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Frequency selective surfaces and metamaterials for high-power microwave applications

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FREQUENCY SELECTIVE SURFACES AND METAMATERIALS FOR HIGH-POWER
MICROWAVE APPLICATIONS
by
Chien-Hao Liu
A dissertation submitted in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
(Electrical and Computer Engineering)
at the
UNIVERSITY OF WISCONSIN–MADISON
2014
Date of final oral examination: 5/16/2014
The dissertation is approved by the following members of the Final Oral Committee:
Nader Behdad, Associate Professor, Electrical and Computer Engineering
John Booske, Professor, Electrical and Computer Engineering
Zhenqiang Ma, Professor, Electrical and Computer Engineering
David Anderson, Professor, Electrical and Computer Engineering
Peter Timbie, Professor, Department of Physics
UMI Number: 3625097
All rights reserved
INFORMATION TO ALL USERS
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a note will indicate the deletion.
UMI 3625097
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c Copyright by Chien-Hao Liu 2014
⃝
All Rights Reserved
i
To my wonderful parents Chun-Hung Liu and Chun-Fang Yao
For their endless love, support and encouragement
ii
ACKNOWLEDGMENTS
Pursuing a PhD degree is painful when struggling with tasks, enjoyable when presenting your work to the public, and rewarding when receiving credits for valuable contributions. It is like running a marathon accompanied by persistence, bitterness, hard work,
frustration, and with so many people’s assistance, moral support and love. I would like
to take this opportunity to acknowledge them and extend my gratitude for helping me to
make this PhD thesis possible.
I would like to start with the person who made the big difference in my life, my advisor,
Prof. Nader Behdad. First of all, I express my gratitude to him for providing an opportunity
to work with him and economic support as a research assistantship throughout my PhD.
As an excellent mentor, he has taught me the spirit of conducting academic research, the
ability to overcome the difficulties in my research, and the attitude of being meticulous
about everything. Under his supervision and guidance, I acquire the ability to execute
multiple projects, work independently, and show professional presentation. I am deeply
grateful to him for his encouragement to pursue a high-quality PhD degree and his support
in applying for faculty positions and job searching at the time closer to graduation. To
me, he is more like a senior friend than a scientist and I feel privileged to have him as my
mentor.
I express heartfelt gratitude to the members of my doctoral committee, Professors John
H. Booske, Zhenqiang (Jack) Ma, David T. Anderson, and Peter Timbie. Their valuable
iii
suggestions have been extremely beneficial in improving the overall quality of my research and thesis. My special thanks to Prof. Booske and John E. Scharer for giving me
access to their facilities for high-power microwave (HPM) experiments. I also gratefully
acknowledge Professors Andreas Neuber (Texas Tech University), Y. Y. Lau (University
of Michigan), and John Verboncoeur (Michigan State University) for helpful discussions
in studying the cause of simultaneous breakdown phenomenon in metamaterials.
My sincere thanks also go to my lab seniors, Dr. Mudar Al-Joumayly, Bin Yu, Dr.
Suzette Aguilar, Dr. Meng Li, who taught me the lab cultures and shared their experiences
during the initial days of my stay in the lab. I would like to acknowledge my fellow
labmates, Dr. Yazid Yusuf, Dr. Arash Rashidi, John Brady, Amir Reza Masoumi, Amin
Momeni, Hung Luyen, Tonmoy Bhatacharjee, Kasra Ghaemi, and Ting-Yen Shih, for their
suggestions in dealing with the issues I encountered in research and assistance in most
experiments I conducted in the lab. My special words of thanks go to Dr. Li for all his
help in academic research and his accompanying me everyday from the time I arrived the
lab in the morning until I left the lab at night throughout the first three years of my PhD
life. I express my heartfelt thanks to Ting-Yen Shih who has shared his experiences in
antenna areas and reviewed most of my documents including this thesis on short notice.
I extend my grateful thanks to my colleagues at Madison, Yung-Ta Sung, Xun Xiang, Brian Kupczyk, Joel D. Neher, and Paul Carrigan, for their assistance in the HPM
experiments. Special words of thanks to Yung-Ta Sung who helped me construct the vacuum systems for HPM experiments and assisted me in doing these experiments. I would
like to acknowledge Joel D. Neher for his helps in building up the magnetron system and
correcting my English in academic writing and professional presentations.
I am grateful to my other colleagues at Madison, Tyler Rowe, Daniel Enderich, Matt
Kirley, Ryan Jacobs, for their valuable suggestions and inspiring discussions at regular
iv
meetings. I’d also thank Fuqiang Gao, Daniel Ramirez, Timothy ”TJ” Colgan, and Owen
Mays for practicing my academic presentation for the student paper competition.
My acknowledgement will never be completed without speical mention of my friends
at Madison, Huan-Yang Chen, Szu-Yi Chen, Kai-Wen Hsu, Panpan Xue, Yan Li, and
Cheng-Hsien Lee. My sincere thanks to Dr. Huan-Yang Chen for his suggestions in
searching for my ideal adviser and beneficial aid in the courses I took during my first
year in Madison. I express heartfelt thanks to Kai-Wen Hsu for always being there and
bearing with me during the good and bad times throughout my PhD life. Special thanks are
given to Stephanie Anderson for being my English conversation patenter and improving
my presentation skills.
I would like to acknowledge all my teachers whom I learnt since my childhood. I
won’t be here without their guidance, supports, and blessing. My special thanks to my
dear teacher, Ching-Hsia Li, who first taught me that hard working is the only way leading
you to success. I extend my sincere thanks to my M.S. adviser, Prof Yuan-Fang Chou
at National Taiwan University, for his inspirations to continue my academic research and
training to make me have strong background in engineering. I would like to acknowledge
my best friends, Yen-Hung Lin and Dr. Ching-Yuan Chang, for their moral support and
more than 10 years of friendships. I really appreciate their helpful academic discussion
and felt privileged to be associated with them during my life.
Last but not least, I would like to acknowledge the people who mean the world to me,
my parents, my grandparents, brothers, sister in law, aunties, and uncles. It is your constant
love and endless support, great sacrifices, and guidance that made this accomplishment
possible. I consider myself as the luckiest person in the world to have you in my life.
DISCARD THIS PAGE
v
TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
1.2
1.3
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1
1
4
4
6
8
9
10
11
11
12
13
13
Electromagnetic Wave Tunneling Through Cascaded ϵ-Negative Metamaterial Layers Sandwiched by Double-Positive Layers . . . . . . . . . . . . .
14
1.4
2
2.1
2.2
Motivation . . . . . . . . . . . . . . . . . . .
Proposed Approach . . . . . . . . . . . . . .
Literature Review . . . . . . . . . . . . . . .
1.3.1 Periodic Structures and Metamaterials
1.3.2 ϵ- and µ-Negative Metamaterials . . .
1.3.3 Microwave-Induced Breakdown . . .
Thesis overview . . . . . . . . . . . . . . . .
1.4.1 Chapter 2 . . . . . . . . . . . . . . .
1.4.2 Chapter 3 . . . . . . . . . . . . . . .
1.4.3 Chapter 4 . . . . . . . . . . . . . . .
1.4.4 Chapter 5 . . . . . . . . . . . . . . .
1.4.5 Chapter 6 . . . . . . . . . . . . . . .
1.4.6 Chapter 7 . . . . . . . . . . . . . . .
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1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generalized Synthesis Procedure . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Topology and equivalent circuit model . . . . . . . . . . . . . . .
14
16
16
vi
Page
2.2.2
Tunneling through a single BENG slab sandwiched by two DPS
slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Tunneling through two or more ENG slabs sandwiched by DPS
layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Synthesis Procedure Verification . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Tunneling through a single ENG slab sandwiched by two high-ϵr
DPS slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Tunneling through four ENG slabs sandwiched by five high-ϵr
DPS slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 The Effect of Loss . . . . . . . . . . . . . . . . . . . . . . . . .
Tunneling Through µ-Negative (MNG) Layers . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
34
36
38
Electromagnetic Wave Tunneling Through Cascaded ϵ- and µ-Negative
Metamaterial Slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.1
3.2
3.3
39
41
45
2.3
2.4
2.5
3
25
29
29
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Synthesis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Transmission Through A Single MING Layer Sandwiched by Two
BENG Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Transmission Through N ENG Layers Separated from One Another by N − 1 MNG Layers (N ≥ 3) . . . . . . . . . . . . . . .
Verification of the Synthesis Procedure Using Full-Wave EM Simulations
3.4.1 Transmission Through A Single MNG Layer Sandwiched by Two
ENG Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 EM Wave Tunneling Through Four MNG Layers Sandwiched by
Five ENG Slabs . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 The Effect of Loss . . . . . . . . . . . . . . . . . . . . . . . . .
EM Wave Tunneling Through the Dual Multi-Layer MNG-ENG Structure
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
59
61
63
High-power microwave filters and frequency selective surfaces exploiting
electromagnetic wave tunneling through ϵ-negative layers . . . . . . . . . .
65
4.1
65
3.4
3.5
3.6
4
22
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
49
53
53
vii
Appendix
Page
4.2
4.3
4.4
4.5
5
Impact of Microwave Induced Breakdown on the Responses of High-Power
Microwave Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
5.3
5.4
6
A tri-layer spatial filter composed of an ENG layer sandwiched by two
DPS layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Practical implementation of the DPS-ENG-DPS filter . . . . . . . . . . .
4.3.1 Emulating Plasma . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 High Power Measurements . . . . . . . . . . . . . . . . . . . . .
Enhancing the Power Handling Capability of the DPS-ENG-DPS filter . .
4.4.1 High-power capable DPS-ENG-DPS filter using cutoff waveguides
4.4.2 High-power capable DPS-ENG-DPS filter using a perforated metal
layer with sub-wavelength holes . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Metasurface Design and Principles of Operation . . . . . . . . . . .
5.2.1 Design Procedure . . . . . . . . . . . . . . . . . . . . . . .
Fabrication and Experimental Characterization . . . . . . . . . . .
5.3.1 Fabrication and Low-Power Measurements . . . . . . . . .
5.3.2 High Power Measurements In Atmospheric Air Environment
5.3.3 High Power Measurements In Argon . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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68
72
72
75
80
80
84
87
88
88
90
90
100
100
101
107
109
Investigating the Physics of Simultaneous Breakdown Events in High-PowerMicrowave (HPM) Metamaterials with Multi-Resonant Unit Cells and Discrete Nonlinear Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.1
6.2
6.3
6.4
6.5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
A Metasurface with a High-Contrast, Three-Stage Nonlinear Response . . 113
Device Fabrications and Measurement . . . . . . . . . . . . . . . . . . . 118
6.3.1 Fabrication and Low-Power Characterization . . . . . . . . . . . 118
6.3.2 High Power Measurements . . . . . . . . . . . . . . . . . . . . . 120
High-Power Measurements With a Physical Barrier Between the Resonators123
6.4.1 Effect of UV Radiation . . . . . . . . . . . . . . . . . . . . . . . 125
6.4.2 Effect of VUV Radiation . . . . . . . . . . . . . . . . . . . . . . 125
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
viii
Appendix
Page
7
Investigating the Effective Range of VUV-Mediated Breakdown in HighPower Microwave Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . 132
7.1
7.2
7.3
7.4
8
Introduction . . . . . . . . . . . . . . . . .
Experimental Approach . . . . . . . . . . .
Results and Analysis . . . . . . . . . . . .
7.3.1 Device design and simulation results
7.3.2 Measurement Results . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . .
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132
135
138
138
138
145
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.1
Developing a plasma reconfigurable high-power metasurface . . . . . . . 147
LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
DISCARD THIS PAGE
ix
LIST OF TABLES
Table
2.1
2.2
2.3
3.1
3.2
3.3
3.4
4.1
4.2
4.3
Page
Physical and geometrical parameters of the DPS-ENG-DPS tri-layer structure
examined in Section 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
Parameters of the fifth-order coupled-resonator filter of the type shown in Fig.
2.2(e) that models the 9-layer structure discussed in Section 3.3.2. . . . . . .
31
Physical and geometrical parameters of the 9-layer composite ENG-DPS structure examined in Section 3.3.2. . . . . . . . . . . . . . . . . . . . . . . . . .
32
Normalized quality factors and coupling coefficients of the coupled-resonator
bandpass filters examined in this paper. . . . . . . . . . . . . . . . . . . . .
54
Physical parameters and dimensions of the tri-layer structures studied in Section 3.3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
Equivalent circuit values of the fifth-order coupled-resonator filter of the type
shown in Fig. 3.2(e), which is studied in Section 3.3.2. . . . . . . . . . . . .
59
Physical parameters of the nine-layer ENG-MNG composite structure examined in Section 3.3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
Physical parameters of the tri-layer structure composed of two DPS and one
ENG layers examined in Section 7.3. . . . . . . . . . . . . . . . . . . . . .
69
Physical and geometrical parameters of the DPS-ENG-DPS filter examined
in Section 7.3.2. The photograph of the device is shown in Fig. 4.5. . . . . .
78
Physical parameters of DPS-ENG-DPS filter examined in Section 4.4-A. The
photograph of the device is shown in Fig. 4.10. . . . . . . . . . . . . . . . .
78
x
Table
4.4
5.1
5.2
5.3
Page
Physical parameters of DPS-ENG-DPS structure examined in Section 4.4-B.
The photograph of the device is shown in Fig. 4.14. . . . . . . . . . . . . . .
78
Physical parameters of the waveguide version of the unit cell of the metasurface shown in Fig. 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
Physical parameters of the waveguide version of the unit cell of the metasurface shown in Fig. 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
Physical parameters of the waveguide version of the metasurafce shown in
Fig. 5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
5.4
Comparison of the measured power reflection and transmission coefficients
of the metasurface shown in Fig. 5.2(a) with the simulated ones. . . . . . . . 100
5.5
Comparison of the measured power reflection and transmission coefficients
of the metasurface shown in Fig. 5.3(a) with the simulated ones. . . . . . . . 100
5.6
Comparison of the measured power reflection and transmission coefficients
of the metasurface shown in Fig. 5.4(a) with the simulated ones. . . . . . . . 101
5.7
Breakdown power level for the three devices discussed in this paper when
they operate in Air at atmospheric pressure level and in Argon at 600 torr. . . 106
6.1
Comparison of the power reflection and transmission coefficients of the device shown in Fig. 6.3(a) derived from full-wave EM simulations and HPM
measurements before breakdown occurs and when both resonators breaks
down simultaneously. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2
Comparison between the simulated and measured power transmission and
reflection coefficients of the metasurface . . . . . . . . . . . . . . . . . . . . 127
7.1
The dimensions of the unit cells of the metasurfaces used in our experiments
with different separation distances. . . . . . . . . . . . . . . . . . . . . . . . 139
7.2
Experimental results for the two cases of one breakdown and two breakdowns
when illuminated with 100 pulses. . . . . . . . . . . . . . . . . . . . . . . . 140
xi
Appendix
Table
7.3
Page
Experimental results in nitrogen gas and at atmospheric pressure level for the
two cases of one breakdown and two breakdowns when illuminated with 100
pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
DISCARD THIS PAGE
xii
LIST OF FIGURES
Figure
2.1
(a) 3D topology of a multi-layer structure composed of N high-permittivity
dielectric slabs separated from each other with N − 1 ϵ-negative layers. (b)
Side view of the structure. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page
17
2.2
(a) The transmission-line model of the structure shown in Fig. 2.1 for a
vertically-incident TEM wave. (b) The transmission-line model of Fig. 2.2(a)
is converted to this equivalent circuit model by substituting appropriate lumpedelement circuit models for the DPS and BENG transmission lines. (c) The
simplified version of the circuit shown in Fig. 2.2(b). (d) The T inductive
networks of the circuit shown in Fig. 2.2(c) are converted to π-equivalent
networks. (e) Simplified version of the network shown in Fig. 2.2(d). This
circuit is a coupled-resonator filter of order N . . . . . . . . . . . . . . . . . 19
2.3
(a) Equivalent circuit model for a short section of an BENG slab. (b) A thick
BENG slab can be modeled with an inductive ladder network, which is obtained by cascading the basic inductive T network shown in part (a). (c) It can
be shown theoretically that the ladder inductive network of part (b) is equivalent to a T inductive network whose inductance values are given by (2.1) and
(2.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
xiii
Figure
2.4
2.5
2.6
2.7
2.8
2.9
Page
(a) A transmission line circuit for modeling the response of a tri-layer structure composed of an BENG layer sandwiched between two DPS layers. This
model can be used for a vertically-incident TEM wave. (b) The transmission
line model of part (a) is converted to this equivalent circuit model by replacing the transmission lines with their equivalent circuit models. (c) Simplified
version of the network shown in part (b). (d) The three inductors forming a T
network in the circuit shown in part (c) are converted to an equivalent π network to obtain this circuit. This circuit is a second-order coupled-resonator
filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
The transmission and reflection coefficients of the three different tri-layer
structures examined in Section 3.4.1 and those of an ideal coupled-resonator
filter with a second-order bandpass response. The results of the tri-layer structures are obtained using full-wave EM simulations in CST Studio. . . . . . .
