. Both gaps in two rings have a gap size of 1mm and the ring width is 1mm.
Figure 3-6 : (a) comparison of the enhancement of the second harmonic power (b) the
dependence of the second harmonic power to the inner ring size.
21mm inner ring
1.45 1.50 1.55 1.60 1.65 1.70-90
-80
-70
-60
-50
-40
Frequency(GHz)
Se
co
nd
ha
rmo
nic
po
we
r (d
B)
20mm inner ring
22mm inner ring (b)
47
When the inner ring size changes, it effects the second harmonic signal as shown in
Figure 3-6(b) and we observe maximum second harmonic power when the outer ring
frequency (𝑓1) is twice as the inner ring frequency (𝑓2) as illustrated in Figure. 3-6(a). However
when the condition 𝑓2 = 2 𝑓1 does not satisfy, for 20mm and 22mm inner ring sizes we see
two peaks in second harmonic signal and the power of the signal gets weaker as shown in
Figure 3-6(b). In the case of satisfied condition of 𝑓2 = 2 𝑓1 , we have an enhancement of
70, when we compare a single split ring with a double split ring. Furthermore, it can be seen
that the when the second harmonic signal power increases, in similar manner third harmonic
power also increases as shown in Figure 3-7. Throughout this experiment we have used 0dBm
(1mW) as the probe signal power.
Figure 3-7 : Third harmonic signal power comparing SRR and DSRR structures
2.1 2.2 2.3 2.4
1E-4
1E-3
0.01
Frequency (GHz)
Thir
d H
arm
onic
Pow
er
SRR
DSRR
48
3.4 Applying an external reverse bias voltage to the unit cell.
The main characteristic of a varactor diode is its capacitance dependance on reverse bias
voltage.
Figure 3-8 : Using an external bias voltage to tune the resonance frequencies to find the
maximum second harmonic
A double split ring resonator structure has two distinguished resonance frequencies which is
produced by inner ring and the outer ring of the strucutre. Resonance frequency is given by
𝜔𝑟𝑒𝑠𝑜𝑛𝑎𝑛𝑐𝑒 = 1 √𝐿𝐶⁄ It can seen that by applying an external dc bias voltage across the
diode, we can obtain its tunability effect for a wide range of frequencies.
0.70 0.75 0.80 0.85 0.90 0.95-10
-8
-6
-4
-2
0
2
4
-1.5V
1V
3V
0V
2V
Frequency(GHz)
Inner Ring resonance
Outer Ring resonance
Second Harmonic Signal
49
Figure 3-9 : (a) S21 (b) Second harmonic (c) third harmonic tuning using an external signal
power
Pump power(dBm)
VN
A F
requency (
MH
z)
S21
-30 -20 -10 0700
750
800
850
900
950
1000
0
0.2
0.4
0.6
0.8
1
Pump power(dBm)
VN
A F
requency (
MH
z)
SHG
-30 -20 -10 01400
1500
1600
1700
1800
1900
2000
0.5
1
1.5
2
2.5
x 10-7
Pump power(dBm)
VN
A F
requency (
MH
z)
THG
-30 -20 -10 0
2200
2400
2600
2800
3000
0.5
1
1.5
2
x 10-7
(a)
(b)
(c)
50
3.5 Using a back-plate to increase the harmonic power
A cavity can be created using the metamaterial structure and a metal plate behind the
metamaterial sample. The nonlinear metamaterial acts as a source of the second harmonic
signal and that signal creates cavity modes inside the cavity. Figure 3-10 shows the behavior
of the second harmonic signal with the change of distance of the metal back-plate. .
Figure 3-10 : Using a back-plate to increase the second harmonic power
The wavelength of the harmonic signal is approximately 18cm and we observe that the
minimums of the second harmonic power occurs at n λ and maximum power of the second
harmonic is at n λ/2.
10 15 20 25 30 35 40 45 50 55
0
5
10
15
20
25
30
35
40
Distance from the sample (cm)
Maxiu
m v
alu
e o
f th
e S
econd H
arm
onic
Pow
er
51
3.6 The arrays of DSRR unit cells
These varactor loaded nonlinear metamaterial unit cells can be considered as point sources of
second harmonic signals which we demonstrate that these point sources has constructive and
destructive interference when we keep them at specific positions.
Figure 3-11 : Interference of second harmonic signal from two unit cells of DSRR
Two unit cells of similar DSRR structures were arranged as shown in Figure 3-11(a). A horn
antenna from the direction indicated by x-axis excited the unit-cells and the second harmonic
signal was collected from the direction of y using another horn antenna. Figure 3-11b shows
the interference pattern when we change the relative distance in the y direction of the two unit
cells. Second harmonic signal emits from the DSRR structure is 1668MHz, which corresponds
to a wavelength of 17.98cm. In Figure 3-11(b), solid line indicates where the separation of the
unit cells in x direction is 0 cm and dashed line indicates that the separation of the unit cells in
X
Y
(b) (a)
52
x direction is 36cm. In both cases, relative change is in the y direction, and the dashed line
represents the x distance between the unit-cells is 36cm. The Figure 3-11(b) shows two
different orientations with respect to varactor diode. One with the same orientation of the
varactor diode in both unit cells and the other is 180 differences between the varactor diodes.
Equation 3 gives the path difference of the two coherent second harmonic signals generated
by two DSRR unit cells
∆𝑙 = (2𝑙𝑦− 𝑙𝑥 )
𝜆𝐹 2π (3-1)
Where ∆𝑙the path difference of the two signals is generated by DSRR cells, 𝑙𝑦 is the distance
between the unit cells in y direction, 𝑙𝑥 is the distance between the unit cells in x direction and
𝜆𝐹 is the wavelength of the fundamental frequency.
The fundamental wavelength of the signal is around 36cm and the second harmonic
wavelength is 18cm. when separation of unit cells in x direction is either 0cm or 36cm, we can
obtain the same interference pattern for both orientations of the diode as shown in
Figure 3-11(b) (red solid line and dashed line indicates both cases). When the diodes are in the
same orientation for λ/2 (9cm), we observe destructive interference and for λ (18cm), we
observe a constructive interference. Similarly, for the opposite orientation of the diodes we
observe constructive interference at a distance of λ/2 (9cm) and destructive interference at λ
(18cm) which agrees with the Equation (3-1). Studies have shown that for non-symmetric
structures, second harmonic signal originates from the side arms and they are in phase. The
53
SRR and DSRR structures are also non-symmetric and its second harmonic field is produced
in the side arms of the structures are radiates to the far field.
Figure 3-12 : Second harmonic of an array of DSRR unit cells
By arranging unit cells in an array as shown in Figure 3-12(a), can increase the total second
harmonic signal with constructive interference of second harmonic waves from individual unit
cells. Figure 3-12(b) shows interference of second harmonic signals from 6 unit cells of SRR
and DSRR compared with their respective single unit cells. Distance between the unit cells in
y direction 𝑙𝑦= 9cm and distance between the unit cells in x direction 𝑙𝑥 = 36cm and the
direction of the diodes of the two unit cells in the middle is kept opposite to the other unit
cells. Array is constructed such that in x direction, unit cell orientation is in the same direction
and in y direction unit cell direction is in opposite direction. When we compare the second
harmonic signal power generated by six DSRR unit cells to the power of 1 DSRR unit cell the
X
Y
1.5 1.6 1.7 1.8 1.9
1E-5
1E-4
1E-3
0.01 1 SRR
1 DSRR
6 SRR
6 DSRR
Opposite Oreintation
6 DSRR
Same Oreintation
Frequency (GHz)
Second H
arm
onic
Pow
er
(b) (a)
54
enhancement factor is 8. It indicates that not only interference of the second harmonic
generated by DSRR unit cells is effecting the enhancement in the array structure but also these
unit cells act as mirrors when they are at resonance and form a cavity inside the array.
