1

Electromagnetically Induced Transparency

(EIT) in Plasmonic System

Guo BingSheng

A0116512U

Supervisor: Professor Ong Chong Kim

Mentor: Loo Yoke Leng (PhD student)

An honours thesis submitted to the Department of Physics, National University

of Singapore in partial fulfilment of the requirements for the Degree of Bachelor

of Science with Honours in Physics

AY2016/17

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Abstract

This thesis proposes a plasmonic metamaterial that is able to mimic Electromagnetically

Induced Transparency (EIT) in the reflectance spectrum. It has the properties of being tunable

and polarisation independent. Moreover, the metamaterial has demonstrated to also possess

slow wave property, with group refractive index of 56; and refractive-index-based sensing

capability, with merit figure of 6.1. The metamaterial comprises of a cross-slot structure as

the bright resonator, and a group of 4 spiral structures as the dark resonator. In the strong

coupling configuration, the plasma frequency and coupling constant of the metamaterial are

calculated to be approximately 5.4 × 1010 𝑟𝑎𝑑/𝑠 and 9.8 × 109 𝑟𝑎𝑑/𝑠 respectively. While

the respective damping constants of the bright resonator and quasi-dark resonator are

calculated to be approximately 4.6 × 1010 𝑟𝑎𝑑/𝑠 and 1.9 × 1010 𝑟𝑎𝑑/𝑠.

3

Acknowledgments

I would like to thank all my NUS physics lecturers who have inculcated a sense of

appreciation and intuition for the subject in me. I would also like to thank my fellow peers

who have accompanied and helped me along the way in this academic journey.

I would like to thank my supervisor Professor Ong Chong Kim for offering me this

project, and giving me valuable advice and guidance not just pertaining to this work but also

in other aspects of life. I would also like to thank my mentor Loo Yoke Leng who is currently

pursuing her Doctorate degree in physics. Without her guidance on the CST studio software,

timely encouragement, and valuable advice, it would have been more than an uphill battle for

me.

Special thanks to Wang HaiGang who is currently pursuing his Master’s degree in

physics, and my mentor, for selflessly forking out their precious time to provide me with the

much needed technical support in conducting the experiment.

Last but not least, I would like to thank all my beloved family members who have always

been there to support me in all possible ways throughout this arduous academic journey. I

thank God for His grace and mercy. To Him be all the glory!

“The fear of the LORD is the beginning of wisdom, and knowledge of the Holy One is

understanding.” -Proverbs 9:11

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Table of Figures

Figure 1: Coupling between a 3-level atomic system and 2 coherent laser sources (pump

source and scanning source). ................................................................................................... 10

Figure 2: Two coupled harmonic oscillators for modelling the quantum system in section 2.1.

.................................................................................................................................................. 13

Figure 3: Simulated power distribution of bright resonator, plotted against detuned frequency,

with damping ratio of 3, at coupling strength of a) 0, b) 2.5 and c) 5 𝑟𝑎𝑑/𝑠. ........................ 15

Figure 4: Simulated power distribution of bright resonator, plotted against detuned frequency

with damping ratio of 1, at coupling strength of a) 0, b) 2.5 and c) 5 𝑟𝑎𝑑/𝑠. ........................ 15

Figure 5: Simulated power distributions of an RLC circuit, each at a different damping

constant, plotted against the detuned frequency. ..................................................................... 17

Figure 6: RLC circuit with an oscillatory time dependent voltage source. ............................. 17

Figure 7: Bright resonator made of copper with thickness of 17 𝑢𝑚, placed on the 1 𝑚𝑚

thick FR-4 substrate. ................................................................................................................ 19

Figure 8: Simulated a) absorbance, b) reflectance and c) transmittance of the bright resonator,

plotted against frequency of incident TEM wave, at various polarization angle 𝑝ℎ𝑖 (in units of

degree)...................................................................................................................................... 20

Figure 9: Simulated a) z-component of 𝐻 field distribution, b) current distribution, and c) z-

component of 𝐸 field distribution at polarization 0° and 7.8 GHz. ......................................... 21

Figure 10: Simulated a) z-component of 𝐻 field distribution, b) current distribution, and c) z-

component of 𝐸 field distribution at polarization 45° and 7.8 GHz. ....................................... 21

Figure 11: Simulated a) absorbance and b) transmittance of FR-4 substrate, plotted against

frequency of incident TEM wave, at substrate thickness (𝑠𝑡) of 1 and 5𝑚𝑚. ........................ 22

Figure 12: Simulated data of a) absorbance, b) reflectance, and c) transmittance of bright

resonator with substrate, plotted against frequency of incident TEM wave, at various

substrate thickness (𝑠𝑡) in units of 𝑚𝑚. .................................................................................. 23

Figure 13: a) Spiral structure’s dimensions and top view of the incident TEM fields. b) to d)

illustrate its simulated optical properties in the frequency range of 20 to 25 GHz, with

electric dipole resonance occuring at 21.2 GHz. e) to g) illustrate its simulated optical

properties in the range of 5 to 10 GHz, with magnetic dipole resonance occuring at 7.9 GHz.

.................................................................................................................................................. 24

Figure 14: Simulated a) z-component of 𝐻 field distribution, b) current distribution, and c) z-

component of 𝐸 field distribution at 𝑡ℎ𝑒𝑡𝑎 0° and 21.2 GHz. ................................................ 26

Figure 15: Simulated a) z-component of 𝐻 field distribution, b) current distribution, and c) z-

component of 𝐸 field distribution at 𝑡ℎ𝑒𝑡𝑎 45° and 7.9 GHz. ................................................ 26

5

Figure 16: a) The intial position of each spiral structure with respect to the cross-slot

structure.They are all equally separated from the centre. The couple strength can be increased

by shifting the structures towards the centre of the cross-slot structure. The resulted position

is shown in b). .......................................................................................................................... 27

Figure 17: Simulated data of a) absorbance, b) transmittance and c) reflectance of the

proposed plasmonic metamaterial, plotted against frequency of incident TEM wave at various

values of separation 𝑆 (in units of 𝑚𝑚). .................................................................................. 27

Figure 18: Simulated distribution of 𝐻 field (z-componet) in the a) cross-slot structure (bright

resonator) and b) spiral structures (quasi-dark resonator), in the weak coupling regime with

𝑆 = 8.6 𝑚𝑚 and, polarization angle 45°. ................................................................................ 28

Figure 19: Simulated distribution of 𝐻 field (z-componet) in the a) cross-slot structure (bright

resonator) and b) spiral structures (quasi-dark resonator), in strong coupling regime with 𝑆 =

6.6 𝑚𝑚 and, polarization angle 45°. ........................................................................................ 28

Figure 20: Simulated a) reflectance, b) absorbance and c) transmittance of the proposed

plasmonic metamaterial, plotted against frequency of incident TEM wave, at polarization

angle ranging from -45 to 45 𝑝ℎ𝑖 (in unit of degree). .............................................................. 30

Figure 21: Simulated a) reflectance of the proposed plasmonic metamaterial, plotted against

frequency of incident TEM wave at 3 different environment’s refractive index, with 𝑆 fixed

at 8.1 𝑚𝑚. Simulated b) Qualty factor and reflectance plotted against seperation 𝑆 of the

proposed plasmonic metamaterial............................................................................................ 31

Figure 22: Simulated real and imaginary parts of effective refractive index of the proposed

plasmonic metamaterial, plotted against frequency of incident TEM wave, in the strong

coupling regime with 𝑆 = 6.6 𝑚𝑚. ......................................................................................... 32

Figure 23: Simulated real and imaginary parts of the refractive index of FR-4 substrate,

plotted against frequency of incident TEM wave. ................................................................... 33

Figure 24: Simulated real and imaginary parts of relative permittivity 𝜖𝑟 of the proposed

plasmonic metamaterial, plotted against the frequency of incident TEM wave, in the strong

coupling regime, with 𝑆 = 6.6 𝑚𝑚. ........................................................................................ 33

Figure 25: a) Periodic array of cross-slot structures (bright resonator), and b) spiral structures

(quasi-dark resonator). The size of each unit cell is 22 𝑚𝑚 by 22𝑚𝑚. There are a total of 81

unit cells in the proposed plasmonic metamaterial. ................................................................. 36

Figure 26: Schematic of the experimental set-up. ................................................................... 36

Figure 27: a) Dielectric lenses and the b) sample holder placed in between them. ................. 37

Figure 28: a) Antenna port connected to the b) network analyser which is connected to the PC

via the grey local area network (LAN) cable. .......................................................................... 37

Figure 29: Experimental (EXP) a) absorbance, b) reflectance and c) transmittance of

proposed plasmonic metamaterial (𝑎𝑡 𝑝ℎ𝑖 = 0°), plotted against frequency of incident TEM

wave at different separation 𝑆. ................................................................................................. 38

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Figure 30: Simulated (SIM) a) absorbance, b) reflectance and c) transmittance of proposed

plasmonic metamaterial (at 𝑝ℎ𝑖 = 0°), plotted against frequency of incident TEM wave at

different separation 𝑆. .............................................................................................................. 38

