Tunable Metamaterials
Muhammad Faisal Khan
Thesis submitted for the partial fulfillment of
Degree of Doctor of Philosophy (PhD) in Electronic Engineering
Ghulam Ishaq Khan Institute of Engineering Sciences and Technology
Topi, Swabi, Khyber Pakhtunkhwa, Pakistan.
May 2011
2
Certificate of Approval
It is certified that the research work presented in this thesis, entitled “Tunable Metamaterials” has
conducted by Mr. Muhammad Faisal Khan under the supervision of Dr. M. Junaid Mughal.
Dr. M. Junaid Mughal (Advisor), Dr. M. Junaid Mughal (Dean),
Associate Professor, Associate Professor,
Faculty of Electronic Engineering. Faculty of Electronic Engineering.
i
Dedication
Dedicated to my loving family.
ii
Acknowledgements
First of all it is obligatory to thank Almighty ALLAH Who has ever helped me and being
very kind, consequently, such thesis has been completed successfully due to His exceptional
compassion. I offer my praises to Hazrat MUHAMMAD (Peace Be Upon Him), whose life
is a glorious model for the whole humanity.
This research work has been done in Ghulam Ishaq Khan Institute of Engineering Sci-
ences and Technology from Fall 2006 to Spring 2011, under the supervision of Dr. M. Junaid
Mughal who guided me very well, gave good suggestions and always boost up my moral.
I am also thankful to other faculty members for their help and guidance in my studies
and research. I would also like to acknowledge Higher Education Commission, Pakistan for
providing me scholarship for MS and PhD studies.
It is unfair if I don’t mention the names of Mr. Habeel Ahmad (Assistant Professor) and
Mr. Muhammad Bilal (Assistant Professor), School of Electrical Engineering and Computer
Science, National University of Sciences and Technology (NUST) Islamabad, who provided
me access to the Agilent E8362B vector network analyzer which I used to test my proposed
resonators.
I consider myself very lucky as I had sincere seniors, cooperative colleagues and nice
juniors during my stay at Ghulam Ishaq Khan Institute. It is not possible to mention the
names of all those persons here. I am extremely thankful to all of them for establishing
pleasant academic atmosphere and wonderful hostel life in the institute.
Raheel Quraishi, my batchmate, is an excellent personality. I don’t have words to thank
him for his support during my stay at the institute.
I am grateful to my parents and other family members who encouraged me throughout
my research work and their prayers have brought me to the completion of this thesis.
Muhammad Faisal Khan
May, 2011
iii
Declaration
I declare that this is my own work and has not been submitted in any form for another
degree or diploma at any university or other institution for tertiary education. Information
derived from published or unpublished work of others has been acknowledged in the text
and a list of references is given.
Muhammad Faisal Khan
May 2011
iv
Abstract
Metamaterials (MTM) are composite materials that provide some unique characteristics
which are not available by nature. But there is one major drawback or limitation of MTM
which is its fixed and narrow working frequency region. To overcome this limitation, there
are few techniques, presented in this thesis, to vary the working frequency band of MTM.
As MTM are artificial materials, so there are various resonators available in literature,
presented for MTM. In this thesis, three resonators, split ring resonator (SRR), S-shaped
resonator (SSR) and single side paired S-ring resonator (SSPSRR) have been chosen from
literature and different techniques to shift their resonant frequencies are presented.
For SRR, a tunable case i.e., inner ring shorted split ring resonator (IRS-SRR) is pre-
sented in this thesis. The analytical expression of the effective permeability of IRS-SRR is
calculated and for the verification of the analytical work, the experimental results are also
included in this thesis.
The analytical expression for the effective permeability of SSR is also presented in this
thesis. Two tunable or modified cases of SSR, i.e., bottom metallic strips shorted S-shaped
resonator (BSSR) and top-bottom metallic strips shorted S-shaped resonator (TBSSR) are
presented in this thesis. The analytical expressions of the effective permeabilities of BSSR
and TBSSR are also calculated. For verification of the analytical work of SSR, BSSR and
TBSSR, experimental results are included in this thesis.
Similar to SSR, the analytical expression of the effective permeability of SSPSRR is
presented in this thesis and the experimental results are also included to verify the analytical
work. Two tunable cases of SSPSRR, i.e., bottom metallic strips shorted single side paired S-
ring resonator (B-SSPSRR) and top-bottom metallic strips shorted single side paired S-ring
resonator (TB-SSPSRR), are also presented in this thesis along with experimental results.
Other than the tunable cases of various resonators, rotation of resonators is presented
in this thesis as a technique to vary the working frequency of MTM. This technique has
been applied on SSR, SSPSRR and their respective tunable cases in this thesis and the
v
experimental results are presented for justification of this technique.
There are number of devices (made using MTM), are available in literature. All these
devices can be upgraded to work as tunable devices using the techniques presented in this
thesis.
vi
List of Publications
Published research papers
1. M. F. Khan and M. J. Mughal, “Effective permeability of inner ring shorted split
ring resonator”, Microwave and Optical Technology Letters, pp. 624-627, vol. 50, no.
3, March 2008.
2. M. F. Khan, M. J. Mughal and M. Bilal, “Effect of rotation of Bottom metallic strips
shorted S-shaped resonator on its working frequencies”, 8th International Bhurban
Conference on Applied Sciences and Technology (IBCAST 2011), 10-13 January 2011,
Islamabad, Pakistan.
3. M. F. Khan and M. J. Mughal, “Tunable Metamaterials by varying the Inductance
and Capacitance of S-shaped Resonator”, 3rd IEEE International Symposium on Mi-
crowave, Antenna, Propagation and EMC Technologies for Wireless Communications
(MAPE 2009), 27-29 Oct 2009, pp. 140-143, Beijing, China.
4. M. F. Khan and M. J. Mughal, “Modified Single Side Paired S-ring Resonators”,
3rd International Congress on Advanced Electromagnetic Materials in Microwaves and
Optics, 30 Aug-4 Sept 2009, pp. 522-524, London, United Kingdom.
5. M. F. Khan and M. J. Mughal, “Design of Tunable Metamaterials by varying the
Height of Rings of S-shaped Resonator”, 3rd International Conference on Electrical
Engineering (ICEE 2009), 9-11 April 2009, pp. 1-4, Lahore, Pakistan.
6. M. F. Khan and M. J. Mughal, “Tuned S-shaped Resonators”, IEEE 2007 Inter-
national Symposium on Microwave, Antenna, Propagation and EMC Technologies
for Wireless Communications (MAPE 2007), 16-17 August 2007, pp. 1017-1019,
Hangzhou, China.
vii
Submitted research papers
1. M. F. Khan, M. J. Mughal and M. Bilal, “Effective permeability of an S-shaped
resonator”, under review.
2. M. F. Khan, M. J. Mughal and M. Bilal, “Rotation - A technique to tune the working
frequency of Left handed materials”, under review.
viii
Contents
1 Introduction 1
2 Metamaterials 3
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Material Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Negative Refraction and Backward waves . . . . . . . . . . . . . . . . . . . . 5
2.3.1 Negative Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.2 Backward waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Effective Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Negative Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Negative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.7 Propagation of electromagnetic waves in Metamaterial (Left handed material) 15
2.8 Published tuning techniques for Metamaterials . . . . . . . . . . . . . . . . . 17
2.8.1 Split ring resonator (SRR) . . . . . . . . . . . . . . . . . . . . . . . . 18
2.8.2 S-shaped resonator (SSR) . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8.3 Single side paired S-ring resonator (SSPSRR) . . . . . . . . . . . . . 20
2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Inner ring shorted split ring resonator (IRS-SRR) - A tunable case of split
ring resonator (SRR) 22
ix
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Split ring resonator (SRR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Published research work related to split ring resonator (SRR) . . . . . . . . 24
3.4 Contribution of this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Inner ring shorted split ring resonator (IRS-SRR) - Analytical solution of the
effective permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.7 Comparison of analytical and experimental results . . . . . . . . . . . . . . . 39
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 S-shaped resonator (SSR) and its modified cases 41
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Published research work related to S-shaped resonator (SSR) . . . . . . . . . 41
4.3 Contribution of this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 S-shaped resonator (SSR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.4.1 Comparison of the analytical work of this chapter with Chen’s work . 48
4.5 Modified cases of S-shaped resonator (SSR) . . . . . . . . . . . . . . . . . . 49
4.5.1 Bottom metallic strips shorted S-shaped resonator (BSSR) . . . . . . 49
4.5.2 Top-bottom metallic strips shorted S-shaped resonator (TBSSR) . . . 51
4.6 Analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.7 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.8 Comparison of the analytical and the experimental results . . . . . . . . . . 61
4.9 How to get S-shaped resonator (SSR) of desired working frequency . . . . . . 62
4.10 Effect of rotation of S-shaped resonator (SSR) and its modified cases on their
respective left handed (LH) frequency bands . . . . . . . . . . . . . . . . . . 63
4.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
x
5 Single side paired S-ring resonator (SSPSRR) and its tunable cases 69
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Published research work related to single side paired S-ring resonator (SSPSRR) 70
5.3 Contribution of this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4 Single side paired S-ring resonator (SSPSRR) . . . . . . . . . . . . . . . . . 71
5.4.1 Analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4.3 Comparison of the analytical and the experimental results . . . . . . 79
5.5 Tunable cases of single side paired S-ring resonator . . . . . . . . . . . . . . 80
5.5.1 Bottom metallic strips shorted single side paired S-ring resonator (B-
SSPSRR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.5.2 Top-bottom metallic strips shorted single side paired S-ring resonator
(TB-SSPSRR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.6 Comparison of left handed (LH) frequency bands of single side paired S-ring
resonator (SSPSRR) and its tunable cases . . . . . . . . . . . . . . . . . . . 85
5.7 Effect of rotation of single side paired S-ring resonator (SSPSRR) and its
tunable cases on their respective left handed (LH) frequency bands . . . . . 86
5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6 Conclusions and Future works 93
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
xi
List of Figures
2.1 Material classifications [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Positive and negative refraction. . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Series capacitance-shunt inductance transmission line. . . . . . . . . . . . . . 9
2.4 (i) Array of thin conducting wires. (ii) Unit cell [12]. . . . . . . . . . . . . . 12
2.5 Split ring resonator (SRR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6 Unit cell (i) 1D case, (ii) 2D case [12]. . . . . . . . . . . . . . . . . . . . . . . 14
2.7 S-shaped resonator (SSR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8 Single side paired S-ring resonator (SSPSRR) [17]. . . . . . . . . . . . . . . . 21
3.1 Split ring resonator (SRR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Effective permeability versus frequency plot of SRR having r 2 = 9mm, r 1 =
8mm, c = 0.5mm, d = 0.5mm, e = 0.5mm, a = 9.3mm, σ1 = 200 and l =
1.05mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Inner ring shorted split ring resonator (IRS-SRR) where e is the split length
of outer ring, c is the copper thickness, d is the distance between both rings,
r2 and r1 are the outer radius of outer and inner rings respectively, r is the
inner radius of outer ring and θ is the angle of split part of outer ring. . . . . 26
3.4 Effective permeability versus frequency plot of IRS-SRR with parameters
same as given in the caption of Figure 3.2. . . . . . . . . . . . . . . . . . . . 31
xii
3.5 Effective permeability versus frequency plots of IRS-SRR for different split
lengths (e). The remaining parameters are same as given in the caption of
Figure 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.6 Experimental setup, showing two ultra wide band vivaldi antennas, connected
to the network analyzer: (a) Angular view. (b) Front view. . . . . . . . . . 34
3.7 (a) Fabricated SRR having split length (e) = 0.5mm. Experimental results
of FR4 and SRR: (b) Transmission (S21) power. (b) Group delay. (b) Trans-
mission (S21) phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.8 (a) Fabricated IRS-SRR having split length (e) = 0.5mm. Experimental
results of FR4 and IRS-SRR having split length e = 0.5mm: (b) Transmission
(S21) power. (b) Group delay. (b) Transmission (S21) phase. . . . . . . . . . 36
3.9 (a) Fabricated IRS-SRR having split length (e) = 1mm. Experimental results
of FR4 and IRS-SRR having split length e = 1mm: (b) Transmission (S21)
power. (b) Group delay. (b) Transmission (S21) phase. . . . . . . . . . . . . 37
3.10 (a) Fabricated IRS-SRR having split length (e) = 2.5mm. Experimental
results of FR4 and IRS-SRR having split length e = 2.5mm: (b) Transmission
(S21) power. (b) Group delay. (b) Transmission (S21) phase. . . . . . . . . . 38
4.1 S-shaped resonator (SSR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Equivalent circuit of SSR [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Comparison of the analytical expressions of the effective permeability of SSR
presented in [30] with Eq. 4.21. The dimensions of SSR are ca1 = 4mm, ca2
= 5.4mm, a = 5.2mm, b = 2.8mm, t (copper thickness) = 0.018mm, h =
0.4mm, d = 0.5mm, εr = 9 and w1 = w2 = 2mm [32]. . . . . . . . . . . . . . 48
4.4 (a) Bottom metallic strips shorted S-shaped resonator (BSSR). (b) Equivalent
circuit of BSSR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 (a) Top-bottom metallic strips shorted S-shaped resonator (TBSSR). (b)
Equivalent circuit of TBSSR. . . . . . . . . . . . . . . . . . . . . . . . . . . 51
xiii
4.6 Effective permeability versus frequency plot of SSR using the analytical ap-
proach having a = 7.5mm, b = 4mm, t (copper thickness) = 0.018mm, h =
0.5mm, d = 1.464mm, εr = 4.6, w1 = w2 = 3mm. . . . . . . . . . . . . . . . 54
4.7 Effective permeability versus frequency plot of BSSR using the analytical
approach with dimensions same as given in the caption of Figure 4.6. . . . . 55
4.8 Effective permeability versus frequency plot of TBSSR using the analytical
approach with dimensions same as given in the caption of Figure 4.6. . . . . 55
4.9 (a) Fabricated SSR, having area 24mm x 21.5mm and the remaining dimen-
sions are given in the caption of Figure 4.6. Experimental results of SSR and
FR4: (b) Transmission (S21) power. (c) Group delay. (d) Transmission (S21)
phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.10 (a) Fabricated BSSR, having area 24mm x 21.5mm and the remaining dimen-
sions are given in the caption of Figure 4.7. Experimental results of BSSR
and FR4: (b) Transmission (S21) power. (c) Group delay. (d) Transmission
(S21) phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.11 (a) Fabricated TBSSR, having area 24mm x 21.5mm and the remaining di-
mensions are given in the caption of Figure 4.8. Experimental results of
TBSSR and FR4: (b) Transmission (S21) power. (c) Group delay. (d) Trans-
mission (S21) phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.12 90o rotated S-shaped resonator (SSR). . . . . . . . . . . . . . . . . . . . . . 63