3.7 A model for the DSRR structure
A varactor loaded split ring resonator can be moddeled as a RLC circuit as shown in Figure 3-
13(a) and Figure 3-13(b) and in DSRR there is an extra capacitance added to the circuit in
parallel to the exsiting capacitance due to the second ring in the structure.
Figure 3-13 : (a) Equivalent circuit of SRR structure (b) Equivalent circuit of DSRR structure
In varactor loaded DSRR, the external signal provides the driven source which can be modeled
as a virtual voltage source in the equavalent cicuit model.
𝑉𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙(𝑡) = 𝑉0 cos𝜔𝑡
𝐿𝑑𝐼
𝑑𝑡+ 𝑅𝐼 + 𝑉𝐶 + 𝑉𝐷 = 𝑉0 cos𝜔𝑡
Where L is the total inductance of the DSRR, including self-inductance and mutual inductance,
R is the effective resistance of the unit cell and 𝑉𝐷 is the voltage across the varactor diode.
(a) (b) (c)
55
Varactor loaded split ring resonator structure produces a second harmonic signal
corresponding to its fundamental frequency 𝜔0 = 1 √𝐿𝐶⁄ . Since the capacitance of the
varactor diode depends on the reverse bias voltage across the diode is given by the following
expression,
𝑉𝐷 = 1
𝐶 ∫ 𝐼 𝑑𝑡 =
𝑡
0
𝑉𝑃 [ 1 − (1 −𝑄
𝐶° 1 − 𝑀
𝑉𝑃)
1(1−𝑀)⁄
]
= 𝑞 + 𝑎𝑞2 + 𝑏𝑞3 + 𝑐𝑞4 +⋯
Where higher order terms contributes to the harmonics generated by the structure. In the
letter we only consider second harmonic generated by the varactor loaded SRR and DSRR
structures. The amplitude of the second harmonic signal can be obtained using the circuit
model of the SRR with the use of perturbation method under weak nonlinearity[51] and by
considering only fundamental and second order terms.
The amplitude of the oscillations are given by
�̃�(2𝜔) =𝛼𝑉0
2
𝜔3𝑍(2𝜔)𝑍(𝜔)2 (3-2)
When the circuit resonates at 𝜔 = 𝜔0 and the impedance 𝑍(𝜔) will be minimized resulting
an increase of amplitude of the oscillations. In similar manner by minimizing 𝑍(2𝜔) will
further enhance the amplitude of oscillations and enhance the effect of second harmonic
generation in the metamaterial.
56
CHAPTER FOUR
CHIRAL METAMATERIALS
4.1 Summary
In this chapter, we have presented a 2D-HMM which exhibits very versatile properties,
including nondispersive optical activity, broadband chirality, and a negative refractive index.
For our chiral metamaterial based on a 2D continuous mesh of helical structures, we have
provided evidence through a retrieval method that the negative index arises from chirality and
exhibits the important features of broad bandwidth, very low loss, and high transmission. The
negative index FOM reaches 91 in the region above the plasma frequency. Finally, we solved
an equivalent wave equation for circular polarized waves in a bi-isotropic effective medium
representing our 2D-HMM and demonstrated positive and negative refractions for LCP and
RCP waves.
4.2 Introduction
Negative refraction is a counter-intuitive phenomenon that has attracted ample attentions due
to the potential for unprecedented “reversed” electromagnetic (EM) properties and the
capability of making a “perfect” lens with resolution unaffected by the diffraction limit for
wavelength[52, 53]. Negative index materials (NIMs) are usually obtained by superimposing
electric and magnetic responses, which separately drive the permittivity and permeability to be
57
simultaneously negative in the same frequency range. While negative permittivity can be
obtained by non-resonant structures such as an array of continuous metallic wire, negative
permeability is usually achieved by using structures with strong magnetic resonances, e.g. split-
ring resonators [3, 54, 55] or short-wire pairs [56-59]. The resonant nature of the NIM results
in very high loss and narrow bandwidth, both of which have been long-lasting obstacles
limiting the potential of NIMs in practical applications. Alternatively, it was proposed that a
negative index can also be achieved through a chiral route [60, 61], where negative permeability
is not required. For chiral media, the refractive index is 𝑛± = √𝜖𝜇 ± 𝜅, where “+” and “−”
refer to the right-handed (RCP) and left-handed (LCP) circularly polarized light, respectively,
and 𝜅 is the chirality parameter. This implies that strong chirality is sufficient to achieve
negative index for one circular polarization. The chirality,𝜅, arises from the molecular
asymmetry of the chiral medium. In chiral materials, e.g. liquid crystals, sugar solutions and
DNA, the molecules lack internal mirror symmetry, and thereby exhibit chirality when
circularly polarized light propagates through it. The important effects of chirality are optical
activity (the ability to rotate the polarization plane of light) and circular dichroism (the
difference in the transmission of LCP and RCP incident light). Chiral metamaterials are
analogously designed to create optical activity in almost any desirable frequency range. Due to
strong electric/magnetic moments induced by resonant magnetic/electric fields, chiral
metamaterials exhibit much stronger optical activity[62-65], circular dichroism[66-72],
asymmetric transmission[73-78], perfect absorption[79] and repulsive Casimir force[80].
Recently, negative refraction due to chirality has been demonstrated through bi-layer twisted
resonators[81, 82] and 3D Split-ring resonators[83] at GHz and THz frequencies, respectively.
58
More recently, chiral NIMs have also been demonstrated by a number of designs such as a
cluster of four “U”-shaped split-ring resonators[84, 85] and conjugated gammadion
resonators[86]. In all these works, strong chirality is induced by electric and magnetic
resonances, which thereby inevitably results in high loss and narrow bandwidth. Very recently,
chirality has been observed in a non-resonant meshed helical structure[87], which exhibits
broadband and nondispersive optical activity due to its Drude-like response. Inspired by this
non-resonant design, we demonstrate numerically and experimentally that a negative index can
be obtained over a broad bandwidth from a 2D helical metamaterial (2D-HMM) with
extremely low loss. Our structure consists of interconnected double-layer clockwise helical
metallic wires forming a continuous 2D mesh grid. The continuous currents flowing in the
wire produce a Drude-like response, which leads to a nearly zero value for√𝜖𝜇. The non-
resonant 𝜅 induced by the current flowing along the helical path brings the index to negative
values for LCP incident waves. Using a retrieval method for bi-isotropic medium, we
calculated the effective material parameters from both experimental and simulation data. Our
results show that the refractive index for LCP, 𝑛−, is negative in an extremely broad range,
from DC to the plasma frequency. More importantly, the Figure of merit (FOM) for 𝑛−
reaches as high as 91, indicating very low absorption. The non-resonant broadband low-loss
architecture paves a new route towards practical application of NIMs.