Figure 31: Experimental (EXP) a) reflectance, b) absorbance and c) transmittance of

proposed plasmonic metamaterial (at 𝑆 = 7.8 𝑚𝑚), plotted against frequency of incident

TEM wave at different polarization angle. .............................................................................. 39

Figure 32: Simulated (SIM) a) reflectance, b) absorbance and c) transmittance of proposed

plasmonic metamaterial (at 𝑆 = 7.8 𝑚𝑚), plotted against frequency of incident TEM wave at

different polarization angle. ..................................................................................................... 39

Figure 33: Optical properties of proposed plasmonic metamaterial (at 𝑆 = 7.8 𝑚𝑚, and

𝑝ℎ𝑖 = 0°) plotted against frequency, when TEM wave is first incident on the a) bright

resonator, or b) quasi-dark resonator. ...................................................................................... 40

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Contents 1 Introduction ..................................................................................................................... 8

1.1 Plasmonic Metamaterial .......................................................................................... 8

1.2 Motivation and Objective ........................................................................................ 8

1.3 Overview ................................................................................................................. 9

2 Fundamentals ................................................................................................................ 10

2.1 Electromagnetically Induced Transparency (EIT) ................................................ 10

2.2 Classical Model of EIT .......................................................................................... 13

2.3 𝑅𝐿𝐶 Circuit as an Analogue of a Plasmonic Resonator ........................................ 16

3 Proposed Plasmonic Structure ...................................................................................... 19

3.1 Bright Resonator (cross-slot structure) .................................................................. 19

3.2 Effect of FR-4 Substrate ........................................................................................ 22

3.3 Quasi-Dark Resonator (spiral structure) ................................................................ 23

3.4 Coupling and Tunability ........................................................................................ 26

3.5 Polarization Independence ..................................................................................... 29

3.6 Refractive-index-based Sensing ............................................................................ 30

3.7 Slow Wave Capability; Damping Constants and Coupling Constant ................... 32

4 Experiment .................................................................................................................... 36

4.1 Set-up and Procedures ........................................................................................... 36

4.2 Comparison between Experimental and Simulated Results .................................. 37

5 Conclusion .................................................................................................................... 41

6 References ..................................................................................................................... 42

7 Annex A ........................................................................................................................ 44

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Chapter 1

1 Introduction

1.1 Plasmonic Metamaterial

In recent decade, intense research effort has been dedicated to the field of metamaterial,

an artificial engineered material which usually comprises a mixture of metallic and non-

conducting components. Such research interest in the scientific community derives from the

metamaterial’s ability to exhibit special properties, which do not exist in conventional

material. For instance, there is no natural material that possesses a negative refractive index,

even though such property has been proven to be theoretically possible by Veslago in 1968

[1]. D.R Smith et al. however managed to achieve a negative refractive index in his designed

metamaterial which comprises of inter-spaced conducting split-rings and wires [2] in 1999.

Furthermore, metamaterial can be engineered such that it possesses the ability to mimic

certain physical effect at a lower cost. For instance, Electromagnetically Induced

Transparency (EIT).

EIT is originally a quantum phenomenon. It has attracted much interest due to its abilities

in slowing down the group velocity of an incident wave, and inducing transparency in an

otherwise opaque sample at resonant frequency. In order for EIT to be clearly observable in

the quantum system, the sample has to be kept under cryogenic temperature during the

experiment [3]. However, with metamaterial, similar effect can be achieved at room

temperature.

The work in this thesis mainly focuses on achieving EIT-like effect in plasmonic

metamaterial.

1.2 Motivation and Objective

Many papers have demonstrated the ability of plasmonic metamaterial in mimicking EIT-

like effect using different kind of structures. Most ([4], [5], [6]) involve breaking the

symmetry of the structures to produce the effect. Such structures are typically tunable as the

coupling strength between the bright resonator and dark resonator is adjustable. However, the

visibility of EIT-like effect in such structures is usually highly dependent on the polarization

state of incident fields. Some ([7], [8]) however have succeeded in achieving polarization

independence in the structure but at the expense of tunability. Thus far, only a few ([9], [10])

have managed to achieve both properties, with the EIT-like effect manifested in the

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transmittance spectrum. Furthermore, to the best of my knowledge, structure possessing both

the abovementioned properties, and exhibiting EIT-like effect in the reflectance spectrum has

yet to be achieved.

It is thus the goal of this work to design a polarization independent and tunable structure

that is able to demonstrate EIT-like effect in the reflectance spectrum.

1.3 Overview

In chapter 2 the fundamentals involve in this work will be sufficiently discussed. In

chapter 3, the developmental process of the proposed plasmonic metamaterial will be

presented; followed by a discussion of its various properties, based on the results obtained

from simulation. In chapter 4, the experiment set-up and procedures will be presented,

followed by a comparison between the experimental and simulated results. Lastly in the

conclusion, a summary of the work and results will be presented, followed by a brief

discussion on areas of improvement and possible future work.

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Chapter 2

2 Fundamentals

2.1 Electromagnetically Induced Transparency (EIT)

The equations in this section are referenced from [11], [12] and [13]. Consider the atomic

system as shown in Figure 1. The system is in the lowest energy level 𝐸1 initially.

Furthermore, two coherent lasers, known as the pump source (with energy corresponds to the

difference between 𝐸1 and 𝐸2) and scanning source (with energy corresponds to the

difference between 𝐸2 and 𝐸3) are introduced into the system.

In the perturbative regime, the effective Hamiltonian of the above system follows equation

(2.1-A). Here ��𝑎 is the dominant Hamiltonian that describes the atom in the absence of the

fields, while ��𝑝 and ��𝑠 are the perturbation Hamiltonian that describe the coupling of the

atom with the pump and scanning source via electric dipole interaction respectively.

(2.1-A)

In order to determine the eigenstates and eigenvalues of the effective Hamiltonian ,

equation (2.1-A) is first inserted with identities leading to equation (2.1-B), followed by

expressing it in the matrix form as shown in equation (2.1-C).

(2.1-B)

Figure 1: Coupling between a 3-level atomic system and 2 coherent laser

sources (pump source and scanning source).

𝑖ℏ𝜕|Ψ⟩

𝜕𝑡= ��|Ψ⟩, 𝑤𝑖𝑡ℎ �� = ��𝑎 + ��𝑠 + ��𝑝

𝑖ℏ𝜕

𝜕𝑡∑𝐶𝑛(𝑡)|𝜓𝑛⟩

𝑛

=∑|𝜓𝑛⟩⟨𝜓𝑛|��(𝑡)|𝜓𝑚⟩𝐶𝑚𝑛.𝑚

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(2.1-C)

The energy terms in the Hamiltonian are then determined by equations (2.1-D) to (2.1-F).

𝑛 is the number that represents the state of the system, while 𝑑 is the electric dipole moment

of the atom. Respectively, ��, Ω and 𝜔 are the electric field, Rabi frequency and angular

frequency of the corresponding laser source (the respective subscript p and s stand for pump

source and scanning source). Substituting with these equations, the effective Hamiltonian

follows equation (2.1-G)

(2.1-D)

(2.1-E)

(2.1-F)

(2.1-G)

After applying rotating-wave approximation to equation (2.1-G) and expressing it in the

dressed-state representation, all the diagonal terms become 𝐸2 while the phase factor of all

the off-diagonal terms are dropped. For simplicity, 𝐸2 is set to zero as the reference point of

energy. The resulting Hamiltonian then follows equation (2.1-H), which is time independent

and thus allows for the determination of its eigenstates and eigenvalues.

(2.1-H)

There are a total of three sets of eigenvalues and eigenstates, as shown in equations (2.1-I)

to (2.1-K). The eigenvalue for eigenstate |0⟩ is zero, while for eigenstate | + 𝐸⟩ and

eigenstate | − 𝐸⟩, they are 𝐸+ and 𝐸− respectively.

𝑖ℏ(

𝐶1𝐶2𝐶3

) = (𝐸1 𝐸12 𝐸13𝐸21 𝐸2 𝐸23𝐸31 𝐸32 𝐸3

)(𝐶1𝐶2𝐶3

) = 𝐸 (𝐶1𝐶2𝐶3

)

𝐸𝑛 = ⟨𝑛|��𝑎|𝑛⟩

|𝐸12| = ⟨1|��𝑝(𝑡)|2⟩ = ⟨1| − 𝑑. ��𝑝(𝑡)|2⟩ = ℏΩ𝑝 cos(𝜔𝑝𝑡)

|𝐸23| = ⟨2|��𝑠(𝑡)|3⟩ = ⟨2| − 𝑑. ��𝑠(𝑡)|3⟩ = ℏΩ𝑠 cos(𝜔𝑠𝑡)

��(𝑡) = (

𝐸1 ℏ𝛺𝑝 𝑐𝑜𝑠(𝜔𝑝𝑡) 0

ℏΩ𝑝∗ cos(𝜔𝑝𝑡) 𝐸2 ℏ𝛺𝑠 𝑐𝑜𝑠(𝜔𝑠𝑡)

0 ℏΩ𝑠∗ 𝑐𝑜𝑠(𝜔𝑠𝑡) 𝐸3

)

�� =

(

0

ℏ

2𝛺𝑝 0

ℏ

2Ω𝑝∗ 0

ℏ

2𝛺𝑠

0ℏ

2Ω𝑠∗ 0 )

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(2.1-I)

(2.1-J)

(2.1-K)

For describing the evolution of the atomic system, the initial state |1⟩ is projected with the

time evolution unitary operator. After which, the identity involving the summation of the

Hamiltonian’s eigenstates is inserted, yielding equation (2.1-L).