4.13 Experimental setup for 90o rotated S-shaped resonator (SSR). . . . . . . . . 64
4.14 Experimental results of rotated SSR and FR4: (a) Transmission (S21) power.
(b) Group delay. (c) Transmission (S21) phase. . . . . . . . . . . . . . . . . . 65
4.15 Experimental results of rotated BSSR and FR4: (a) Transmission (S21) power.
(b) Group delay. (c) Transmission (S21) phase. . . . . . . . . . . . . . . . . . 66
4.16 Experimental results of rotated TBSSR and FR4: (a) Transmission (S21)
power. (b) Group delay. (c) Transmission (S21) phase. . . . . . . . . . . . . . 67
xiv
5.1 Single side paired S-ring resonator (SSPSRR) [17]. . . . . . . . . . . . . . . . 70
5.2 Equivalent circuit of SSPSRR [17]. . . . . . . . . . . . . . . . . . . . . . . . 72
5.3 Effective permeability versus frequency plot of SSPSRR, where SSPSRR is
taken from: (a) [17]. (b) [34]. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4 Effective permeability versus frequency plot of SSPSRR, with and without
considering mutual inductance (M24). . . . . . . . . . . . . . . . . . . . . . . 77
5.5 (a) Fabricated SSPSRR. Experimental results of SSPSRR and FR4: (b)
Transmission (S21) power. (c) Group delay. (d) Transmission (S21) phase. . . 78
5.6 (a) Bottom metallic strips shorted single side paired S-ring resonator (B-
SSPSRR). (b) Equivalent circuit of B-SSPSRR. . . . . . . . . . . . . . . . . 80
5.7 (a) Fabricated B-SSPSRR. Experimental results of B-SSPSRR and FR4: (b)
Transmission (S21) power. (c) Group delay. (d) Transmission (S21) phase. . . 82
5.8 (a) Top-bottom metallic strips shorted single side paired S-ring resonator
(TB-SSPSRR). (b) Equivalent circuit of TB-SSPSRR. . . . . . . . . . . . . . 83
5.9 (a) Fabricated TB-SSPSRR. Experimental results of TB-SSPSRR and FR4:
(b) Transmission (S21) power. (c) Group delay. (d) Transmission (S21) phase. 84
5.10 (a) Fabricated SSPSRR. Experimental results of SSPSRR and FR4: (b)
Transmission (S21) power. (c) Group delay. (d) Transmission (S21) phase. . . 85
5.11 90o rotated SSPSRR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.12 Experimental setup for 90o rotated SSPSRR. . . . . . . . . . . . . . . . . . . 88
5.13 Experimental results of rotated SSPSRR and FR4: (a) Transmission (S21)
power. (b) Group delay. (c) Transmission (S21) phase. . . . . . . . . . . . . . 89
5.14 Experimental results of rotated B-SSPSRR and FR4: (a) Transmission (S21)
power. (b) Group delay. (c) Transmission (S21) phase. . . . . . . . . . . . . . 90
5.15 Experimental results of rotated TB-SSPSRR and FR4: (a) Transmission (S21)
power. (b) Group delay. (c) Transmission (S21) phase. . . . . . . . . . . . . . 91
xv
Acronyms
MTM
LHM
RH
LH
SRR
IRS-SRR
SSR
BSSR
TBSSR
SSPSRR
B-SSPSRR
TB-SSPSRR
Metamaterial
Left handed material
Right handed
Left handed
Split ring resonator
Inner ring shorted split ring resonator
S-shaped resonator
Bottom metallic strips shorted S-shaped resonator
Top-bottom metallic strips shorted S-shaped resonator
Single side paired S-ring resonator
Bottom metallic strips shorted single side paired S-ring resonator
Top-bottom metallic strips shorted single side paired S-ring resonator
xvi
Chapter 1
Introduction
Metamaterials (MTM) are artificially fabricated structures which exhibit some unique prop-
erties that do not exist by nature. In 1968, Veselago theoretically investigated a material
named Left handed material (LHM) which possessed negative values of the electric permit-
tivity (ε) and the magnetic permeability (µ) simultaneously [1]. It must be noted that in
nature, there is no such material which has simultaneous negative values of permittivity and
permeability. So LHM is included in the list of MTM and in this thesis, the terms MTM
and LHM are considered as the same entity.
MTM has few astonishing characteristics like negative refractive index, backward waves,
negative phase velocity etc. In this medium, there exists antiparallel phase and group
velocities where the group velocity (vg) moves forward and the phase velocity (vp) moves
backward i.e., backward waves propagate in LHM.
A wave, coming from a conventional medium, after entering MTM, is bent to opposite
side as compared to the conventional refraction and this phenomenon is termed as negative
refraction and MTM is said to possess negative refractive index.
Due to these unusual characteristics, this medium is used for designing of different devices
such as antennas, radars, filters etc. [2-4]. In some applications, it is required for these
devices to vary their working frequency, as these devices are made using MTMs. Therefore,
1
varying the working frequency of devices means varying the working frequency of MTMs,
so research has been going on to vary MTM’s working frequency.
In this thesis, different tuning techniques for MTM are discussed and presented. Analyt-
ical and experimental results are included for the justification of presented idea. This thesis
is divided into six chapters. Basic physics of metamaterial and related literature survey are
reported in Chapter 2. Chapter 3 is about the analytical and experimental work related to
Split ring resonator (SRR). Tunable and modified cases of S-shaped resonator (SSR) and
Single side paired S-ring resonator (SSPSRR) along with their analytical and experimen-
tal results are described in Chapter 4 and 5 respectively. All the conclusions and future
recommendations are summarized in Chapter 6.
2
Chapter 2
Metamaterials
2.1 Introduction
In 1968, Victor Veselago initiated a new topic of research named as Left handed material
(LHM) which exhibits negative values of the electric permittivity (ε < 0) and the magnetic
permeability (µ < 0) simultaneously [1]. Backward waves propagation and negative refrac-
tive index are few astonishing characteristics of this material [1]. There is no such material
exists by nature, which has simultaneous negative values of permittivity and permeability,
so the said material must be an artificial and man-made composite medium. Veselago inves-
tigated LHM theoretically and no physical realization of LHM was presented for next three
decades. Pendry in 1996 and 1999 presented thin wires structure for negative permittivity
and Split ring resonators (SRR) for negative permeability, respectively [5, 6]. Finally Smith
et al. [7] in 2000, experimentally verified the first ever LHM using Pendry’s work of [5] and
[6].
3
Figure 2.1: Material classifications [12].
2.2 Material Classification
On the basis of the electric permittivity (ε) and the magnetic permeability (µ), there are
four possible classes of materials, as shown in Figure 2.1:
1. When ε > 0 and µ > 0
These are conventional materials, also named as double positive materials (DPS) or
right handed materials (RHM) and dielectrics are its examples. The propagation of
electromagnetic waves is possible in such materials.
2. When ε < 0 and µ > 0
The materials included in this category can be named as epsilon-negative materials
(ENG). This characteristic of having negative value of permittivity is shown by many
plasmas in a particular frequency region. Metals like Gold, Silver etc., show this behav-
ior in the infrared and visible frequency domains. The propagation of electromagnetic
4
waves is not possible in ENG.
3. When ε > 0 and µ < 0
The materials of this category are named as mu-negative materials (MNG). Few gy-
rotropic materials exhibit this behavior in certain frequency bands. The propagation
of electromagnetic waves is not possible in such materials.
4. When ε < 0 and µ < 0
The materials of this category are termed as double negative materials (DNG) or
Left handed materials (LHM). There is no such material exists by nature which can
be included in this category, but such material is physically realizable by making
composite structures. The propagation of electromagnetic waves is possible in such
materials.
2.3 Negative Refraction and Backward waves
MTM (LHM) exhibits some unique characteristics like negative refraction and propagation
of backward waves etc. In the scientific community, these two characteristics had been
under discussion for many years before Veselago’s work of 1968, as both these properties
are very useful for different devices. Few researchers, Schuster, Kock and Rotman, did
some quality work regarding these characteristics before 1968 and then Veselago suggested
that these properties can be found in a medium having negative values of permittivity and
permeability [1, 8, 9].
2.3.1 Negative Refraction
When a wave is incident from one medium to another medium, it enters and propagates in
the second medium with an angle different than incidence angle and the process is known
as Refraction. This process can be divided into two types, positive refraction and negative
5
Figure 2.2: Positive and negative refraction.
refraction. In the above definition of refraction, if the first and the second mediums are dou-
ble positive (DPS) materials or conventional materials, positive or conventional refraction
takes place which is shown as solid line of Medium 2 in Figure 2.2. If the second medium
is double negative material (DNG) (i.e., MTM), the refracted wave is bent on the opposite
side of the normal, (shown as dotted line in Figure 2.2) and this the phenomenon is called
negative refraction. According to Snell’s law of refraction,
n1 sin θi = n2 sin θr, (2.1)
θr = sin−1
(n1
n2
sin θi
), (2.2)
where, n1, n2 are the refractive indices of two mediums and θi, θr are the incident and
refracted angles, respectively. If n2 is negative, then θr is also negative which is noticed in
Eq. 2.2. As the permittivities and the permeabilities of both mediums are not same, so
6
some part of the wave will be reflected back and,
θi = θrefl. (2.3)
History
Negative refraction was first discussed, to the best of our knowledge, in 1904 by Schuster who
gave the idea that negative refraction can occur at the interface of forward and backward
wave media [8]. In 1944, Mandelshtam presented his work about negative refraction that for
an incidence angle θ1 at the interface of conventional material and MTM (labeled as Medium
1 and Medium 2 in Figure 2.2). Snell’s law gives two solutions, one is conventional solution
with refraction angle (θ2) and other is the unusual solution with refraction angle (π − θ2)
[10]. Mandelshtam also mentioned that in MTM, energy must propagate away from the
boundary, constant phase point moves towards the boundary and the propagation direction
of the refracted wave makes an angle π − θ2 with the normal [10]. No further research was
carried out by Mandelshtam on this unusual phenomenon. In 1959, Silin discussed negative
refraction in connection with periodic slow wave structures [10]. Finally in 1968, Veselago
discussed negative refraction thoroughly and introduced the idea that a medium with ε < 0,
µ < 0 can exhibit this unique characteristic [1].
2.3.2 Backward waves
There are two velocities, observed in a material i.e., phase velocity and group velocity. The
velocity of phase of any one frequency component of a wave is termed as the phase velocity
and is given by,
vp =ω
k=
ω
ω√
µε=
1√µε
, (2.4)
7
where, ω is the wave’s angular frequency and k is the wavenumber. The velocity of all
frequency components as a whole in a material is known as group velocity,
vg =∂ω
∂k. (2.5)
For a material, if ω and k are directly proportional to each other, then phase and group
velocities are exactly equal to each other. In most of the mediums, phase and the group
velocities propagate in same direction and the wave is termed as forward wave, but the wave
which propagates with antiparallel phase and group velocities is called backward wave.
History
Negative refraction and the propagation of backward waves are related to the term arti-
ficial dielectric; this term was used before the introduction of the term MTM. Artificial
dielectric is a model of original dielectric and it can be obtained by arranging the artificial
structures in 3-dimension [11]. Jagadis Chunder Bose worked on the concept of artificial
dielectrics and conducted microwave experiments on twisted structures in 1898 [12]. Nowa-
days, these twisted structures are named as artificial chiral elements. In 1914, Lindman also
worked on artificial chiral media and embedded randomly oriented small wire helices in a
host medium [12]. It was artificial dielectric due to which backward waves were physically
realized. Most of the research was carried out about backward waves during second world
war. Winston E. Kock of Bell Laboratories introduced the term artificial dielectric in 1948
for designing electromagnetic structures of practical dimensions that could give the same
response as natural solids due to electromagnetic radiation [9]. These low loss and light
weight artificial dielectrics were appreciated as there was need of replacement of natural
dielectrics for designing different microwave devices. Kock added mixture of metal spheres
in a matrix, formed dielectric lens and in this way, Kock suggested dielectric lens, lighter in
weight [9]. Kock’s metallic lenses phase advancement and behaved like an effective media
having positive refractive index less than unity [9]. Kock also noticed that like natural
8
C C C
L L L
Figure 2.3: Series capacitance-shunt inductance transmission line.
dielectrics, the wavelength of the incoming electromagnetic wave (i.e., wavelength in the
effective medium) must be much larger than the lattice spacing of the artificial dielectric for
capturing the refraction characteristics and the diffraction occurs when the wavelength in
the medium is comparable to the lattice spacing and the scattering occurs when wavelength
is much smaller than the lattice spacing [9]. Kock noted that each conducting element of
the artificial dielectric, act as an electric dipole and establishes a net dipole moment [9].
Brillouin and Pierce in 1946 and 1950 respectively, used the term backward waves for
describing antiparallel phase and group velocities using series-capacitance/shunt-inductance
circuit model, shown in Figure 2.3 [10]. This same circuit model was used by Eleftheriades
in 2002 to get MTM [13]. After publication of Kock, Brillouin and Pierce work, the research
continued on artificial dielectrics in 1950s and 1960s. In 1957, the simultaneous negative
values of permittivity and permeability i.e., double negative material (or MTM), was first
discussed in 1957 by Sivukhin who suggested that such material would be a backward wave
medium, but Sivukhin was unlucky as he mentioned that such medium is not known [10].
9
Sivukhin did not do much research on this medium and in 1968, Veselago presented this
medium using theoretical approach and discussed different terms and laws like Snell’s law,
Fermat principle etc., while applying them to MTM. Using these laws, Veselago determined
that this material possesses negative group delay, backward waves, negative refractive index
and other astonishing properties [1].