4.3 Experimental Design
The unit cell is composed of four double-layer cyclone-shaped structures (DCS) made from
copper (Figure 4-1). The bottom layer of the DCS is formed by mirroring the top layer with
59
respect to the y-axis. Along the clockwise direction, each DCS is rotated by π/2 from its
adjacent neighbor. The array of unit cells forms a C4 square lattice with fourfold rotational
symmetry, which ensures a pure optical activity effect (i.e. without conversion from LCP to
RCP waves or verse versa). The metallic structures on the front and back surfaces of the
dielectric substrate are connected via metallized holes (inner surface coated with copper),
forming a continuous metallic meshed grid in the xy-plane. In the experiments, a 40 x 40-unit
cell 2D-HMM sample was fabricated on a double-sided copper-clad F4BM-2 circuit board via
a standard etching process. The geometrical parameters of the unit cell are as follows: a = 3
mm, b = 1.3 mm, d = 0.6 mm, g = 0.3 mm, l = 1.7 mm, r = 1.5 mm, t = 3.07 mm, and w = 0.9
mm. The metal copper cladding is 0.035 mm in thickness and has a conductivity of =
5.8×107 S/m. The permittivity of the F4BM-2 substrate is 2.65 + 0.001*i.
Figure 4-1 : Design of the 2D-HMM (a) Schematic diagram and (b) photograph of the chiral
metamaterial.
4.4 Mechanism of Chirality of 2D-HMM
The constitutive equations describe how chiral media interact with EM waves. Specifically, the
electric displacement vector, �⃗⃗⃗�, for a bi-isotropic medium is given by
60
�⃗⃗⃗� = 𝜖0𝜖�⃗⃗� + 𝑖𝜅�⃗⃗⃗�/𝑐 = 𝜖0�⃗⃗�+𝜖0𝜒𝑒 �⃗⃗� + 𝑖𝜅�⃗⃗⃗�/𝑐
Comparing this equation with the general definition,
�⃗⃗⃗� = 𝜖0�⃗⃗� + �⃗⃗�
�⃗⃗� - The polarization vector
We interpret the latter two terms as the polarization induced by the electric field and the
polarization induced by the magnetic field,
�⃗⃗�𝐸 = 𝜖0𝜒𝑒 �⃗⃗�
�⃗⃗�𝑀 = 𝑖𝜅�⃗⃗⃗�/𝑐
Similarly, the magnetic field is given by
�⃗⃗� = 𝜇0𝜇�⃗⃗⃗� − 𝑖𝜅�⃗⃗�/𝑐 = 𝜇0�⃗⃗⃗� + 𝜇0𝜒𝑚�⃗⃗⃗� − 𝑖𝜅�⃗⃗�/𝑐 = 𝜇0(�⃗⃗⃗� + �⃗⃗⃗�𝑀 + �⃗⃗⃗�𝐸)
Where, �⃗⃗⃗�𝑀 = 𝜒𝑚�⃗⃗⃗� and �⃗⃗⃗�𝐸 = (−𝑖𝜅/𝑐𝜇0)�⃗⃗� are the magnetizations induced by the magnetic
and electric fields, respectively.
Figures 4-2(a) and Figure 4-2(b) show the simulated distribution and a schematic drawing of
the current flowing in the 2D-HMM structure. They show that the currents on the top (black
solid lines) and bottom (blue dash lines) layers flow along opposite directions. As a result,
strong magnetization, �⃗⃗⃗�, is induced antiparallel to the direction of the electric field, �⃗⃗�, of the
61
incident EM wave. Therefore, the electric field, �⃗⃗�, contributes to the magnetic field, �⃗⃗�, as
discussed above, which explains the origin of the chirality of the 2D-HMM.
We also calculated the polarization vector, �⃗⃗� =1
𝑖𝜔𝑉∫ 𝐽𝑑𝜏
The magnetization vector, �⃗⃗⃗� =1
2𝑐𝑉∫(𝑟 × 𝐽)𝑑𝜏
Using the volume current density distribution, 𝐽, obtained from simulations. Figure 4-2(c)
shows the y-component of �⃗⃗� and �⃗⃗⃗� when the electric field,�⃗⃗�, of the incident wave is linearly
polarized along the y-direction (and �⃗⃗⃗� is along the x-direction). Based on the above discussion,
we find 𝑃𝑦 and 𝑀𝑦 are both excited by the �⃗⃗�-field of the incident wave, while 𝑃𝑥 and 𝑀𝑥 are
both excited by the �⃗⃗⃗�-field.
Figure 4-2 : (a) Surface current distribution of 2D-HMM at 3.4 GHz. (b) Schematic drawing
of the current. (c) Calculated polarization vector, P, and magnetization vector, M. Here, Px
(Py) and Mx (My) are the x(y)-components of the polarization vector and the magnetization
vector, respectively.
Po
lariza
tio
n (
C/m
2)
Ma
gn
etiz
atio
n (A
m2/k
g)
2 3 4 5 6 7 80
1x10-7
2x10-7
3x10-7
4x10-7
5x10-7
P
Frequency (GHz)
0
1x10-8
2x10-8
3x10-8
4x10-8
5x10-8
6x10-8
M
𝑃𝑥 (𝑃𝑀)𝑃𝑦 (𝑃𝐸) 𝑀𝑥 (𝑀𝑀)
𝑀𝑦 (𝑀𝐸)
a b c
Frequency (GHz)
62
The excitations of strong 𝑀𝑦 (which is 𝑀𝐸) and 𝑃𝑥 (which is 𝑃𝑀) show that the 2D-HMM
must be described by a bi-isotropic medium, and thereby explain the mechanism of chirality.
Figure 4-3 : Induced current paths inside the material
4.5 Equivalent Wave Equation in Bi-Isotropic Medium
In the following, we derive the equivalent wave equation for bi-isotropic media from
Maxwell’s equations:
BE
t
(4-1)
DH
t
(4-2)
Using the continuity equations, we can obtain the following wave equation from equations.
4-1 and 4-2:
E
B k
M M
63
2 2
2 2
0 0 0 02 22
E HE i
t tn
ò ò (4-3)
2 2
2 2
0 0 0 02 22
E HH i n
t t
ò ò ò (4-4)
where 𝑛 = √𝜖𝜇. Equations 4-3 and 4-4 are not readily solvable by common numerical EM
solvers, because the �⃗⃗� and �⃗⃗⃗�-fields are coupled. To get a decoupled form of the wave
equations, Lindell et. al. introduced the wavefields, 1
2E E i H , where
òis the
impedance. By using equations. 4-3 and 4-4, we can obtain an equivalent wave equation:
2
20E E
t
ò (4-5)
where 1n
ò ò and 1
n
. For time-harmonic fields, �⃗⃗�± = �⃗⃗�±(𝑟)𝑒
𝑖𝜔𝑡 , Eq. 4-5
can be written as:
2
0 0E k E ò (4-6)
Where 𝑘0 = 𝜔/𝑐 is the wave number in the vacuum. Equation 4-6 has the exact same form
as the wave equations for isotropic media that are used in many numerical solvers. For
instance, Comsol Multiphysics can solve the following equation:
2
0
0
10
iE k E
ò
ò (4-7)
For isotopic dielectric media, 𝜇 has no spatial dependence and 𝜎 = 0, so that Eq. 4-7 can be
written as:
2
0 0E k E ò (4-8)
64
Equation 4-8 has exactly the same form as Equation 4-6. Therefore, we can solve equation 4-
6 to study the propagation and refraction of EM waves in bi-isotropic media using Comsol
Multiphysics, where 𝜖± and 𝜇± are calculated from the values for 𝜖, 𝜇, and 𝜅 of 2D-HMM
obtained by the retrieval method.