(2.1-L)

In the absence of the scanning source, with Ω𝑠 = 0, equation (2.1-L) becomes,

|𝛹(𝑡)⟩=1

2 [𝑐𝑜𝑠 (

𝛺𝑝

2𝑡) (

100) − 𝑖 𝑠𝑖𝑛 (

𝛺𝑝

2 𝑡) (

010)] .

This implies that the electron in the atom is only transiting between energy state 𝐸1 and 𝐸2

when the scanning source is turned off. This results in a typical Lorentzian distribution in the

absorption spectrum of the atom when damping, due to spontaneous emission, is factored into

consideration. While for the case when the scanning source is turned on, and under the

condition that Ω𝑠 ≫ Ω𝑝, equation (2.1-L) becomes,

𝐸± = ±

ℏ

2Ω𝑚, 𝑤𝑖𝑡ℎ 𝛺𝑚 = √|𝛺𝑝|

2+ |𝛺𝑠|2

|±𝐸⟩ =1

√2|Ω𝑝|2+ 2Ω𝑚2

(

Ω𝑝±Ω𝑚Ω𝑠∗),

|0⟩ =1

√|Ω𝑝|2+ |Ω𝑠|

2

(

Ω𝑠0−Ω𝑝

∗)

|Ψ(t)⟩ = exp (−

𝑖��𝑡

ℏ) |1⟩

= exp (−𝑖𝐸−𝑡

ℏ) | − 𝐸⟩⟨−𝐸| 1⟩

+ exp (−𝑖𝐸+𝑡

ℎ) |+𝐸⟩⟨+𝐸|1⟩ + |0⟩⟨0|1⟩

=𝛺𝑝∗

|𝛺𝑝|2+ |𝛺𝑚|2

[cos (Ω𝑚2𝑡) (

𝛺𝑝0𝛺𝑠∗)

− 𝑖 sin (Ω𝑚2 𝑡) (

0𝛺𝑚0)] +

𝛺𝑠∗

|𝛺𝑝|2+ |𝛺𝑚|2

(

𝛺𝑠0−𝛺𝑝

∗)

13

|Ψ⟩ ≈ (100)

which is independent of time. This suggests that large transparency is being induced in the

atom. In other words, the laser fields from the pump source passes through the atom without

being significantly absorbed at all time. This leads to a significant dip at the resonant

frequency of the atom’s absorption spectrum. Such phenomenon is termed as

Electromagnetically Induced Transparency (EIT). The effect of EIT will lead to a large delay

in the group velocity of the incident wave, which will be demonstrated in section 3.6.

2.2 Classical Model of EIT

The equations in this section are referenced from [14]. Even though EIT is inherently a

quantum mechanical effect, it can be modelled by a classical system. One such system is the

two coupled harmonic oscillators [14] as illustrated in Figure 2. The atom is modelled as the

oscillator in the blue region. The blue region symbolises an environment such as water, that

leads to higher damping experience by an oscillator as compared to when it is in the white

region which symbolises an environment such as air. It is coupled to a driving source with

force amplitude 𝐹, to represent the pump source in the quantum system. The scanning source

is modelled as the harmonic oscillator in the white region, which indeed can be done so

according to quantum field theory if the laser source is coherent. The two oscillators are

connected via a spring (with proportionality constant 𝑘0) to model the coupling between the

atom and the scanning source.

Figure 2: Two coupled harmonic oscillators for modelling the

quantum system in section 2.1.

For simplicity, we set the spring constant 𝑘 and mass 𝑚 of one oscillator to be equal to

that of the other oscillator. In the quantum analogue, this will imply that |1⟩ and |3⟩ are in the

same energy level. Equations (2.2-A) and (2.2-B) describe the motion of the left oscillator

14

and right oscillator respectively. 𝛾 and 𝜔𝑑 represent the damping constant of the

corresponding oscillator and angular frequency of the driving source respectively.

(2.2-A)

(2.2-B)

Using equation (2.2-C) as the Ansatz, where 𝐴 is the constant to be solved using equations

(2.2-A) and (2.2-B), the final form of 𝑥1 follows equation (2.1-D). Differentiating equation

(2.2-D) with respect to time and multiply it with the time varying force 𝐹 exp(−𝑖𝜔𝑑𝑡), the

power experienced by the left oscillator over one period follows equation (2.2-E).

(2.2-C)

(2.2-D)

(2.2-E)

From equation (2.2-E), after setting the factor 𝐹2

𝑚 to be 10 𝑁𝑚𝑠−2, the real part of 𝑃1 is

plotted against the detuned frequency (𝜔𝑑 − 𝜔) at different coupling strength 𝜔0 for each

particular value of damping ratio as illustrated in Figure 3 and Figure 4. Clearly from the

graphs, the conditions for EIT to be observable are: the coupling strength between the

oscillators must be strong, and the damping constant of the right oscillator must be lower than

that of the left oscillator. Furthermore from Figure 2, the left oscillator which must have a

larger damping constant, must be connected directly to the driving source and thus it is

commonly termed the bright resonator. While the right oscillator which must have a lower

damping constant, can only to be driven by the left oscillator but not the driving source, and

therefore it is commonly termed the dark resonator. Both resonators must also resonate at

approximately the same frequency.

The dip at the resonant frequency in Figure 3 and Figure 4 is due to the destructive

interference between the normal modes of the coupled oscillators. This implies that in the

𝑑2[𝑥1(𝑡)]

𝑑𝑡2+ 𝛾1

𝑑[𝑥1(𝑡)]

𝑑𝑡+ 𝜔2𝑥1(𝑡) − 𝜔0

2𝑥2(𝑡) =𝐹

𝑚exp(−𝑖𝜔𝑑𝑡)

𝑤ℎ𝑒𝑟𝑒 𝜔2 =

𝑘

𝑚 𝑎𝑛𝑑 𝜔0

2 =𝑘0𝑚

𝑑2[𝑥2(𝑡)]

𝑑𝑡2+ 𝛾2

𝑑[𝑥2(𝑡)]

𝑑𝑡+ 𝜔2𝑥2(𝑡) − 𝜔0

2𝑥1(𝑡) = 0

𝑥1(𝑡) = 𝐴 exp(−𝑖𝜔𝑑𝑡)

𝑥1(𝑡) =

(𝜔2 − 𝜔𝑑2 − 𝑖𝛾2𝜔𝑑)𝐹 exp(−𝑖𝜔𝑑𝑡)

𝑚[(𝜔2 − 𝜔𝑑2 − 𝑖𝛾1𝜔𝑑)(𝜔2 − 𝜔𝑑

2 − 𝑖𝛾2𝜔𝑑) − 𝜔04]

𝑃1(𝜔) =

−𝑖(𝜔2 − 𝜔𝑑2 − 𝑖𝛾2𝜔𝑑)𝐹

2𝜔𝑑

𝑚[(𝜔2 − 𝜔𝑑2 − 𝑖𝛾1𝜔𝑑)(𝜔2 − 𝜔𝑑

2 − 𝑖𝛾2𝜔𝑑) − 𝜔04]

15

strong coupling configuration, the bright resonator becomes stationary while the dark

resonator continues to oscillate. This concept applies to the proposed plasmonic metamaterial

and will be discussed again in section 3.4.

Figure 3: Simulated power distribution of bright

resonator, plotted against detuned frequency, with

damping ratio of 3, at coupling strength of a) 0, b)

2.5 and c) 5 𝑟𝑎𝑑/𝑠.

Figure 4: Simulated power distribution of bright

resonator, plotted against detuned frequency with

damping ratio of 1, at coupling strength of a) 0,

b) 2.5 and c) 5 𝑟𝑎𝑑/𝑠.