2.4 Effective Medium
In natural dielectrics, the electromagnetic interactions with the atoms or molecules give
some responses i.e., the electric permittivity (ε) and the magnetic permeability (µ). These
basic parameters achieve some meaning when there is a spatial order and the wavelength of
the applied field is much larger than the lattice spacing. To get desired material parameters,
access to the scatterer i.e., atoms or molecules is required. Electromagnetic scatterers can
be fabricated and they can react to the applied field just like atoms or molecules of a crystal
lattice. If the wavelength is much larger than the periodicity of these scatterers, then they
will behave like an effective medium and show dielectric properties [9]. This theory was the
base of the introduction of artificial dielectrics in 1940s.
2.5 Negative Permittivity
Permittivity of a material is the resistance offered by the material to an electric field. Every
medium has positive value of permittivity but this scenario was changed when Veselago
discussed about negative permittivity and negative permeability for MTM in 1968. But
no physical realization of negative permittivity and negative permeability was available in
literature at that time. After many years of research, negative permittivity was achieved in
microwave region using metals as effective plasma medium.
Plasma, fourth state of matter, is an ionized gas which is energized so much that few
of its electrons become free from their nuclei but still travel with their respective nucleus.
10
It is a collection of charged particles that respond to electromagnetic fields strongly and
collectively. Plasma is described using the permittivity function [9],
ε = 1− f 2p
f 2, (2.6)
where, fp and f are the plasma and signal frequencies respectively. According to Eq.
2.6, plasma’s permittivity attains negative value below the plasma frequency and in this
frequency region, the electromagnetic wave incident on the plasma is reflected back [9].
This approach is used to achieve negative permittivity. To get the same effect as that of
plasma in order to fabricate MTM, the periodic structures made up of metals were used.
In 1950s, a wire medium was presented as material for microwave lenses [10]. This
medium was similar to thin wire medium, presented by Pendry in 1996, to achieve negative
permittivity for MTM. In 1962, Walter Rotman tried to model plasma using Kock’s artificial
dielectrics. Rotman noticed that a medium having index of refraction less than unity can be
used to model plasma where the medium has permeability near that of free space [9]. With
all these considerations, dielectric medium with thin wire rods along the applied electric
field was chosen which behaved like plasma.
Electromagnetic structures in the microwave region consist of metals i.e., plasmas. The
natural plasma frequencies of metals occur in the ultraviolet region of the electromagnetic
spectrum [9]. So to get negative permittivity in the microwave region, a decrease in the
plasma frequencies of metals was required. This problem was solved by Pendry who used
a structure similar to that of Rotman [5, 9]. This structure was a mesh of thin conducting
wires which were arranged in a periodic lattice [5]. The plasma frequency of this structure
is reduced to microwave region due to two reasons, firstly because of reduction in effective
electron concentration in the volume of this structure due to spatial confinement of electrons
to thin wires and secondly, due to the enhancement of effective mass of electrons confined to
thin wires because of self inductance of the wire array [9]. In this way, negative permittivity
is achieved at microwave frequencies for MTM.
11
Figure 2.4: (i) Array of thin conducting wires. (ii) Unit cell [12].
Figure 2.4 shows the array of thin conducting wires which is another form of Pendry’s
structure for negative permittivity. When the lattice constant (a) is much smaller than the
wavelength, the array of wires can be thought as an effective plasma medium which can be
described by permittivity function [12],
εreff,z = ε′reff,z − jε′′reff,z = 1− f 2p
f 2 − jγf, (2.7)
where εreff,z is the effective relative permittivity in the z -direction, f and fp are the signal
and plasma frequencies respectively, generally depends upon the geometry of the system,
i.e., lattice constant and the wire radius, and γ represents the losses [12]. It must be noted
that Eq. 2.7 is valid only when there is no component of wave vector parallel to the wires
[12].
12
Figure 2.5: Split ring resonator (SRR).
2.6 Negative Permeability
Permeability is the characteristic of a material to resist the formation of magnetic field
within itself. Conventionally, every medium has positive value of permeability but the case
is different for MTM which possesses negative value of permeability.
After the theoretical introduction of MTM in 1968, it was a big challenge to get negative
permeability as it does not exist in nature. To achieve negative permeability, Veselago
assumed a gas of magnetic charges which could give the magnetic plasma frequency and
below this plasma frequency, the permeability could achieve negative value [9]. But the non-
existence of magnetic charges was the problem. Pendry in 1996 used thin wires to achieve
plasma effect and the negative permittivity, but for the case of permeability, no such thing
exists in nature that could give magnetic plasma frequency.
Finally, the breakthrough was achieved in 1999 when Pendry et al. presented microstruc-
tured material, Split ring resonator (SRR), shown in Figure 2.5 and 2.6, consisting of purely
13
Figure 2.6: Unit cell (i) 1D case, (ii) 2D case [12].
nonmagnetic materials, to get artificial magnetic plasma [6]. Pendry’s structure was similar
to that of a parallel plate capacitor wrapped around a central axis. SRR shows strong
electric fields and a very large capacitance between the rings. Currents are induced in both
rings by applying the magnetic field perpendicular to the plane of SRR [9].
The effective permeability is given by [12],
µeff = µ′eff − jµ′′eff = 1− f 2mp − f 2
o
f 2 − f 2o − jγf
, (2.8)
where f is the frequency of the signal, fmp is the magnetic plasma frequency, fo is the
resonant frequency of SRR and γ represents the losses. Note that fmp and fo depends on
the lattice constant (a) and different geometric parameters of SRR like radii of inner and
outer rings, gap between the rings, copper width, length of split parts of both rings.
SRR also shows bianisotropy which means that it is anisotropic particle. If the magnetic
field vector of the plane wave is perpendicular to the SRR, the currents are induced in the
14
rings and negative permeability is achieved, but if the magnetic field vector is parallel to
SRR, the currents are not induced and so no effect on the permeability [12].
It is interesting to know that the name of SRR was not new in the electromagnetic
community before its usage for MTM. In 1952, different forms of SRR were presented in a
textbook by Schelkunoff and Friis but these SRRs were not considered as artificial inclusions
of a magnetic medium, as done by Pendry in 1999 [10].
2.7 Propagation of electromagnetic waves in Metamaterial (Left
handed material)
As metamaterial (MTM) exhibits negative values of permittivity and permeability, so it has
to be checked whether the propagation of electromagnetic waves is possible in this medium
or not. So, starting with the Maxwell’s equations,
∇× E = −∂B
∂t−Ms, (2.9)
∇×H =∂D
∂t+ Js, (2.10)
∇ ·D = ρe, (2.11)
∇ ·B = ρm, (2.12)
where E (V/m) is the electric field intensity, H (A/m) is the magnetic field intensity, D
(C/m2) is the electric flux density, B (W/m2) is the magnetic flux density, Ms (V/m2) is the
(fictitious) magnetic current density, Js (A/m2) is the electric current density, ρe (C/m3) is
the electric charge density and ρm (C/m3) is the (fictitious) magnetic charge density.
In a linear non-dispersive medium [10],
D = εoE + P = εo(1 + χe)E = εE, (2.13)
15
B = µoH + M = µo(1 + χm)H = µH, (2.14)
where P = εoχe and M = µoχm are the electric and magnetic polarizations respectively,
χe and χm are the electric and magnetic susceptibilities respectively. For time dependence
e−iωt, Maxwell equations can be written as,
∇× E = iωµH−Ms, (2.15)
∇×H = −iωµE + Js, (2.16)
∇ ·D = ρe, (2.17)
∇ ·B = ρm, (2.18)
and
D = εE, (2.19)
B = µH, (2.20)
Consider a plane wave,
E(r) = Eoeik.r, (2.21)
H(r) =Eo
ηeik.r, (2.22)
where η = |E|/|H| is the wave impedance. Taking curl of both sides of Eq. 2.21,
∇× [E(r)] = ∇× [Eoeik·r], (2.23)
Using the identity,
∇× [f(r)g(r)] = f [∇× g(r)] + [∇f(r)]× g(r), (2.24)
16
as
∇× Eo = 0, (2.25)
so Eq. 2.23 becomes,
∇× [E(r)] = (∇eik·r)× Eo, (2.26)
As
∇eik·r = ikeik·r. (2.27)
So ∇ can be replaced by ik. In that case, Eq. 2.15 becomes [10],
ik× E = iωµH, (2.28)
k× E = ωµH. (2.29)
Similarly
k×H = −ωεE. (2.30)
From Eq. 2.29 and 2.30, it is clear that E, H and k forms the right handed triad. For left
handed medium i.e., ε, µ < 0,
k× E = −ω|µ|H, (2.31)
and
k×H = ω|ε|E, (2.32)
From Eq. 2.31 and 2.32, it is confirmed that E, H and k form the left handed triad and so
the propagation of electromagnetic wave is possible through MTM (LHM).
2.8 Published tuning techniques for Metamaterials
Due to unique properties of MTM, it has been utilized in different fields. In microwave
regime, MTM has fixed bandwidth or working frequency which is its limitation. This fact
17
compels researchers to work for tunable MTM, i.e., a MTM whose working frequency can
be varied according to user’s will.
There are some resonators available in literature other than Split ring resonator (SRR)
which also exhibit negative permeability, like Spiral resonator, Omega resonator, S-shaped
resonator (SSR), Single side paired S-ring resonator (SSPSRR) etc. [14-17]. Omega res-
onator, SSR and SSPSRR are the resonators which provides negative permittivity and
negative permeability simultaneously, so thin wire is not required to achieve negative per-
mittivity for this resonators. In this section, published tuning techniques of only SRR, SSR
and SSPSRR are discussed, as they are part of the remaining thesis.
2.8.1 Split ring resonator (SRR)
Split ring resonator (SRR), shown in Figure 2.5, was presented in 1999 by Pendry to get
negative value of permeability [6]. Using thin wires and SRR, first MTM was experimentally
verified in 2000 by Smith et al. [7]. In 2004, Shamonin et al. presented Singly split double
ring (SSDR), which is a special case of SRR [18]. The inner ring of SRR is shorted in
this modified case, so there is only a single split which is in the outer ring. The resonant
frequencies of SSDR were calculated for different lengths of outer split and inter ring spacing.
The researchers did not use the word tunable in their work, but this can be said as a tuning
technique for MTM.
In 2005-06, Ozbay et al. determined the resonant frequencies of SRR by varying its
different geometric parameters [19, 20]. These geometric parameters include the split width,
inter ring spacing and metal width. External capacitors were also placed at the split width
of outer ring of SRR in this work and the change in the resonant frequency was observed.
This work can be included as tuning techniques for SRR, although it must be remembered
that geometric parameters of SRR can not be changed once it gets fabricated. Same research
group presented another tuning option in 2007 where external capacitors were placed at the
splits of inner and outer ring and also between the inter ring spacing of SRR [21].
18
The working frequency region of MTM can also be changed by shortening inner and
outer rings of SRR one by one. This work has been done by Faisal et al. in 2006, although
this work is little similar to Shamonin’s work of 2004 [22].
Few research groups loaded varactors in SRR, in the year 2006, to achieve tunability for
the working frequency of metamaterials [23, 24]. It was a good approach as capacitance
of SRR can be varied in this way which finally resulted in the variation of the resonant
frequency of SRR.
In 2006, Varadan et al. presented their work in which the effect of gap orientation of
SRR on its resonant frequency was observed [25]. In 2007, Varadan et al. presented another
research work in which the permittivity and thickness of the substrate, over which SRR was
placed, were varied one by one [26]. From the basic knowledge of capacitance, it is known
that permittivity of the substrate and distance between the plates are directly and inversely
proportional to the capacitance respectively. This basic approach was used by Varadan et
al. and techniques to vary the resonant frequency of MTM had been presented.
Tunable SRR has also been presented by introduction of metallic strip in SRR and
also using control of modes in SRR [27, 28]. An interesting way of changing the resonant
frequency of SRR was presented by Xu et al. in 2009 who etched broadside coupled SRR on
two substrates and varied the gap between the substrate to cause variation in the working
frequency of SRR [29].
2.8.2 S-shaped resonator (SSR)
S-shaped resonator (SSR), shown in Figure 2.7, was presented by Chen et al. in 2004 [16].
Chen et al. also presented the analytical solution for the effective permeability of SSR in
2005 [30]. Crankled SSR, a modified case of SSR, was presented in 2006 by Chen et al. in
which the length of center metallic strips were increased [31]. In 2007, same research group
introduced another modified case of SSR in which top metallic strips and bottom metallic
strips were shorted [32, 33].
19
Figure 2.7: S-shaped resonator (SSR).
2.8.3 Single side paired S-ring resonator (SSPSRR)
Single side paired S-ring resonator (SSPSRR), shown in Figure 2.8, was presented in 2007
by Wang et al. [17]. SSPSRR is made by etching two S-shaped metallic patterns only on one
side of the substrate, so its fabrication is much easier than SSR. The installation of varactors
or switches etc. between the metallic strips is also quite easy in SSPSRR as compared to
SSR. A tunable case of SSPSRR was also presented in 2007 in which the varactor was
installed between the center metallic strips [34].
2.9 Summary
Basic theoretical concepts of Metamaterials (MTM) are described in this chapter. The his-
tory that made the ground for MTM and the relevant research work, available in literature,
are also presented in this chapter.
In today’s world, with new fabrication techniques, it is easy to acquire constitutive
parameters of a medium that do not exist naturally or are not easily available. MTM can
20
Figure 2.8: Single side paired S-ring resonator (SSPSRR) [17].
be made by embedding various inclusions, having novel geometric shapes, in a host medium.
In composite media, the macroscopic effective permittivity and permeability are affected by
the electric and magnetic moments which are induced due to interaction of electromagnetic
waves with inclusions [12]. As MTM is fabricated using inclusions, so there is quite flexibility
for the designer due to geometric parameters of the inclusion and also of the host medium.