4.6 Results
Figure 4-4 shows simulated and measured transmission coefficients for LCP and RCP incident
waves along the normal direction. We performed numerical simulations using CST Microwave
Studio[88], which utilizes a finite integration technique to obtain the solutions of Maxwell’s
equations. In our simulations, LCP and RCP waves propagate through the unit cell shown in
Figure 4-2(a) (periodic boundary conditions), and the corresponding transmission coefficients,
𝑇𝐿𝐿 , 𝑇𝑅𝐿 , 𝑇𝐿𝑅 , and 𝑇𝑅𝑅 are calculated. In experiments, the transmission of the fabricated sample
was measured using two horn antennas connected to a vector network analyzer. The co-
polarized and cross-polarized transmission coefficients for linearly polarized waves, 𝑇𝑥𝑥 and
𝑇𝑦𝑥, are measured when the receiving horn antenna is oriented horizontally and vertically,
respectively, while the transmitting antenna is horizontally fixed. The transmission coefficients
are then converted to a circular basis (see methods). Due to the fourfold rotational symmetry
of the 2D-HMM, the cross-polarization transmissions, 𝑇𝑅𝐿 and 𝑇𝐿𝑅 , are exactly zero, which
confirms the expected result for pure optical activities. Figures 4-4(a) and Figure 4-4(b) show
the amplitude and the phase of the co-polarized transmission coefficients, 𝑇𝐿𝐿 and 𝑇𝑅𝑅 ,
respectively. Our measurements show excellent agreement with simulations in the frequency
range between 3 and 8 GHz. Slight discrepancy is observed below 2.8 GHz due to diffraction
65
effects as the wavelength approaches the overall sample size (240 mm). For the chosen
geometric parameters, the phase of 𝑇𝐿𝐿 constantly leads that of 𝑇𝑅𝑅 by an amount of ~𝜋/2 in
the range from 2 to 8 GHz. As a result, the polarization azimuthal rotation angle,
arg( ) arg( )
2
RR LLT T
, keeps at a nearly constant value of about 45° within the entire
measured frequency region, showing nondispersive optical activity. Ellipticity, defined as
2 2
2 2
1arctan
2
RR LL
RR LL
T T
T T
, quantifies the polarization state of an EM wave as the ratio of the
semi-minor to the semi-major axes of the polarization ellipse. Figure 4-4(d) displays nearly
zero ellipticity (𝜂 = 0) of the transmitted wave when an incident linearly polarized wave
propagates through the metamaterial, revealing the unchanged polarization state. Note that
𝜂 = 0 within the entire frequency range is remarkable as compared to very large 𝜂 at resonance
frequencies in resonator-based chiral metamaterials[62, 65, 78, 81-83, 85] due to the large
difference between absorptions of LCP and RCP waves. Figure 4-4(a) shows that the
transmission reaches its peak value of nearly unity at 3.4 GHz and that the transmission is
more than 80% in the wide range from 2.9 GHz to 4.0 GHz (a relative bandwidth of 32% of
the center frequency), indicating high transmission within a broad frequency range. The inset
of Figure 4-4(c) plots the trace of the tip of the electric field vector of the transmitted wave,
�⃗⃗�𝑡/𝐸𝑖 = (𝑇𝑥𝑥�̂� + 𝑇𝑦𝑥�̂�), as the phase increases from 0 to 2𝜋 at 3.4 GHz, which essentially
depicts the polarization status of the transmitted wave as a horizontally polarized incident
wave (with �⃗⃗�𝑖 field along 0° direction) propagates through the sample. It is clearly seen that
66
the linear polarization is very well maintained in the transmitted wave with an azimuthal
rotation of 45°.
Figure 4-4 : Transmission and optical Activity. The amplitude (a) and phase (b) of simulated
(solid curves) and measured (symbols) transmission coefficients for LCP (red) and RCP (blue)
incident waves along the normal direction. (c) Polarization azimuthal rotation angle θ, and (d)
Ellipticity η. Inset in (c) shows simulated (black) and measured (red) polarization states of a
transmitted wave at 3.4 GHz.
Due to the fourfold rotational symmetry, our 2D-HMM can be described as a bi-isotropic
medium[89] with the following constitutive equations:
0
0
/
/
i c
i c
D E
B H
òò
where 𝜖, 𝜇 and 𝜅 are the effective permittivity, permeability and chirality, respectively. 𝜖, 𝜇
and 𝜅 for our 2D-HMM can be calculated from both simulated and measured complex
2 3 4 5 6 7 80
10
20
30
40
50
2 3 4 5 6 7 8-8
-6
-4
-2
0
2
4
6
8
2 3 4 5 6 7 80.0
0.2
0.4
0.6
0.8
1.0
2 3 4 5 6 7 8-120
-90
-60
-30
0
30
60
90
120
150
𝑇𝐿𝐿 (Experiment)
𝑇𝑅𝑅 (Experiment)
𝑇𝐿𝐿 (Simulation)
𝑇𝑅𝑅 (Simulation)
,
a
,(d
egre
e)
Frequency (GHz)
b
(Simulation)
(Experiment) 𝜅 𝑘0 𝑑
(d
egre
e)
𝜂(d
egre
e)
𝜂 (Simulation)
𝜂 (Experiment)
𝜅 𝑘0 𝑑
Frequency (GHz)
d
c
45◦90◦
0◦
135◦
180◦
225◦
270◦
0.6 0.8 1.0
0.40.2
315◦
67
transmission and reflection coefficients by a retrieval method [65, 81, 82] for bi-isotropic
media. As shown in Figure 4-5, the effective parameters calculated from simulations and
experimental data show excellent agreement except for a longer wavelengths (2 GH to 3 GHz)
where the measurement shows slight inaccuracy (Figure 4-4(a)).
Figure 4-5 : Effective medium parameters. The effective permittivity 𝜖 (a), permeability 𝜇 (b),
chirality (c), and refractive indices, 𝑛 (d), 𝑛+ (e) and 𝑛− (f) of 2D-HMM calculated from
simulated (solid lines) and measured (symbols) transmission and reflection coefficients of
circular polarization. The red and blue colors represent the real and imaginary parts of the
effective parameters, respectively.