30
The transmission and reflection coefficients of the 9-layer structure examined
in Section 3.4.2 and those of an ideal coupled-resonator filter with a fifth-order
bandpass response. For the 9-layer structure, the results obtained using both
full-wave EM simulations in CST Studio as well as the analytically calculated
results using the wave transfer matrix method are shown. . . . . . . . . . . .
33
Percentage of total power loss as a function of fractional bandwidth for a
number of multi-layer structures of the type shown in Fig. 2.1 and discussed
in Section 3.4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
(a) 3D topology of a multi-layer structure composed of N high-permeability
slabs separated from each other with N −1 µ-negative layers. This structure is
the dual of the structure shown in Fig. 2.1. (b) Side view of the structure. (c)
A coupled-resonator filter model composed of series LC resonators coupled
to each other using parallel capacitors. This filter can be used to model the
frequency response of the structure shown in Fig. 2.8(a). . . . . . . . . . . .
36
Transmission and reflection coefficients of the tri-layer structure discussed in
Section 3.5 and its dual structure discussed in Section 3.3.1. As can be seen,
the two results match perfectly as expected. . . . . . . . . . . . . . . . . . .
37
xiv
Appendix
Figure
3.1
3.2
3.3
Page
(a) 3D topology of the generalized electromagnetic wave tunneling problem
considered in this paper. This multi-layer structure consists of N − 1 MING
layers sandwiched by N BENG layers. Each layer extends to infinity along x
and y directions but has finite thickness along the z direction. (b) Side view
of the multi-layer structure. . . . . . . . . . . . . . . . . . . . . . . . . . .
42
(a) Transmission line model of the multi-layer structure shown in Fig. 3.1(a)
valid for a normally incident TEM wave. (b) The transmission line model
of part (a) can be transformed to this equivalent circuit model by substituting
lumped-element equivalent circuit models for the BENG and MING layers.
(c) The equivalent LC circuit model of part (b) is simplified to this circuit by
ignoring the series inductors Lis used in the equivalent circuit model of BENG
layers. This is justified when BENG layers are thin. (d) The capacitive T
networks highlighted in part (c) can be converted to capacitive π networks
highlighted in this figure after a number of T to π transformations. (e) After
combining the adjacent parallel inductors of the equivalent circuit network
of part (d), the equivalent circuit model of an N th -order coupled-resonator
bandpass filter is obtained. . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
(a) The equivalent circuit model for a short length of an ϵ-negative material
with a length of ∆ℓ → 0. (b) The equivalent circuit model for a short length
of a µ-negative material with a length of ∆ℓ → 0. (c) The equivalent circuit
model of a thick MING layer can be obtained by cascading the equivalent
circuit model shown in part (b) in the form of the ladder capacitive network
shown in this figure. (d) Through successive π to T transformations, the
capacitive ladder network shown in part (c) can be converted to a single T
network. This T network model is valid for an MING layer with any thickness. 46
xv
Appendix
Figure
3.4
3.5
3.6
3.7
3.8
Page
(a) The transmission line model of a tri-layer structure composed of a single
MING layer sandwiched by two BENG layers valid for a vertically-incident
TEM wave. (b) The transmission line model of part (a) can be transformed to
this circuit by substituting the appropriate lumped-element equivalent circuit
models for the MING and BENG layers as shown in Fig. 3.3. (c) The network
of part (c) is simplified by ignoring the series inductors L1s and L2s . This can
be done if the thickness of the BENG layers are small. (d) The capacitive
T network of part (c) is changed to an equivalent π network to achieve this
equivalent circuit model. This circuit is a classical second-order bandpass
coupled-resonator filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
The reflection (left) and transmission (right) coefficients of the tri-layer structure studied in Section 3.4.1 for different ϵM N G values obtained using fullwave EM simulations in CST Microwave Studio. The desired (ideal) filter
response is also shown as reference. The physical parameters of the structures
are provided in Table 3.2 (Case 1 - Case 4). These results show the effect of
the dielectric constant of the MNG layer of a tri-layer structure (ENG-MNGENG) on the accuracy of the synthesis procedure described in Section 3.3.1. .
55
The transmission and reflection coefficients of the tri-layer structure composed of one MNG layer sandwiched by two ENG layers discussed in Section
3.4.1 obtained using full-wave EM simulations in CST Studio. The desired
(ideal) filter response is also shown for comparison. The physical parameters
of the structures are given in Table 3.2 (Case 4-Case 6). . . . . . . . . . . . .
57
The transmission and reflection coefficients of the multi layer structure composed of four MNG layers sandwiched by five ENG layers discussed in Section 3.4.2 obtained from full-wave EM simulations in CST Studio. The desired (ideal) filter response is also shown for comparison. The physical parameters of this structures are given in Table 3.4. . . . . . . . . . . . . . . .
58
The percentage of the total power lost in the multi-layer structure shown in
Fig. 3.1 as a function of the number of ENG layers, N , and the fractional
bandwidth, δ. The results are obtained for multi-layer structures exhibiting a
maximally flat bandpass response centered at 2.4 GHz. It is assumed that the
ENG and MNG materials are lossy with νEN G /ω = νM N G /ω = 0.005. . . .
60
xvi
Appendix
Figure
3.9
Page
(a) 3D topology of the multi-layer structure that is dual of the one shown in
Fig. 3.1. In this case, the structure is composed of N MNG layers separated
from each other by N − 1 ENG layers. (b) Side view of the dual structure.
(c) Following a procedure similar to that described in Section 7.3, it can be
shown that the multi-layer structure of parts (a)-(b) can be modeled with a
coupled-resonator filter composed of series LC resonators coupled to each
other using parallel inductors. . . . . . . . . . . . . . . . . . . . . . . . . .
61
3.10 The transmission and reflection coefficients of the tri-layer structure discussed
in Section 3.4.1 and its dual structure examined in Section 3.5. These results
are obtained from full-wave EM simulation in CST Studio and show perfect
agreement. The physical parameters of the structures are presented in Table
3.2 (Case 4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
(a) 3D topology of a tri-layer structure composed of an ϵ-negative layer sandwiched by two high-permittivity dielectric layers. Each layer extends to infinity in x and y directions. (b) Side view of the structure. . . . . . . . . . .
68
The reflection and transmission coefficients of the tri-layer structure shown
in Fig. 4.1 with physical and geometrical parameters provided in Table 4.1.
The transparency window is centered at 9.382 GHz and it has a factional
bandwidth of 50%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
The electric and magnetic fields within the tri-layer DPS-ENG-DPS device
studied in Section 7.3. (a) Normalized magnitude. (b) Phase. The results
are obtained assuming that a TEM wave, propagating along the +z axis, is
normally incident on the structure. The magnitudes of the total |E| and |H|
are normalized respectively to the magnitudes of the incident electric and
magnetic fields. The reference phase plane (zero phase) is at z = -2 mm. . .
70
The power transmission coefficients ,|T |2 , of the tri-layer structure studied in
Section 7.3 as a function of frequency and (a) Plasma frequency of the ENG
layer, fEN G , (b) the thickness of the ENG layer, hEN G , and (c) the thickness
of DPS layers, hDP S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
4.1
4.2
4.3
4.4
xvii
Appendix
Figure
4.5
4.6
4.7
4.8
4.9
Page
The tri-layer structure shown in Fig. 4.1 is implemented using a section of
cutoff waveguide sandwiched by two high-ϵr dielectric slabs. The waveguide
that emulates the ENG layer operates below its cutoff frequency and has an
opening of ac × bc and a thickness of hc . Both dielectric slabs have the dimensions of aDP S × bDP S and thickness of hDP S . This structure is placed
inside a standard WR-90 rectangular waveguide and its frequency response is
measured using a calibrated vector network analyzer. . . . . . . . . . . . . .
73
The measured and simulated transmission coefficients of the tri-layer structure shown in Fig. 4.5. Measurement results are obtained using a calibrated
vector network analyzer and the simulation results are obtained using fullwave EM simulations in CST Microwave Studio. . . . . . . . . . . . . . . .
75
Experimental test-bed used to measure the transmission and reflection coefficients of the DPS-ENG-DPS filters under high-power excitation levels. The
device under test (DUT) is illuminated at various power levels to detect the
level at which air breakdown occurs. The magnetron source generates a short
pulse with a single frequency of 9.382 GHz, a duration of 1 µs and a peak
power level of 25 kW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
Normalized reflection and transmission coefficients of the DPS-ENG-DPS
structure examined in Section 7.3.2. The measurements are conducted in
time domain using the experimental setup shown in Fig. 4.7. (a) Incident
power level of 4.2 kW. (b) Incident power level of 4.4 kW. As can be seen, air
breakdown occurs at about 200 ns after the start of the high-power pulse. . .
77
The magnitude of electric field distribution over the cross section of the waveguide version of the tri-layer structure shown in Fig. 4.5 at (a) input port of the
standard WR-90 waveguide (b) the interface between the dielectric substrate
and the waveguide operating below the cutoff frequency. In both cases, the
incident power level is 1 Watt. . . . . . . . . . . . . . . . . . . . . . . . . .
79
xviii
Appendix
Figure
Page
4.10 A modified waveguide version of the tri-layer structure shown in Fig. 4.5
is composed of a section of cutoff waveguide with rectangular cross section
sandwiched by two high-ϵr dielectric substrates. This rectangular waveguide
operates below its cutoff frequency and has an opening of am × bm and a
thickness of hm . The height of this cutoff waveguide (bm ) is increased compared to the height of the cutoff waveguide of Fig. 4.5(b) to increase the
power handling capability of the ENG layer. Both dielectric slabs have the
dimensions of aDP S × bDP S and a thickness of hDP S . . . . . . . . . . . . .
80
4.11 Magnitude of the electric field distribution in the cross section of the DPSENG-DPS structure shown in Fig. 4.10. (a) E-field distribution in the cross
section of the input waveguide (outside of the device) for an incident power
level of 1 W. (b) E-field distribution in the cross section at the boundary between the DPS layer and the cutoff waveguide emulating the ENG layer (the
boundary closer to the input) for an incident power level of 1 W. . . . . . . .
81
4.12 The simulated and measured transmission coefficients of the DPS-ENG-DPS
filter shown in Fig. 4.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4.13 Time-domain transmission and reflection coefficients of the DPS-ENG-DPS
filter shown in Fig. 4.10. The measurements are conducted using the setup
shown in Fig. 4.7 at a power level of 25 kW. Observe that the tri-layer structure maintains its transparency when excited with a high power signal and
does not break down. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
4.14 An alternative method for implementation of the tri-layer structure shown in
Fig. 4.1. In this case, the structure is composed of a metallic sheet perforated
with small sub-wavelength holes sandwiched by two dielectric substrates.
The sub-wavelength holes have an opening of awg × bwg and are separated
from each other by a distance of wwg . The perforated sheet has a thickness of
hwg and it emulates the ENG layer. Both dielectric slabs have the dimensions
of aDP S × bDP S and a thickness of hDP S . . . . . . . . . . . . . . . . . . .
84
xix
Appendix
Figure
Page
4.15 Magnitude of the electric field distribution in the cross section of the DPSENG-DPS structure shown in Fig. 4.14. (a) E-field distribution in the cross
section of the input waveguide (outside of the device). (b) E-field distribution
in the cross section at the boundary between the DPS layer and the perforated
metallic sheet that emulates the ENG layer (the boundary closer to the input).
Both figures are obtained for an incident power level of 1 W. . . . . . . . . .
85
4.16 The measured and simulated transmission coefficients of the DPS-ENG-DPS
structure shown in Fig. 4.14. . . . . . . . . . . . . . . . . . . . . . . . . .
85
4.17 Time-domain transmission and reflection coefficients of the DPS-ENG-DPS
tri-layer structure shown in Fig. 4.14. The measurements are conducted using
the setup shown in Fig. 4.7 at a power level of 25 kW. Observe that the trilayer structure maintains its transparency when excited with a high power
signal and does not break down. . . . . . . . . . . . . . . . . . . . . . . . .
86
5.1
5.2
(a) 3D topology of a single-layer metasurface illuminated by a transverse electromagnetic (TEM) wave. (b) One possible unit cell of the metasurface composed of a series LC resonator. (c) A different unit cell of the metasurface
composed of two meandered inductors and a gap capacitor. For the same resonant frequency, this LC resonator has a higher quality factor compared to
the one shown in part (b). (d) Another possible unit cell of the metasurface
composed of two series LC resonators in parallel with each other. The two
resonators have the same resonant frequency but they have different loaded
quality factors. The resonator on the left has a higher Q than the one on the
right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
(a) 3D topology of the waveguide version of the unit cell of a metasurafce
composed of two inductive strips separated from each other by a capacitive
gap of 0.2 mm. The physical dimensions are shown in Table 5.1. (b) The
magnitude of electric field intensity in the cross section of the device before
breakdown. (c) The magnitude of the electric field intensity in the cross section of the device obtained when the capacitive gap is loaded using a resistiveinductive impedance with a value of 32+j0.6 Ω. Assuming that the discharge
is localized to the capacitive gap, this field distribution demonstrates the field
distribution within the device after breakdown occurs. . . . . . . . . . . . .
93
xx
Appendix
Figure
5.3
5.4
5.5
Page
(a) 3D topology of the waveguide version of the unit cell of a metasurafce
composed of two meander inductive strips separated from each other by a
capacitive gap of 0.2 mm. The physical dimensions are shown in Table 5.2.