21
Chapter 3
Inner ring shorted split ring resonator
(IRS-SRR) - A tunable case of split
ring resonator (SRR)
3.1 Introduction
Split ring resonator (SRR) is the first resonator, presented to achieve negative value of
permeability for MTM. After the introduction of SRR as a way to achieve negative per-
meability, number of resonators have been presented for the same task and still more to
come. Due to the requirement of tunable MTM, different techniques for tuning the working
frequency of SRR (or tunable SRR) have been presented. One of the tunable cases of SRR
is the inner ring shorted split ring resonator (IRS-SRR) which is discussed in this chapter.
3.2 Split ring resonator (SRR)
Before the discussion of IRS-SRR, it is better to understand SRR which consists of two
C-shaped copper rings having splits on opposite sides of each other, as shown in Figure
22
Figure 3.1: Split ring resonator (SRR).
3.1. After the introduction of LHM in 1968, it was not possible to achieve negative value
of permeability for next three decades and finally in 1999, Pendry presented the idea that
negative value of permeability can be achieved using microstructured SRR [6].
Pendry also calculated the expression for the effective permeability of SRR. For this
calculation, 2-dimensional array of split ring resonator configured cylinders is analyzed and
then each cylinder is cut into slices to find the effective permeability of a single SRR which
is given by [6],
µeff = 1−πr2
a2[1 + i 2lσ1
ωrµo− 3lc2o
πω2(ln 2cd )r3
] , (3.1)
where r is the radius of inner ring of SRR, c is the metal thickness, d is the distance between
two rings, a is the lattice constant of cylinders, l is the distance between two consecutive
slices of SRR and σ1 is the resistance of unit length of SRR. Figure 3.2 shows the simulated
result of the effective permeability of SRR with respect to frequency where it is observed
23
2.7 2.72 2.74 2.76 2.78 2.8 2.82 2.84 2.86 2.88 2.9−400
−300
−200
−100
0
100
200
300
400
500
Frequency (GHz)
Eff
ectiv
e Pe
rmea
bilit
y
Figure 3.2: Effective permeability versus frequency plot of SRR having r2 = 9mm, r1 = 8mm, c= 0.5mm, d = 0.5mm, e = 0.5mm, a = 9.3mm, σ1 = 200 and l = 1.05mm.
that the permeability achieves negative value at 2.8GHz for the geometric parameters given
in the caption of Figure 3.2.
3.3 Published research work related to split ring resonator (SRR)
SRR was presented for achieving negative value of permeability for MTM. Large number
of research papers are available in literature, related to SRR. In this section, only those
research works are included which are closely related to the scope of this chapter.
Pendry introduced SRR as a way to get negative permeability for MTM and also pre-
sented SRR’s expression for the effective permeability in 1999 [6]. First MTM was experi-
mentally verified by Smith et al. in 2000 using thin wires for negative permittivity and SRR
for negative permeability [7]. In 2004, Shamonin et al. presented single split double ring
(SSDR) which is modified form of SRR and calculated its resonant frequencies [18]. In 2005,
24
E. Ozbay et al. studied different geometric parameters of SRR and presented the effect of
SRR’s geometric parameters on the resonant frequency [19]. This research group presented
another work for tuning the resonant frequency of SRR by placing external capacitors at the
split of outer ring, at the split of inner ring and between outer and inner rings [21]. Another
research group worked on this tuning technique of SRR in 2006 by shortening outer and
inner rings of SRR one by one [22]. Few research groups loaded varactors in SRR to tune its
working or resonant frequency i.e., by changing its capacitance [23, 24]. In 2007, Varadan
et al. used the basic definition of capacitance and noticed tuning of the resonant frequency
of SRR by varying the permittivity and the thickness of the substrate [26].
3.4 Contribution of this chapter
The literature survey related to SRR has been discussed in Section 2.8 and 3.3. The reso-
nant frequencies of inner ring shorted split ring resonator (IRS-SRR) have been calculated
analytically by Shamonin in [18], but the expression for the effective permeability of IRS-
SRR has not been calculated by Shamonin or any other research group and it is presented
in this chapter.
3.5 Inner ring shorted split ring resonator (IRS-SRR) - Analytical
solution of the effective permeability
IRS-SRR is that tunable or modified case of SRR in which the inner ring of SRR is shorted
i.e., the capacitance at the split of inner ring is zero and so there is only one split in IRS-SRR
which is in the outer ring, as shown in Figure 3.3.
To calculate an expression for the effective permeability of inner ring shorted split ring
resonator (IRS-SRR), same steps which have been used by Pendry in [6], are used in this
section. The effective permeability of IRS-SRR is calculated first by finding the net electro-
25
Figure 3.3: Inner ring shorted split ring resonator (IRS-SRR) where e is the split length of outerring, c is the copper thickness, d is the distance between both rings, r2 and r1 are the outer radiusof outer and inner rings respectively, r is the inner radius of outer ring and θ is the angle of splitpart of outer ring.
motive force (emf) around the circumference of IRS-SRR configured cylinder. The cylinder
is then cut into slices to find the capacitance between the rings and finally all these calcu-
lations are put in the formula for the effective permeability. This work has been published
in [35].
Consider a 2-dimensional array of IRS-SRR configured cylinders in x -y plane, each cylin-
der of length a along z -axis. The cylinders are excited by y-polarized plane wave, having
propagation vector along x -axis. Due to the conducting boundary, current (j ) per unit
length flows in each cylinder. Using the continuity equation for calculating this current,
∫
s
J.dS = −∂q
∂t, (3.2)
where, J is the current density in A/m2 and q is the charge in coulombs.
26
∫ a
z=0
∫ r2
ρ=r
JaΦ.aΦdρdz = −(−iω)(CoV ), (3.3)
where, Co is the total capacitance between two cylinders, V is the voltage due to charges
and r is the radius, as shown in Figure 3.3. By further calculation,
Ja(r2 − r)(2π − θ)
(2π − θ)= iωCoV, (3.4)
V =j
iωCo(2π − θ), (3.5)
where, θ is the angle of the split part of outer cylinder. Co can be written in terms of per
unit length capacitance C as,
Co = (2πr − e)C, (3.6)
where, e is the length of split part of outer cylinder. So Eq. 3.5 becomes,
V =j
iωC(2πr − e)(2π − θ). (3.7)
Because of two capacitances, one described in Eq. 3.6 and other one existing at the split
part of outer cylinder, the expression for voltage becomes,
V =2j
iωC(2πr − e)(2π − θ). (3.8)
Using Faraday’s law of electromagnetic induction to find the net emf around the circumfer-
ence of outer cylinder,
emf = −∂φ
∂t, (3.9)
where, φ represents the magnetic flux and is given by,
27
φ =
∮
s
B.dS, (3.10)
where, B is the magnetic flux density, so
emf = iωπr2µoH. (3.11)
If Ho is the applied external field, parallel to the cylinders then the total field inside each
cylinder is,
H =
(Ho + j − πr2
a2j
), (3.12)
i.e., it is the sum of applied field, field due to current and depolarizing fields with sources
at the remote ends of the cylinders [6]. So using this magnetic field,
emf = iωπr2µo
(Ho + j − πr2
a2j
). (3.13)
Net emf (emfnet) around the circumference of outer cylinder is,
emfnet = iωπr2µo
(Ho + j − πr2
a2j
)− j(2πr − e)σ +
2j
iωC(2πr − e)(2π − θ), (3.14)
where, σ is the resistance of unit length of the conductive part and j(2πr − e)σ is the emf
across outer cylinder. The net emf must be balanced, so
0 = iωπr2µo
(Ho + j − πr2
a2j
)− j(2πr − e)σ +
2j
iωC(2πr − e)(2π − θ), (3.15)
and the current is found to be,
28
j =−Ho[(
1− πr2
a2
)+ i (2πr−e)σ
ωµoπr2
]− 2
πr2ω2µoC(2πr−e)(2π−θ)
. (3.16)
The average magnetic field over a line lying outside the cylinders is [6],
Havg = Ho − πr2
a2j, (3.17)
Substituting Eq. 3.16 in Eq. 3.17,
Havg = Ho
[1 + i (2πr−e)σ
ωµoπr2 − 2πr2ω2µoC(2πr−e)(2π−θ)
][(
1− πr2
a2
)+ i (2πr−e)σ
ωµoπr2 − 2πr2ω2µoC(2πr−e)(2π−θ)
] , (3.18)
As the effective permeability is defined in [6] as,
µeff =Bavg
µoHavg
, (3.19)
so using Eq. 3.18,
µeff = 1−πr2
a2[1 + i (2πr−e)σ
ωµoπr2 − 2πω2µoCr2(2πr−e)(2π−θ)
] . (3.20)
The capacitance per unit area between two sheets is,
C =εo
d=
1
dc2oµo
, (3.21)
so Eq. 3.20 becomes,
µeff = 1−πr2
a2[1 + i (2πr−e)σ
ωµoπr2 − 2dc2oπω2r2(2πr−e)(2π−θ)
] . (3.22)
Now each cylinder is cut in slices where the distance between two consecutive slices be l
such that l < r. For this inner ring shorted split ring, the electric flux density (D) is given
29
by,
D = Dφaφ, (3.23)
If v is the induced emf between the plates,
v = −∫
c
E.dl = −∫ (2π−θ)
0
Dφ
εo
rdφ, (3.24)
so,
Dφ = − vεo
r(2π − θ). (3.25)
The charge density on the external plate is,
ρs = Dn = −Dφ =vεo
r(2π − θ), (3.26)
and the total charge on the plates can be calculated as,
Q =
∫ρsds =
∫ a
z=0
∫ r
r=r1
vεo
r(2π − θ)drdz, (3.27)
so,
C =Q
v=
aεo
(2π − θ)
(ln
r
r1
), (3.28)
The capacitance between unit lengths of two parallel sections of metallic strips is then given
as,
C =εo
(2π − θ)ln
r
r − d=
1
(2π − θ)µoc2o
(ln
r
r − d
), (3.29)
Substituting Eq. 3.29 in 3.20 and therefore the effective permeability of inner ring shorted
split ring resonator (IRS-SRR) is calculated as,
30
Figure 3.4: Effective permeability versus frequency plot of IRS-SRR with parameters same as givenin the caption of Figure 3.2.
µeff = 1−πr2
a2[1 + i (2πr−e)σl
ωµoπr2 − 2lc2oπω2r2(2πr−e)(ln r
r−d)
] . (3.30)
The resonant frequency of IRS-SRR is given by,
ω2o =
2lc2o
πr2(2πr − e)(ln r
r−d
) . (3.31)
Eq. 3.30 is simulated and the result is shown in Figure 3.4 where it is observed that negative
value for permeability is achieved at 2.567GHz for IRS-SRR. By comparing Figure 3.2 and
3.4, it can be noticed that the working frequency of SRR is varied from 2.8GHz to 2.567GHz
by shortening the inner ring i.e., tuning of the resonant frequency is achieved. If a variable
capacitor is placed at the split part of inner ring instead of switch, then a further variation
in the working frequency can be achieved. This work has been done by Aydin et al. in [21].
31
Figure 3.5: Effective permeability versus frequency plots of IRS-SRR for different split lengths (e).The remaining parameters are same as given in the caption of Figure 3.2.
Shamonin et al. has also analyzed IRS-SRR (as mentioned in Section 3.4) and determined
its resonant frequency by considering this resonator as combination of distributed elements
[18]. This research group calculated the resonant frequency of IRS-SRR for different split
lengths (e). For verification of the analytical expression of the effective permeability of
IRS-SRR (Eq. 3.30), the effective permeability of IRS-SRR for different split lengths (e) is
calculated using Eq. 3.30 and is shown in Figure 3.5. Comparison between the analytical
work of this chapter and Shamonin’s work is given in Table 3.1 where difference of 0.3GHz is
noticed for 0.1mm split length and for other split lengths, the results of both approaches are
found in good agreement with each other. Thus, the analytical expression for the effective
permeability of IRS-SRR, calculated in this chapter is verified by the Shamonin’s work of
[18].
32
Table 3.1: Comparison of the resonant frequencies of IRS-SRR, calculated using the analyticalwork of this chapter (Eq. 3.30) and Shamonin’s work for different split lengths (e) of outer ringwith r = 8.5mm and the remaining parameters are same as given in the caption of Figure 3.2.
e Shamonin results. Analytical results.[18]
(mm) (GHz) (GHz)0.1 2.25 2.5580.5 2.5 2.5671.0 2.618 2.582.5 2.62 2.617
3.6 Experimental results
Experiments have been performed using Agilent vector network analyzer E8362B and a
pair of ultra wide band (UWB) vivaldi antennas, to verify the analytical results. The
experimental setup is shown in Figure 3.6 where two vivaldi antennas are placed on the test
bed and are connected to the network analyzer. The design of UWB antennas is taken from
[36]. For calibration of the experimental setup, calibration procedures have been used as
per standards. Each experiment has been revised three times and for confirmation, every
experiment has again repeated after a month.
The structures of SRR and IRS-SRR for different split lengths (e) are fabricated using
the LPKF ProtoMat S-100 machine with the dimensions given in the caption of Figure 3.2.
Experimental results of FR4, a conventional right handed material (RHM) which is the most
commonly used as insulating material in large number of printed circuit boards (PCB), area
31mm x 26mm and thickness 1.464mm, are also included in this section as it will help in
differentiating between right hand (RH) and left hand (LH) frequency bands.
The experimental results of transmission (S21) power, group delay and transmission (S21)
phase of SRR and IRS-SRR for different split lengths (e), are shown in Figures 3.7-3.10.
As SRR is used to get negative permeability and it does not provide negative permittivity
at that frequency band, so there is no transmission at the resonant frequency of SRR.
Therefore, transmission (S21) power results give a dip at the resonant frequency of SRR.
33
(a) (b)
Figure 3.6: Experimental setup, showing two ultra wide band vivaldi antennas, connected to thenetwork analyzer: (a) Angular view. (b) Front view.
As negative group delay and negative phase are included in the characteristics of MTM, so
if these two properties are observed in a frequency band, it confirms the existence of LH
frequency band.