2 3 4 5 6 7 8-5
-4
-3
-2
-1
0
1
2
3
4
5
Effe
ctive
Re
factive
In
de
x
Frequency (GHz)
2 3 4 5 6 7 8-1
0
1
2
3
4
5
Effe
ctive
Re
factive
In
de
x
Frequency (GHz)
2 3 4 5 6 7 8-2
-1
0
1
2
3
4
Effe
ctive
Re
factive
In
de
x
Frequency (GHz)
2 3 4 5 6 7 8-1
0
1
2
3
4
5
Re
(e
ff
Frequency (GHz)
2 3 4 5 6 7 8-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Re
(e
ff
Frequency (GHz)
2 3 4 5 6 7 8-30
-20
-10
0
10
20
Re
(e
ff
Frequency (GHz)
𝑓𝑝𝑓0
e
d 𝜖 Sim (𝜖) Sim
a
𝑓𝑝
Frequency (GHz)
c f
𝜖 Exp (𝜖) Exp
Frequency (GHz)
b
68
In our calculations of effective parameters, we added 0.5 mm of air space on both sides of the
2D-HMM, so the thickness of the effective medium is 𝑑 = (𝑡 + 1) = 4.07 . As
shown in Figure 4-5(a), the effective permittivity exhibits a Drude-like response, where the
real part, (𝜖), increases from negative to positive values as frequency increases. The low
plasma frequency arises from the continuity of the metallic mesh of 2D-HMM, which can be
regarded as an effective plasma medium with diluted electron concentration[90]. The plasma
frequency, 𝑓𝑝 = 3.35 GHz (for (𝜖) = 0), can be tuned by adjusting the density of the
metallic mesh through geometric parameters, e.g. 𝑎 and 𝑡. The imaginary part, (𝜖), is nearly
zero (<0.1 in simulations) in the entire range from 2 to 8 GHz. For plasma media, (𝜖) plays
distinct roles for frequencies both above and below the plasma frequency. Above 𝑓𝑝, a plasma
medium behaves as a regular dielectric medium with refractive index (𝑛) > 0, where the
nonzero (𝜖) contributes to (𝑛) and results in the absorption of electrical energy from
an EM wave. Below 𝑓𝑝, a plasma medium behaves as a conductor, and (𝜖) describes the
conductivity, 𝜎 = 𝜔 (𝜖). The effective permeability in Figure 4-5(b) remains nearly
constant, (𝜇) ≈ 0.5 showing a diamagnetic behavior[91] and (𝜇) ≈ 0 indicating
negligible loss of magnetic energy. The large chirality, (𝜅), reveals non-resonant hyperbolic
dispersion as shown in Figure 4-5(c), leading to nearly constant giant azimuthal rotation =
(𝜅eff) 𝑘0𝑑 ≈ 45°. The imaginary part, | (𝜅)|, is less than 1 × 10−3 (simulation) over
the entire frequency range from 2 to 8 GHz, resulting in nearly zero ellipticity 𝜂 = (𝜅)
𝑘0𝑑 (Figure 4-4(d)). Figure 4-5(d) shows the ordinarily defined refractive index, 𝑛 = √𝜖𝜇.
Below the plasma frequency (𝑓𝑝=3.35 GHz), the negative values of (𝜖) lead to purely
69
imaginary 𝑛 (i.e. (𝑛) = 0), which prohibits any wave propagation. Above 𝑓𝑝, 𝑛 is like that
of a typical dielectric medium with (𝑛)>0 and nearly zero (𝑛). For chiral media, the
eigensolutions of wave equations are LCP and RCP plane waves. The refractive indices are
given by 𝑛± = 𝑘±/𝑘0 = 𝑛 ± 𝜅, where 𝑘+ and 𝑘− are wave numbers for RCP and LCP waves,
respectively, and 𝑘0 = 2𝜋/𝜆 is the wave number in a vacuum. Note that the propagation and
refraction of EM waves in chiral media are determined by 𝑛± rather than 𝑛, where the latter
only provides the convenience to understand the effects of 𝜖 and 𝜇.With strong chirality,
|𝜅| > |𝑛|, a negative refractive index for one circular polarization can be achieved. The
difficulty here is that if Re(𝑛) = (√𝜖𝜇) is not very close to zero, then very strong chirality
is required. However, in previous chiral NIMs[81-83], chirality is achieved via electric or
magnetic resonances, which unavoidably lead to a large value of Re(𝑛). For our 2D-HMM,
(𝑛) decreases monotonically as frequency decreases and remains at zero below 𝑓𝑝. (𝜅)
shows the opposite trend, which increases monotonically and reaches the same value
(𝜅)= (𝑛)=1.98 at 𝑓0 = 4.70 GHz. Therefore, as shown in Figure 4-5(f), below 𝑓0 we
observed a negative refractive index for LCP waves, i.e. (𝑛−) < 0, due to (𝑛) < (𝜅).
Compared to NIMs due to simultaneous 𝜖 and 𝜇 using magnetic resonators[3, 54-59] or due
to strong 𝜅 using chiral resonators[81-83], our 2D-HMM shows a negative index in an ultra-
wide range from 4.70 GHz down to DC. The negative index, (𝑛−)(Figure 4-5(f)), is a slowly
varying smooth function of frequency in contrast to the strong dispersions present in
resonator-based NIMs. This feature is important to improving the robustness of practical
NIM devices since the index of refraction changes very little when the operating frequency
70
drifts slightly. Another important feature is that 𝑛− is a continuous function as shown in Figure
4-5(f). The negative index in previous chiral metamaterials [81-83] usually occurs when 𝜅
exhibits discontinuity and jumps between positive and negative values. This happens when the
phase of LCP or RCP transmission, Φ(𝑇𝐿𝐿) or Φ(𝑇𝑅𝑅) reaches ±𝜋 and is forced to remain
within the range of (−𝜋, 𝜋). However, the adjacent branches, i.e.(𝜋, 3𝜋), are also possible
solutions for Φ(𝑇𝐿𝐿) or Φ(𝑇𝑅𝑅). The selections of branches lead to ambiguous 𝜅 and 𝑛±
values. For our 2D-HMM, both 𝜅 and 𝑛 are continuous functions of frequency, and thereby
avoid possible ambiguities of 𝑛±.
In the following, we discuss the loss for the negative index of 𝑛−. Since (𝜅) is nearly zero,
the imaginary part of the refractive index, (𝑛−), which describes the absorption of an EM
wave, is dominated by (𝑛). Below 𝑓𝑝, (𝑛) increases monotonically as frequency
decreases (Figure 4-5(d)). Above 𝑓𝑝, (𝑛) is nearly zero (<0.01) due to extremely small
(𝜖) (<0.07) and (𝜇) (<0.002). In the range between 𝑓𝑝 and 𝑓0 (green highlighted region
in Figure 4-5(f), (𝑛−) is negative and (𝑛−) is extremely small, and therefore exhibits a
very high Figure of merit, 𝐹𝑂𝑀 = | (𝑛)/ (𝑛)|. As shown in Figure 4-6, the FOM in this
region reaches a maximum value of 91 in simulations (or 131.4 in experiment data), which is
significantly higher than typical FOMs of 3~10 for other low-loss negative index
metamaterials [92-94]. The high FOM shows good consistency with extremely low loss in the
2D-HMM. As shown in Figure 4-6, the absorption of an LCP wave, 𝐴𝐿𝐿 = 1 − |𝑇𝐿𝐿|2 −
|𝑅𝐿𝐿|2 reaches a peak of 0.019 at 𝑓𝑝=3.38 GHz where the transmission (Figure 4-4(a)) is also
71
close to its maximum. This level of absorption is extremely small compared to resonator-based
NIMs [81, 83, 92, 95, 96] (𝐴 = 0.1~0.5). Note that the peak of 𝐴𝐿𝐿 in the green highlighted
region is due to higher transmission, as shown in Figure 4-4(a), which allows more EM wave
energy to enter the metamaterial. As the frequency moves away from the peak, most of the
energy of the EM wave is reflected and therefore does not contribute to the loss. The
normalized absorption,𝐴𝐿𝐿′ = 𝐴𝐿𝐿/(1 − |𝑅𝐿𝐿|
2), calculates the absorption of EM wave
energy that enters the metamaterial. As shown in Figure 4-6, 𝐴𝐿𝐿′ reaches minimum values in
the green highlighted region where the FOM is high. The region with high FOM overlaps with
the region with transmission peaks as shown in Figure 4-6. Therefore, in the negative refractive
index region between 𝑓𝑝 and 𝑓0 the metamaterial exhibits an exceptional combination of high
transmission, very low loss, and broad bandwidth.