(b) The magnitude of electric field intensity in the cross section of this device
before breaking down. (c) The magnitude of the electric field intensity in the
cross section of the device obtained when the capacitive gap loaded with a
resistive-inductive impedance of 32 + j0.6 Ω. Assuming that the discharge is
localized to the capacitive gap, this figure demonstrates the field distribution
within the device after breakdown occurs. . . . . . . . . . . . . . . . . . .
94
(a) 3D topology of the waveguide version of the unit cell of a metasurface
composed of two parallel LC resonators. Both LC resonators have similar
topology shown in Fig. 5.2(a) and maintain the same capacitive gap of 0.2
mm. The physical dimensions are shown in Table 5.3. (b) The magnitude
of electric field intensity in the cross section of this device before breaking
down. (c) The magnitude of the electric field intensity in the cross section
of the device obtained when the capacitive gap of the high-Q LC resonator
(on the left) is loaded with a resistive-inductive impedance of 32 + j0.6 Ω.
(d) The magnitude of the electric field intensity in the cross section of the
device obtained when the capacitive gap of the high-Q LC resonator (on the
right) is loaded with a resistive-inductive impedance of 32 + j0.6 Ω. (e) The
magnitude of the electric field intensity in the cross section of the device obtained when the capacitive gaps of both LC resonators are loaded with the
resistive-inductive impedances of 32 + j0.6 Ω. . . . . . . . . . . . . . . . .
96
Photographs of the fabricated unit cells of the metasurfaces shown in (a) Fig.
5.2, (b) Fig 5.3, and (c) Fig. 5.4. . . . . . . . . . . . . . . . . . . . . . . . . 103
xxi
Appendix
Figure
Page
5.6
(a) The measured and simulated transmission coefficients of the metasurfaces shown in Fig. 5.2 and 5.3 when no breakdown occurs. (b) Simulated
transmission coefficient of the metasurfaces shown in Fig. 5.2 and 5.3 when
breakdown occurs. These results are obtained based on the assumption that
breakdown event in these devices creates a localized discharge confined to
the region of the capacitive gap. (c) Measured and simulated transmission
coefficients of the metasurface shown in Fig. 5.4. (d) Simulated transmission
coefficient of the metasurface shown in Fig. 5.4 for two different possible
breakdown scenarios: the high-Q resonator (on the left) breaks down only
and both resonators break down together. . . . . . . . . . . . . . . . . . . . 105
5.7
Experimental test-bed used in the high-power measurements. The magnetron
source generates a short pulse with a single frequency of 9.382 GHz, a duration of 1 µs, and a peak power level of 25 kW. . . . . . . . . . . . . . . . . 106
5.8
Normalized, time-domain power transmission and reflection coefficients in
measured in air and at atmospheric pressure level for the device shown in Fig.
5.2 illuminated with a peak power of (a) 18.3 kW and (b) 20.2 kW, the device
shown in Fig. 5.3 illuminated with a power of (c) 7.6 kW and (d) 10.1 kW,
and the device shown in Fig. 5.4 illuminated with a peak power of (e) 20.2
kW and (f) 21 kW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.9
Normalized, time-domain power transmission and reflection coefficients in
measured in Argon and at pressure of 600 torr for the device shown in Fig.
5.2 illuminated with a peak power of (a) 15.9 kW and (b) 16.5 kW, the device
shown in Fig. 5.3 illuminated with a power of (c) 6.1 kW and (d) 7.6 kW, and
the device shown in Fig. 5.4 illuminated with a peak power of (e) 16.5 kW
and (f) 18.3 kW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.1
Different metamaterial topologies and applications. (a) 3D topology of arrays
of split ring resonators cascaded sequentially to achieve negative permeability. (b) Topology of an active negative index metamaterial powered by an
electron beam [1]. (c) A metamaterial based Cerenkov maser composed of
metallic rings used to obtain a desired engineered dielectric constant [2]. (d)
3-D topology of metamaterial-based structure used to facilitate plasma generation within a distributed discharge limiter [3]. . . . . . . . . . . . . . . . . 114
xxii
Appendix
Figure
Page
6.2
(a) Topology of the unit cell of a metasurface, which was examined in [4].
The unit cell is composed of two resonators with different quality factors
but the same resonant frequencies. The resonator on the left has a higher Q
and lower breakdown threshold power level than the one on the right. (b)
Simulated time-domain transmission coefficients of this device under three
operational conditions (no breakdown, high-Q resonator breaks down, and
both resonators breakdown simultaneously). . . . . . . . . . . . . . . . . . 115
6.3
(a) Topology of the unit cell of the modified metasurface examined in this
work. The unit cell is composed of two resonators with considerably different resonant frequencies. The resonator on the left has a resonant frequency
of 9.112 GHz and the one on the right has a resonant frequency of 7.516
GHz. Under high-power excitation at 9.382 GHz, the resonator on the left
is expected to breakdown at a lower threshold power level due to its higher
local field enhancement factor. (b) Simulated time-domain transmission coefficients of this device under three operational conditions (no breakdown,
high-Q resonator breaks down, and both resonators breakdown simultaneously). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4
(a) The simulated and measured transmission coefficients of the device shown
in Fig. 6.3 under low-power excitation. (b) The simulated transmission coefficients of the device as a function of frequency for situations where either
one (the high-frequency) or both resonators break down. . . . . . . . . . . . 117
6.5
(a) Normalized time-domain power reflection and transmission coefficients
of the metasurface shown in Fig. 6.3 when the metasurface is illuminated
with a peak power level of 4.2 kW. No breakdown is observed at this power
level. (b) As the power level is increased to 4.4 kW, breakdown occurs in both
resonators making the metasurface transparent. . . . . . . . . . . . . . . . . 118
xxiii
Appendix
Figure
Page
6.6
(a) Statistical distribution of the rise time of the breakdown at the power level
of 4.4 kW for a total of 100 pulses. The calculated mean, µ, and standard
deviation, σ, of the distribution are 31 nsec and 4 nsec, respectively. (b)
Statistical distribution of the delay time of the breakdown at the power level
of 4.4 kW for a total of 100 pulses. The calculated mean, µ, and standard
deviation, σ, of the distribution are 511 nsec and 140 nsec, respectively. The
dashed lines represent the corresponding normal distribution fitting curves for
these statistical distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.7
(a) The setup used to physically isolate the two resonators of the unit cell of
the metasurface under investigation in this paper during the high-power microwave conditions. A physical barrier with the dimension of 20 mm × 10
mm × 1 mm is placed between two resonators to minimize the chances of
having any physical interaction between the two resonators during the discharge process. (b) Photograph of the setup (with one waveguide removed)
showing the physical barrier composed of 20 layers of UV opaque Kynarr
films placed between the two resonators. (c) The transmission spectrum of a
single UV opaque Kynarr film with a thickness of 50 µm [5] (Reproduced
with permission from Arkema Inc.). Although it is not shown in this spectrum, this material also blocks VUV photons with wavelengths below 200nm
[6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.8
(a) Normalized time-domain power reflection and transmission coefficients
of the metasurface measured using the setup shown in Fig. 6.7(a) when it is
illuminated with a power level of 4.2 kW. At this power level, no breakdown
is observed. (b) As the power level is increased to 4.4 kW, breakdown occurs. The values of the power transmission and reflection coefficients indicate
that breakdown occurs at only single (left) resonator in this case. (c) As the
power level is increased to 7.6 kW, the transmission and reflection coefficients
change and the metasurface becomes more transparent. This scenario and the
higher measured transparency are consistent with both resonators breaking
down simultaneously. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
xxiv
Appendix
Figure
6.9
Page
(a) A UV transparent window with the dimensions of 20 mm × 10 mm × 1
mm is placed between two resonators to allow the propagation of any potential
UV photons that may be generated from the discharge in one resonator to the
the location of the second resonator. (b) Photograph of the setup showing the
Fused Silica window placed between the two resonators. (c) The estimated
transmission spectrum of Fused Silica with a thickness of 1 mm extracted
from the transmission spectrum of the sample with a thickness of 10 mm [7].
Although it is not shown in this spectrum, Fused Silica is opaque below 156
nm [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.10 A physical barrier embedded with a circular VUV transparent MgF2 window
is placed between two resonators to physically forbid electron diffusion and
allow the penetration of VUV emission. (b) Photograph of a dielectric substrate, Rogers 6010.LM [9], embedded with a circular VUV transparent MgF2
window with the diameter of 10 mm and a thickness of 0.5 mm. (c) The estimated transmission spectrum of a MgF2 window with a thickness of 0.5 mm
extracted from the transmission spectrum of the sample with a thickness of 2
mm [10]. This spectrum demonstrates good transmission for VUV emission
in the range from 115 nm to 130 nm. . . . . . . . . . . . . . . . . . . . . . . 128
6.11 (a) Normalized time-domain power reflection and transmission coefficients of
the device embedded with a VUV transparent MgF2 window when the device
is illuminated with a peak power of 4.2 kW. At this power level, no breakdown
is observed. (b) As the power level is increased to 4.4 kW, breakdown occurs
in both resonators simultaneously as indicated by the measured transmission
and reflection coefficient values. . . . . . . . . . . . . . . . . . . . . . . . . 129
6.12 (a) Statistical distribution of the rise time of the breakdown for the case with
VUV physical barrier at the power level of 4.4 kW for a total of 100 pulses.
The calculated mean, µ, and standard deviation, σ, of the distribution are 32
nsec and 5 nsec, respectively. (b) Statistical distribution of the delay time
of the breakdown for the case with VUV physical barrier at the power level
of 4.4 kW for a total of 100 pulses. The calculated mean, µ, and standard
deviation, σ, of the distribution are 516 nsec and 151 nsec, respectively. The
dashed lines represent the corresponding normal distribution fitting curves for
these statistical distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . 130
xxv
Appendix
Figure
Page
7.1
(a) 3D topology of a single-layer frequency selective surface with a bandstop
response in close proximity to a high-power source. (b) Side view of the
structure under illumination. . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2
(a) 3D topology of the two cascaded unit cells described in Figs. 7.3(a)-(b)
with variable distances. (b) Side view of (a) showing that initial VUV photons
generated at one unit cell propagate and arrive the other unit cell to induce
breakdown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.3
(a)-(b) Unit cells of the two resonators used in a bandstop metasurface composed of two lumped resonators cascaded with variable distances. (c) The
time-domain power transmission coefficients for three different cases (i.e.
no breakdown, one breakdown, and two breakdowns) for the case where the
spacing between the resonators is 0.2 mm. . . . . . . . . . . . . . . . . . . . 137
7.4
Simulated transmission coefficients of the devices as functions of frequency
for different separation distances betweens the two resonators: (a) 6.5 mm,
(b) 13.6 mm, (c) 16.2mm, (d) 16.74mm, (e) 16.92mm, (e) 17.42mm, (f)
18.58mm, and (g) 97 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.5
Measured time-domain transmission coefficients of the devices for different
separation distances between the two resonators: (a) 6.5 mm, (b) 13.6 mm,
(c) 16.2mm, (d) 16.74mm, (e) 16.92mm, (e) 17.42mm, (f) 18.58mm, and (g)
97 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.6
Percentage of the time where two separate breakdown states were observed
for the experiment shown in Fig. 7.2 when the experiments were conducted
in air and at atmospheric pressure levels. The inset shows data points in the
range from 0 mm to 20 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.7
Percentage of the time where two separate breakdown states were observed
for the experiment shown in Fig. 7.2 when the experiments were conducted
in pure nitrogen gas and at atmospheric pressure levels. The inset shows data
points in the range from 0 mm to 20 mm. . . . . . . . . . . . . . . . . . . . 144
xxvi
Appendix
Figure
8.1
Page
3D topology of a high-power metasurface illuminated with an oblique incident wave in W-band for the cases (a) before no microwave-induced breakdown occurs and, (b) after microwave-induced breakdown occurs on the surface.148
xxvii
ABSTRACT
In recent years, metamaterials have received a significant amount of attention for providing engineered artificial properties which do not exist in nature such as high surface
impedance, negative permittivity/permeability, and negative refractive index. However,
under high-power illuminations, metamaterials tend to breakdown and alter their frequency responses. This dissertation includes two parts. First, I investigate the phenomenon
of electromagnetic (EM) waves tunneling through ϵ- and µ-negative metamaterial slabs
and its potential applications in designing high-power filters and frequency selective surfaces without breakdown. The second part is to investigate breakdown events in highpower microwave metamaterials.
In this thesis, I examine EM waves tunneling through multi-layer structures composed
of ϵ-negative (the relative permittivity is negative) materials sandwiched by double positive layers. Conventionally, EM waves can only propagate through ϵ-negative material
under certain circumstance referred to as resonant tunneling. I demonstrate that this EM
waves tunneling phenomenon is analogous to a well-known classic microwave filter theory. Based on this analogy, I proposed a synthesis procedure for designing this kind of
structure from desired responses which are beneficial for developing high-power-capable
spatial filters and microwave FSSs. To verify the proposed procedure, three prototypes of
xxviii
such a device are designed, fabricated and experimentally characterized and it is demonstrated that they can handle extremely high peak power levels.
In the second half of my thesis, I study the impact of breakdown on the responses of
metamaterials by examining several single-layer metasurfaces composed of miniaturized
LC resonators. I demonstrate that the breakdown events, in atmospheric air, can be characterized with a reasonable degree of accuracy by modeling the streaming discharge as
a low-impedance connection path. My recent study shows that breakdown at one location induces breakdown at neighboring locations where the power level is lower than the
breakdown threshold within a multi-resonator unit cell of high-power metamaterials with
discrete nonlinear responses. I examine three candidate mechanisms, energetic electron
diffusion, ultraviolet (UV) radiation, and vacuum ultraviolet (VUV) radiation, to study the
cause of this simultaneous breakdown and demonstrated that this is due to VUV photoemission.
1
Chapter 1
Introduction
1.1
Motivation
Periodic structures are widely used at RF, microwave, and millimeter-wave (MMW)
ranges for various applications [11] –[18]. Recently, the emergence of metamaterials attracts lots of attention for providing engineered artificial properties such as high surface
impedances [19], negative permittivity/permeability [20]-[21], and negative refractive index [22] in RF [23], microwave [24], terahertz [25], and optical [26] frequencies. Most
metamaterials can not handle high-power microwave (HPM) energy because their unit
cells containing sub-wavelength elements tend to cause breakdown and arcing within the
structures which can damage the structures and alter their performances. In this thesis, our
goal is to investigate how to avoid breakdown in metamaterials and how the metamaterials
react when breakdown occurs.
1.2
Proposed Approach
For the first addressed task, we want to design a metamaterial-based structure which
can operate at high-power levels without breaking down. One potential option is to utilize
the tunneling phenomenon of electromagnetic (EM) waves through multi-layer structures
containing single-negative (SNG) layers or double-negative layers (DNG). SNG refers to
2
the medium with single negative constitutive parameters such as epsilon-negative (BENG)
medium that its relative permittivity is negative but its relative permeability is positive and
mu-negative (MING) medium that its relative permittivity is positive but its relative permeability is negative. Previous research demonstrates a complete tunneling of EM waves
in a cascaded BENG slabs paired with MING slabs [27] –[29], a two-layer structure composed of an BENG layer separated from a MING slab with free space [30], a paired SNG
layers and double-positive (DPS) layers (both constitutive parameters are positive) [31]
–[33], and a cascaded DNG-DPS multi-layer structure [34]. This complete transmission
reveals that the EM waves penetrate through this structure and the electric field intensities within the structure have not been amplified. If the electric field intensities within the
structure are larger than the breakdown thresholds, breakdown happens within the structure [35]. Therefore, the multi-layer structures containing negative constitutive parameters
can be used in designing spatial filters and frequency selective surfaces (FSSs) operating at
high-power levels. Various research groups investigate the tunneling phenomenon of EM
waves through these structure referred as resonant tunneling and analyze the responses of
the structures from their physical dimensions.