All the experimental results presented in this section, have been measured with frequency
step size of 6MHz and antenna to antenna distance of 68mm i.e., measurements have been
taken in the near field of antennas. In Figure 3.7, a picture of fabricated SRR is shown along
with its experimental results where the results show that 2.79GHz is the resonant frequency
of SRR as it has lowest value of transmission power, negative group delay and negative
phase at 2.79GHz. Similarly, the experimental results of IRS-SRR of split lengths 0.5mm,
1mm and 2.5mm, shown in Figure 3.8, 3.9 and 3.10 respectively, show that IRS-SRR of
0.5mm split length has resonant frequency at 1.75GHz, IRS-SRR of 1mm split length has
the resonant frequency at 1.77GHz and IRS-SRR of 2.5mm split length has the resonant
frequency at 1.878GHz.
34
(a) (b)
(c) (d)
Figure 3.7: (a) Fabricated SRR having split length (e) = 0.5mm. Experimental results of FR4and SRR: (b) Transmission (S21) power. (b) Group delay. (b) Transmission (S21) phase.
35
(a) (b)
(c) (d)
Figure 3.8: (a) Fabricated IRS-SRR having split length (e) = 0.5mm. Experimental results of FR4and IRS-SRR having split length e = 0.5mm: (b) Transmission (S21) power. (b) Group delay. (b)Transmission (S21) phase.
36
(a) (b)
(c) (d)
Figure 3.9: (a) Fabricated IRS-SRR having split length (e) = 1mm. Experimental results of FR4and IRS-SRR having split length e = 1mm: (b) Transmission (S21) power. (b) Group delay. (b)Transmission (S21) phase.
37
(a) (b)
(c) (d)
Figure 3.10: (a) Fabricated IRS-SRR having split length (e) = 2.5mm. Experimental results ofFR4 and IRS-SRR having split length e = 2.5mm: (b) Transmission (S21) power. (b) Group delay.(b) Transmission (S21) phase.
38
3.7 Comparison of analytical and experimental results
The resonant frequencies of IRS-SRR of different split lengths (e) have been calculated
analytically (Section 3.5) and experimentally (Section 3.6). The comparison of these results
is described in this section and is summarized in Table 3.2. Experimental results of IRS-SRR
for 0.1mm split length (e) are not included due to fabrication issues.
Difference of about 0.7-0.8GHz is observed between the analytical and the experimental
results of IRS-SRR for split lengths 0.5mm, 1mm and 2.5mm. It must be noted that
the effect of substrate is not considered in the analytical results, but it is included in the
experimental results which causes increase in capacitance, i.e., decrease in the resonant
frequency of each resonator. So this is the reason why there is some difference between the
analytical and the experimental results. Fabrication imperfections also play some part in
this difference between the results using both approaches.
3.8 Summary
A tunable or modified case of split ring resonator (SRR) i.e., inner ring shorted split ring
resonator (IRS-SRR) is discussed in this chapter. The expression for its effective permeabil-
ity is calculated analytically and verified by the experimental results. Difference is observed
between the analytical and the experimental results which is due to neglecting the effect of
Table 3.2: Comparison of the analytical and the experimental results of the resonant frequenciesof SRR and IRS-SRR for different split lengths (e) of outer ring.
e Analytical results. Experimental results.(mm) (GHz) (GHz)
SRR 0.5 2.8 2.79IRS-SRR 0.1 2.558 -IRS-SRR 0.5 2.567 1.75IRS-SRR 1.0 2.58 1.77IRS-SRR 2.5 2.617 1.878
39
substrate in the analytical work and due to fabrication imperfections. The work, presented
in this chapter, can be used in designing of various tunable microwave devices.
40
Chapter 4
S-shaped resonator (SSR) and its
modified cases
4.1 Introduction
Split ring resonator (SRR), spiral resonator etc. are the resonators which only provide
negative value of permeability (µ) at some frequencies. But there are few other resonators
that provide negative values of permittivity and permeability within same frequency band.
One of those resonators is S-shaped resonator (SSR), shown in Figure 4.1. SSR is fabri-
cated by etching S-shaped metallic pattern on both sides of the substrate (or dielectric).
In this chapter, SSR is discussed along with its two modified cases i.e., bottom metallic
strips shorted S-shaped resonator (BSSR) and top-bottom metallic strips shorted S-shaped
resonator (TBSSR).
4.2 Published research work related to S-shaped resonator (SSR)
SSR was presented in 2004 by Chen et al. [16]. This research group also presented SSR’s
analytical expression for the effective permeability in 2005 [30]. As SSR single handedly
41
z
y
x
a
b
d
h
w 2
w 1
c a 2
c a 1
Figure 4.1: S-shaped resonator (SSR).
provides negative values of the permittivity and the permeability simultaneously, so it is
very useful as tunable MTM. A modified case of SSR is also available in literature which has
been published in 2007 [32, 33]. There are other research papers which have been published
after 2007, but those papers are not cited in this section as they are not closely related to
the scope of this thesis.
4.3 Contribution of this chapter
As mentioned in the previous section, the expression for the effective permeability of SSR
has been presented in [30], but that expression has a limitation as it includes a periodicity
parameter along x -axis which is undefined for single SSR. This problem is removed in the
expression for the effective permeability, presented in this chapter. The fringing of electric
fields in calculating capacitances have not been considered in [30] which have been taken
into account in the analytical work, presented in this chapter. Two modified cases of SSR
are also discussed in this chapter along with their respective expressions for the effective
permeability. As most of the resonators presented for MTM are anisotropic, so rotation
42
of the incident wave’s polarization or rotation of the resonator can change the working
frequencies of MTM. The effect of rotation of these resonators on their working frequencies
is also included in this chapter and the comparison of un-rotated (or 0o rotated) and 90o
rotated resonators confirm that this rotation of the resonators or change of the incident
wave’s polarization is also a technique to vary the working frequency of MTM.
4.4 S-shaped resonator (SSR)
As mentioned in Section 4.1, SSR is fabricated by etching S-shaped metallic patterns on both
sides of the substrate, so SSR is bit difficult to fabricate but SSR eliminates the requirement
of thin wires for getting negative permittivity as SSR provides negative permittivity and
negative permeability within same frequency band. In this section, the effective permeability
of SSR is calculated using the equivalent circuit model approach [37]. The analytical solution
of the effective permeability of SSR is already available in literature [30], but the expression
calculated in this section is better than that of [30] due to the reason, given in Section 4.3.
To calculate the effective permeability, SSR is excited by a plane wave. Electromotive
forces (emf) are induced in both rings due to time varying magnetic field and emf causes
currents to flow in both rings. These currents are calculated and are used in the formula of
the effective permeability. It must be noted that upper C-shaped part of S-shaped metallic
pattern and another C-shaped part of S-shaped metallic part on the opposite side of the
substrate, form a ring or loop for the current in SSR. Similarly another ring or loop is formed
by the remaining two C-shaped metallic patterns.
Consider a z -polarized plane wave, propagating along y-axis, is incident on single SSR.
Due to time varying magnetic field, emfs are induced in the rings (according to Faraday’s
law of electromagnetic induction). As a result of these emfs, the currents flow in both rings
of SSR, including front and back faces. The capacitance exists at three places in SSR, i.e.,
between top metallic strips, between center metallic strips and between bottom metallic
43
Figure 4.2: Equivalent circuit of SSR [16].
strips. The equivalent circuit of SSR is shown in Figure 4.2 where (U1, U2) are the emf
sources, (R1, R2) are the resistances, (L1, L2) are the inductances, (j1, j2) are the currents
of both rings and (C1, Cm, C2) are the capacitances between the pairs of top, center and
bottom metallic strips of SSR respectively [16].
From Figure 4.2, the mesh equations of upper and lower loops (or rings) are given by,
U1 = −j1
(R1 + iωL1 +
1
iωC1
+1
iωCm
)− j2
(1
iωCm
), (4.1)
and
U2 = j1
(1
iωCm
)+ j2
(R2 + iωL2 +
1
iωC2
+1
iωCm
). (4.2)
Using Faraday’s law and Gauss’s law, emfs of the rings are given as,
44
U1 = iωµoHob(w1 + 2h), (4.3)
U2 = iωµoHob (w2 + 2h) , (4.4)
where w1, w2 are the central heights of upper and lower rings of SSR, h is the copper width,
Ho is the magnetic field and b is the width of SSR, as shown in Figure 4.1. The formula
for capacitance used in [30], includes periodicity parameter along x -axis, which is unknown
when dealing with single SSR and this formula also does not include the fringing of electric
field. So, for the calculation of capacitance, another formula is used in this chapter which
does not take into account any such periodicity parameter and also includes fringing of
electric field as h < d [38],
C1 = Cm = C2 = εbh
d
(1 +
d
πb+
d
πbln
(2πb
d
))(1 +
d
πh+
d
πhln
(2πh
d
)), (4.5)
where ε and d are the permittivity and thickness of substrate respectively. This formula for
capacitance is valid for h < d but not for h << d. The resistances of the rings are given by
[39],
R1 =
(1
σδ
)(2(w1 + b)
h
)(4.6)
and
R2 =
(1
σδ
)(2(w2 + b)
h
). (4.7)
where σ, δ are the conductivity and the skin depth respectively. The value of the term
(1/σδ) for different frequencies is given in [39].
The inductance of each ring of SSR is calculated using the expression for the inductance
45
of rectangular loop of wire having radius (h/2) [40],
Lk =µo
π
[−2(b + mk) + 2
√b2 + m2
k − ALk −BLk + CLk
], (4.8)
where,
mk = wk + 2h, (4.9)
ALk = mk ln
(mk +
√b2 + m2
k
b
), (4.10)
BLk = b ln
(b +
√b2 + m2
k
mk
), (4.11)
CLk = mk ln
(2mk
h/2
)+ b ln
(2b
h/2
), (4.12)
with k = 1,2 as there are two rings. By solving Eq. 4.1 and 4.2, the currents in the rings
are found to be,
j1 =
(U1F − U2
iωC2
−GF − 1ω2C2
2
), (4.13)
j2 =
(−U2G− U1
iωC2
−GF − 1ω2C2
2
), (4.14)
where,
G =
(R1 + iωL1 +
1
iωC1
+1
iωCm
), (4.15)
F =
(R2 + iωL2 +
1
iωC2
+1
iωCm
). (4.16)
When multiple resonators are analyzed, mutual inductances (ML1, ML2) must be considered
and the terms (iωML1) and (iωML2) are added in Eq. 4.15 and 4.16 respectively. The mutual
inductances are given by [37],
ML1 =
(ab
ca1ca2
)L1 (4.17)
46
and
ML2 =
(ab
ca1ca2
)L2. (4.18)
Magnetic dipole moment per unit volume (Md) of SSR is given by [37],
Md =1
ca1ca2dab (j1 − j2) , (4.19)
where a is the height of SSR, ca1 and ca2 are the height and the width of the substrate.
The effective permeability is given by [37],
µeff =
Bµo
Bµo−Md
, (4.20)
substituting Eq. 4.19 in Eq. 4.20, the effective permeability of SSR is found to be,
µeff =1
1− ab(j1−j2)ca1ca2dHo
, (4.21)
where it can be noticed that this formula includes only geometric parameters of SSR i.e.,
no periodicity parameter is involved. The solution of the equivalent circuit of SSR (Figure
4.2) gives the following resonant frequencies,
ωo,SSR =
√(L1 + L2)−
√L2
1 + L22 − L1L2
L1L2C,
√(L1 + L2) +
√L2
1 + L22 − L1L2
L1L2C, (4.22)
where C = C1 = Cm = C2. There are two resonant frequencies of SSR for unequal heights
of the rings (w1 6= w2), as given in Eq. 4.22. If the height of both rings are equal i.e., w1 = w2
and L1 = L2 = L [16], then there is only one resonant frequency which is given by,
47
Figure 4.3: Comparison of the analytical expressions of the effective permeability of SSR presentedin [30] with Eq. 4.21. The dimensions of SSR are ca1 = 4mm, ca2 = 5.4mm, a = 5.2mm, b =2.8mm, t (copper thickness) = 0.018mm, h = 0.4mm, d = 0.5mm, εr = 9 and w1 = w2 = 2mm[32].
ωo,SSR =
√3
LC. (4.23)
4.4.1 Comparison of the analytical work of this chapter with Chen’s work
The comparison of the expression of the effective permeability of SSR, calculated in this
chapter (Eq. 4.21), with the published work of Chen et al. [30] is done in this section. This
comparison is shown in Figure 4.3 where SSR’s effective permeability is calculated using
both expressions. SSR, used in this comparison, is taken from [32] having 6.3GHz as the
working frequency. Chen’s work shows 10.22GHz as the frequency where the permeability
achieves negative value, while 6.3GHz is the calculated resonant frequency of SSR using Eq.
48
4.23. As negative permeability frequency exists near the resonant frequency, so the effective
permeability versus frequency plot using Eq. 4.21 shows a dip at 5.2GHz, but permeability
possesses negative value at 6.3GHz. This dip can be closed to 6.3GHz if higher values of
substrate parameters (ca1, ca2) are used. These parameters are included in the expression
for the effective permeability of SSR while the resonant frequency remains unchanged as the
substrate parameters are not included in the expression of the resonant frequency. Therefore
it is noted that the analytical expression for the effective permeability of SSR (Eq. 4.21),
calculated in this chapter, is more accurate than Chen’s work of [30].
4.5 Modified cases of S-shaped resonator (SSR)
Tunable SSR can be achieved by varying its capacitances. This idea is described in this
section using two modified cases of SSR. In both cases, capacitance is varied by shortening
the metallic strips. If switches or varactors are installed between these metallic strips, then
working frequency of SSR can be varied as per user’s will. Some part of this work has been
published in [41, 42].
4.5.1 Bottom metallic strips shorted S-shaped resonator (BSSR)
Bottom metallic strips shorted S-shaped resonator (BSSR) is one of the modified cases of
SSR, which are discussed in this chapter. In this modified case, the metallic strips of bottom
ring of SSR are shorted, as shown in Figure 4.4a, so there is no capacitance between bottom
metallic strips. For calculating the expression of the effective permeability of BSSR, same
steps are followed, which are used in Section 4.4. The only difference is,
F =
(R2 + iωL2 +
1
iωCm
), (4.24)
and Eq. 4.21 is used as the expression for the effective permeability of BSSR.