Figure 4-6 : Figure of merit for 𝒏−- and absorption for LCP. FOM calculated from simulated
(red solid curve) and measured (red circle) data. Absorption (blue solid curve) and normalized
absorption (blue dashed curve) calculated from simulation.
2 3 4 5 6 7 8
0
50
100
150
200
FO
M
Frequency (GHz)
0.000
0.005
0.010
0.015
0.020
0.025
0.030
AL
L
𝐴𝐿𝐿
𝐴𝐿𝐿′
𝐹𝑂𝑀Fig
ure
of
Me
rit
FOM Sim
FOM Exp
𝐴𝐿𝐿 Sim
𝐴𝐿𝐿′ Sim
Frequency (GHz)
Abso
rptio
n
72
The propagation and refraction of an EM wave in a chiral medium with negative index has
not been depicted through numerical simulation yet, because of a lack of a numerical solver
that can solve the Maxwell’s equations in bi-isotropic media defined by 𝜖, 𝜇, and 𝜅. In the
following, we show a numerical approach to simulate the wave propagation in chiral media by
solving an equivalent wave equation in an isotropic medium based on the wavefield
postulates[89] introduced by Lindell et. al. The wave equation for bi-isotropic medium can be
written as:
2
20E E
t
ò (4-9)
Where, 1n
ò ò , 1
n
,
The wave fields,�⃗⃗�±, are defined as
1
2E E i H
With the impedance 𝜂 = √𝜇
𝜖
The solutions for equation 4-2, E and E , are electric fields for RCP and LCP waves,
respectively. Since equation 4-9 has the same form as the linearly polarized EM wave equation,
the propagation and refraction of circular polarized waves, ( ) ik rE r E e
, in bi-isotropic
media (with material parameter 𝜖, 𝜇, and 𝜅) are the same as for linearly polarized waves in
isotropic media with 𝜖 = 𝜖± and 𝜇 = 𝜇±. We performed numerical simulations to solve
73
Equation 4-9 in an isotropic medium with Comsol Multiphysics, which uses a finite elements
method. In our simulation, we calculated Gaussian beams propagating through a wedge
defined by an isotropic medium with frequency dependent 𝜖± and 𝜇± calculated from𝜖, 𝜇, 𝜅,
and 𝑛 shown in Figure 4-5. Figure 4-7(a) and Figure 4-7(b) show the refractions at the slant
surface of a wedge with refractive indices of 𝑛− = −2.42 + 0.05𝑖 and 𝑛+ = 3.02 + 0.05𝑖,
respectively, at a frequency of 3.37 GHz. The incident Gaussian beam with �⃗⃗� = �̂�𝐸𝑧 is excited
at the bottom edge and propagates into the wedge. The incident angle at the slant surface,
1 = 18°, is same as the angle of the wedge. The refractive beam lies in the same (Figure 4-
7(a)) or the opposite (Figure 4-7(b)) side of the normal direction (red dash line), showing
negative and positive refraction, respectively. The measured refraction angles, 2 = −48.5°
and 68.9°, agree very well with Snell’s law. The FOM for the negative index in Figure 4-7(a)
is 45, and the intensity of the refractive beam reduces slightly. However, as shown in Figure
4-7(c) and Figure 4-7(d), the wave intensity decreases very quickly inside a lossy wedge with a
FOM of 16. As a result, the intensities of refracted beams are significantly reduced. Figure 4-
7(e) clearly shows that the amplitude of the electric field decreases exponentially (𝐸 =
𝐸0𝑒−(𝐹𝑂𝑀) 𝑘𝑑) inside the wedge (green highlight region). Figure 4-7(f) plots the phase of the
EM wave, 𝐸 = 𝐸0𝑒𝑖Re(𝑛±) 𝑘0𝑑, where 𝑘0 is the wave number in a vacuum and 𝑑 is the
propagation distance. The green highlighted region clearly shows that the phase increases for
a positive index, but decreases for a negative index, both as linear functions of 𝑑. Although
our metamaterial are fabricated at microwave frequency, the same design can be scaled down
in size by using existing micro- and nano-fabrication techniques for higher frequency ranges.
74
For instance, at THz frequencies, the 2D-HMM can be fabricated by standard
microfabrication processes to form double-layer planner structure followed by an
electroplating process to connect them via vertical pillars[78]. At infrared regime, our chiral
NIM design can be made by direct layer writing, which has been proved suable for making
similar continuous metallic structures such as helix chiral metamaterials[66].
Figure 4-7 : Wave refraction in effective medium. (a-d) Real part of electric field,�⃗⃗⃗� = �̂�𝑬𝒛,
for a linearly polarized EM wave propagating through an isotropic wedge with a negative (a,c)
and a positive (b,d) refractive index. (e) Amplitude and (f) phase of �⃗⃗⃗� along the propagation
path
0 5 10 15 20 25-20
-10
0
10
20
30
40
Ph
ase
Ez/p
i
Distance/Wavelength
0 5 10 15 20 25
1
10
100
Ez
Distance/Wavelength
f=3.37 GHz𝑛− =-2.42+i0.05
f=3.37 GHz𝑛+ =3.02+i0.05
f=3.35 GHz𝑛− =-2.64+i0.16
f=3.35 GHz𝑛+ =2.84+i0.16
Distance/Wavelength Distance/Wavelength
𝐸𝑧
Φ𝐸𝑧 /𝜋
𝑛− =-2.42+i0.05
𝑛+ =3.02+i0.05
𝑛− =-2.64+i0.16
𝑛+ =2.84+i0.16
d
ba
c
e f
15
10
5
0
-5
-10
-15
(𝑉/𝑚)
1 = 18°
1 1 = 18°
1
1 = 18° 1
1 = 18° 1
FOM~45
FOM~16
75
CHAPTER FIVE
MAGNETICALLY TUNABLE TERAHERTZ METAMATERIALUSING
POLYMERIC MICROACTUATORS
5.1 Introduction
Tunable THz metamaterials has shown great potential to solve the material challenge due to
the so-called “THz gap”. However, the tunable mechanism of current designs relies on using
semiconductors insertions, which inevitably results in high Ohmic loss, and thereby
significantly degrades the performance of metamaterials. In this work, we demonstrate a novel
tunable mechanism based on polymeric microactuators. Our metamaterials are fabricated on
the surface of patterned pillar array of flexible polymers embedded with magnetic
nanoparticles. The transmission spectrum of the metamaterial can be tuned as the pillars are
mechanically deformed though applied magnetic field. We observed and measured several type
of deformation including bending, twisting and compressing when the applied magnetic field
is polarized along different direction with respected to the axis of the magnetic particles.
Compared to previous semiconductor based tunable mechanism, our structure has shown
much lower loss.