In summary, EIT effect can indeed be mimicked by a classical system involving two

coupled resonators if it satisfies the abovementioned requirements. Interestingly, 𝑘𝑜 in

general can be a complex number, which suggests that it is possible for 𝜔𝑜4 in equation

(2.2-E) to be a negative real quantity. In such case, this will lead to the increase of the

0.0

0.2

0.4

0.6

0.8

1.0

-5.0 -2.5 0.0 2.5 5.0

𝑃𝑜𝑤𝑒𝑟𝑃 1/𝑤

𝐷𝑒𝑡𝑢𝑛𝑖𝑛𝑔 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦/𝐻𝑧

𝜔0 = 0

𝛾1𝛾2= 3

3𝑎)

0.0

0.2

0.4

0.6

0.8

1.0

-5.0 -2.5 0.0 2.5 5.0

𝑃𝑜𝑤𝑒𝑟𝑃 1/𝑤

𝐷𝑒𝑡𝑢𝑛𝑖𝑛𝑔 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦/𝐻𝑧

𝜔0 = 0

𝛾1𝛾2= 1

4𝑎)

0.0

0.2

0.4

0.6

0.8

1.0

-5.0 -2.5 0.0 2.5 5.0

𝑃𝑜𝑤𝑒𝑟𝑃 1/𝑤

𝐷𝑒𝑡𝑢𝑛𝑖𝑛𝑔 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦/𝐻𝑧

𝜔0 = 2.5

𝛾1𝛾2= 3

3𝑏)

0.0

0.2

0.4

0.6

0.8

1.0

-5.0 -2.5 0.0 2.5 5.0

𝑃𝑜𝑤𝑒𝑟𝑃 1/𝑤

𝐷𝑒𝑡𝑢𝑛𝑖𝑛𝑔 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦/𝐻𝑧

𝜔0 = 2.5

𝛾1𝛾2= 1

4𝑏)

0.0

0.2

0.4

0.6

0.8

1.0

-5.0 -2.5 0.0 2.5 5.0

𝑃𝑜𝑤𝑒𝑟𝑃 1/𝑤

𝐷𝑒𝑡𝑢𝑛𝑖𝑛𝑔 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦/𝐻𝑧

𝜔0 = 5

𝛾1𝛾2= 3

3𝑐)

0.0

0.2

0.4

0.6

0.8

1.0

-5.0 -2.5 0.0 2.5 5.0

𝑃𝑜𝑤𝑒𝑟𝑃 1/𝑤

𝐷𝑒𝑡𝑢𝑛𝑖𝑛𝑔 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦/𝐻𝑧

𝜔0 = 5

𝛾1𝛾2= 1

4𝑐)

16

resonant peak, which is known as the Electromagnetically Induced Absorption (EIA). This

has been observed in both the quantum and plasmonic system. This thesis, however, will

focus only on EIT.

2.3 𝑹𝑳𝑪 Circuit as an Analogue of a Plasmonic Resonator

The equations in this section are referenced from [14] and [15]. Circuit analysis is often

used as a guide to design a plasmonic resonator such that it possesses certain particular

properties.

Equations (2.3-A) to (2.3-D) are the Maxwell equations. The symbols in these equations

have the usual meaning, with the subscript 𝑓 indicating that the quantity of interest is free;

while respectively, 𝑥𝑒 and 𝑥𝑚 refer to the electric and magnetic susceptibility of a linear and

passive material.

(2.3-A)

(2.3-B)

(2.3-C)

(2.3-D)

𝐿 is the self-inductance which is defined as the division of magnetic flux Φ (induced by

the current in an inductor) by the current i passing through the inductor. After substituting

with equation (2.3-D), 𝐿 obtains the more general form as shown in equation (2.3-E), where

��𝐵 is the unit vector of �� field. Clearly, 𝐿 relates the dependence of a material’s magnetic

response to its structure and the �� field’s orientation.

(2.3-E)

𝐶 is the capacitance defined by the division of charge 𝑞 accumulated, by the voltage 𝑉

across a capacitor. After substituting with equation (2.3-A), 𝐶 obtained the more general form

as shown in equation (2.3-F), where ��𝐸 is the unit vector of the �� field. Clearly 𝐶 relates the

dependence of a material’s electric response to its structure and the �� field’s orientation.

∇. �� =𝜌

𝜖0, ∇. �� =

𝜌𝑓

𝜖, 𝜖 = 𝜖0(1 + 𝑥𝑒)

∇. �� = −

𝜕��

𝜕𝑡

∇. �� = 0

∇ × �� = 𝐽𝑓 + 𝜖

𝜕��

𝜕𝑡, �� =

��

𝜇, 𝜇 = 𝜇0(1 + 𝑥𝑚)

𝐿 =

Φ

i=

∫ ��. 𝑑��𝐴

∫1𝜇 (∇ × ��). 𝑑��′𝐴′

= ∫ ��𝐵. 𝑑��𝐴

∮1𝜇 ��𝐵. 𝑑𝑙𝐿

17

(2.3-F)

The loss in a material can be defined by its resistance 𝑅 times the square of the current

flowing through it. It can be expressed using Poynting’s vector as shown in equation (2.3-G)

which displays the dependence on both the electric and magnetic properties of the material. ��

is the wave’s direction of propagation within the material.

(2.3-G)

The above discussion shows that a linear and passive material such as a typical plasmonic

resonator can be modelled by the components of 𝑅, 𝐿 and 𝐶 which account for its loss,

magnetic and electric properties respectively. More specifically, it can be modelled as a 𝑅𝐿𝐶

circuit that is connected with a time varying voltage source as illustrated in Figure 6. The

voltage source is used to represent the electromagnetic wave incident on the plasmonic

resonator.

Starting with equation (2.3-H) to describe the time evolution of the charge in the 𝑅𝐿𝐶

circuit , with some manipulation, the equation becomes (2.3-I). Clearly, equation (2.3-I)

follows the same physics as in equation (2.2-A), except without the coupling term. Thus by

comparison, the damping constant 𝛾 and resonant frequency 𝑓𝑅 in a 𝑅𝐿𝐶 circuit are

identified be the forms following equation (2.3-J).

(2.3-H)

𝐶 =

𝑞

𝑉=∫ 𝜖��. ��𝑑𝑣𝑣

∫ ��. 𝑑𝑙𝑙

=∮ 𝜖��𝐸 . 𝑑��𝐴

∫ ��𝐸 . 𝑑𝑙𝑙

𝑅𝑖2 = ∫ (�� ×��

𝜇) . 𝑑��

𝐴

= ∫ √𝜖

𝜇𝐸2��. 𝑑��

𝐴

Figure 5: Simulated power distributions of an RLC circuit,

each at a different damping constant, plotted against the

detuned frequency.

Figure 6: RLC circuit with an oscillatory

time dependent voltage source.

𝐿𝑑𝑖

𝑑𝑡+𝑞

𝐶+ 𝑅𝑖 = 𝑉𝑠(𝑡), 𝑖 =

𝑑𝑞

𝑑𝑡

0.0

0.2

0.4

0.6

0.8

1.0

-5.0 -2.5 0.0 2.5 5.0

𝑃𝑜𝑤𝑒𝑟𝑃/𝑊

𝐷𝑒𝑡𝑢𝑛𝑒𝑑 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦/𝐻𝑧

𝜸 = 𝟑

𝜸 = 𝟐

𝜸 = 𝟏

18

(2.3-I)

(2.3-J)

Plotting the power of the circuit against the detuned frequency, the Lorentzian distribution

is obtained for each different damping constant as illustrated in Figure 5. The results indicate

that with higher damping, the power spectrum becomes broader which is the characteristic of

a resonator. This indicates that a 𝑅𝐿𝐶 circuit may indeed be used to model a plasmonic

resonator. It will thus help to serve as a guide in the designing process to achieve certain

property in a plasmonic resonator.

𝑑𝑞2

𝑑𝑡+ (𝑅

𝐿)𝑑𝑞

𝑑𝑡+𝑞

𝐿𝐶=𝑉𝑠(𝑡)

𝐿

𝑤ℎ𝑒𝑟𝑒 𝛾 =

𝑅

𝐿, 𝑓𝑅 =

1

2𝜋√𝐿𝐶

19

Chapter 3

3 Proposed Plasmonic Structure

3.1 Bright Resonator (cross-slot structure)

The first goal is to design the structure of the bright resonator such that its resonance is

independent of the incident traverse electromagnetic (TEM) wave which is propagating along

the normal of the resonator’s surface. In order to achieve that, the structure must have more

than 2 folds of rotational symmetry with respect to its surface’s normal, as proven by Mackay

[16] . Thus the structure of a cross having 4-fold rotational symmetry is chosen.

The next goal is to ensure that the resonator possesses high transmittance at its resonant

frequency in order to demonstrate EIT-like effect in the reflectance spectrum. Furthermore, as

demonstrated in section 2.2, a bright resonator must also have a higher damping constant as

compared to the dark resonator. The bright resonator must therefore be designed so that it is

able to produce an electric dipole resonance, within a considered frequency range. This is

because electric dipole resonance is highly radiative, leading to higher damping constant in a

system [17]. In this work the considered frequency range is between 5.5 to 10 GHz.

However, in the GHz range, a non-complementary copper structure such as a bar is

usually highly reflective at electric dipole resonance, as the resonance occurs below its

plasma frequency [24].

Figure 7: Bright resonator made of

copper with thickness of 17 𝑢𝑚,

placed on the 1 𝑚𝑚 thick FR-4

substrate.