The equivalent circuit of BSSR is same as that of SSR with the absence of capacitance
49
(a)
L 1 C
1
C m
L 2
U 1
U 2 R 2
R 1
(b)
Figure 4.4: (a) Bottom metallic strips shorted S-shaped resonator (BSSR). (b) Equivalent circuitof BSSR.
(C2) due to shortening of bottom metallic strips, as shown in Figure 4.4b. The solution
of this circuit (for unequal heights of the rings (w1 6= w2)) gives the following resonant
frequencies of BSSR,
ωo,BSSR =
√(L1 + 2L2)−
√L2
1 + 4L22
2L1L2C,
√(L1 + 2L2) +
√L2
1 + 4L22
2L1L2C. (4.25)
If w1 = w2 i.e., L1 = L2 = L,
ωo,BSSR =
√3 +
√5
2LC. (4.26)
By comparing the resonant frequencies of BSSR and SSR, the relation between their working
frequencies is found to be,
50
(a)
L 1
C m
L 2
U 1
U 2 R 2
R 1
(b)
Figure 4.5: (a) Top-bottom metallic strips shorted S-shaped resonator (TBSSR). (b) Equivalentcircuit of TBSSR.
ωo,BSSR ≈√
3 +√
5
6ωo,SSR. (4.27)
4.5.2 Top-bottom metallic strips shorted S-shaped resonator (TBSSR)
Top-bottom metallic strips shorted S-shaped resonator (TBSSR) is the second modified
case of SSR which is discussed in this chapter. TBSSR was presented by Cheng et al. in
2007, but its expression for the effective permeability was not calculated which is done in
this section [32, 33]. As the name describes, top and bottom metallic strips are shorted in
TBSSR, as shown in Figure 4.5a.
The analytical expression for the effective permeability of TBSSR is calculated using the
same steps that are described in Section 4.4 with the following differences,
51
G =
(R1 + iωL1 +
1
iωCm
), (4.28)
F =
(R2 + iωL2 +
1
iωCm
), (4.29)
and Eq. 4.21 can be used as the expression for the effective permeability of TBSSR. The
equivalent circuit of TBSSR is shown in Figure 4.5b with the absence of capacitances, C1
and C2 due to shortening of top and bottom metallic strips. The resonant frequency of
TBSSR, calculated by solving the equivalent circuit is found to be,
ωo,TBSSR =
√(L1 + L2)
L1L2C. (4.30)
If w1 = w2 i.e., L1 = L2 = L,
ωo,TBSSR =
√2
LC. (4.31)
By comparing the resonant frequencies of SSR and TBSSR, the relation between the resonant
frequencies of SSR and TBSSR is given by,
ωo,TBSSR ≈√
2
3ωo,SSR, (4.32)
which is also given in [32].
52
4.6 Analytical results
In this section, the analytical results for the effective permeability of SSR and its modified
cases are presented. Each analytical result is given in this section for two different values
of substrate parameters (ca1, ca2) in order to understand the effect of substrate parameters
on the effective permeability. For the dimensions given in the caption of Figure 4.6, the
resonant frequency of SSR (using Eq. 4.23) is found to be 7.47GHz which means that SSR
exhibits negative value of permeability near this frequency. The effective permeability of
SSR (i.e., Eq. 4.21) is simulated with respect to frequency and is shown in Figure 4.6 for
two different values of substrate parameters (ca1, ca2). For ca1 = ca1 = 30mm, the effective
permeability of SSR shows resonant behaviour very close to the resonant frequency, but
for ca1= 8mm and ca1 = 10.75mm, the resonant behaviour is slightly far from the resonant
frequency. This is just due to the reason that (ca1, ca2) are included in the expression for
the effective permeability and are not included in the expression for the resonant frequency,
but it is clear from Figure 4.6 that SSR possesses negative value of permeability near the
resonant frequencies for both cases of ca1 and ca2.
It is given in Chen’s work ([30]) that if w1 = w2, then there is only one resonant frequency
while for w1 6= w2, there are two resonant frequencies and it is also discussed in Section 4.4.
The effective permeability expression of SSR (Eq. 4.21), calculated in this chapter, gives
two negative permeability frequencies each for w1 = w2 and w1 6= w2. When w1 = w2, the
higher negative permeability frequency is considered and lower one is neglected. The higher
negative permeability frequency is due to the resonant frequency given in Eq. 4.23. As
w1 = w2 in this SSR, so only 7.47GHz is taken as the negative permeability frequency.
According to Eq. 4.26, the resonant frequency of BSSR is 6.98GHz. The graphs of
BSSR’s effective permeability versus frequency for different values of substrate’s structural
parameters are given in Figure 4.7 which show that BSSR’s permeability is negative at
6.98GHz. Due to the same reason as given for SSR, 6.98GHz is taken as the negative
permeability frequency.
53
Figure 4.6: Effective permeability versus frequency plot of SSR using the analytical approachhaving a = 7.5mm, b = 4mm, t (copper thickness) = 0.018mm, h = 0.5mm, d = 1.464mm, εr =4.6, w1 = w2 = 3mm.
Figure 4.8 shows the effective permeability versus frequency plot of TBSSR for two
different values of substrate parameters. 6.1GHz is noticed in this figure as a frequency
where negative permeability is achieved for TBSSR as predicted by the resonant frequency
expression of TBSSR i.e., Eq. 4.31.
Therefore, the analytical results of the effective permeability of SSR, BSSR and TBSSR
give 7.47GHz, 6.98GHz and 6.1GHz as the frequencies where the permeability has negative
value.
54
Figure 4.7: Effective permeability versus frequency plot of BSSR using the analytical approachwith dimensions same as given in the caption of Figure 4.6.
Figure 4.8: Effective permeability versus frequency plot of TBSSR using the analytical approachwith dimensions same as given in the caption of Figure 4.6.
55
4.7 Experimental results
To verify the analytical results, experiments have been performed using Agilent E8362B
vector network analyzer, a pair of ultra wide band Vivaldi antennas and the structures of
SSR, BSSR and TBSSR where each structure consists of six cells i.e., three along y-axis and
two along z -axis. The experimental setup has already been discussed in Section 3.6. All
the measurements, given in this section, are taken with a frequency step size of 6MHz and
antenna to antenna distance of 66mm. Experimental results of a simple FR4 (conventional
right handed material), are also included in this section as it will help in distinguishing the
RH and LH frequency bands. These experimental results also include the measurements of
transmission (S21) power, group delay and transmission (S21) phase when there is nothing
placed between both antennas for better understanding of LH frequency bands.
The fabricated structure of SSR along with different measurements of SSR and FR4 are
shown in Figure 4.9. In the experimental results of transmission (S21) power of SSR and
FR4, shown in Figure 4.9b, the existence of LH transmission band is not quite clear as
small structure of SSR is used in this measurement. The transmission group delay of SSR
is shown in Figure 4.9c which describes that SSR is getting lower values at 6.92-7.14GHz
frequency band. It gives the idea that SSR is behaving as LH material in this frequency
band. Figure 4.9d shows the transmission (S21) phase results of SSR and FR4 where the
graph of FR4 is decreasing with increase in frequency, while the graph of SSR increases
in a frequency band (6.92-7.14GHz), while decreases at remaining frequencies. It gives the
idea that SSR behaves as RH material at all frequencies other than 6.92-7.14GHz. So, it is
confirmed that SSR of geometric parameters given in the caption of Figure 4.6 behaves as
LHM at 6.92-7.14GHz frequency band.
56
(a) (b)
(c) (d)
Figure 4.9: (a) Fabricated SSR, having area 24mm x 21.5mm and the remaining dimensions aregiven in the caption of Figure 4.6. Experimental results of SSR and FR4: (b) Transmission (S21)power. (c) Group delay. (d) Transmission (S21) phase.
57
(a) (b)
(c) (d)
Figure 4.10: (a) Fabricated BSSR, having area 24mm x 21.5mm and the remaining dimensionsare given in the caption of Figure 4.7. Experimental results of BSSR and FR4: (b) Transmission(S21) power. (c) Group delay. (d) Transmission (S21) phase.
Different experimental results of BSSR are shown in Figure 4.10 along with its fabricated
picture. Figure 4.10b shows the transmission (S21) results of BSSR and FR4, but the
frequency band where BSSR behaves as LHM, is not clear from this figure due to the
reason given for SSR, discussed earlier in this section. Negative group delay for BSSR
is observed in Figure 4.10c at 6.73-6.95GHz frequency band which gives the idea that in
58
this frequency band, BSSR works as LHM. This is further verified from the experimental
results of transmission phase (shown in Figure 4.10d) where the phase of FR4, continuously
decreases with increase in frequency, while the phase of BSSR increases in 6.73-6.95GHz
frequency region, but decreases in remaining frequencies. This means that negative phase is
observed and backward waves propagate in this frequency region. So, this is the frequency
band where BSSR (of geometric parameters given in the caption of Figure 4.6) works as
LHM.
59
(a) (b)
(c) (d)
Figure 4.11: (a) Fabricated TBSSR, having area 24mm x 21.5mm and the remaining dimensionsare given in the caption of Figure 4.8. Experimental results of TBSSR and FR4: (b) Transmission(S21) power. (c) Group delay. (d) Transmission (S21) phase.
The fabricated structure of TBSSR is shown in Figure 4.11a while different measurements
of TBSSR and FR4 are shown in Figure 4.11b, 4.11c and 4.11d. Similar to SSR and BSSR,
the frequency band at which TBSSR behaves as LHM is not clear from the transmission
results, shown in Figure 4.11b. TBSSR shows lower values of group delay at 6.5-6.74GHz
frequency band, shown in Figure 4.11c. In Figure 4.11d, the behavior of transmission
60
Table 4.1: Comparison of frequencies at which the permeability achieves negative values for SSR,BSSR and TBSSR using the analytical and the experimental results.
Analytical results. Left handed transmission bands.Experimental results.
(GHz) (GHz)SSR 7.47 6.92-7.14BSSR 6.98 6.73-6.95TBSSR 6.1 6.5-6.74
(S21) phase results of TBSSR is different as that of FR4 in 6.5-6.74GHz frequency region
which gives the idea that backward waves propagate in this frequency region. So, due to
negative group delay and propagation of backward waves, it can be said that TBSSR (having
geometric parameters given in the caption of Figure 4.6), behaves as LHM at 6.5-6.74GHz
frequency band.
4.8 Comparison of the analytical and the experimental results
Analytical and experimental results of SSR, BSSR and TBSSR are presented in detail in
Section 4.6 and 4.7 respectively and their comparison is discussed in this section. The
frequencies at which permeability achieves negative value using analytical and experimental
results is summarized in Table 4.1. In the experimental results, LH transmission bands of
SSR and its modified cases are used as the permeability and the permittivity, both have
negative values in LH transmission bands.
For SSR, the analytical result gives 7.47GHz as the negative permeability frequency
which is verified by the experimental result of 6.92-7.14GHz frequency band. BSSR’s an-
alytical result shows 6.98GHz as the negative permeability frequency which is comparable
with LH frequency band noticed which is at 6.73-6.95GHz. In TBSSR, 6.1GHz is observed
as negative permeability frequency in the analytical results and it is verified by the experi-
mental results which show LH frequency band at 6.5-6.74GHz.
Thus, it can be said that the analytical and the experimental results of SSR and its
61
two modified cases (i.e, BSSR and TBSSR) are in good agreement which validates the
analytical expressions for the effective permeability of SSR, BSSR and TBSSR, presented in
this chapter. Small difference between the results of both approaches is due to assumptions
taken in the analytical work and also due to the fabrication imperfections.
4.9 How to get S-shaped resonator (SSR) of desired working fre-
quency
Number of research papers are available in literature where MTM has been utilized in
designing of different devices. If SSR is to be used as MTM, then SSR of desired working
frequency is required. The method of getting SSR of desired working frequency is discussed
in this section.
The resonant frequency of SSR (with equal heights of the rings i.e., w1 = w2) is given in
Eq. 4.23 as,
ωo,SSR =
√3
LC, (4.33)
where C1 = Cm = C2 = C and L1 = L2 = L. The formulae for the capacitance and
the inductance are already given in Eq. 4.5 and Eq. 4.8 respectively where the formula
for capacitance is valid for h < d and not for h << d. These formulae include only
geometric parameters of SSR, so by choosing sufficient values of these parameters, the
working frequency of SSR can be determined. If this frequency is not equal to the desired
frequency, then variation in the value of any one geometric parameter must be tested and
in this way, SSR of desired working frequency is achieved which can be used for desired
purpose.
62
z
y
x
C 2
E
k
H
Figure 4.12: 90o rotated S-shaped resonator (SSR).
4.10 Effect of rotation of S-shaped resonator (SSR) and its mod-
ified cases on their respective left handed (LH) frequency
bands
The frequency regions where SSR and its modified cases behave as LHM are determined by
making plane wave incident on the resonators with E-field parallel and H-field perpendicular
to the resonators. If these resonators are rotated 90o or the polarization of the incident wave
is rotated 90o, as shown in Figure 4.12, then there is shift in LH frequency bands of each
resonator. This idea is taken from the point that SSR is an anisotropic material i.e., change
of direction of applied field causes change in the permittivity and the permeability of SSR.
Some part of this work has been published in [43].
LH frequency bands of un-rotated (or 0o rotated) SSR, BSSR and TBSSR are already
determined in Section 4.7. For finding LH frequency bands of 90o rotated SSR, BSSR and
TBSSR, the experimental setup is shown in Figure 4.13. The experimental results of rotated
SSR, rotated BSSR and rotated TBSSR with FR4 are shown in Figure 4.14, 4.15 and 4.16
63
Figure 4.13: Experimental setup for 90o rotated S-shaped resonator (SSR).
respectively. It is observed from these measurements that rotated SSR behaves as LHM at
9.1-9.69GHz, BSSR at 8.1-8.3GHz, 8.4-8.8GHz and TBSSR at 6.7-6.95GHz, 7.6-8.2GHz.
64
(a) (b)
(c)
Figure 4.14: Experimental results of rotated SSR and FR4: (a) Transmission (S21) power. (b)Group delay. (c) Transmission (S21) phase.
65
(a) (b)
(c)
Figure 4.15: Experimental results of rotated BSSR and FR4: (a) Transmission (S21) power. (b)Group delay. (c) Transmission (S21) phase.