76
Microactuators have been in use to manipulate electromagnetic waves using different methods
such as external currents or voltages, temperature gradient, chemical solutions and
electromagnetic fields. The disadvantages of these methods are associated with tuning range,
response time and accuracy, which limits the applications. Furthermore, for an example, if the
external control method for these active polymeric actuators uses currents or voltages, it must
be physically wired to external electrodes or control devices, which makes the fabrication
process complicated and limits freedom in actuation range. The advantage of magnetically
actuated micro devices, is that it can be controlled without any additional apparatus and can
be controlled remotely as long as the actuation environment is magnetically transparent. In
addition, they can be operated in many different media such as air, vacuum, conducting and
non-conducting liquids. Furthermore, magnetic actuators have an instantaneous response with
compared to temperature gradient actuators. Many of the microstructures use magnetic
particles mixed with or encapsulated by soft materials but there are various types of engineered
polymeric materials, such as electroactive polymers, carbon nanotube composites and liquid
crystal polymers as active actuating components in devices.
In this chapter, we demonstrate a magnetically controllable THz metamaterial structure, which
has magnetic nano-particle, embedded pillars and with the change of the magnetic strength,
the pillars bend differently towards the magnetic field. The pillars are made out of PDMS and
carbonyl iron power is embedded in the pillars as nano magnetic particles. Figure 5-1 shows
the bending of the pillars when applying a magnetic field to the sample. The magnetic nano-
77
particles are aligned vertically in the pillars and bending of the pillars can be reversed easily by
taking away the applied magnetic field.
Figure 5-1 : Behavior of the pillars without the magnetic field and with the magnetic field
5.2 Simulation results
Simulations were carried out using CST and Figure 5-2 shows the designs used in simulations.
The direction of the THz signal shows by the k vector and Figure 5-2 (a) shows the straight
pillars, which indicates that the applied magnetic field is zero.
S
S N
Without Magnetic Field With Magnetic Field
Magnetic Nanoparticles
B = 0
k
TE TM
PDMS
CIP embedded in Si composite
(b) (a)
(a)
78
Figure 5-2 : Simulating Microactuators
Figure 5-2(b) and Figure 5-2(c) shows the bended pillars indicating the applied external
magnetic field. In simulations, instead of carbonyl ion powder embedded PDMS, we used
permittivity ε = 5+0.52i and permeability μ = 0.85+0.1i [97] . Polydimethylsiloxane (PDMS)
is used as the substrate and it has a permittivity of ε = 2.35+0.02i [98]. The results obtained
from simulations are shown in Figure 5-3.
B ≠ 0
k
TE TM
B ≠ 0
(b)
(c)
79
Figure 5-3 : (a) Magnitude (b) Phase of the transmission signal
Simulation results show that behavior of the pillars with the magnetic field and without the
magnetic field for transverse electric (TE) or transverse magnetic (TM) are nearly the same.
Figure 5-3(a) shows the magnitude of the transmitted signal and Figure 5-3(b) simulations
(b)
(a)
80
results exhibit that the magnitude does not change with the frequency but phase of the THz
signal can be manipulated using the external magnetic field.
The images of the pillars taken under a microscope is shown in Figure 5-4. It shows the top
view of the pillars with half circle pillars, which has a length of 28.498 μm in the long axis and
17.394μm in the short axis.
Figure 5-4 : Images of the pillars (Top View), (b) geometrical size (Top View)
Figure 5-5 shows the experimental set up with external permanent magnets placed on the side
of the sample.
Figure 5-5 : Applying the magnetic field to the sample - actual experimental setup
28.498 µm
17.394 µm
(b) (a)
81
The applied magnetic field is horizontal and we changed the magnetic field strength by
changing the distance between the permanent magnets. As shown in Figure 5-6(c), the phase
change of the THz signal increases linearly as the magnetic field strength increases. This
metamaterial device has been shown to modulate THz signal.
The phase shift is given by
∆∅(𝜔) = |∅1(𝜔) − ∅2(𝜔)|
Where, ∅1(𝜔) is the phase of the THz signal with externally applied magnetic field and ∅2(𝜔)
is the phase of the THz signal without applying the magnetic field
5.3 Experimental results
Experiments were carried out using THz-TDS system and the magnetic field was applied using
permanent magnets as shown in Figure 5-5. After obtaining data from THz-TDS, Mathlab is
used to analyze data.
0.2 0.4 0.6 0.8 1 1.20
2
4
6
8
10
12
14
Frequency (THz)
Inte
nsity
Ref-air
VA 0 angle 0.0T
VA 0 angle 0.15T
VA 0 angle 0.25T
VA 0 angle 0.28T
VA 0 angle 0.32T
(a)
82
Figure 5-6: (a) THz signal intensity( b) Transmission (c) Phase change of the THz signal with
magnetic field.
As shown in Figure 5-6(b) the amplitude of the transmitted THz signal does not vary with the
applied magnetic field, but as shown in Figure 5-6(c) the phase change of the transmitted signal
increases with the magnetic field strength. The results obtained from simulations and
experiments agree and it indicates that these microactuators can be used as phase modulators.
0.2 0.4 0.6 0.8 10.5
0.6
0.7
0.8
0.9
1
Frequency (THz)
S2
1
VA 0 angle 0.0T
VA 0 angle 0.15T
VA 0 angle 0.25T
VA 0 angle 0.28T
VA 0 angle 0.32T
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-40
-35
-30
-25
-20
-15
-10
-5
0
Frequency (THz)
Phase (
Deg)
VA (B=0.15T - B=0)
VA (B=0.25T - B=0)
VA (B=0.28T - B=0)
VA (B=0.32T - B=0)
(b)
(c)
83
CHAPTER SIX
CONCLUSION AND FUTURE WORK
6.1 Conclusion
In conclusion varactor loaded split ring resonator (SRR) design, which can be used as a
nonlinear metamaterial demonstrates nonlinear effects such as tenability, hysteresis effect and
higher order harmonic generation. Varactor embedded SRR exhibits power and frequency
dependent broadband tunability and it is realized for not only applying a direct voltage across
the diode but also for external electro-magnetic signals. More importantly, the nonlinear SRR
shows bi-stability with distinct transmission levels, where the transition between bi-states is
controlled by the impulses of pump signal. We demonstrate that the nonlinear hysteresis
behavior can be used to exhibit a switching device in microwave regime. In order to increase
its functionality in other frequencies, a new design is introduced with two distinct rings which
has two distinct resonance frequencies. The double split ring resonator (DSRR) also
demonstrate similar behavior as the SRR but it is broadband. Simulation results were obtained
with advanced designs to achieve non-linearity in terahertz frequency regime to realize
tunability, hysteresis and bi-stable states.
Furthermore, by designing the structure such that inner ring has a frequency twice as outer
ring resonance frequency, we demonstrate the enhancement of second harmonic generation.
84
By using a ground plate, we have exhibit that we can further increase the harmonic power. In
addition, we can stack these unit cells in a pre-calculated manner in order to make a 3D
material which enhance the power of the second harmonic.
In chiral metamaterials, we have presented a 2D-HMM which exhibits very versatile
properties, including nondispersive optical activity, broadband chirality, and a negative
refractive index. For our chiral metamaterial based on a 2D continuous mesh of helical
structures, we have provided evidence through a retrieval method that the negative index arises
from chirality and exhibits the important features of broad bandwidth, very low loss, and high
transmission. The negative index FOM reaches 91 in the region above the plasma frequency.
Finally, we solved an equivalent wave equation for circular polarized waves in a bi-isotropic
effective medium representing our 2D-HMM and demonstrated positive and negative
refractions for LCP and RCP waves.