In order to overcome the issue, the cross-slot structure is selected as the bright resonator

instead. This is because by Babinet’s principle, a complementary structure of the same

material will produce the converse effect at the same frequency ([18], [19]). This implies that

the cross-slot structure will possess high transmittance at the same resonant frequency. The

20

structure which has also been adopted by Paul [7] ( with different dimensions and for

different application) is illustrated in Figure 7. The yellow region represents copper with

thickness of 17𝜇𝑚. It is placed on the FR-4 substrate with thickness of 1 𝑚𝑚 to enhance the

magnitude of the resonator’s absorbance. The rationale for the choice of the substrate’s

thickness will be discussed in the next section.

The simulation results in this work are obtained using CST Microwave Studio. The solver

used is known as the Frequency Domain Solver; and the boundary is set as “unit cell”, so that

for instance a single cross-slot structure will be continuously repeated, forming a periodic

array. Such array is considered as a metamaterial [17].

Figure 8 shows the simulation results obtained for the cross-slot structure at various

polarization angle 𝑝ℎ𝑖 (in units of degree). Clearly from the figure, the structure is

polarization independent and has high transmittance at resonant frequency of 7.8 GHz.

Figure 8: Simulated a) absorbance, b) reflectance and c) transmittance of the bright resonator,

plotted against frequency of incident TEM wave, at various polarization angle 𝑝ℎ𝑖 (in units of

degree).

The z-component of �� field distribution, z-component of �� field distribution, and current

distribution of the bright resonator , are plotted at polarization angle 0° (Figure 9) and

8c) 8b)

8a)

21

45°(Figure 10) at 7.8 GHz. From Figure 9 and Figure 10, it shows that even though the

magnitude of each optical properties is polarization independent, the spatial distributions of

the TEM fields and current within the cross-slot structure are not so. This factor determines in

section 3.4 the manner in which the quasi-dark resonator should be placed for coupling with

the bright resonator to ensure polarization independent of the entire system.

Figure 9: Simulated a) z-

component of �� field distribution,

b) current distribution, and c) z-

component of �� field distribution at

polarization 0° and 7.8 GHz.

Figure 10: Simulated a) z-

component of �� field distribution,

b) current distribution, and c) z-

component of �� field distribution at

polarization 45° and 7.8 GHz.

Interestingly, despite having different spatial fields distributions and current distribution at

different polarization angle, the resonant frequency remains almost unchanged. This can be

understood qualitatively from equation (2.3-J) involving relations between resonant

frequency, self-inductance and capacitance. Comparing Figure 9Figure 10, while the

inductance is twice as large at 45° polarization (due to twice the size of the area with

10a)

9b)

9c)

10b)

10c)

9a)

22

localised �� field as shown in Figure 10a) and 10b)), the capacitance is approximately half as

large (due to twice the width (on average) of the gap between the accumulated charges as

shown in Figure 10b)). As a result, the value of the resonant frequency remains the same.

3.2 Effect of FR-4 Substrate

FR-4 is a glass-reinforced epoxy sheet, commonly used as a dielectric insulator. FR stands

for fire resistance, while the value 4 is the quality grade of the material. As mentioned in the

previous section, the FR-4 substrate (with 1 𝑚𝑚 thickness, and 4.3 dielectric constant) in the

proposed metamaterial is for enhancing the absorbance of the resonator. Additionally, it

lowers the resonant frequency due to increase in the metamaterial’s overall capacitance. The

substrate at the chosen thickness must possess low absorbance and high transmittance in the

absence of any resonators, as this ensures low ohmic loss and thereby minimizes the

impediment of the interaction between the bright resonator and dark (on the opposite surface

of the substrate) resonator.

Figure 11: Simulated a) absorbance and b) transmittance of FR-4 substrate, plotted against

frequency of incident TEM wave, at substrate thickness (𝑠𝑡) of 1 and 5𝑚𝑚.

In the absence of any resonators, as shown in Figure 11, there is a significant difference

between the two substrates of thickness 1 and 5𝑚𝑚 in terms of their absorbance and

transmittance. Moreover, as illustrated in Figure 12, in the presence of the bright resonator on

the substrate, the absorbance distribution varies as the substrate thickness (𝑠𝑡) changes.

Analysing the figures, it is clear that 1 𝑚𝑚 thickness is the favourable choice. When 𝑠𝑡 is

zero the absorbance is so low that the bright resonator is almost transparent. Such low

absorbance lowers the visibility of EIT-like effect in the proposed metamaterial. When 𝑠𝑡 is 1

𝑚𝑚 there is a significant increase in the absorbance, and the resonant frequency is lowered

by approximately 2.5 GHz. At larger thickness, however, the width of the electric dipole

resonance increases due to higher damping cause by ohmic loss in the substrate, and other

11b) 11a)

st = 1 st = 5

st = 1 st = 5

23

higher resonance modes begin to approach the electric dipole resonance mode. These effects

are undesirable, and therefore 1 𝑚𝑚 is indeed the optimal 𝑠𝑡 for the proposed metamaterial.

Figure 12: Simulated data of a) absorbance, b) reflectance, and c) transmittance of bright

resonator with substrate, plotted against frequency of incident TEM wave, at various substrate

thickness (𝑠𝑡) in units of 𝑚𝑚.

3.3 Quasi-Dark Resonator (spiral structure)

As mentioned in section 2.2, one of the condition for EIT-like effect to be observable is

that the dark resonator must have a lower damping constant as compared to the bright

resonator. For that to occur, according to equation (2.3-J), a plasmonic dark resonator must

have a relatively large 𝐿. Secondly it can only be excited by the bright resonator but not the

driving source (incident TEM wave). However, to improve the prospect of achieving EIT-like

effect in a plasmonic system, the second requirement will be slightly relaxed in this work

without obscuring the visibility of the phenomenon. Hence, for satisfying the second

requirement, upon interacting with the incident TEM wave at the considered frequency range

(5.5 to 10 GHz), a quasi-dark resonator must possess relatively low absorbance and high

transmittance. It is described as quasi-dark because it interacts weakly with the incident TEM

wave. Finally, both resonators must also resonate at approximately the same frequency.

12a)

12c) 12b)

24

Figure 13: a) Spiral structure’s dimensions and top view of the incident TEM fields. b) to d)

illustrate its simulated optical properties in the frequency range of 20 to 25 GHz, with electric

dipole resonance occuring at 21.2 GHz. e) to g) illustrate its simulated optical properties in the

range of 5 to 10 GHz, with magnetic dipole resonance occuring at 7.9 GHz.

13e) 13b)

13a)

13c)

13d)

13f)

13g)

theta = 0 theta = 15 theta = 30 theta = 45

theta = 0 theta = 15 theta = 30 theta = 45

theta = 0 theta = 15 theta = 30 theta = 45

25

The proposed quasi-dark resonator in this work has a spiral structure, which has also been

adopted by Meng (with different dimensions and quantity) [9]. It is illustrated in Figure 13a).

This structure is capable of achieving magnetic dipole resonance only in the present of a

perpendicular component of �� field that is parallel to the structure surface’s normal. This is

illustrated in Figure 13e) to 13g), with 𝑡ℎ𝑒𝑡𝑎 in units of degree. The magnetic dipole resonant

frequency is 7.9 GHz which is approximately the same as the electric dipole resonant

frequency of the cross-slot structure, and thus fulfilled the third condition mentioned above.

Furthermore, by comparing the absorbance spectrum between the spiral structure and cross-

slot structure in the frequency range of 5.5 to 10 GHz, clearly the bandwidth is narrower for

the spiral structure. This implies that the damping constant of the spiral structure is lower and

hence satisfies the first condition mentioned above. From Figure 13, when 𝑡ℎ𝑒𝑡𝑎 is zero in

the frequency range of 5.5 to 10 GHz, both the absorbance and reflectance are particularly

low. This is due to the fact that the spiral structure’s electric dipole resonance occurs at a

much higher frequency of 21.2GHz. Therefore, the second condition mentioned above is also

achieved.

In order to understand qualitatively the cause for the large frequency gap between the

electric dipole resonance and magnetic dipole resonance of the spiral structure, the fields

distributions and current distribution are plotted as illustrated in Figure 14 (for electric dipole

resonance at 𝑡ℎ𝑒𝑡𝑎 zero) and Figure 15 (for magnetic dipole resonance at 𝑡ℎ𝑒𝑡𝑎 45°). It can

be observed from Figure 14b) that the direction of current flow in the left end and right end of

the spiral structure are parallel, while that is antiparallel in Figure 15b). This leads to much

larger accumulation of charges in the structure, which translates to much higher capacitance

for the case of magnetic dipole resonance, as illustrated by Figure 14 Figure 15. Moreover, by

comparing Figure 14 and Figure 15, clearly the spiral structure also possess much higher self-

inductance at magnetic dipole resonance. Thus, from equation (2.3-J), the magnetic dipole

resonance must occur at a significantly lower frequency, as compared to the electric dipole

resonance of the spiral structure.