66
(a) (b)
(c)
Figure 4.16: Experimental results of rotated TBSSR and FR4: (a) Transmission (S21) power. (b)Group delay. (c) Transmission (S21) phase.
The comparison of LH frequency bands of un-rotated (or 0o rotated) and 90o rotated
resonators is summarized in Table 4.2. This comparison gives the idea that LH frequency
bands are shifted to higher frequencies when they are rotated. Therefore, this rotation of
resonators is verified as a technique to vary the working frequencies of SSR and its modified
cases. As most of the resonators presented for LHM are anisotropic, so this technique can
also be used for all of them.
67
Table 4.2: Comparison of the experimental results of LH frequency bands of un-rotated (or 0o
rotated) and 90o rotated of SSR, BSSR and TBSSR.
Un-rotated Rotated(GHz) (GHz)
SSR 6.92-7.14 9.1-9.69BSSR 6.73-6.95 8.1-8.3, 8.4-8.8TBSSR 6.5-6.74 6.7-6.95, 7.6-8.2
4.11 Summary
S-shaped resonator (SSR) and its two modified cases i.e., bottom metallic strips shorted
S-shaped resonator (BSSR) and top-bottom metallic strips shorted S-shaped resonator (TB-
SSR) are discussed in this chapter. The analytical expressions for the effective permeabil-
ities of these resonators are presented and verified by the experimental results. Utilizing
anisotropic nature of SSR, BSSR and TBSSR, rotation of resonators is presented as a tuning
technique for the working frequency which can also be used for other resonators as well.
68
Chapter 5
Single side paired S-ring resonator
(SSPSRR) and its tunable cases
5.1 Introduction
Single side paired S-ring resonator (SSPSRR), shown in Figure 5.1, is one of those resonators,
presented for metamaterial (MTM), that provides negative values of electric permittivity
(ε) and magnetic permeability (µ) within same frequency band. SSPSRR was presented
in 2007 by the same research group that presented S-shaped resonator (SSR) in 2004 [17,
16]. SSPSRR consists of two S-shaped metallic patterns on one side of the substrate (or
dielectric), so it can be fabricated easily as compared to SSR. Due to metallic patterns on
one side, lumped active elements (e.g. diodes or varactors) can be installed easily between
the metallic strips (for tuning the working frequency) which is difficult in SSR. These are
two main advantages of SSPSRR over SSR and other such type of resonators.
In this chapter, SSPSRR and its two tunable cases i.e., bottom metallic strips shorted
single side paired S-ring resonator (B-SSPSRR) and top-bottom metallic strips shorted
single side paired S-ring resonator (TB-SSPSRR) are discussed.
69
z
y
x
d 1
c a 1
d 4
h
l 1
l 2
l 3
d 3
d 2
c a 2
Figure 5.1: Single side paired S-ring resonator (SSPSRR) [17].
5.2 Published research work related to single side paired S-ring
resonator (SSPSRR)
SSPSRR was presented in 2007 by Ran et al. [17]. There are three capacitances in SSPSRR
which exist between top metallic strips, between center metallic strips and between bottom
metallic strips. Ran et al. placed varactor between the center metallic strips and presented
a tunable case of SSPSRR [34]. After this work, few more research papers have also been
published but they are out of scope of this thesis.
5.3 Contribution of this chapter
The numerical and the experimental results for LH frequency bands of SSPSRR are already
available in literature. A tunable case of SSPSRR has also been presented in which a
varactor is installed between the center metallic strips of the resonator [34]. This tunable
case can only change the electric resonant frequency of SSPSRR and there is no effect on
70
the magnetic resonant frequency, so the LH frequency band does not change.
In this chapter, the analytical expression for the effective permeability of SSPSRR is
presented and is verified by the experimental results. This analytical expression also verifies
the experimental and the numerical results of [17] and [34]. Two tunable cases of SSPSRR
i.e., bottom metallic strips shorted single side paired S-ring resonator (B-SSPSRR) and top-
bottom metallic strips shorted single side paired S-ring resonator (TB-SSPSRR) are also
presented in this chapter along with their experimental results which shows that the change
of LH frequency bands is possible using these tunable cases.
Rotation of incident wave’s polarization or rotation of resonator, has been presented as
a tuning technique in Section 4.10 of this thesis. In this chapter, this technique is presented
for SSPSRR and its tunable cases.
5.4 Single side paired S-ring resonator (SSPSRR)
SSPSRR has already been described in Section 5.1 and 5.2. In this section, the analytical
expression for the effective permeability of SSPSRR is calculated using equivalent circuit
model approach [37].
The process of exciting SSPSRR is same as that of SSR, discussed in previous chapter.
Consider a z -polarized plane wave, incident on a single SSPSRR. Electromotive force (emf)
is induced in both rings of SSPSRR due to time varying magnetic field and these emfs cause
currents to flow in both rings. The currents are calculated and are used in the basic formula
of the effective permeability.
The equivalent circuit of SSPSRR is shown in Figure 5.2 where C1 is the capacitance
between top metallic strips and also between bottom metallic strips, C2 is the capacitance
between center metallic strips, L is the total inductance of SSPSRR, (R1, R2) are the
resistances of the rings and (U1, U2) are the electromotive forces (emf) [17]. From Figure
5.2, the mesh equations are,
71
L/2 C 1
C 2
L/2
U 1
U 2
C 1
R 1
R 2
Figure 5.2: Equivalent circuit of SSPSRR [17].
U1 = −j1
(R1 +
iωL
2+
1
iωC1
+1
iωC2
)− j2
(1
iωC2
)(5.1)
and
U2 = j1
(1
iωC2
)+ j2
(R2 +
iωL
2+
1
iωC1
+1
iωC2
), (5.2)
where j1 and j2 are the currents in upper and lower loops (or rings) respectively. Using
Faraday’s law and Gauss’s law, the emfs of the rings are given as,
U1 = iωµoHol2l4, (5.3)
U2 = iωµoHo(h− l2)l4, (5.4)
where l4 = (l1 + d4/2), ω is the frequency, µo is the permeability of free space, Ho is the
magnetic field and h, l1, l2, d4 are different geometric parameters of SSPSRR, shown in
72
Figure 5.1. The resistances of the rings are [39],
R1 =
(1
σδ
)((2l4 + 2h)
d3
), (5.5)
R2 =
(1
σδ
)((2l4 + 2h)
d3
), (5.6)
where σ is the conductivity and δ is the skin depth.
The inductance of SSPSRR is calculated as the self inductance of wire having rectangular
cross-section, consisting of four segments and is given by [44],
L =N∑
m=1
Lm +N∑
m=1
N∑o=m+1
2kmoMmo, (5.7)
where kmo = 0 if segment m and o are perpendicular to each other, kmo = 1 if segment m
and o have same current direction and kmo = −1 if segment m and o have opposite current
direction. The self inductance of single segment of wire (Lm) is given by,
L1 = L3 =µo
2π
[l4 ln
(2l4
d3 + t
)+
l42
+ 0.2235(d3 + t)
]; l4 >> (d3 + t), (5.8)
L2 = L4 =µo
2π
[h ln
(2h
d3 + t
)+
h
2+ 0.2235(d3 + t)
]; h >> (d3 + t), (5.9)
where (L1, L3) and (L2, L4) are parallel segments of wire. Mutual inductance is considered
between two parallel wire segments of equal length where the length of segments is greater
than distance between the segments. So,
M24 =µol
2π
[ln
(2l
d
)− 1 +
d
l
], (5.10)
and M13 is not considered as l4 < h.
The capacitance between the metallic strips is not calculated analytically in this chapter
but a commercially available software named CoventorWare is used for this purpose.
73
By solving Eq. 5.1 and 5.1, the currents in the rings are found to be,
j1 =
(U1F − U2
iωC2
−GF − 1ω2C2
2
), (5.11)
j2 =
(−U2G− U1
iωC2
−GF − 1ω2C2
2
), (5.12)
where,
G =
(R1 +
iωL
2+
1
iωC1
+1
iωC2
), (5.13)
F =
(R2 +
iωL
2+
1
iωC1
+1
iωC2
). (5.14)
Magnetic dipole moment per unit volume (Md) of SSPSRR is given by [37],
Md =1
ca1ca2d1
(j1 − j2) hl4, (5.15)
where ca1, ca2 and d1 are the width, height and thickness of the substrate respectively, as
shown in Figure 5.1. The effective permeability is given by [37],
µeff =
Bµo
Bµo−Md
, (5.16)
substituting Eq. 5.15 in Eq. 5.16, the effective permeability of SSPSRR is found to be,
µeff =1
1− (j1−j2)hl4ca1ca2d1Ho
. (5.17)
By solving the equivalent circuit of SSPSRR (shown in Figure 5.2), the magnetic resonant
frequency of SSPSRR is found to be,
ωo,SSPSRR =
√2
LC1
. (5.18)
74
(a) (b)
Figure 5.3: Effective permeability versus frequency plot of SSPSRR, where SSPSRR is taken from:(a) [17]. (b) [34].
Ran et al. has also mentioned this expression as the resonant frequency of SSPSRR in [34].
For verification of the expression of the effective permeability of SSPSRR (Eq. 5.17), two
different SSPSRRs are chosen from [17] and [34] and their respective effective permeabilities
are calculated using Eq. 5.17 which are shown in Figure 5.3a and 5.3b respectively with and
without considering M24. Using CoventorWare [45], the values of capacitances, C1 and C2
are found to be 2.314e-2 pF and 4.057e-2 pF respectively for SSPSRR of [17] and 7.448e-2
pF and 5.526e-2 pF respectively for SSPSRR of [34]. In [17], LH frequency band of SSPSRR
is calculated experimentally and is found to be 10-11GHz while Eq. 5.18 (expression of the
resonant frequency) gives 11.12GHz as the resonant frequency when M24 is not considered
and 13.1GHz when M24 is considered.
Similarly in [34], LH frequency band of SSPSRR is calculated as 2.93-3.02GHz and Eq.
5.18 gives 3.62GHz as the resonant frequency when M24 is not considered and 4.3GHz when
M24 is considered. Figure 5.3a and 5.3b show the effective permeability versus frequency
plots for these two SSPSRRs where the resonant behaviors are noticed little away from
the resonant frequencies but it is observed that these two SSPSRRs have negative value of
75
Table 5.1: Comparison of frequencies at which SSPSRR achieves negative value of permeability,calculated using the Eq. 5.18 for two different SSPSRRs, one taken from [17] and other from [34].
Left handed frequency bands. Analytical results. Analytical results.Published results. M24 = 0. M24 6= 0.
(GHz) (GHz) (GHz)[17] 10.0-11.0 11.12 13.1[34] 2.93-3.02 3.62 4.3
permeability at the resonant frequency. The resonant behavior of the effective permeability
can be further closed to the resonant frequency if higher values of substrate parameters (ca1,
ca2) are used.
The comparison of the published works of [17] and [34] with the analytical work of this
chapter is summarized in Table 5.1 where it is noticed that the analytical results, presented
in this chapter, are in good agreement with published results for M24 = 0.
5.4.1 Analytical results
The analytical results of the effective permeability of SSPSRR are presented in this section.
These results are calculated using Eq. 5.17 and is shown in Figure 5.4 with and without
considering mutual inductance (M24). The dimensions of SSPSRR are d1 = 1.464mm, d2 =
1mm, d3 = 0.47mm, d4 = 2.9mm, l1 = 3.21mm, l2 = 13.9mm, l3 = 8.9mm, h = 25mm, εr =
4.0, ca1 = 9.92mm and ca2 = 29mm. Copper height (t) over the substrate (not shown in
Figure 5.1) is 0.035mm.
Using CoventorWare [45], the value of capacitance between top metallic strips and be-
tween bottom metallic strips are found to be 8.825e-2 pF each and the capacitance between
center metallic strips is found to be 8.237e-2 pF. The resonant frequency of SSPSRR, cal-
culated using Eq. 5.18, is 3.16GHz. Although the resonant behavior is noticed little away
from the resonant frequency which is due to the substrate geometric parameters (ca1, ca2).
If higher values of ca1, ca2 are taken, the resonant behavior (or frequency at which perme-
ability is negative) can be observed much closer to the resonant frequency. In this SSPSRR,
76
Figure 5.4: Effective permeability versus frequency plot of SSPSRR, with and without consideringmutual inductance (M24).
3.16GHz is the frequency where permeability has negative value whether (ca1, ca2) have high
values or lower values.
5.4.2 Experimental results
Experimental results are presented in this section to verify the analytical result of SSPSRR,
calculated in Section 5.4.1. The experimental setup has already been discussed in Section
3.6. The structure of SSPSRR, consisting of three resonators along y-axis, is fabricated,
shown in Figure 5.10a, having an area of 24.5mm x 29mm and the dimensions of each
resonator are given in Section 5.4.1.
77
(a) (b)
(c) (d)
Figure 5.5: (a) Fabricated SSPSRR. Experimental results of SSPSRR and FR4: (b) Transmission(S21) power. (c) Group delay. (d) Transmission (S21) phase.
The experimental results of the antenna pair, i.e., transmission results without placing
anything between the two antennas, are first determined. Then the experimental results of
FR4 (area 31mm x 26mm and thickness 1.464mm) are measured by placing FR4 between the
antennas as FR4 results help in distinguishing between RH and LH frequency bands. After
that SSPSRR is placed between the antennas and its experimental results are determined.
Similar to the experimental results presented earlier in this thesis, the experimental
78
Table 5.2: Comparison of the frequencies at which permeability achieves negative value for SSPSRRusing analytical and experimental results.
Analytical result. Left handed frequency band.Experimental result.
(GHz) (GHz)3.16 3.63-4.0
results of transmission (S21) power, group delay and transmission (S21) phase are presented
in this section. As MTM (LHM) exhibits negative group delay and negative transmission
phase i.e., propagation of backward waves, so on the basis of these two characteristics, LH
frequency band can be identified.