In this thesis, we want to explain the tunneling phenomenon of EM waves through
the multi-layer structures containing negative constitutive parameters with a classic microwave filter theory. We demonstrate that a multi-layer structure composed of cascaded
ϵ-negative metamaterial layers sandwiched by double-positive layers is analogous to an
inductively-coupled, coupled-resonator filter and a multi-layer structure composed of cascaded ϵ- and µ-negative metamaterial slabs is analogous to a capacitively-coupled, coupledresonator filter. With these analogies, we propose a generalized synthesis procedure to
synthesize these kind of structures that their physical dimensions can be calculated from
a priori known desired responses. The proposed synthesis procedure is beneficial for designing high-power spatial filers and FSSs. The proposed synthesis method is verified
3
with the full-wave EM simulation in CST Studio. In addition, we investigate the tunneling
phenomenon of EM waves through a dual multi-layer structure composed of µ-negative
metamaterial layers sandwiched by double-positive layers and a dual multi-layer structure
composed of cascaded µ- and ϵ-negative metamaterial slabs.
To respond the second addressed task, we want to experimentally examine the metamaterials with high-power experiments and observe the responses of the metamaterials
due to the occurrence of breakdown. Recently, various research groups investigate the
applications of metamaterials in designing high-power microwave (HPM) sources [36]
–[41], improving interaction between an electron beam and the surrounding propagating
EM waves [1], and developing particle accelerators, [42] –[44]. However, to the best of our
knowledge, few researchers have investigated the breakdown events within metamaterials
and their responses when breakdown occurs.
In this thesis, we investigate the breakdown events with three different metasurfaces
composed of single-resonator unit cell and multi-resonator unit cell with bandstop responses as our testing devices. These structures are designed to be opaque at low power
levels and become transparent when breakdown occurs at high power levels. This is an
useful diagnosis tool for us to identify wether breakdown occurs or not. Our hypothesis
is that when breakdown occurs within the embedded miniature resonators of each unit
cell, electrons and charged particles gather together near the gap regions and bridge the
gaps. These localized discharges short-circuit this structure with relative low impedance.
To verify our hypothesis, three different metasurfaces are designed and placed in a rectangular waveguide environment for high power experiments. The physical dimensions of
these devices are designed and optimized in full-wave EM simulations in CST Studio. In
these simulations, we estimate the responses of these structures after breakdown occurs
by modeling the localized discharges with low-impedance paths. The waveguide versions
of the unit cell of the metasurfaces are fabricated out with a 0.1 mm think stainless steel
4
sheet via chemical etching and examined with a high-power magnetron which generates a
single-frequency pulse at 9.382 GHz with a duration of 1 µs and a power level of 25 kW.
By comparing the simulated and measured responses of these metasurfaces, we can verify our hypothesis and accurately predict the responses of metasurfaces after breakdown.
These are useful for switching or continuously tuning the responses of metamaterials and
sub-wavelength periodic structures operating at HPM levels.
1.3
Literature Review
As mentioned in the previous section, the goal of this thesis is to design high-power
frequency selective surfaces by exploiting EM waves tunneling through ϵ- and µ-negative
metamaterials and the effect of microwave induced breakdown on the responses on metamaterials and sub-wavelength periodic structures. In the past decades, lots of research has
been conducted in the area of periodic structure, metamaterials, and RF/microave induce
breakdown. In this section, we present a brief literature review of the research work which
has been done in the past.
1.3.1
Periodic Structures and Metamaterials
Periodic structures are composed of infinite building blocks arranged sequentially in
either one-, two-, or three-dimensions. Periodic structures are commonly found in nature
such as crystals or crystalline solids containing atoms, molecules or ions in a periodic arrangement. Due to the periodicity, periodic structures provide the well-known band-gap
structures that black the EM waves with certain wavelengths and pass the others only when
the dimensions of each building block are comparable to the wavelength of the EM waves.
If the dimensions of each building block are much greater or less than the wavelengths of
the propagating EM waves, the periodic structures act as effective homogenous media (e.g.
5
metamaterial) and do not demonstrate the filtering responses. The filtering characteristics
of periodic structures are derived from the fact that if the dimension of the unit cell of the
periodic structure is half of the wavelength or an integer number of half the wavelength,
the incident waves interfere with the reflective waves forming standing waves within the
periodic structures which block the propagation of the EM waves and store their energy
within the periodic structures. For EM waves with other wavelengths, they can propagate
through periodic structures and these propagating waves are named Bloch waves. To analyze the propagations of waves in periodic structures, Floquet’s Theorem is commonly
used to obtain the dispersion relation of the periodic structures (ω-β diagram). Floquet’s
Theorem states that for a propagating wave in a periodic structure at a steady state frequency, the difference of fields between one location and the other location one period
away is only a phase shift. Periodic structures are widely used in the areas of semiconductor [45], wireless communication systems [46], and military applications. In the past
decades, the implementation of periodic structures was limited by the fabrication technique and tolerance. With the modern microfabrication techniques, various researchers
experimentally fabricate the periodic structures in nano-meter scales [47].
Metamaterials are created by embedding engineered artificial inclusions with various
shapes in the host media. Similar to periodic structures, these embedded inclusions are
arranged periodically within the host media. However, metamaterials demonstrate special
characteristics such as high-impedance surfaces, negative permittivity/permeability, and
negative index compared to periodic structures. These characteristics are derived from the
mechanism that when the EM waves interact with the inclusions, the multi-reflection and
scattering will affect the macro electromagnetic properties of the overall media such as
the relative permittivity and permeability. By definition, the physical dimensions of these
embedded inclusions must be much smaller than the wavelength of the propagating EM
waves. Therefore, these host media embedded with the sub-wavelength inclusions act as
6
effective homogeneous media and their electromagnetic responses can be represented by
the effective relative permittivity and permeability [48]. These properties also depends on
the the shapes, sizes, arrangement, alignment, density, and composition of the inclusions.
This provides lot of flexibilities to engineer the electromagnetic responses of metamaterials which are different from the properties of the host media or do not exist in nature.
Recently, the metamaterial with negative relative permittivity and negative relative permeability attract lots of attentions in RF, microwave, terahertz, and optical frequencies. A
substantial amount of research has been conducted in the applications of metamaterials
in designing non-high-power superlens [49]–[50], frequency selective surface, and highimpedance surfaces [19].
1.3.2 ϵ- and µ-Negative Metamaterials
In general, we use relative permittivity, ϵr and relative permeability, µr to describe the
macro electromagnetic responses of media under the influences of external electric and
magnetic fields. Based on the vales of the ϵr and µr , media are classified in four different
categories: double-positive (DPS), epsilon-negative (ENG), mu-negative (MNG), doublenegative (DNG) materials [51] –[53].
A medium with relative permittivity and relative permeability greater than zero (ϵr and
µr > 0) is defined as DPS material. Most materials existing in nature such as dielectrics
and ferro-magnets fall within this category. When EM waves propagate through DPS materials, the phase velocity (the propagation of the perturbation) and the group velocity (the
propagation of the energy) are in the same direction. This phenomenon is often referred
to as forward-wave propagation and the DPS materials are sometimes named right-handed
materials. In addition, the refractive index of a DPS material is usually used
to describe the scattering and the diffraction phenomenon defined as n =
√
ϵr µr .
7
The second category is the ENG material with relative permittivity less than zero and
relative permeability greater than zero (ϵr < 0 and µr > 0). Due to negative ϵr , the propagation constant and the impedance becomes imaginary and inductive, respectively. Therefore, the EM waves propagating within ENG materials become evanescent fields and the
amplitudes decay along the propagation directions. In other words, EM waves can not
penetrate through ENG materials. ENG materials exist in nature in terms of cold plasmas
and cutoff waveguides. For a lossless cold plasma, the relative permittivity can be described by a Drude model and it becomes negative when the operating frequencies of the
EM waves are below the plasma frequency. When operating above the plasma frequency,
cold plasmas act as ordinary dielectric media through which EM waves propagate without
decaying their amplitudes. For example, metals act as ENG materials and the corresponding plasma frequency is at the optical frequency range. They are opaque at the microwave,
infrared, and visible frequency range (below the plasma frequency) and become transparent above the optical frequency (above the plasma frequency). This is why x-rays can
penetrate through metallic objects. Similarly, if the operating frequencies of the EM waves
are below the frequency of the transverse electric (TE) mode insides waveguides, the EM
waves become evanescent fields as well. The lowest frequency of the propagating TE
modes are defined as the cutoff frequency of the waveguide and this waveguide acts as an
ENG material. As stated in the previous section, engineered, artificial metamaterials can
act as ENG materials. Wire grids and complementary split ring resonators (CSRRs) are
commonly used as the sub-wavelength inclusions in designing these ENG metamaterials.
The third kind is the MNG material with the relative permittivity greater than zero and
the relative permeability less than zero (ϵr > 0 and µr < 0). In nature, MNG materials are
usually composed of ferrimagnetic materials and their relative permeabilities are described
by Lorenz model. Since the Lorenz model has a resonant-like behavior, the relative permeability of MNG material is negative within a narrow frequency range. This disadvantage
8
limits the applications of MNG materials. The other example of MNG materials are cutoff
waveguides operating below transverse magnetic (TM) mode. Since the TM modes inside waveguides are not the dominant propagation modes, the challenge of implementing
MNG materials is to intelligently suppress the dominant modes insides waveguides. MNG
materials can also be implemented with metamaterials composed of sub-wavelength split
ring resonators (SRRs).
The final category is the DNG material with the relative permittivity and the relative
permeability less than zero (ϵ < 0 and µ < 0). DNG materials do not exist in nature but
they can be physically realized by the combination of ENG and MNG materials. With
the negative ϵr and µr , the phase velocities of the DNG materials are oriented towards the
opposite direction of the group velocities of the DNG materials. In other words, the energy of the wave is propagating forwards and the perturbation of the wave is propagating
backwards. This phenomenon is referred to as backward-wave propagation and the DNG
materials are named left-handed materials. Recently, the backward-wave characteristics
of DNG materials are commonly used in designing slow-wave strcutures for TWT amplifiers. The other significant characteristic of DNG material is the nagative refractive index
√
defined as n = − ϵr µr . When refraction of EM waves occurring at the interface between
a DPS materaial and a DNG material, the direction of the refractive wave within a DNG
material is contrast to that within a DPS material. This peculiar characteristic is commonly
used in designing superlenses and invisible cloaks [54].
1.3.3
Microwave-Induced Breakdown
Breakdown is usually considered as a process of electron avalanche when the total
number of electrons exceeds 108 . Under strong EM waves illuminations, the seed electron
starts to fluctuate along the direction of the applied electric fields and collide with neutrons
or surfaces for generating the secondary electrons. Then, these secondary electrons again
9
collide with electrons and the surfaces to generate more electrons and so on until the
value of the total electrons is above the critical number, 108 . In general, breakdown tend
to occur when the applied electric field intensities are higher than the electrical strength
of the dielectric materials (or gas) surrounding the device. However, it is a challenge
to accurately predict the occurrence of breakdown because it depends on the electrical
strength of the dielectric materials (or gas) surrounding the device, humidity, and the ratio
of pressure to temperature. The field enhancement factor (FEF) is usually used to estimate
the power levels at which breakdown is expected to occur [55]. In addition, FEF can be
used to compare the relative power handling capabilities of different devices. The FEF is
defined as the ratio of electric field intensity inside the device to the electric field intensity
of the incident EM wave. A lower FEF indicates a higher breakdown level for the device
and a higher transient power handling capability [56]. For a fair comparison between
different dielectric materials with different breakdown field levels, we assume that the
dielectric media surrounding the different devices are the same throughout this proposal.
1.4
Thesis overview
The goal of this thesis includes two parts. First, we want to design spatial filters and
frequency selective surfaces operating at high-power levels by exploiting the phenomenon
of electromagnetic (EM) waves tunneling through ϵ- and µ-negative metamaterial slabs in
the first three chapters. The second part is to investigate the breakdown events in highpower microwave metamaterials and periodic structures in the last two chapters. This
thesis is divided into the following chapters. In Chapter 2, we study the phenomenon
of EM waves tunneling through cascaded ϵ-negative metamaterial layers sandwiched by
double-positive layers. We explore the analogy between the composite ENG-DPS multilayer structure and a classic microwave filter theory. In Chapter 3, we apply the same
10
approach to analyze the tunneling phenomenon of EM waves through cascaded ϵ- and µnegative metamaterial slabs. In Chapter 4, we utilize the proposed method in Chapter 1
to design high-power frequency selective surfaces and experimentally characterize their
power handling capabilities. In chapter 5, we experimentally investigate the impact of
microwave induced breakdown on the responses of high-power microwave metamaterials.
In chapter 6, we experimentally examine the cause of simultaneous breakdown within a
multi-resonator unit cell of a high-power metamaterial with discrete nonlinear responses.
The detailed overview of each chapter is provided as below.
1.4.1
Chapter 2
We examine the electromagnetic (EM) wave tunneling and filtering characteristics of
multi-layer structures composed of an arbitrary number of ϵ-negative (ENG) metamaterial
layers sandwiched by very thin double-positive (DPS) layers with high dielectric constant
values. We explain the phenomenon of EM wave tunneling through this propagation barrier by drawing an analogy between this problem and a generalized coupled resonator
system. Using this analogy, we demonstrate that a multi-layer structure composed of N
DPS layers (N ≥ 2 is an integer number) that sandwich N − 1 ENG layers can not only be
made transparent in a frequency range where the ENG layers are normally opaque but also
be designed to provide a desired spectral filtering characteristics. Furthermore, we present
an analytical method for synthesizing such multi-layer spectral filters from the characteristics of their desired responses. The proposed synthesis procedure can be used to develop
spatial filters operating throughout the microwave and mm-wave frequency bands as well
as multi-layer metallo-dielectric filters at optical wavelengths. We demonstrate the validity of the proposed analytical synthesis method using full-wave numerical electromagnetic
simulations. Finally, using the duality principle, we expand these analytical results to the
11
problem of EM wave tunneling through multiple µ-negative layers surrounded by thin
DPS layers with high relative permeability values.
1.4.2
Chapter 3
We examine the close relationship that exists between the phenomenon of electromagnetic (EM) wave tunneling through stacks of single-negative metamaterial slabs and
classical microwave filter theory. In particular, we examine the propagation of EM waves
through a generalized multi-layer structure composed of N ϵ-negative layers separated
from each other by N −1 µ-negative layers, where N ≥ 2 is a positive integer. We demonstrate that, if certain conditions are met, this multi-layer structure can act as a capacitivelycoupled, coupled-resonator filter with an N th -order bandpass response. Exploiting this
relationship, we develop a generalized, analytical synthesis method that can be used to
determine the physical parameters of this structure from its a priori known frequency response. We present several design examples in conjunction with numerical EM simulation
results to demonstrate the validity of this analogy and examine the accuracy of the proposed synthesis procedure.