In summary, the spiral structure has a relatively lower damping constant as compared to

the cross-slot structure. Furthermore its magnetic dipole resonance occurs only when a

perpendicular component of �� that is parallel to its surface’s normal is present. Its electric

dipole resonance has also shown to be occurring only at a much higher frequency. The spiral

structure, therefore, indeed satisfies all the conditions to be considered as a quasi-dark

26

resonator. Additionally, in the GHz range since it is a non-complementary copper structure, it

will lead to the increase in reflectance when its magnetic dipole resonance mode is activated

by for instance, the cross-slot structure (bright resonator) via coupling (in section 3.4).

Figure 14: Simulated a) z-

component of �� field distribution,

b) current distribution, and c) z-

component of �� field distribution at

𝑡ℎ𝑒𝑡𝑎 0° and 21.2 GHz.

Figure 15: Simulated a) z-

component of �� field distribution,

b) current distribution, and c) z-

component of �� field distribution at

𝑡ℎ𝑒𝑡𝑎 45° and 7.9 GHz.

3.4 Coupling and Tunability

In this section, the coupling between the cross-slot structure and spiral structure will be

discussed. In order to ensure polarization independence of the proposed metamaterial, its

quasi-dark resonator will have to comprise of 4 spiral structures instead of one. (Refer to

section 3.5.) Furthermore, for coupling to occur, the induced fields from each resonator must

be able to interact with each other directly. Hence, the quasi-dark resonator is placed on the

layer just opposite to the cross-slot structure as illustrated in Figure 16. The coupling strength

15a)

14b)

14c)

15b)

15c)

14a)

27

is adjustable and can be increased by shifting the 4 spiral wire structures towards the centre of

the cross-slot structure.

Figure 16: a) The intial position of each spiral structure with respect to the cross-slot

structure.They are all equally separated from the centre. The couple strength can be increased

by shifting the structures towards the centre of the cross-slot structure. The resulted position is

shown in b).

Figure 17: Simulated data of a) absorbance, b) transmittance and c) reflectance of the proposed

plasmonic metamaterial, plotted against frequency of incident TEM wave at various values of

separation 𝑆 (in units of 𝑚𝑚).

Each spiral structure is placed in such manner because the induced magnetic field from the

cross-slot structure is concentrated at its edges as illustrated in Figure 9 and 10. The

16b) 16a)

17c) 17b)

17a)

28

simulated absorbance, transmittance and reflectance spectra are plotted as illustrated in

Figure 17. The symbol 𝑆 here refers to the separation (in units of 𝑚𝑚) of each spiral structure

from the centre of the proposed metamaterial. From Figure 17, when 𝑆 is decreased, the dip

between the resonant peak in the absorbance spectrum increases, (similar to Figure 3 in

section 2.2), while the peak in the reflectance spectrum increases. This effect is due to the

increase in the coupling strength between the two resonators as demonstrated in section 2.2.

Hence the simulated data shows that the proposed metamaterial can indeed exhibit EIT-like

effect in the reflectance spectrum, and concurrently possesses the property of being tunable in

terms of coupling strength.

Figure 18: Simulated distribution of �� field (z-componet) in the a) cross-slot structure (bright

resonator) and b) spiral structures (quasi-dark resonator), in the weak coupling regime with 𝑆 =

8.6 𝑚𝑚 and, polarization angle 45°.

Figure 19: Simulated distribution of �� field (z-componet) in the a) cross-slot structure (bright

resonator) and b) spiral structures (quasi-dark resonator), in strong coupling regime with 𝑆 = 6.6

𝑚𝑚 and, polarization angle 45°.

In order to understand the physical significance of the results obtained in Figure 17, the z-

component of �� field distributions are plotted at 𝑆 = 8.6 (weak coupling) and 6.6 𝑚𝑚

(strong coupling), both at polarization angle 45° as demonstrated in Figure 18 and Figure 19

respectively. When the coupling strength is weak, there is significant localisation of �� field

18b)

19a) 19b)

18a)

29

(z-component) within the bright resonator and quasi-dark resonator. However, in the case of

strong coupling, there is only negligible localisation of �� field within the bright resonator,

while large localisation of the field is observed in the quasi-dark resonator. The vanishing ��

field in the bright resonator is caused by the destructive interference between the incident

TEM fields and the induced fields from the quasi-dark resonator. The physics is similar to the

case of destructive interference between the normal modes of the two coupled harmonic

oscillators in section 2.2. Therefore, since the cross-slot structure (bright resonator) which is

highly transmissive at resonance, is being excited strongly in the weak coupling regime, this

must result in high transmittance of the metamaterial. Conversely, since only the spiral

structures (quasi-dark resonator) which is highly reflective at resonance, is being excited

strongly in the strong coupling regime, this then must result in high reflectance of the

metamaterial.

3.5 Polarization Independence

In order to ensure polarization independent of the proposed plasmonic metamaterial, the

first condition is to ensure that the optical properties of its bright resonator remain constant at

all polarization angle of the incident TEM fields. This has been achieved by selecting a

structure comprises of 4-fold rotationally symmetry as demonstrated in section 3.1. Similarly,

the same requirement applies to its quasi-dark resonator. Since the electric dipole resonant

frequency of its quasi-dark resonator is far from the considered frequency range, and hence

only interacts weakly with the incident TEM fields, it is also polarization independent.

Furthermore, the isotropic FR-4 substrate is clearly also polarization independent.

The second condition requires the magnitude of the metamaterial’s optical properties to be

independent of the spatial distribution of the localised fields within the bright resonator. As

illustrated in section 3.1, the spatial distribution of the localised �� field varies as the

polarization angle of the incident TEM fields changes. A total of 4 spiral structures are

therefore required, with each being positioned around the edge of the cross-slot structure.

This ensures maximum portion of the induced �� field from its bright resonator is coupled to

its quasi-dark resonator as the spatial distribution of localised fields varies. As a consequence,

the change in the magnitude of the metamaterial’s optical properties is minimized as the

polarization angle varies.

30

Figure 20: Simulated a) reflectance, b) absorbance and c) transmittance of the proposed

plasmonic metamaterial, plotted against frequency of incident TEM wave, at polarization angle

ranging from -45 to 45 𝑝ℎ𝑖 (in unit of degree).

The simulated magnitude of each optical property is plotted as shown in Figure 20. 𝑝ℎ𝑖

represents the polarization angle of the incident TEM fields in units of degree. The range of

angle that needs to be tested is only between -45° to 45° instead of -90° to 90° due to the

symmetry of the proposed metamaterial.

The slight variation of optical magnitude is caused by the change in the strength of the

induced fields from the quasi-dark resonator at different polarization angle. This induced

fields are the result of the weak electric dipole interaction with the incident TEM wave. The

effect however is negligible, and does not obscure the EIT-like effect or shift the resonant

frequency significantly. The existence of the anomaly sharp peaks in Figure 20a) to 20c) is

the results of errors from simulation, as will be verified by comparing with experimental data

in chapter 4. The proposed plasmonic metamaterial, hence is indeed polarization independent.

3.6 Refractive-index-based Sensing

The commonly suggested application of a plasmonic structure which mimics EIT-like

effect is in the use as a refractive-index-based sensor [9] to detect change in its environment’s

20b) 20c)

20a)

31

refractive index . The factors measuring the quality of a sensor are: the magnitude and

bandwidth of the resonant peak that is used for sensing, and its sensitivity which is defined as

the shift in wavelength per unit change of refractive index.

In order to determine the optimal coupling strength of the proposed metamaterial for

sensing, the simulated Quality factor (Q factor) and reflectance are plotted against 𝑆 in Figure

21b). The commonly used definition of Quality factor follows equation (3.6-A) [20]. 𝑓𝑅 is the

resonant peak frequency, while ∆𝑓𝐹𝐻𝑊𝑀 is the resonance full width at half maximum

(FWHM).

(3.6-A)

From Figure 21b), the Q factor and the reflectance curves intersect at 𝑆 = 8.1 𝑚𝑚,

representing the position of quasi-dark resonator that gives the optimal coupling strength for

sensing.

Figure 21: Simulated a) reflectance of the proposed plasmonic metamaterial, plotted against frequency of

incident TEM wave at 3 different environment’s refractive index, with 𝑆 fixed at 8.1 𝑚𝑚. Simulated b)

Qualty factor and reflectance plotted against seperation 𝑆 of the proposed plasmonic metamaterial.

3.6-B)

The reflectance spectrum at 𝑆 = 8.1 𝑚𝑚 is plotted at 3 different refractive index of the

proposed metamaterial’s environment as illustrated in Figure 21a). The figure demonstrates

that a slight increment of the refractive index resulted in significant shift of the resonant peak.

From the simulated data, the Figure of Merit (FOM) of the proposed metamaterial is

calculated to be 6.1. FOM is defined by equation (3.6-B) [9], where ∆𝜆𝑅 is the shift in the

wavelength of the resonant peak, ∆𝑛 is the change of the environment’s refractive index, and

∆𝜆𝐹𝑊𝐻𝑀 is the FWHM of the resonance in terms of wavelength. The proposed metamaterial

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

50.0

6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0

Refle

cta

nce

Qu

ali

ty F

acto

r

Separation S /mm

Q Factor

Reflectance

𝑄 =

𝑓𝑅∆𝑓𝐹𝐻𝑊𝑀

𝐹𝑂𝑀 =

∆𝜆𝑅∆𝑛 × ∆𝜆𝐹𝑊𝐻𝑀

21a) 21b)

n = 1.0 n = 1.1 n = 1.2

32

performs fairly, in terms of sensing capability, as compared with other plasmonic structures

with FOM ranging from 2.86 to 10.1 [9].