The experimental results of transmission (S21) power, group delay and the transmission
(S21) phase are shown in Figure 5.5b, 5.5c and 5.5d respectively. As number of cells (or
resonators) used in this experiment are less, so LH frequency band is not clear from trans-
mission (S21) power results (i.e., Figure 5.5b). The group delay results of SSPSRR and FR4
are shown in Figure 5.10c where SSPSRR’s group delay achieves negative value at 3.63-
4.0GHz frequency band. Transmission (S21) phase results are shown in Figure 5.5d where
it is noticed that SSPSRR’s phase behaves unconventionally in the frequency region where
negative group delay also exists. So it can be said that 3.63-4.0GHz is the LH frequency
band for this SSPSRR.
5.4.3 Comparison of the analytical and the experimental results
The comparison of the analytical and the experimental results of the frequency where per-
meability achieves negative value for SSPSRR (having dimensions given in Section 5.4.1) is
summarized in Table 5.2. In the experimental result, LH frequency band is given i.e., the
frequency region where permittivity and permeability, both are negative. Due to fabrication
imperfections and the assumptions taken in the analytical work, small difference is observed
between the analytical and the experimental results but it can be said that the results are
in good agreement with each other and so the expression for the effective permeability of
79
z
y
x
(a)
L/2 C 1
C 2
L/2
U 1
U 2
R 1
R 2
(b)
Figure 5.6: (a) Bottom metallic strips shorted single side paired S-ring resonator (B-SSPSRR).(b) Equivalent circuit of B-SSPSRR.
SSPSRR is verified.
5.5 Tunable cases of single side paired S-ring resonator
There are two tunable cases of SSPSRR, presented in this section. Change in the working
frequencies of these cases can be achieved by shortening the metallic strips i.e., varying
the capacitance. If varactors are installed instead of the shortened metallic part, then the
working frequency of SSPSRR can be further varied. This is the basic idea of these tunable
cases. In either of these cases, the capacitance between center metallic strips is not varied
as it had already been done in [34] and no change of LH frequency band was achieved.
To notice the tunability, the experimental results of both these cases are included in this
section. Some part of this work has been published in [46].
80
5.5.1 Bottom metallic strips shorted single side paired S-ring resonator (B-
SSPSRR)
Bottom metallic strips shorted single side paired S-ring resonator (B-SSPSRR) is that tun-
able case of SSPSRR in which bottom metallic strips of SSPSRR are shorted, as shown in
Figure 5.6a. The equivalent circuit of B-SSPSRR is shown in Figure 5.6b which is similar
to that of original SSPSRR with the absence of capacitance (C 1) in the bottom or lower
loop.
Experimental results
Fabricated structure of B-SSPSRR is shown in Figure 5.7a and the experimental results of
transmission (S21) power, group delay and transmission (S21) phase are shown in Figure
5.7b, 5.7c and 5.7d respectively. 3.63-4.0GHz, 7.6-9GHz, 10.55-10.7GHz, 10.8-11.1GHz,
12.1-12.45GHz are the frequency bands where negative group delay and unconventional
behavior of transmission phase are observed, so these are the LH frequency bands of B-
SSPSRR.
81
(a) (b)
(c) (d)
Figure 5.7: (a) Fabricated B-SSPSRR. Experimental results of B-SSPSRR and FR4: (b) Trans-mission (S21) power. (c) Group delay. (d) Transmission (S21) phase.
5.5.2 Top-bottom metallic strips shorted single side paired S-ring resonator
(TB-SSPSRR)
Top-bottom metallic strips shorted single side paired S-ring resonator (TB-SSPSRR) is the
second tunable case of SSPSRR, presented in this chapter. In TB-SSPSRR, the metallic
strips of top and bottom metallic rings are shorted, as shown in Figure 5.8a and the equiva-
82
z
y
x
(a)
L/2
C 2
L/2
U 1
U 2
R 1
R 2
(b)
Figure 5.8: (a) Top-bottom metallic strips shorted single side paired S-ring resonator (TB-SSPSRR). (b) Equivalent circuit of TB-SSPSRR.
lent circuit of TB-SSPSRR is shown in Figure 5.8b where C1 is not present due to shortening
of top and bottom metallic strips.
Experimental results
TB-SSPSRR, fabricated on FR4 substrate with dimensions given in Section 5.4.1, is shown
in Figure 5.9a. The experimental results of transmission (S21) power, group delay and
transmission (S21) phase are shown in Figure 5.9b, 5.9c and 5.9d respectively, where negative
group delay and negative transmission phase are observed in 3.6-3.7GHz, 3.74-4GHz, 7.6-
9GHz, 10.3-10.65GHz, 11.2-11.6GHz frequency bands, so these are the LH frequency bands
of TB-SSPSRR.
83
(a) (b)
(c) (d)
Figure 5.9: (a) Fabricated TB-SSPSRR. Experimental results of TB-SSPSRR and FR4: (b) Trans-mission (S21) power. (c) Group delay. (d) Transmission (S21) phase.
84
(a) (b)
(c) (d)
Figure 5.10: (a) Fabricated SSPSRR. Experimental results of SSPSRR and FR4: (b) Transmission(S21) power. (c) Group delay. (d) Transmission (S21) phase.
5.6 Comparison of left handed (LH) frequency bands of single
side paired S-ring resonator (SSPSRR) and its tunable cases
The comparison of LH frequency bands of SSPSRR, B-SSPSRR and TB-SSPSRR is sum-
marized in Table 5.3 where the measurements are taken from 2GHz to 12GHz. As in Figure
85
Table 5.3: Comparison of LH frequency bands of SSPSRR, B-SSPSRR and TB-SSPSRR, calculatedexperimentally.
LH frequency bands.(GHz)
SSPSRR 3.63-4.0, 7.6-9, 10.55-10.7, 10.8-11.2B-SSPSRR 3.63-4.0, 7.6-9, 10.55-10.7, 10.8-11.1, 12.1-12.45TB-SSPSRR 3.6-3.7, 3.74-4, 7.6-9, 10.3-10.65, 11.2-11.6
5.5, the experimental results of SSPSRR are given from 2GHz to 5GHz which only gives
first LH frequency band, so for this comparison, the experimental results of SSPSRR from
2GHz to 12GHz are shown in Figure 5.10 which gives 3.63-4GHz, 7.6-9GHz, 10.55-10.7GHz,
10.8-11.2GHz as LH frequency bands.
The comparison (Table 5.3) shows that first two LH frequency bands are same for these
three resonators and change in frequency bands is noticed at higher frequencies. This
variation of LH frequency bands verifies that change in the capacitance causes shift of
LH frequency band in SSPSRR. Although this change of LH frequency bands is not quite
big which is due to small value of capacitance between metallic strips. If varactors are
installed between strips, then large variation of capacitance causes sufficient change in the
LH frequency bands.
5.7 Effect of rotation of single side paired S-ring resonator (SSP-
SRR) and its tunable cases on their respective left handed
(LH) frequency bands
There are number of resonators, presented for MTM, but most of them are anisotropic i.e.,
their constitutive parameters (ε, µ) are dependent on the direction of applied field. This
gives the idea that if the polarization of the incident wave is changed or the resonators are
rotated (as shown in Figure 5.11), change in the LH frequency band can be observed which
means that this approach can be used a technique to tune the working frequency of MTM.
86
This technique has also been presented for S-shaped resonator (SSR) in Section 4.10 of this
thesis.
LH frequency bands of 0o rotated (or un-rotated) SSPSRR and its tunable cases have
already been determined earlier in this chapter. The experimental setup for determining
the results of 90o rotated SSPSRR is shown in Figure 5.12. The experimental results of
transmission (S21) power, group delay and transmission (S21) phase of 90o rotated SSPSRR
and its tunable cases are shown in Figure 5.13, 5.14 and 5.15. With the existence of negative
group delay and negative phase, LH frequency bands are identified and are found to be 9.7-
9.9GHz, 14.25-14.75GHz for rotated SSPSRR, 11.35-12.05GHz, 14.25-14.75GHz for rotated
B-SSPSRR and 7.05-7.15GHz, 10.25-10.4GHz, 10.5-10.85GHz for rotated TB-SSPSRR.
The comparison of LH frequency bands of un-rotated and rotated SSPSRR, B-SSPSRR
and TB-SSPSRR is summarized in Table 5.4 where it is noticed that change in LH frequency
bands occur as expected and the frequency bands are shifted to higher frequencies in rotated
resonators as compared to the un-rotated case. It is also observed in this comparison that
the bandwidth of LH frequency bands is lesser in the rotated resonators as compared to the
un-rotated case.
With this comparison, it is confirmed that rotation can be used as a technique to vary
the working frequencies of SSPSRR, B-SSPSRR and TB-SSPSRR.
87
z
y
x
E
k
H
Figure 5.11: 90o rotated SSPSRR.
Figure 5.12: Experimental setup for 90o rotated SSPSRR.
88
(a) (b)
(c)
Figure 5.13: Experimental results of rotated SSPSRR and FR4: (a) Transmission (S21) power. (b)Group delay. (c) Transmission (S21) phase.
89
(a) (b)
(c)
Figure 5.14: Experimental results of rotated B-SSPSRR and FR4: (a) Transmission (S21) power.(b) Group delay. (c) Transmission (S21) phase.
90
(a) (b)
(c)
Figure 5.15: Experimental results of rotated TB-SSPSRR and FR4: (a) Transmission (S21) power.(b) Group delay. (c) Transmission (S21) phase.
91
Table 5.4: Comparison of LH frequency bands of SSPSRR, B-SSPSRR and TB-SSPSRR with andwithout 90o rotation, calculated using the experimental results.
Un-rotated Rotated(GHz) (GHz)
SSPSRR 3.63-4.0, 7.6-9, 10.55-10.7, 10.8-11.2 9.7-9.9, 14.25-14.75B-SSPSRR 3.63-4.0, 7.6-9, 10.55-10.7, 10.8-11.1, 12.1-12.45 11.35-12.05, 14.25-14.75TB-SSPSRR 3.6-3.7, 3.74-4, 7.6-9, 10.3-10.65, 11.2-11.6 7.05-7.15, 10.25-10.4, 10.5-10.85
5.8 Summary
Single side paired S-ring resonator (SSPSRR) is discussed in this chapter along with its
two tunable cases i.e., bottom metallic strips shorted single side paired S-ring resonator (B-
SSPSRR) and top-bottom metallic strips shorted single side paired S-ring resonator (TB-
SSPSRR). The expression for the effective permeability of SSPSRR is calculated analytically
and verified by the experimental results. The effect of rotation of SSPSRR and its tunable
cases is also presented in this chapter which justifies that the rotation can be used as a
technique to vary the working frequencies of MTM.
92
Chapter 6
Conclusions and Future works
6.1 Conclusions
Different techniques to vary the working frequency of metamaterials (MTM) are presented in
this thesis. As MTM are artificial materials and there are a number of resonators available
in literature for MTM, so these tuning techniques are applied on some of the important
resonators which include split ring resonator (SRR), S-shaped resonator (SSR) and single
side paired S-ring resonator (SSPSRR).
In the third chapter of this thesis, inner ring shorted split ring resonator (IRS-SRR),
a tunable case of SRR, is presented. The aim of this chapter is to present the idea that
the working or resonant frequency of SRR can be changed by varying its capacitance. To
verify this approach, the capacitance between the split of inner ring of SRR is shorted (and
the structure is named as IRS-SRR), which causes change in the working frequency. The
expression for the effective permeability of IRS-SRR has also calculated in this chapter an-
alytically. For the verification of the analytical results, IRS-SRR is fabricated, its resonant
frequency has been determined experimentally and the experimental results verify the ana-
lytical work. For further variation in the resonant frequency, varactors or variable capacitors
can be installed in place of switches between the split of inner ring and in this way, more
93
flexibility can be achieved in changing the working frequency of SRR.
In the fourth chapter, the same idea of varying the working frequency has been im-
plemented on S-shaped resonator (SSR). Two modified cases of SSR i.e., bottom metallic
strips shorted S-shaped resonator (BSSR) and top-bottom metallic strips shorted S-shaped
resonator (TBSSR), have also presented in this thesis. In both these modified cases, capac-
itance is shorted using vias and shift of LH frequency band is observed. To describe this
approach, the expression for the effective permeabilities of SSR, BSSR and TBSSR have
been calculated. The experimental results have also included in this thesis to verify the
analytical results and the results using both approaches are found in good agreement with
each other.
Most of the resonators, presented for MTM, are anisotropic in nature i.e., their con-
stitutive parameters (permittivity (ε), permeability (µ)) are dependent on the direction of
applied field. This means that if the direction of the applied field is changed, variation in
the (ε, µ) is observed or in other words, shift of LH frequency band is possible. Due to this
variation of LH frequency band, this approach can be also used as a technique to tune the
working frequency of MTM. The change of the direction of applied field can be achieved
by rotating the polarization of the applied field or by rotating the resonator. This idea has
been implemented on SSR, BSSR and TBSSR and the results confirm that this approach
can be used as a technique to vary the working frequency of MTM.
The idea of varying the capacitance to shift the LH frequency band has also implemented
in this thesis on another resonator named single side paired S-ring resonator (SSPSRR). Two
tunable cases of SSPSRR i.e., bottom metallic strips shorted single side paired S-ring res-
onator (SSPSRR) and top-bottom metallic strips shorted single aide paired S-ring resonator
(TB-SSPSRR), have also been presented. The expression for the effective permeability of
SSPSRR is calculated analytically and is verified by the experimental results. B-SSPSRR
and TB-SSPSRR’s experimental results are also included in this chapter to observe the
change in LH frequency bands. Rotation tuning technique which has been presented for
94
SSR and its modified cases earlier in this thesis, has also been presented for SSPSRR and
its tunable cases and variation in LH frequency bands is noticed.
6.2 Future Work
Various techniques to vary the working frequency of MTM have been presented in this thesis
which can be used in designing of tunable microwave devices.
Rotation of resonator (or change of incident wave polarization) is presented in this thesis
as a tuning technique on the basis of the experimental results. Analytical work of this tuning
technique has not been done yet which can be addressed in future.
Two tunable cases of SSPSRR are presented in this thesis. Both these cases are obtained
by removing the capacitance. If varactors are installed instead of the shortened metallic
part, more variation of LH frequency band with high flexibility can be achieved. This can
be another useful future work. The analytical work of the tunable cases of SSPSRR is not
presented in this thesis which can also be done in future.
95
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