1.4.3
Chapter 4
In this chapter, we experimentally investigate the phenomenon of electromagnetic
(EM) wave tunneling through ϵ-negative (ENG) metamaterial layers surrounded by doublepositive (DPS) layers. Initial experiments are conducted by using a rectangular waveguide,
which operates below its cutoff frequency to emulate an ENG layer. This ENG layer is then
sandwiched by two dielectric substrates with relatively high dielectric constants and it is
shown that the entire setup acts as a classical microwave filter with a second-order bandpass response. The power handling capability of this filter is examined experimentally
using a high-power magnetron source with a frequency of 9.382 GHz, a pulse duration of
12
1 µs, and a peak power of 25 kW. Based on the results of this experiment, two methods
for improving the power handling capability of these multi-layer structures are proposed.
In particular, it is demonstrated that emulating the ENG layers with thin perforated metallic sheets with sub-wavelength holes significantly enhances their peak power handling
capability. A prototype of such a device is designed, fabricated, and experimentally characterized and it is demonstrated that it can handle extremely high peak power levels. The
results presented in this work are expected to be useful in designing microwave filters and
frequency selective surfaces that can handle extremely high peak power levels.
1.4.4
Chapter 5
We investigate the effect of microwave-induced breakdown on the frequency responses
of a class of metamaterials composed of planar sub-wavelength periodic structures. When
breakdown occurs in such a structure, its frequency response changes based on the nature
of the plasma created within its unit cell. We examine how the frequency responses of such
periodic structures change as a result of creation of microwave-induced discharges within
their unit cells. To do this, we examine single-layer metasurfaces composed of miniature LC resonators arranged in a two-dimensional periodic lattice. These metasurfaces
are engineered to be opaque at microwave frequencies when operated at low power levels
but can be made transparent if a localized discharge is created within the LC resonators.
By measuring their transmission and reflection coefficients under high-power excitation in
different conditions, the impact of breakdown on the frequency responses of these devices
is determined. Several prototypes of such structures are examined both theoretically and
experimentally. It is demonstrated that when breakdown occurs in air and at atmospheric
pressure levels, the responses of such periodic structures can be predicted with a reasonable degree of accuracy. Additionally, when the unit cell of the metasurface is composed
13
of two different resonators, breakdown is always observed to occur in both resonators despite their different topologies and local field enhancement factors. In such structures, the
discharge in one resonator appears to be mediated by the one in the other.
1.4.5
Chapter 6
In the previous chapter, we observed simultaneous breakdown discharges at two separate sites within a multi-resonator metamaterial unit cell, even though the electric field
intensities at one of the resonator sites should have been well below the threshold intensity
required for breakdown. Here, we investigate three candidate mechanisms for the simultaneous breakdown discharges: energetic electrons, ultraviolet (UV) radiation, and vacuum
ultraviolet (VUV) radiation. Experiments inserting different dielectric barriers between
the two resonators of a multi-resonator unit cell were able to selectively isolate the coupling influence of the candidate mechanisms. It was established that VUV radiation from
the discharge at the resonator with a lower electric field breakdown threshold causes simultaneous breakdown at the other resonator where the field intensities are otherwise too
low to induce breakdown.
1.4.6
Chapter 7
In this section, we present the results of an experimental study of the effective range of
this physical phenomenon for periodic structures that operate in air and pure nitrogen gas
at atmospheric pressure levels. It is demonstrated that this phenomenon is highly likely
to happen in radii of approximately 16-17 mm from the location of the initial discharge
under these test conditions. The results of this study are significant in designing metamaterials and periodic structures for high-power microwave applications as they suggest that
a localized discharge created in such a periodic structure with a periodicity less than 16-17
mm can spread over a large surface and result in a distributed
14
Chapter 2
Electromagnetic Wave Tunneling Through Cascaded ϵ-Negative
Metamaterial Layers Sandwiched by Double-Positive Layers
2.1
Introduction
Materials with negative relative permittivity (ϵr ) or negative relative permeability (µr )
values do not allow electromagnetic (EM) waves to propagate over long distances. EM
waves are evanescent in these materials and exponentially decay along the direction of
propagation. Under certain circumstances, however, these conventionally opaque materials can be made completely transparent and allow total transmission of EM waves. This
behavior is commonly referred to as tunneling of EM waves due to its analogous nature to
tunneling of electrons through potential barriers. This phenomenon has been investigated
by different researchers over the past three decades for its various practical microwave,
mm-wave, and optical applications. In 1984, Tai et al. examined complete transmission of
EM waves through a plasma-dielectric sandwich structure at microwave frequencies. The
structure examined in this work consists of three conventional dielectric slabs separated
from one another by two plasma layers [57]. Tai et al. demonstrated that such a structure
allows for complete transmission of EM waves within a frequency range where the relative
permittivity of the plasma is negative. Since the transmission frequency can be controlled
by changing the plasma frequency of the plasma layers, this structure was proposed as a
15
tunable microwave filter capable of handling high power levels. Total transmission of light
through other conventionally opaque layers such as metals [58] and sub-wavelength holes
[59] has also been investigated.
With the emergence of the field of metamaterials in recent years, the idea of EM wave
tunneling through metamaterial layers having negative permittivity (ϵ-negative or BENG)
or negative permeability (µ-negative or MING) values has received significant attention
[27] –[34], [60]– [61]. EM wave tunneling through various different multi-layer structures composed of one or more types of single-negative (SNG) materials has been studied.
Examples include tunneling through BENG layers paired with MING ones [27] –[29], tunneling through an BENG layer separated from an MING layer with free space [30], and
tunneling through multi-layer structures composed of SNG and double-positive (DPS) layers [31] –[33]. However, most of the studies conducted in this area to date have primarily
focused on the analysis problem (or the forward problem). In this regard, the inverse problem (or the synthesis problem) has not received significant attention. In a given structure
that allows EM waves to tunnel through, by solving the inverse problem, one can determine the physical parameters of the structure that will result in an a priori known desired
response. This synthesis problem is of significant practical importance when such a structure is to be designed for a given application. For example, the structures reported in Refs.
[31], [33], [57], [61] can all be used as spatial filters at microwave or optical frequency
bands. In such applications, usually the desired transmission characteristics of the filter
are design parameters that are known a priori. The design task then reduces to determining the physical parameters of the structure that result in the desired response (i.e., the
synthesis problem).
In this work, we examine the EM wave tunneling and filtering characteristics of a generalized multi-layer structure composed of multiple ϵ-negative layers sandwiched by DPS
layers. The structure considered here is composed of N , double-positive dielectric layers
16
with high relative permittivity values that sandwich N − 1 BENG layers. N is an arbitrary positive integer and N ≥ 2. We offer a simple physical explanation for tunneling
of EM waves through this structure by treating it as a system composed of N individual resonators that are electrically coupled to each other. Using this analogy we will not
only explain the physical reasons that lead to resonant tunneling of EM waves through
this structure but also demonstrate that such structures can act as spectral filters for electromagnetic waves. Furthermore, we present an analytical procedure for determining the
physical parameters of such multi-layer filters from their a priori known desired transfer
functions. This synthesis procedure allows for determining the physical parameters of the
different DPS and BENG layers, which will result in the desired (a priori known) transfer
function. The proposed synthesis procedure can be used to develop spatial filters operating
throughout the microwave and mm-wave frequency bands as well as multi-layer metallodielectric filters at optical wavelengths. In what follows, we first discuss the relationship
between the proposed multi-layer DPS-BENG structure and a coupled-resonator structure.
Then, we describe the proposed analytical synthesis procedure in detail. Subsequently, we
present two design examples and verify the validity of the proposed analytical synthesis
formulas using full-wave EM simulations. Finally, we use the principle of duality and generalize these results to the problem of EM wave tunneling through multiple MING layers
sandwiched between DPS layers with high relative permeability values.
2.2
Generalized Synthesis Procedure
2.2.1
Topology and equivalent circuit model
Figure 2.1(a) shows the three dimensional (3D) topology and Fig. 2.1(b) shows the
side view of the generalized tunneling problem considered here. The structure consists of
N double-positive dielectric slabs separated from each other by N − 1 ϵ-negative slabs.
17
ε-Negative Layers
High-εr
Dielectric
Layers
1
hENG
2
hENG
ENG
Layers
N-2
hENG
N-1
ENG
h
z
y
x
1
hDPS
2
hDPS
N-2 N-1
hENG
hENG
2
1
hENG
hENG
N-1
hDPS
N
hDPS
2
1
hDPS
hDPS
N-1
N
hDPS
hDPS
High-εr Dielectric
Layers x
(a)
(b)
z
Figure 2.1 (a) 3D topology of a multi-layer structure composed of N high-permittivity
dielectric slabs separated from each other with N − 1 ϵ-negative layers. (b) Side view of
the structure.
The slab dimensions are infinite along the x and y directions and have finite thicknesses
along the z direction. The DPS slabs have high dielectric constant values (ϵr ) and their
thicknesses are assumed to be small compared to the wavelength of the EM waves within
them. The BENG slabs are assumed to be non-magnetic (i.e., µr = 1) and the dielectric
constant of each BENG layer is modeled with a lossless Drude model. Here, we assume
that each slab (whether BENG or DPS) is isotropic, linear, and homogeneous. The different BENG (or DPS) layers are not necessarily identical to each other.
Figure 2.2(a) shows a transmission-line model that can be used to compute the transmission and reflection coefficients of a vertically incident transverse electromagnetic wave
from the structure shown in Fig. 2.1. In this model, the BENG layers are modeled with
−1
transmission lines with lengths of h1EN G , h2EN G , · · · , hN
EN G and characteristic impedances
N −1
1
2
of ZEN
G , ZEN G , · · · , ZEN G . Since the relative permittivity of each BENG layer is mod-
eled with a lossless Drude model, the characteristic impedances of the BENG layers
18
will be purely imaginary in the frequency range of interest. The DPS layers are modeled with transmission lines with lengths of h1DP S , h2DP S , · · · , hN
DP S , and characteristic
1
2
N
impedances of ZDP
S , ZDP S , · · · , ZDP S . The semi-infinite spaces on the both sides of the
structure are modeled with semi-infinite transmission lines with characteristic impedances
of Z0 = 377Ω. The transmission line model of Fig. 2.2(a) can be converted to the
one shown in Fig. 2.2(b) by using lumped element circuit models for each transmission line section. In this case, the thin DPS layers are modeled with two series inductors
i
i
i
i
(LiDP S = µ0 hiDP S /2) and one parallel capacitor (CDP
S = ϵ0 ϵDP S hDP S ) where ϵDP S is
the relative permittivity of ith DPS dielectric layer (i = 1, 2, ..., N ). This representation is
valid as long as the thickness of the DPS layer is relatively small. The model is most accurate if the thickness of the layer is less than λ/12, where λ is the wavelength of the EM
wave inside the DPS layer, and it begins to loose its accuracy as the thickness is increased
beyond this value[62].
The dielectric constant of each ϵ-negative layer is modeled with a lossless Drude model
and the relative permittivity of the ith BENG layer (i = 1, 2, ..., N − 1) is represented by
2
i
ϵiEN G = 1 − (ωEN
G /ω) where
i
ωEN
G
2π
is the electric plasma frequency of the ith BENG
layer. Below its plasma frequency, the dielectric constant of such a dielectric layer is negative and EM waves propagating through it are evanescent[27]. Reference [27] demonstrates that a short section of such an BENG slab with a length of ∆ℓ can be modeled
with an inductive network composed of two series inductors with inductance values of
1
µ µ ∆ℓ
2 0 r
(H)1 and one parallel inductor with an inductance value of
1
2
2
ϵ0 (ωEN
G −ω )∆ℓ
(H)
as shown in Fig. 2.3(a). A thick section of such an BENG layer will simply result in an
1
In this expression, µ0 = 4π × 10−7 (H/m) is the permeability of free space and µr is the relative
permeability of the BENG layer. In this work, we assume that µr = 1 for the BENG layers.
19
(a)
(b)
1
hDPS
1
hENG
2
hDPS
2
hENG
3
hDPS
N-1
hDPS
N-1
hENG
N
hDPS
1
ZDPS
1
ZENG
2
ZDPS
2
ZENG
3
ZDPS
N-1
ZDPS
N-1
ZENG
N
ZDPS
1
LDPS
Ls1
2
LDPS
Ls2
Z0
Z0 C1
Lp1
L’s1 L’s1
(c) Z0 C1
Lp1
Lp2
C2
(e)
Z0
C3
Lp2
C2
LpN-1
C3
LpN-1
CN-1
C3
CN-1
L2*
LN-1*
L1,2
L2,3
LN-1,N
Resonator 1
C2
L2
Resonator 2
CN
Z0
CN
Z0
CN
Z0
LN-1,N
L1*
C1 L 1
N
LDPS
L’sN-1 L’sN-1
L2,3
C2
LsN-1
CN-1
L’s2 L’s2
L1,2
(d) Z0 C1
N-1
LDPS
3
LDPS
Z0
C3 L 3
Resonator 3
CN-1
LN-1
LN
CN Z 0
Resonator N-1 Resonator N
Figure 2.2 (a) The transmission-line model of the structure shown in Fig. 2.1 for a
vertically-incident TEM wave. (b) The transmission-line model of Fig. 2.2(a) is
converted to this equivalent circuit model by substituting appropriate lumped-element
circuit models for the DPS and BENG transmission lines. (c) The simplified version of
the circuit shown in Fig. 2.2(b). (d) The T inductive networks of the circuit shown in Fig.
2.2(c) are converted to π-equivalent networks. (e) Simplified version of the network
shown in Fig. 2.2(d). This circuit is a coupled-resonator filter of order N .
inductive ladder network similar to the one shown in Fig. 2.3(b). Through successive application of π to T transformations2 , it can be shown that such an inductive ladder network
2
a π to T transformation converts a circuit composed of two parallel elements separated from each other
with one series element to an equivalent circuit that is composed of two series elements separated from each
other by a parallel element.
20
1 μ Δl
2 0
Ls
1
ε0 (ωENG
2
Ls
Lp
ω2 ) Δl
Δl
(a)
hENG
(b)
hENG
(c)
Figure 2.3 (a) Equivalent circuit model for a short section of an BENG slab. (b) A thick
BENG slab can be modeled with an inductive ladder network, which is obtained by
cascading the basic inductive T network shown in part (a). (c) It can be shown
theoretically that the ladder inductive network of part (b) is equivalent to a T inductive
network whose inductance values are given by (2.1) and (2.2).
is equivalent to a simple T network composed of three inductors as shown in Fig. 2.3(c).