3.7 Slow Wave Capability; Damping Constants and Coupling Constant

In this section the discussion is on the slow wave property of the proposed plasmonic

metamaterial, in the strong coupling configuration at 𝑆 = 6.6 𝑚𝑚. By substituting the

simulated values of reflection and transmission coefficients in section 3.5 (before converting

to reflectance and transmittance) into the retrieval method ([21], [22], [23]), the simulated

effective refractive index of the proposed metamaterial is obtained and plotted against the

frequency of the incident TEM wave. The results are illustrated in Figure 22.

Figure 22: Simulated real and imaginary parts of effective refractive index of the proposed

plasmonic metamaterial, plotted against frequency of incident TEM wave, in the strong

coupling regime with 𝑆 = 6.6 𝑚𝑚.

The group refractive index, 𝑛𝑔 of a material is inversely proportional to the group velocity

of the incident wave propagating through the material with a particular angular frequency of

𝜔𝑑. Hence it is commonly used to measure the strength of slow wave property in a material

[20]. 𝑛𝑔 follows equation (3.7-A) [19], where 𝑘 is the wavenumber of the incident wave in

vacuum. From which, 𝑛𝑔 of the proposed plasmonic metamaterial is calculated to be 56 at

7.85 GHz using the real part index values in Figure 22.

(3.7-A)

Comparing to the case when a system only consists of FR-4 substrate, which has 𝑛𝑔 of

2.09 at 7.85 GHz as illustrated in Figure 23, the group velocity of incident wave propagating

within the proposed metamaterial is approximately 27 times slower at the same frequency.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Ref

ract

ive

Ind

ex /

n

Frequency /GHz

Real (n)

Imaginary (n)

𝑛𝑔 = 𝑅𝑒(𝑛) + 𝜔𝑑

𝜕𝑅𝑒(𝑛)

𝜕𝜔𝑑

33

This suggests that the proposed plasmonic metamaterial has relatively good slow wave

property at 7.85 GHz.

Figure 23: Simulated real and imaginary parts of the refractive index of FR-4 substrate, plotted

against frequency of incident TEM wave.

Once the refractive index distribution is known, the relative permittivity 𝜖𝑟 is determined

using equation (3.7-B) [21], where 𝑧 is the metamaterial’s impedance which can be

calculated by using the reflection and transmission coefficients [21]. 𝜖𝑟 of the proposed

metamaterial is then plotted against the frequency of incident TEM wave as illustrated in the

Figure 24. The real part of 𝜖𝑟 reaches zero at the metamaterial’s plasma frequency [24]. From

Figure 24, the angular plasma frequency 𝜔𝑝 is determined to be approximately 5.4 ×

1010 𝑟𝑎𝑑/𝑠.

(3.7-B)

Figure 24: Simulated real and imaginary parts of relative permittivity 𝜖𝑟 of the proposed

plasmonic metamaterial, plotted against the frequency of incident TEM wave, in the strong

coupling regime, with 𝑆 = 6.6 𝑚𝑚.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

(Im

ag

ina

ry)

Ref

ract

ive

Ind

ex

/ n

(Rea

l) R

efra

ctiv

e In

dex

/ n

Frequency /GHz

Real (n)

Imaginary (n)

-90

-60

-30

0

30

60

90

120

150

180

210

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Rel

ati

ve

Per

mit

tiv

ity

Frequency /GHz

𝐈𝐦𝐚𝐠𝐢𝐧𝐚𝐫𝐲 (𝛜𝐫)

𝐑𝐞𝐚𝐥 (𝛜𝐫)

𝜖𝑟 = 𝑛/𝑧

34

The coupling constant at 𝑆 = 6.6 𝑚𝑚 is calculated to be 9.8 × 109 𝑟𝑎𝑑/𝑠 using equation

(3.7-C) [9], where 𝜔𝑑+ and 𝜔𝑑− are the respective angular frequency of the right absorbance

peak and left absorbance peak. While 𝜔 and 𝜔0 are the resonant angular frequency and

coupling constant respectively.

(3.7-C)

The estimated values of damping constants 𝛾1(bright resonator) and 𝛾2 (quasi-dark

resonator) can be calculated by solving the simultaneous equations of (3.7-K) and (3.7-L).

Since 𝜖𝑟 is complex in general, one data point is sufficient for determining 𝛾1 and 𝛾2. For

simplicity the data point located at approximately the plasma frequency is selected for the use

of calculation. The selected data point has the value of 0.03088 + 4.688𝑖. After substituting

the values of 𝜔𝑝 and 𝜔0 into equations (3.7-K) and (3.7-L), using the NSolve function in

Mathamatica, 𝛾1 and 𝛾2 are calculated to be 4.6 × 1010 𝑟𝑎𝑑/𝑠 and 1.9 × 1010 𝑟𝑎𝑑/𝑠

respectively. The damping constant 𝛾2 of the quasi-dark resonator is smaller than 𝛾1 of the

bright resonator as expected.

(3.7-D)

The development of equations (3.7-K) and (3.7-L) begins with the of polarization 𝑃 of the

proposed metamaterial at equation (3.7-D), where 𝐸 is the electric field amplitude of the

incident wave. Following equation (3.7-E), 𝑃 can also be expressed in terms of electric dipole

moments per unit volume ([19], [24]), where 𝑒 and 𝑛 are the electron charge and electron

concentration respectively. 𝑥1 follows equation (2.2-D) in section 2.2. Here the contribution

to 𝑃 from the quasi-dark resonator is assumed to be negligible as it only interacts weakly with

the incident TEM wave in the considered frequency range.

(3.7-E)

(3.7-F)

(3.7-G)

𝜔0 =

√𝜔2 − 𝜔𝑑−2 +√𝜔𝑑+

2 − 𝜔2

2

𝑃 = 𝜖0𝑥𝑒𝐸 exp(−𝑖𝜔𝑑𝑡) = 𝜖0(𝜖𝑟 − 1)𝐸 exp(−𝑖𝜔𝑑𝑡)

𝑃 ≈ 𝑛𝑒𝑥1

𝜖𝑟 =𝑛𝑒𝑥1

𝜖0𝐸 exp(−𝑖𝜔𝑑𝑡)+ 1

𝜖𝑟 =

𝑛𝑒2

𝜖0𝑚[

(𝜔2 − 𝜔𝑑2 − 𝑖𝛾2𝜔𝑑)

(𝜔2 − 𝜔𝑑2 − 𝑖𝛾1𝜔𝑑)(𝜔2 − 𝜔𝑑

2 − 𝑖𝛾2𝜔𝑑) − 𝜔04] + 1

35

After manipulating with equations (3.7-D), (3.7-E) and (3.7-F), equation (3.7-G) is

obtained. The pre-factor of the first term of equation (3.7-G) is identified as the square of

plasma angular frequency as shown in equation (3.7-H).

(3.7-H)

(3.7-I)

(3.7-J)

(3.7-K)

(3.7-L)

After substituting the pre-factor with 𝜔𝑝2, 𝜖𝑟 follows equation (3.7-I). From which, after

rationalizing the denominator and with further simplification, the respective real and

imaginary parts of 𝜖𝑟 obtain the form of equations (3.7-K) and (3.7-L). These equations are

derived based on the model discussed in section 2.2. The damping constants which are

calculated from these equations, are in the reasonable range of magnitude, and hence

demonstrates the correctness of the suggested model (two coupled harmonics oscillators) in

section 2.2 for the proposed plasmonic metamaterial.

𝜔𝑝2 =

𝑛𝑒2

𝜖0𝑚

𝜖𝑟 =

𝜔𝑝2(𝜔2 − 𝜔𝑑

2 − 𝑖𝛾2𝜔𝑑)

(𝜔2 − 𝜔𝑑2 − 𝑖𝛾1𝜔𝑑)(𝜔2 − 𝜔𝑑

2 − 𝑖𝛾2𝜔𝑑) − 𝜔04 + 1

𝐷 = (𝜔2 − 𝜔𝑑2)

𝑅𝑒(𝜖𝑟) = 1 +

𝜔𝑝2𝐷(𝐷2 − 𝛾1𝛾2𝜔𝑑

2 − 𝜔04) + 𝛾2𝜔𝑑𝜔𝑝

2𝐷(𝛾2 + 𝛾1)𝜔𝑑(𝐷2 − 𝛾1𝛾2𝜔𝑑

2 − 𝜔04)2 − (𝐷(𝛾2 + 𝛾1) 𝜔𝑑)2

𝐼𝑚(𝜖𝑟) =

𝜔𝑝2𝐷2(𝛾2 + 𝛾1)𝜔𝑑 − 𝛾2𝜔𝑑𝜔𝑝

2(𝐷2 − 𝛾1𝛾2𝜔𝑑2 − 𝜔0

4)

(𝐷2 − 𝛾1𝛾2𝜔𝑑2 − 𝜔0

4)2 − (𝐷(𝛾2 + 𝛾1)𝜔𝑑)2

36

Chapter 4

4 Experiment

4.1 Set-up and Procedures

Figure 25: a) Periodic array of cross-slot structures (bright resonator), and b)

spiral structures (quasi-dark resonator). The size of each unit cell is 22 𝑚𝑚 by

22𝑚𝑚. There are a total of 81 unit cells in the proposed plasmonic metamaterial.