Polymeric microactuators are used to tune the phase of the THz signal by using external
magnetic fields. These pillar arrays of flexible polymers embedded with magnetic nanoparticles
demonstrate an easy mechanism to achieve phase modulation. The experimental results agree
with experimental results with low loss compared to other structures and broadband
responses.
85
6.2 Future work
6.2.1 Wave front control by nonlinear metamaterial in terahertz frequency.
In preliminary studies, we have experimentally demonstrated that we can control transmission
as well as phase of a varactor loaded nonlinear metamaterial by applying an external dc bias
voltage. In this project, we propose that by carefully designing an arrray structure where we
can individually address each element using an external voltage source and circuitry to
manipulate voltage appropriately to tune phase and magnitude of each element, we can focus
the terahertz beam to a focal point.
Figure 6-1 : Focusing terahertz beam to a focal point
-10 -5 0 5 10-10
-5
0
5
10
x (mm)
y (
mm
)
Phase delay ()
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
-0.04 -0.02 0 0.02 0.04-0.04
-0.02
0
0.02
0.04
x (mm)
y (
mm
)
Intensity
1
2
3
4
5
6
7
-0.04
-0.02
0
0.02
0.04
-0.04
-0.02
0
0.02
0.040
1
2
3
4
5
6
7
8
x (mm)y (mm)
Inte
nsity
1
2
3
4
5
6
7
86
Furthermore, each element will act as a point source in this design and according to huygens
and fresnel principle, we can demonstrate computationally that with phase manipulation,
waves can be focused to a point as illustrated in Figure 6-1.
6.2.2 Second Harmonic generation using phase matching
The nonlinear effects of natural materials are limited by small susceptibility values and
nonlinear effects involving frequency mixing are further restricted by the phase-matching
condition required by momentum conservation.
Figure 6-2 : Second harmonic generation using phase matching in a crystal
Figure 6-2 illustrates phase matching of SHG in a birefringence crystal. The incident wave is
chosen to travel at an angle θ with respect to the crystal optical-axis, such that the extraordinary
refractive index ne (ω), of the incident ω wave equals the ordinary refractive index no(2ω) of the
generated 2ω wave. It is required that the phase matching has to be achieved by carefully
aligning the incident signal along a particular direction, θ, with respect to the optical axis of a
nonlinear birefringent crystal. In this project, we propose that the DSRR design can also
87
enhance SHG at microwave frequencies by satisfying phase-matching condition in a simple
way without alignment.
Figure 6-3 : Refractive index matching through satisfying phase matching condition
Varactor loaded DSRR structure was used and by carefully designing the sizes of the outer
and inner rings, we can achieve electric resonances at ω0 and 2ω0, respectively. Figure 6-3
(a),Figure 6-3(b) shows the extracted effective refractive index from the preliminary simulation
data; assuming the nonlinear effects have a small impact on the linear refractive index. Around
the resonance frequencies, we achieve the required phase-matching condition n(ω)=n(2ω). Due
to the flexibility of manipulating the refractive index by changing the geometric design of the
DSRR, the proposed Metamaterial design relieves the precise alignment requirement in
natural crystals.
88
6.2.3 Enhanced second harmonic generation in THz frequency regime.
Enhancement of nonlinear process such as second harmonic generation was demonstrated
for microwaves by embedding varactor diodes and SRRs under preliminary results. In this
project our goal is to enhance second harmonic generation in teraherts frequency regime by
using a specially designed structures. It has been theoritically proven that fundamental
frequency and second harmonic frequency have different polarizatons[48]. By using this
theoritical explanation we can design a U-shaped SRR with w2= 2w1, where magnetic
resonance at w1 with Ex polarization and electric resonance at w2 with Ey polarization. In this
approach, incident wave is polarized along Ex direction, the second harmonic will be excited
along y-direction and it will enhance second harmonic. Figure 6-4(b) shows frequency
matching condition for different polarizations and Figure 6-4(a) and Figure 6-4(c) show the
surface current density observed at those resonance frequencies.
Figure 6-4 : Enhancing second harmonic using U shaped SRRs
Linear current of electric resonance
at 2 =2
1
Circular current of
magnetic resonance at 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency
T
c
)
b
) a
)
89
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LIST OF PUBLICATIONS
1. Khagendra Bhattarai, Sinhara R Silva, Kun Song, Augustine Urbas, Sang Jun Lee, Zahyun
Ku, Jiangfeng Zhou. “Metamaterial Perfect Absorber Analyzed by a Meta-cavity Model
Consisting of Multilayer Metasurfaces” Scientific Reports, 2017. (Equal contribution)
2. Kun Song, Zhaoxian Su, Min Wang, Sinhara Silva, Khagendra Bhattarai, Changlin
Ding, Yahong Liu, Chunrong Luo, Xiaopeng Zhao, Jiangfeng Zhou. “Broadband angle-
and permittivity-insensitive nondispersive optical activity based on planar chiral
metamaterials”. Scientific Reports, 2017.
3. Khagendra Bhattarai, Sinhara R. Silva, Augustine Urbas, Sang Jun Lee, Zahyun Ku,
Jiangfeng Zhou. . “Angle-dependent Spoof Surface Plasmons in Metallic Hole Arrays at
Terahertz Frequencies” IEEE Journal of Selected Topics in Quantum Electronics, 2016.
4. Khagendra Bhattarai, Zahyun Ku, Sinhara Silva, Jiyeon Jeon, Jun Oh Kim, Sang Jun Lee,
Augustine Urbas, Jiangfeng Zhou. “Mushroom capped plasmonic optical cavity based
refractive index sensor fabricated by interferometric lithography”, Advanced Optical
Materials, 2015.
103
5. Zahyun Ku, Woo-Yong Jang, Jiangfeng Zhou, Jun Oh Kim, Ajit V. Barve, Sinhara Silva,
Sanjay Krishna, S. R. J. Brueck, Robert Nelson, Augustine Urbas, Sangwoo Kang, and Sang
Jun Lee. . “Analysis of subwavelength metal hole array structure for the enhancement of
back-illuminated quantum dot infrared photodetectors” Optics Express, 2013.
6. Kun Song, Sinhara Silva, Clayton Fowler, Zhaoxian Su, Yahong Liu, Changlin Ding,
Chunrong Luo, Sang Jun Lee, Zahyun Ku, Augustine Urbas, Xiaopeng Zhao, Jiangfeng
Zhou. “Broadband Negative Refractive Index with Extraordinary Figure-of-Merit from
2D Helical Chiral Metamaterial”. (Equal contribution, Submitted: Nature Communication)
7. Clayton Fowler, Sinhara R Silva, Jiangfeng Zhou. “High Efficiency Ambient RF Energy
Harvesting by a Metamaterial Perfect Absorber”(Submitted: Advanced optical materials)
8. Clayton Fowler, Sinhara R Silva, Jiangfeng Zhou. “A Highly Efficient Polarization-
Independent Metamaterial Perfect Absorber Based Rectenna for Low-intensity RF Energy
Harvesting” (Submitted: Applied Physics Letters)
9. Sinhara R Silva, K. Bhattarai, A. Shields, Jiangfeng Zhou, “Enhancing harmonic
generation using nonlinear metamaterials”. (In preparation)
10. Sinhara R Silva, K. Bhattarai, A. Shields, Jiangfeng Zhou “Tunable Bistable Nonlinear
Metamaterials as a switch in Microwave frequencies”. (In preparation)