Therefore, an BENG slab with an arbitrary thickness (not necessarily thin) can be modeled
accurately with the inductive network shown in Fig. 2.3(c). The values of these inductors
can be obtained by calculating the wave transfer matrix of the BENG slab and that of the
inductive network shown in Fig. 2.3(c) and equating the two. Doing this, the inductance
values of Fig. 2.3(c) are calculated, in terms of the parameters of the BENG slab, as:
Z
( √0
)
Lip = − √
i
i
i
ω ϵEN G sin ω µ0 ϵ0 ϵEN G hEN G
( ( √
))
i
i
i
i
Ls = Lp cos ω µ0 ϵ0 ϵEN G hEN G
(2.1)
(2.2)
Note that both of these inductors are frequency dependent. Using this equivalency, the ith
BENG slab in Fig. 2.1 is modeled with two series inductors Lis and one parallel inductor
Lip as shown in Fig. 2.2(b). The equivalent circuit shown in Fig. 2.2(b) is converted to that
shown in Fig. 2.2(c) by combining the series inductors located between parallel branches.
Using a T to π transformation, the T inductive networks highlighted in Fig. 2.2(c) can
21
be converted to the π inductive networks highlighted in Fig. 2.2(d). After combining the
adjacent parallel inductors (L∗i ) of the network shown in Fig. 2.2(d), the circuit is simplified to the one shown in Fig. 2.2(e). The equivalent circuit model shown in Fig. 2.2(e)
is a classical example of an inductively-coupled, coupled-resonator filter with an N th order bandpass response [63]. Therefore, the problem of EM wave tunneling through the
multi-layer BENG-DPS structure shown in Fig. 2.1 can be analyzed using the microwave
filter theory. Using microwave filter theory, filters of the type shown in Fig. 2.2(e) can
be synthesized to provide a given transfer function. Using such a synthesis procedure, all
physical and geometrical parameters of the different constituting layers of the structure
shown in Fig. 2.1 can be determined from its desired frequency response. In this paper,
we will use the well-established microwave filter terminology for identifying the shape of
the transfer function. For example a Butterworth (Chebyshev) filter response represents a
maximally flat (equal ripple) transmission coefficient within the transmission window of
the structure[63]. Our synthesis procedure is based on determining the element values of
the equivalent circuit of Fig. 2.2(e) and relating them to the physical and geometrical parameters of different DPS and BENG layers of the structure shown in Fig. 2.1. To simplify
the design procedure and the analytical formulas, we assume that the structure shown in
Fig. 2.1 is symmetric with respect to a plane passing through its mid section. This assumption means that DPS layers 1 is identical to DPS layer N , DPS layer 2 is identical to
DPS layer N − 1, and so on. Similarly, BENG layer 1 is identical to BENG layer N − 1,
BENG layer 2 is identical to BENG layer N − 2, and so on. This assumption does not
constitute a major limitation of the proposed synthesis procedure, since many response
types of practical interest will inherently result in a symmetrical structure.
22
2.2.2
Tunneling through a single BENG slab sandwiched by two DPS
slabs
Let us first consider the problem of EM wave tunneling through a single BENG slab
surrounded by two DPS slabs. The transmission line model for this problem, which is valid
for a vertically incident TEM wave, is shown in Fig. 2.4(a). Using a procedure similar to
the one described in Section 2.2.1, the transmission line model shown in Fig. 2.4(a) can
be converted to the equivalent circuit model of the second-order, coupled-resonator filter
shown in Fig. 2.4(d) (the intermediate steps are shown in Figs. 2.4(b)-2.4(c)). For the
circuit shown in Fig. 2.4(d), the capacitance values C1 and C2 are determined from:
q1
ω0 Z0 δ
q2
C2 =
ω0 Z0 δ
C1 =
(2.3)
(2.4)
ω0 = 2πf0 , where f0 represents the center frequency of the transmission window. δ =
(fhigh − flow )/f0 is the fractional bandwidth of the transmission window, where fhigh and
flow represent the frequencies at which the magnitude of the transmission coefficient drops
from 1 to
√1 .
2
Analogous to the microwave filter theory, we consider fhigh − flow to be
the frequency bandwidth (BW ) over which the structure is transparent. Finally, q1 and q2
are the normalized quality factors of the parallel LC resonators of the filter (as shown in
Fig. 2.4(d)). The values of q1 and q2 are determined from the desired transfer function of
the filter (e.g. Butterworth response, Chebyshev response, etc.) and they are tabulated in
most microwave filter design handbooks (e.g. see pp. 341-380 in Ref. [63]). For example,
for a second-order (N = 2) coupled-resonator filter having a maximally-flat transmission
coefficient (also known as Butterworth), q1 = q2 = 1.4142. Once the capacitor values are
determined, the coupling inductor L1,2 can be calculated from:
L1,2 =
1
√
C1 C2
ω02 k1,2 δ
(2.5)
23
(a)
(b)
1
hDPS
1
hENG
2
hDPS
1
ZDPS
1
ZENG
2
ZDPS
1
LDPS
Ls1
2
LDPS
Z0
Z0 C1
Lp1
L’s1
(c)
Z0
Z0
C2
L’s1
Z0 C1
Lp1
C2
Z0
L1,2
(d)
Z0
C1 L 1
Resonator 1
C2
L2 Z0
Resonator 2
Figure 2.4 (a) A transmission line circuit for modeling the response of a tri-layer
structure composed of an BENG layer sandwiched between two DPS layers. This model
can be used for a vertically-incident TEM wave. (b) The transmission line model of part
(a) is converted to this equivalent circuit model by replacing the transmission lines with
their equivalent circuit models. (c) Simplified version of the network shown in part (b).
(d) The three inductors forming a T network in the circuit shown in part (c) are converted
to an equivalent π network to obtain this circuit. This circuit is a second-order
coupled-resonator filter.
where k1,2 is the normalized coupling coefficient between the first and the second resonators of the coupled-resonator filter shown in Fig. 2.4(d). Similar to q1 and q2 , the value
of k1,2 is determined from the filter’s desired response type[63]. For a second-order filter with a maximally-flat response, k1,2 = 0.70711. Finally, the inductance values of the
coupled-resonator filter L1 and L2 are calculated from:
L1(2) =
ω02
(
C1(2)
1
)
√
− δk1,2 C1 C2
(2.6)
24
After determining all parameters of the circuit shown in Fig. 2.4(d), the element values
of the equivalent circuit models shown in Figs. 2.4(b)-2.4(c) can be easily determined. In
particular, a π to T transformation can be used to obtain the inductance values of L1p and
L1s in Fig. 2.4(b) from L1 , L2 , and L1,2 . Doing this results in:
L1 L1,2
1
− µ0 h1DP S
L1 + L2 + L1,2 2
L1 L2
L1p =
L1 + L2 + L1,2
L1s =
(2.7)
(2.8)
Using the calculated inductance and capacitance values, the parameters of the transmissionline model shown in Fig. 2.4(a) can then be determined. In this design procedure, the
1(2)
dielectric constants of the two DPS layers, ϵDP S , can be chosen arbitrarily. Using the
1(2)
arbitrarily chosen ϵDP S values, the thicknesses of the two DPS layers can be calculated
from:
1(2)
hDP S =
C1(2)
(2.9)
1(2)
ϵ0 ϵDP S
Since the ENG layer is modeled with a lossless Drude model, its dielectric constant can
be uniquely identified by its electric plasma frequency. The plasma frequency and the
thickness of the ENG slab can be calculated from:
v
Z2
1u
u 2
1
(0
fEN G =
tω +
2π
L1 L1 2 +
)
(2.10)
(
)
1
cos−1 1 + LL1s
p
=√
)
(
2
1
µ0 ϵ0 ω 2 − (2πfEN
)
G
(2.11)
s
h1EN G
p
L1s
L1p
The plasma frequency and the thickness values calculated from (2.10) and (2.11) are frequency dependent. This stems from the fact that the inductance values of the inductive T
network used to model the plasma are frequency dependent as demonstrated by (2.1)-(2.2).
To obtain constant values for the thickness and the plasma frequency of the ENG slab, a
25
natural choice is to substitute ω0 instead of ω in (2.10) and (2.11). Doing this will result in
fixed values for the plasma frequency and the thickness of the ENG layers. As we show in
Section 3.4, this choice does not significantly affect the accuracy of the proposed synthesis procedure within the frequency range of interest. Furthermore, the values predicted by
(2.10) and (2.11) are solely determined by the desired frequency response of the structure
and the model used to obtain them is valid for both thick and thin ENG layers.
2.2.3
Tunneling through two or more ENG slabs sandwiched by DPS
layers
As shown in Section 2.2.1, the multi-layer ENG-DPS structure shown in Fig. 2.1 is
analogous to a bandpass coupled-resonator filter. The order of the response of this filter,
N , is always one more than the number of ENG layers. In the case of the single-layer
ENG slab sandwiched by two DPS layers examined in Section 3.3.1, the structure acts as
a second-order bandpass filter. As the number of the ENG layers increases, so does the
order of the filter. A higher-order filter provides a sharper transmission response and a
higher out of band rejection. As the order of the filter (or the number of ENG and DPS
layers) increases, the transfer function of the system approaches a brick wall function. In
this section, we present the generalized synthesis procedure for the structure shown in Fig.
2.1 when it has two or more ENG layers. For a multi-layer structure containing two or
more ENG layers (a filter with order N ≥ 3), the synthesis procedure takes advantages of
the equivalency between the transmission line model of the problem, shown in Fig. 2.2(a),
and the coupled-resonator filter shown in Fig. 2.2(e). For a given desired transfer function,
we will first determine the element values of the filter circuit shown in Fig. 2.2(e). We will
then relate these values to the parameters of the transmission-line model of the multi-layer
structure shown in Fig. 2.2(a) and ultimately to the physical parameters of the different
ENG and DPS layers that constitute the multi-layer barrier structure.
26
With reference to the circuit model shown in Fig. 2.2(e), the capacitance values of the
first and last resonators of the coupled-resonator filter can be determined from:
C1(N ) =
q1(N )
ω0 Z0 δ
(2.12)
where q1 (qN ) is the normalized quality factor of the first (last) resonator of the filter. For
the symmetric structure considered here q1 = qN and their values are determined from
the desired transfer function of the structure. Values corresponding to different transfer
function types are tabulated in most microwave filter handbooks (e.g. see pp. 341-380 in
Ref. [63]). Then the capacitance value of the second resonator is chosen such that:
C2 = CN −1 <
C1
(δk1,2 )2
(2.13)
where k1,2 is the normalized coupling coefficient between the first and the second resonator
of the filter shown in Fig. 2.2(e). k1,2 values are also determined uniquely from the desired
transfer function type and are tabulated in microwave filter handbooks [63]. Notice that
(2.13) only provides a range for C2 values as opposed to a specific value. Thus, the exact
value of C2 can be chosen arbitrarily as long as the condition specified in (2.13) is satisfied.
The capacitance values of the remaining capacitors, except C N +1 (for odd N ) are chosen
2
to satisfy the following condition:
Ci = CN −i+1
√
)2
(√
Ci−1 − ki−2,i−1 δ Ci−2
<
ki−1,i δ
N
i = 3, 4, · · · ,
N : Even.
2
N −1
i = 3, 4, · · · ,
N : Odd.
2
(2.14)
where ki,j represents the normalized coupling coefficient between the ith and j th resonators
of the coupled-resonator filter shown in Fig. 2.2(e). ki,j values are also determined from
the desired filter response and their values are known a priori[63]. Finally, the middle
27
capacitor C i+1 of an odd-order coupled-resonator filter can be obtained from:
2
(
(
))
C i+1 = C i+1 −1 − C i+1 −2 − (· · · − (C2 − C1 ) · · · )
2
2
2
(
(
))
+ C i+1 +1 − C i+1 +2 − (· · · − Ci ) . . . )
2
2
i = 3, 5, 7, · · · , N
N : Odd.
(2.15)
The conditions specified by equations (3.18)-(2.14) indicate that there are more than one
solutions for some of the capacitors of a coupled-resonator filter with an order of N ≥ 5.
This stems from the fact that the coupled-resonator filter shown in Fig. 2.2(e) is an underdetermined system. Thus, for a given response, more than one circuit exists that yields
the desired transfer function. This, however, offers a degree of flexibility in choosing
circuit element values that result in more desirable physical parameters for the multi-layer
structure shown in Fig. 2.1. After determining all the capacitance values, the inductance
values of the first and last resonators can be calculated from:
L1 = LN =
ω02
(
1
)
√
C1 − k1,2 δ C1 C2
(2.16)
The values of the remaining parallel inductors are calculated from:
Li =
ω02
(
1
)
√
√
Ci − ki−1,i δ Ci−1 Ci − ki,i+1 δ Ci Ci+1
i = 2, 3, · · · , N − 1.
(2.17)
Finally, the values of the coupling inductors can be obtained from:
Li,i+1 =
1
√
Ci Ci+1
ω02 ki,i+1 δ
i = 1, 2, · · · , N − 1.
(2.18)
This completes the design of the coupled-resonator filter shown in Fig. 2.2(e). Using the
element values calculated from (3.18)-(2.18), the element values of all of the equivalent
28
circuits shown in Figs. 2.2(b)-2.2(d) can be determined. With reference to Fig. 2.2(d), the
values of inductors L∗i can be calculated from:
L∗1 = L1
(2.19)
L∗N −1 = LN
(2.20)
L∗i =
L∗i−1 Li
L∗i−1 − Li
i = 2, · · · , N − 2.
(2.21)
Via a π to T transformation, the series and parallel inductance values of each T network
shown in Fig. 2.2(c) can be calculated from:
)
L∗i Li,i+1
1 ( i
i+1
−
µ
h
+
h
o
DP
S
DP
S
2L∗i + Li,i+1 8
L∗i L∗i
i
Lp =
2L∗i + Li,i+1
Lis =
(2.22)
(2.23)
i = 1, 2, · · · , N − 1.
The parameters of the transmission line model shown in Fig. 2.2(a) can then be determined
from the element values calculated using the previous equations. In this design procedure,
the dielectric constant of each of the dielectric layers can be chosen arbitrarily. Once the
dielectric constant of the ith dielectric layer is chosen, its thickness can be calculated from:
hiDP S =
Ci
ϵ0 ϵiDP S
i = 1, 2, · · · , N − 1.
(2.24)
The electric plasma frequency and thickness of the ith ENG layer can be calculated from:
v
u
1
Z02
u 2
i
)
(
fEN
=
+
(2.25)
tω0
G
Lis
2π
i
i
L L 2+
s
(
hiEN G
p
)
i
Lip
cos−1 1 + LLis
p
=√
(
)
2
2
i
µ0 ϵ0 ω0 − (2πfEN G )
i = 1, 2, · · · , N − 1.
(2.26)
29
Table 2.1 Physical and geometrical parameters of the DPS-ENG-DPS tri-layer structure
examined in Section 3.4.
h1DP S = h2DP S
hEN G
fEN G
ϵ1,2
DP S =200
337.1 µm
2.997 mm
40.311 GHz
ϵ1,2
DP S =500
134.8 µm
3.359 mm
37.178 GHz
ϵ1,2
DP S =1000
67.4 µm
3.481 mm
36.243 GHz
In general, the plasma frequency and the thickness values calculated from (2.25) and (2.26)
should be frequency dependent. However, to obtain constant values for the thickness and
the plasma frequency of the ENG slabs, ω0 has been substituted for ω in (2.25) and (2.26).