Three design templates (with 𝑆 = 7.4 𝑚𝑚, 𝑆 = 7.8 𝑚𝑚 and 𝑆 = 8.2 𝑚𝑚), of the

proposed metamaterial in chapter 3, are sent to Interhorizon Corporation for fabrication of

samples via lithography with estimated manufacturing tolerance of ±20 𝑢𝑚. Figure 25 shows

an example of a fabricated sample’s top and bottom layers, with each layer consisting of 81

unit cells.

Figure 26 illustrates the schematic of the experimental set-up. The incident TEM wave is

emitted from Port 1 which also measures the reflecting signal, while Port 2 measures the

transmitting signal of the sample. The network analyser measures the signals received from

Port 1 and 2, and based on which the reflection and transmission coefficients are calculated.

The measured data is then sent to the PC for extraction. The dielectric lenses are for

collimating the incident and outgoing wave such that far field condition is fulfilled.

Figure 26: Schematic of the experimental set-

up.

25a) 25b)

37

Figure 27: a) Dielectric lenses and the b) sample holder placed in between them.

Before the experiment is conducted, calibration is done to account for the background

noise and signal loss through the antennas’ cables. The procedures involved in the calibration

and the use of the “85071” software to extract data are listed in Annex A. These procedures

are suggested by Temasek Laboratory of NUS.

The chamber is covered with anechoic cone-shape foam as illustrated in Figure 27 to

prevent wave that is reflected by the walls, from being collected by the antennas’ ports. The

experiment is conducted in Temasek Laboratory of NUS.

Figure 28: a) Antenna port connected to the b) network analyser which is connected to the PC

via the grey local area network (LAN) cable.

4.2 Comparison between Experimental and Simulated Results

The experimental results agree well with the simulated data in general as illustrated from

Figure 29 to Figure 32. The discrepancy between the two sets of results is due to the

difficulty in estimating the ohmic loss in the metamaterial accurately in the simulation.

Furthermore, depending on the fabrication process, each sample’s properties will vary

accordingly. Despite the sources of error, it can be observed from the experimental results

27a) 27b)

28a) 28b)

38

that the EIT-like effect, polarization independence and tunability are all clearly manifested in

the proposed metamaterial. Moreover, the cause of the sharp anomaly peaks in the simulated

data have verified to be the result of errors in the simulation process by comparing with the

experimental results.

Figure 29: Experimental (EXP) a) absorbance, b)

reflectance and c) transmittance of proposed

plasmonic metamaterial (𝑎𝑡 𝑝ℎ𝑖 = 0°), plotted

against frequency of incident TEM wave at different

separation 𝑆.

Figure 30: Simulated (SIM) a) absorbance, b)

reflectance and c) transmittance of proposed

plasmonic metamaterial (at 𝑝ℎ𝑖 = 0°), plotted

against frequency of incident TEM wave at different

separation 𝑆.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Frequency /GHz

Absorbance (EXP)

S=7.4

S=7.8

S=8.2

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Frequency /GHz

Absorbance (SIM)

S=7.4

S=7.8

S=8.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Frequency /GHz

Reflectance (EXP)

S=7.4

S=7.8

S=8.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Frequency /GHz

Reflectance (SIM)

S=7.4

S=7.8

S=8.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Frequency /GHz

Transmittance (EXP)

S=7.4

S=7.8

S=8.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Frequency /GHz

Transmittance (SIM)

S=7.4

S=7.8

S=8.2

29a)

29b)

29c)

30a)

30b)

30c)

39

Figure 31: Experimental (EXP) a) reflectance, b)

absorbance and c) transmittance of proposed

plasmonic metamaterial (at 𝑆 = 7.8 𝑚𝑚), plotted

against frequency of incident TEM wave at different

polarization angle.

Figure 32: Simulated (SIM) a) reflectance, b)

absorbance and c) transmittance of proposed

plasmonic metamaterial (at 𝑆 = 7.8 𝑚𝑚), plotted

against frequency of incident TEM wave at different

polarization angle.

In the absence of coupling, the bright resonator is highly transmissive, while the quasi-

dark resonator only interacts weakly with the incident TEM wave in the considered frequency

range. Hence, one might expect the EIT-like effect to vanish regardless of the value of 𝑆 if

the incident TEM wave is propagating towards the quasi-dark resonator first instead of the

bright resonator. However, experimental results suggest otherwise as illustrated in Figure 33.

The reason is because the gap between the 2 resonators is only 1 𝑚𝑚, while the wavelength

of the induced �� field of the bright resonator falls between 55 to 30 𝑚𝑚. Thus the bright

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0Frequency /GHz

Reflectance (EXP)

0°

-45°

-30°

-15°

45°

30°

15°

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0Frequency /GHz

Reflectance (SIM)

0°

-45°

-30°

-15°

45°

30°

15°

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Frequency /GHz

Absorbance (EXP)

0°

-45°

-30°

-15°

45°

30°

15°

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0Frequency /GHz

Absorbance (SIM)

0°

-45°

-30°

-15°

45°

30°

15°

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0Frequency /GHz

Transmittance (EXP)

0°

-45°

-30°

-15°

45°

30°

15°

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0Frequency /GHz

Transmittance (SIM)

0°

-45°

-30°

-15°

45°

30°

15°

31a)

31b)

31c)

32a)

32b)

32c)

40

resonator will still be able to interact strongly with the quasi-dark resonator despite the

change in the propagating direction of the incident TEM wave.

Figure 33: Optical properties of proposed plasmonic metamaterial (at 𝑆 = 7.8 𝑚𝑚, and 𝑝ℎ𝑖 = 0°) plotted

against frequency, when TEM wave is first incident on the a) bright resonator, or b) quasi-dark resonator.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Frequency/GHz

Magnitude

(incident on brigth resonator)

Absorbance

Reflectance

Transmittance

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Frequency/GHz

Magnitude

(incident on quasi-dark resonator)

Absorbance

Reflectance

Transmittance

33a) 33b)

41

5 Conclusion

The proposed plasmonic metamaterial has demonstrated from both experimental and

simulated data to be capable of mimicking EIT-like effect in the reflectance spectrum as well

as being tunable and polarization independent. The concepts in designing the structure of the

proposed metamaterial are mainly based on the classical model of two coupled harmonic

oscillators, 𝑅𝐿𝐶 circuit and Babinet’s principle. The metamaterial has also shown to possess

slow wave property, with group refractive index 𝑛𝑔 of 56; and a fair refractive-index-based

sensing property with FOM of 6.1. In the strong coupling configuration, the plasma

frequency and coupling constant of the metamaterial are calculated to be approximately 5.4 ×

1010 𝑟𝑎𝑑/𝑠 and 9.8 × 109 𝑟𝑎𝑑/𝑠 respectively. While the respective damping constants of its

bright resonator and quasi-dark resonator are calculated to be approximately 4.6 ×

1010 𝑟𝑎𝑑/𝑠 and 1.9 × 1010 𝑟𝑎𝑑/𝑠.

There is however still room for improvement in the proposed plasmonic metamaterial in

terms of its abovementioned properties. This may be done by optimising the parameters of

each spiral structure so as to increase the number of coils without changing the resonant

frequency. This will reduce the quasi-dark resonator’s damping constant and hence increases

the Q factor of the metamaterial. The size of the cross-slot structure may be further reduced

by first adding a ‘cap’ on each of its 4 edges (forming two orthogonal “I” shape

complementary structure) to increase its capacitance. After which, the length L1 of the

structure can then be reduced to keep the resonant frequency constant. The reduction in the

size of the cross-slot structure will therefore enhance the tunability of the metamaterial.

Furthermore, in the strong coupling configuration, the two resonant peaks in the absorbance

spectrum can be made more symmetric by tuning the structure of each resonator such that

their resonant frequency become much closer to each other.

Apart from optimisation, future work includes discovering applications for the proposed

metamaterial that require polarization independence, tunability, EIT-like effect in the

reflectance spectrum, and the top layer of metamaterial having a complementary structure.

42

6 References

[1] V.G. Veselago. The electrodynamics of substances with simultaneously negative values of

𝜖 and μ. Soviet Physics Uspekhi 10(4), 509 (1968).

[2] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz. Composite medium

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7 Annex A

The following lab manual, which listed the steps for calibration and data extraction, is

prepared by Temasek Laboratory of NUS.

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