2.3
Synthesis Procedure Verification
2.3.1
Tunneling through a single ENG slab sandwiched by two high-ϵr
DPS slabs
The synthesis procedure presented in Sections 3.3.1 and 3.3.2 is verified using two
examples. In the first example, tunneling through a single ENG barrier surrounded by two
DPS slabs is examined. In this case, we desire the structure to be transparent within the
microwave range with a transmission window centered at f0 = 5 GHz, a transmission
bandwidth of BW = 1.0 GHz (i.e. δ = BW/f0 = 0.2), and a maximally flat transmission
coefficient within this band. As shown in Section 3.3.1, this structure is equivalent to a
second-order coupled-resonator bandpass filter. In such a filter, a maximally flat bandpass
response is characterized by q1 = q2 = 1.4142 and k1,2 = 0.7071. Using these q and
k1,2 values in conjunction with f0 = 5.0 GHz and δ = 0.2 in equations (3.6)-(2.11), the
physical parameters of the this tri-layer structure are obtained and presented in Table 3.2.
Since the dielectric constants of the two DPS layers can be chosen arbitrarily, Table 3.2
30
T: Ideal
T: εDPS=1000
T: εDPS=500
T: εDPS=200
R: Ideal
R: εDPS=1000
R: εDPS=500
R: εDPS=200
Figure 2.5 The transmission and reflection coefficients of the three different tri-layer
structures examined in Section 3.4.1 and those of an ideal coupled-resonator filter with a
second-order bandpass response. The results of the tri-layer structures are obtained using
full-wave EM simulations in CST Studio.
shows three different rows containing the parameters of the tri-layer structure obtained for
1
2
ϵ1,2
DP S values of 200, 500, and 1000 (ϵDP S = ϵDP S ).
Using the parameters of the DPS and ENG layers provided in Table 3.2, the transmission and reflection coefficients of the tri-layer structure can be easily calculated. This
can be done either analytically (e.g. using the wave transfer matrix technique) or numerically (e.g. by using the Finite Difference Time Domain (FDTD) or Finite Element
Method (FEM)). Figure 2.5 shows the calculated transmission and reflection coefficients
of the tri-layer structures identified in Table 3.2 obtained using full-wave electromagnetic
simulations in CST Microwave Studio. Additionally, Fig. 2.5 also shows the transmission and reflection coefficients of an ideal coupled resonator filter having a second-order
maximally-flat bandpass response with f0 = 5 GHz and δ = 0.2. As can be observed,
31
Table 2.2 Parameters of the fifth-order coupled-resonator filter of the type shown in Fig.
2.2(e) that models the 9-layer structure discussed in Section 3.3.2.
Parameter
C1 = C5
C2 = C4
C3
L1 = L5
Value
0.3814 pF
0.7628 pF
0.7628 pF
4.2088 nH
Parameter
L2 = L4
L3
L1,2 = L4,5
L2,3 = L3,4
Value
2.2066 nH
2.3194 nH
7.2042 nH
6.2177 nH
there is a good agreement between the responses of the the tri-layer structures and that
of the ideal coupled-resonator filter. Figure 2.5 also shows that this agreement improves
considerably as the dielectric constants of the two DPS layers increase. This is due to the
fact that the accuracy of the lumped element circuit models used for the DPS slabs (see
Fig. 2.2(b) and 2.4(b)) increases as their thicknesses is reduced. Since using a higher ϵDP S
value results in a thinner DPS slab according to (2.9), the accuracy of the model increases
with increasing ϵDP S values and results in a better agreement between the response of the
tri-layer structure and that of the ideal second-order coupled-resonator bandpass filter. In
a practical application, however, DPS materials with very high dielectric constant values
may not be readily available. In such cases, DPS slabs with lower dielectric constant values which are commercially available (with typical ϵr values ranging from 20-100) may be
used. Using the proposed synthesis procedure, the physical parameters of the multi-layer
structure can be determined. These values can then be fined tuned using full-wave numerical EM simulations to enhance the agreement between the actual response of the structure
and the desired filter response following a procedure similar to the one described in Ref.
Behdad2. Alternatively, instead of using a DPS slab with a very high ϵr value, a material
32
Table 2.3 Physical and geometrical parameters of the 9-layer composite ENG-DPS
structure examined in Section 3.3.2.
1
5
Parameter
hDP S = hDP S h2DP S = h4DP S
h3DP S
ϵDP S
Value
215.4 µm
430.7 µm
430.7 µm
Parameter
h1EN G = h4EN G
1
4
fEN
G = fEN G
h2EN G = h3EN G
2
3
fEN
G = fEN G
Value
3.54 mm
22.3947 GHz
3.21 mm
22.0943 GHz
200
with a very high effective dielectric constant value may be synthesized artificially. Generally, such artificial dielectrics are synthesized using two- or three-dimensional periodic
structures utilizing sub-wavelength metallic particles[64].
2.3.2
Tunneling through four ENG slabs sandwiched by five high-ϵr
DPS slabs
The second example that we consider in this section is the case of tunneling of EM
waves through a multi-layer barrier structure consisting of four ENG layers and five DPS
layers. As discussed in Section 3.3.2, this problem is equivalent to a coupled-resonator
filter of the type shown in Fig. 2.2(e) with a fifth-order bandpass response. We seek
to determine the specific physical parameters of different ENG and DPS layers that will
result in a transparency window in the microwave region centered at f0 = 5.0 GHz with a
bandwidth of BW = 2.0 GHz (i.e. δ = 0.4). Additionally, this time we will consider an
equal-ripple transmission coefficient with 0.5 dB ripple within the transmission window.
Such a transmission coefficient is referred to as the Chebyshev response in microwave
filter theory [63] and is characterized by normalized quality factor and coupling values
of q1 = q5 = 1.8068, k1,2 = k4,5 = 0.6519, and k2,3 = k3,4 = 0.5341 [63]. Using
these values in conjunction with f0 = 5.0 GHz and δ = 0.4 in equations (3.18)-(2.26),
the parameters of different constituting layers of this 9-layer structure can be determined.
33
T: Ideal
T: Analytical
T: Full Wave
R: Ideal
R: Analytical
R: Full Wave
Figure 2.6 The transmission and reflection coefficients of the 9-layer structure examined
in Section 3.4.2 and those of an ideal coupled-resonator filter with a fifth-order bandpass
response. For the 9-layer structure, the results obtained using both full-wave EM
simulations in CST Studio as well as the analytically calculated results using the wave
transfer matrix method are shown.
The parameters of the equivalent circuit model of this filter are provided in Table 3.3. In
this design example, more than one equivalent circuit exists for the filter shown in Fig.
2.2(e) that will result in the desired frequency response, as described in Section 3.3.2.
Furthermore, since the dielectric constant of each DPS layer is a free design parameter, we
have chosen a dielectric constant of ϵDP S = 200 for all five DPS layers. Using this ϵDP S
value in conjunction with the element values of the equivalent circuit model provided in
Table 3.3, the parameters of the different constituting layers of this structure are calculated
and provided in Table 3.4.
The transmission and reflection coefficients of this structure are calculated both analytically and using full-wave EM simulations in CST Microwave Studio and the results
are presented in Fig. 2.6. Figure 2.6 also shows the ideal response of a coupled-resonator
34
filter with a fifth-order bandpass response having f0 = 5.0 GHz and δ = 0.4. As can be
observed, the analytical solution of the problem, obtained using the wave transfer matrix
method, agrees perfectly with the full-wave EM simulation results obtained in CST Microwave Studio. Additionally, a good agreement between these results and the response
of an ideal microwave filter is observed. The small discrepancies observed between these
results can primarily be attributed to two main sources. These include the approximate
LC model used for the short section of a DPS dielectric slab and substitution of ω0 for
ω in equations (2.25) and (2.26). This latter step was necessary to obtain frequencyindependent values for the thickness and the plasma frequency of the ENG layers. However, as can be seen from the results presented in Figs. 2.5 and 2.6, these approximations
do not appear to affect the response of the structure significantly.
2.3.3
The Effect of Loss
Perfect tunneling of EM waves through the structure shown in Fig. 2.1 is possible only
when the structure is lossless. In reality, however, both the DPS and the ENG layers that
constitute the structure will have some loss. The effect of the loss of the DPS and the ENG
layers can be taken into account by considering the complex constituting parameters of
the structure (ϵr and µr ). Assuming that µr = 1 for all the layers, the dielectric losses of
the DPS layers can be taken into account by considering their loss tangents (tan(δ)). The
dielectric losses of the ENG layers can be taken into account using a lossy Drude model:
ϵr = 1 −
2
ωEN
G
ω 2 + iωνEN G
(2.27)
where νEN G is the collision frequency. When the structure shown in Fig. 2.1 is illuminated with an EM wave, some part of the power is reflected back towards the direction
of incidence and some part of the wave’s power tunnels through the structure. In a lossy
Total Power Loss [%]
35
1 ENG Layer
2 ENG Layers
3 ENG Layers
4 ENG Layers
Fractional Bandwidth (δ) [%]
Figure 2.7 Percentage of total power loss as a function of fractional bandwidth for a
number of multi-layer structures of the type shown in Fig. 2.1 and discussed in Section
3.4.3.
structure, the difference between the power of the incident wave and the sum of the power
of the reflected and transmitted waves is the total power loss. In addition to tan(δ) of the
DPS layers and vEN G of the ENG layers, the total power loss of this structure is also a
function of the total number of the layers and the fractional bandwidth of the transmission
window. In particular, for fixed tan(δ) and vEN G values, loss increases when the total
number of layers increases or when the fractional bandwidth of the transmission window
decreases. This behavior is demonstrated in Fig. 2.7 for a number of different structures
of the type shown in Fig. 2.1. In all of these structures, the DPS layers are assumed to
have ϵDP S and tan(δ) values of 500 and 0.001 respectively. Additionally, for all ENG
layers, it is assumed that vEN G /ω = 0.01. All of these structures are synthesized to have
a maximally-flat transmission coefficient centered at f0 = 5.0 GHz and their losses are
calculated at this center frequency. Figure 2.7 shows the total power loss within these
structures as a function of the number of ENG layers and the fractional bandwidth (δ) of
36
µ-Negative Layers
High-µr,
Magnetic
Layers
1
hMNG
2
MNG
h
µ-Negative
N-2
hMNG
Layers
N-1
hMNG
z
1
hDPS
2
hDPS
N
hDPS
y
x
Z0
C2 L 2
C1,2
N-2 N-1
hMNG
hMNG
N-1
hDPS
2
1
hDPS
hDPS
N-1
N
hDPS
hDPS
High-µr, Magnetic
Layers x
z
(a)
C1 L1
2
1
hMNG
hMNG
(b)
C3 L 3
C2,3
CN-1 LN-1
CN L N
CN-1,N
Z0
(c)
Figure 2.8 (a) 3D topology of a multi-layer structure composed of N high-permeability
slabs separated from each other with N − 1 µ-negative layers. This structure is the dual
of the structure shown in Fig. 2.1. (b) Side view of the structure. (c) A coupled-resonator
filter model composed of series LC resonators coupled to each other using parallel
capacitors. This filter can be used to model the frequency response of the structure shown
in Fig. 2.8(a).
the transmission window. As can be observed, loss increases when δ decreases or the total
number of layers is increased. The same behavior is also observed in microwave filters
where the total loss of the filter increases when the bandwidth is decreased or when the
order of the filter is increased[65].
2.4
Tunneling Through µ-Negative (MNG) Layers
The principle of duality can be used to generalize the results obtained in Sections
7.3-3.4 to the problem of tunneling of EM waves through multiple layers of materials
37
1.0
1.0
T: ENG
0.8
T: MNG
R: ENG
R: MNG
0.6
0.6
0.4
0.4
0.2
0.2
0.0
Ref. Coeff. |R|
Trans. Coeff. |T|
0.8
0.0
2
3
4
5
6
7
8
Frequency [GHz]
Figure 2.9 Transmission and reflection coefficients of the tri-layer structure discussed in
Section 3.5 and its dual structure discussed in Section 3.3.1. As can be seen, the two
results match perfectly as expected.
with negative permeability values (i.e., µ-negative or MNG). In particular, in the multilayer barrier structure shown in Fig. 2.1, if we exchange the relative permittivity with the
relative permeability (i.e. ϵ → µ, µ → ϵ), the transmission and reflection coefficients of
the structure will be unchanged. Doing this will convert the structure shown in Fig. 2.1
to the one shown in Fig. 2.8(a)-2.8(b). Here, multiple MNG layers are surrounded by
thin DPS layers with very high relative permeability values. Using a procedure similar
to what was followed in Section 2.2.1, we can show that this problem can be modeled
with a coupled-resonator filter topology shown in Fig. 2.1(c). This time, the resonators
constituting the filter are series LC resonators that are coupled with each other through
parallel capacitors. The filter topology shown in Fig. 2.1(c) is dual of the one shown in
Fig. 2.2(e). To demonstrate the duality between the structures shown in Fig. 2.1(a) and
2.8(a), we examine the dual problem of one of the examples presented in Section 3.3.1. In
this case, a tri-layer structure composed of a single MNG layer surrounded by two DPS
38
layers is examined. The two DPS layers are identical and have relative permittivity and
permeability values of ϵr = 1 and µr = 200. The MNG layer has a relative permittivity
of ϵr = 1 and its relative permeability is modeled with a lossless Drude model with a
magnetic plasma frequency of fM N G = 40.311 GHz. The physical dimensions of the
layers are provided in the first row of Table 3.2 (corresponding to ϵDP S = 200). This
structure is simulated using full-wave EM simulations in CST Microwave Studio and its
transmission and reflection coefficients are calculated and presented in Fig. 2.9 along with
those of its dual structure (shown in Fig. 2.5). As can be seen, the two results are identical
as expected.
2.5
Conclusions
We examined the problem of EM wave tunneling through multiple ENG layers paired
with high-dielectric-constant DPS layers. We demonstrated that the problem of EM wave
tunneling through this multi-layer barrier structure is analogous to the classical problem
of coupled-resonator microwave filters. Through this analogy, we developed an analytical
method for synthesizing these structures from their desired transfer functions. The validity
of the proposed synthesis procedure was demonstrated through two design examples. Using the principle of duality, we demonstrated that tunneling of EM waves through multiple
layers of µ-negative materials can also be treated using a similar approach.
39
Chapter 3
Electromagnetic Wave Tunneling Through Cascaded ϵ- and
µ-Negative Metamaterial Slabs
3.1
Introduction
Electromagnetic (EM) waves cannot propagate over several skin depths
1
in materi-
als exhibiting negative permittivity or negative permeability values (also known as singlenegative (SNG) materials). EM waves that enter SNG materials exponentially decay along
the direction of propagation and rapidly loose their energy. However, when SNG slabs are
used in a suitably-designed multi-layer structure, they can be made completely transparent
and allow for the total transmission of electromagnetic waves. Complete transmission of
EM waves through these propagation barriers is often referred to as EM wave tunneling
due to its analogous nature to quantum tunneling problems. EM wave tunneling through
such propagation barriers has been extensively investigated over the years2 . Early investigations of this phenomenon focused on the propagation of EM waves through naturally
occurring SNG materials such as plasmas [57] or metallic films [58]. With the emergence of the field of metamaterials in recent years, however, major advancements have
Skin depth is the distance that an EM wave travels in a lossy material and its value reduces to e−1 or
36.8%.
2
In earlier investigations, the keyword tunneling is used less frequently.
1
40
been made in synthesizing materials with any desired permittivity or permeability values. Therefore, the idea of EM wave tunneling through metamaterial layers having negative permittivity (ϵ-negative or BENG) or negative permeability (µ-negative or MING)
values has received significant attention in the past decade [27], [29] –[34], [60] –[62],
[66] –[72]. Studies conducted in this area have examined various multi-layer structures
composed of BENG, MING, and/or double-positive (DPS) materials. Examples include
tunneling through BENG layers paired with MING ones [27] –[29], tunneling through an
BENG layer separated from an MING layer with free space [30], and tunneling through
multi-layer structures composed of SNG and double-positive (DPS) layers [31] – [33].
Despite the large number of studies conducted in this area to date, few of them provide any in-depth examination of the close relationship that exists between such tunneling
problems and classical microwave filters. Such an examination is important for two reasons. First, it provides a simple and easy-to-understand method for explaining a rather
complicated physical phenomenon. Additionally, by exploiting this analogy, a synthesis procedure can be developed3 and used for determining the physical parameters of the
multi-layer structure that will result in an a priori known desired frequency response. In
this paper, we examine the problem of electromagnetic wave tunneling through a generalized multi-layer structure composed of multiple cascaded BENG and MING layers. The
structure considered here is composed of N , BENG layers that are separated from one another with N − 1 MING layers. N is an arbitrary positive integer and N ≥ 2. We propose
an equivalent circuit model for this multi-layer composite structure and demonstrate that
if certain conditions are met, this structure becomes equivalent to capacitively-coupled,
coupled-resonator bandpass filter of order N . By establishing this relationship, we develop an analytical procedure for determining the physical parameters of structure that
3
The synthesis procedure will be unique to the specific tunneling problem considered.
41
will result in a given a priori known transfer function. We verify the validity of the proposed synthesis procedure using full-wave numerical EM simulations and examine the
conditions that must be met to ensure that this analogy remains valid. Finally, we extend
the proposed synthesis procedure to multi-layer structures composed of N MING slabs
sandwiching N − 1 BENG slabs through the use of the duality principle.
3.2
Problem Definition
Figure 3.1(a) shows the three dimensional (3D) topology of the problem considered
in this paper. The structure consists of planar MING layers separated from each other
by very thin BENG layers. Each layer extends to infinity along the x and y directions
but has finite thickness along z direction. In our theoretical analysis in this paper, we
assume that each layer is isotropic, linear, and homogeneous. The relative permittivity of
2
2
each BENG layer is described by a lossless Drude model (ϵEN G = 1 − ωEN
G /ω ) where
ωEN G / (2π) is the electric plasma frequency. The relative permeability of the BENG layer,
µEN G , is assumed to be constant. Similarly, the relative permeability of each MING layer
2
2
is described by a lossless Drude model, µM N G = 1 − ωM
N G /ω where ωM N G / (2π) is
the magnetic plasma frequency. The relative permittivity of the MING layers, ϵM N G , is
assumed to be constant. The BENG (MING) layers become opaque at frequency bands
that fall below their electric (magnetic) plasma frequencies where the relative permittivity
(relative permeability) values are negative. Additionally, the different BENG (MING)
layers are not necessarily identical to each other and can have different thicknesses and
plasma frequencies.
To examine the propagation of electromagnetic waves through this multi-layer structure, we adopt a simple circuit-based analysis method. Figure 3.2(a) shows a transmissionline-based equivalent circuit model for this structure which is valid for a vertically incident
42
ENG
Layers
MNG
Layers
ENG Layers
2
1
hENG
hENG
N-1
N
hENG
hENG
x
x
z
z
y
1
MNG
h
2
hMNG
N-1
N-2
hMNG
hMNG
N
N-1
hENG
hENG
2
1
hENG
hENG
(a)
2
1
hMNG
hMNG
N-2 N-1
hMNG
hMNG
MNG Layers
(b)
Figure 3.1 (a) 3D topology of the generalized electromagnetic wave tunneling problem
considered in this paper. This multi-layer structure consists of N − 1 MING layers
sandwiched by N BENG layers. Each layer extends to infinity along x and y directions
but has finite thickness along the z direction. (b) Side view of the multi-layer structure.
transverse electromagnetic (TEM) wave. Here, the MING layers are modeled with transN −1
mission lines with lengths of h1M N G , h2M N G , · · · , hM
N G and characteristic impedances of
N −1
1
2
ZM
N G , ZM N G , · · · , ZM N G . The BENG layers are modeled with transmission lines with
1
N
lengths of h1EN G , · · · , hN
EN G and characteristic impedances of ZEN G , · · · , ZEN G . The
semi-infinite spaces on the both sides of the composite BENG-MING multi-layer structure are modeled with semi-infinite transmission lines with characteristic impedances of
Z0 , where Z0 = 377Ω. The transmission line model shown in Fig. 3.2(a) can be converted
to a lumped-element equivalent circuit model consisting of only capacitors and inductors
by substituting lumped-element models for the BENG and MING layers. A thin BENG
layer with an overall thickness of ∆ℓ can be modeled with an inductive T network composed of two series inductors with inductance values of 21 µ0 µEN G ∆ℓ (H) separated by one
43
parallel inductor with an inductance value of
1
2
2
ϵ0 (ωEN
G −ω )∆ℓ
(H) as shown in Fig. 3.3(a)
[27].
Similarly, a thin MING layer with a length of ∆ℓ can be represented by a capacitive
T network composed of two series capacitors with capacitance values of
2
2
2
µ0 (ωM
N G −ω )∆ℓ
(F) separated from one another by one parallel capacitor with a capacitance value of
ϵ0 ϵM N G ∆ℓ (F) as shown in Fig. 3.3(b). The equivalent circuit model of a thick MING
layer can be obtained by repeating the T network shown in Fig. 3.3(b) as shown in Fig.
3.3(c). This ladder network can be converted to a simple capacitive T network composed
of three inductors shown in Fig. 3.3(c) via consecutive applications of π to T transformation4 . Therefore, an MING layer with arbitrary thickness can be represented by a capacitive network composed of two series capacitors and one parallel capacitor. The capacitor
values are obtained by equating the wave transfer matrices of the transmission line model
of the thick MING layer and the capacitive T network shown in Fig. 3.3(d). These capacitor values are expressed in terms of the parameters of the MING layer:
( √
)
√
ϵiM N G sin ω ϵ0 ϵiM N G µ0 µiM N G hiM N G
√
Cpi =
ωZ0 µiM N G
Cpi
( √
)
Csi =
cos ω ϵ0 ϵiM N G µ0 µiM N G hiM N G − 1
(3.1)
(3.2)
Note that the values of the above two capacitors depend on frequency. Using the equivalent circuit models for the BENG and MING layers shown in Figs. 3.3(a) and 3.3(d),
the transmission-line model shown in Fig. 3.2(a) can be converted to the lumped-elementbased equivalent circuit model shown in Fig. 3.2(b). Since the BENG layers constituting
this structure are thin, the values of the series inductors L1s , L2s , ..., LN
s are small and they
can be ignored to further simplify the structure’s equivalent circuit model as shown in Fig.
4
A π to T transformation can convert a circuit composed of one series capacitor sandwiched by two
parallel capacitors to a circuit composed of one parallel capacitor sandwiched by two series capacitors.
44
Z0
1
hENG
1
hMNG
2
hENG
2
hMNG
3
hENG
1
ZENG
1
ZMNG
2
ZENG
2
ZMNG
3
ZENG
1
s
1
s
N-1
hENG
N-1
ZENG
N-1
hMNG
N
hENG
N-1
ZMNG
N
ZENG
Z0
(a)
L
C
Z0 L1
2
s
L
Cp1
Ls3
2
s
C
Cp2
L2
L3
LN-1
s
LN-1
CsN-1
CpN-1
LNs
LN
Z0
LN
Z0
LN
Z0
LN
Z0
(b)
1
s
2
s
C
Z0 L 1
CN-1
s
C
Cp1
L2
L3
Cp2
LN-1
CpN-1
(c)
C1,2
Z0 L 1
C2,3
L2
C1*
CN-1,N
L3
C2*
LN-1
*
CN-1
(d)
C1,2
Z0 L1
C1
C2,3
C2
L2
CN-1,N
C3
L3
LN-1
CN-1 CN
(e)
Figure 3.2 (a) Transmission line model of the multi-layer structure shown in Fig. 3.1(a)
valid for a normally incident TEM wave. (b) The transmission line model of part (a) can
be transformed to this equivalent circuit model by substituting lumped-element
equivalent circuit models for the BENG and MING layers. (c) The equivalent LC circuit
model of part (b) is simplified to this circuit by ignoring the series inductors Lis used in
the equivalent circuit model of BENG layers. This is justified when BENG layers are
thin. (d) The capacitive T networks highlighted in part (c) can be converted to capacitive
π networks highlighted in this figure after a number of T to π transformations. (e) After
combining the adjacent parallel inductors of the equivalent circuit network of part (d), the
equivalent circuit model of an N th -order coupled-resonator bandpass filter is obtained.
45
3.2(c). Utilizing a T to π transformation, the T networks highlighted in Fig. 3.2(c) can
be converted to the π networks highlighted in Fig. 3.2(d). After combining the adjacent
parallel capacitors (Ci∗ ) at each node of this circuit, it can be converted to the equivalent
circuit model shown in Fig. 3.2(e), which is a classical, bandpass coupled-resonator filter
of order N . Based on the analysis presented above, the structure shown in Fig. 3.1 is
expected to behave as a coupled-resonator bandpass filter if the constituting BENG layers
are thin and the frequency of operation is in a range where the BENG and MING layers
have negative permittivity and permeability values respectively. In the subsequent section,
we will present an analytical synthesis procedure that relates the desired transmission response of this filter to the physical and geometrical parameters of the multi-layer structure
shown in Fig. 3.1.
3.3
Synthesis Procedure
3.3.1
Transmission Through A Single MING Layer Sandwiched by Two
BENG Layers
We will first examine a simple tri-layer structure composed of one MING layer surrounded by two very thin BENG layers and derive analytical synthesis formulas for this
structure. The equivalent circuit model of this tri-layer structure and the analogy between
its equivalent circuit model and a second-order, coupled-resonator bandpass filter is illustrated in Figs. 3.4(a)-(d). Using this analogy a synthesis procedure is developed that
allows for obtaining the physical and geometrical parameters of this structure from the
characteristics of its desired transmission coefficient. This synthesis procedure is based
on determining the parameters of the equivalent circuit model of the structure shown in
Fig. 3.4(d), for a given a priori known response, and relating them to the physical and
46
1 μ μ Δl
2 0 ENG
2
μ0 (ωMNG
ω2 ) Δl
2
1
ε0 (ωENG
2
ε0εMNG Δl
ω2 ) Δl
Δl
Δl
(a)
(b)
Cs
Cs
Cp
hMNG
hMNG
(c)
(d)
Figure 3.3 (a) The equivalent circuit model for a short length of an ϵ-negative material
with a length of ∆ℓ → 0. (b) The equivalent circuit model for a short length of a
µ-negative material with a length of ∆ℓ → 0. (c) The equivalent circuit model of a thick
MING layer can be obtained by cascading the equivalent circuit model shown in part (b)
in the form of the ladder capacitive network shown in this figure. (d) Through successive
π to T transformations, the capacitive ladder network shown in part (c) can be converted
to a single T network. This T network model is valid for an MING layer with any
thickness.
geometrical parameters of the BENG and MING layers. For the second-order coupledresonator filter shown in Fig. 3.4(d), the inductance values L1 and L2 are determined from
the following equations:
δZ0
ω 0 q1
δZ0
L2 =
ω 0 q2
L1 =
(3.3)
(3.4)
Where δ = ∆f /f0 is the fractional bandwidth of the transmission window. ∆f represents
the 3 dB bandwidth of the transmission window (i.e., the difference between frequencies
47
Z0
1
hENG
1
hMNG
2
hENG
1
ZENG
1
ZMNG
2
ZENG
C1s
Z0
Z0
L1
Cp1
(a)
Ls1
Z0
L1
Cs1
Z0
L2
Z0
(c)
L2s
Cp1
L2
C1,2
Z0
L2
Z0
L1
(b)
C1
C2
(d)
Figure 3.4 (a) The transmission line model of a tri-layer structure composed of a single
MING layer sandwiched by two BENG layers valid for a vertically-incident TEM wave.
(b) The transmission line model of part (a) can be transformed to this circuit by
substituting the appropriate lumped-element equivalent circuit models for the MING and
BENG layers as shown in Fig. 3.3. (c) The network of part (c) is simplified by ignoring
the series inductors L1s and L2s . This can be done if the thickness of the BENG layers are
small. (d) The capacitive T network of part (c) is changed to an equivalent π network to
achieve this equivalent circuit model. This circuit is a classical second-order bandpass
coupled-resonator filter.
where the magnitude of transmission coefficient decreases from 1 to
√1 )
2
and f0 is the
center frequency of operation (ω0 = 2πf0 ). After determining the inductance values, the
value of the coupling capacitor C1,2 can be obtained from:
C1,2 =
ω02
δk1,2
√
L1 L2
(3.5)
where k1,2 is the normalized coupling coefficient between the first and second resonators
of the coupled-resonator filter shown in Fig. 3.4(d). The values of the normalized quality
factors, q1 and q2 , and the coupling coefficient, k1,2 , are provided in most filter design
handbooks [63]. Finally, the capacitance values of C1 and C2 are calculated from:
C1(2) =
1
ω02 L1(2)
−
ω02
δk1,2
√
L1 L2
(3.6)
48
After determining all lumped element values of coupled-resonator filter shown in Fig.
3.4(d), the capacitance values of the capacitive T network shown in Fig. 3.4(c) are obtained after converting the capacitive π to T transformation from:
C1 C1,2 + C2 C1,2 + C1 C2
C1
C
C
+
C
1
1,2
2 C1,2 + C1 C2
Cp1 =
C1,2
Cs1 =
The electric plasma frequencies of the exterior ENG layers are obtained from:
√
1
1
1(2)
fEN G =
ω02 +
1(2)
2π
ϵ0 hEN G L1(2)
(3.7)
(3.8)
(3.9)
1(2)
In this case, the thickness of the ENG layers hEN G are assumed to be free design parameters which can be chosen by the designer. The magnetic plasma frequency and the
thickness of the MNG layer can be calculated from:
v
(
)
u
Cp1
1
u
1 t 2 ϵM N G 2 + Cs1
1
ω0 +
fM
=
NG
2π
Z02 Cs1 Cp1
(
)
Cp1
−1
cosh
1 + C1
s
h1M N G = √
(
)
2
1
2
µ0 ϵ0 ϵ1M N G (2πfM
)
−
ω
0
NG
(3.10)
(3.11)
Since the equivalent capacitive T network modeling the MNG layer has frequency-dependent
capacitors (as demonstrated in (3.1)-(3.2)), the magnetic plasma frequency and the thickness of the MNG layer predicted by (3.10) and (3.11) are frequency dependent as well. To
obtain constant values for magnetic plasma frequency and thickness of the MNG layer, ω
is replaced by ω0 in (3.10) and (3.11).
49
3.3.2
Transmission Through N ENG Layers Separated from One Another by N − 1 MNG Layers (N ≥ 3)
The synthesis procedure for the generalized structure shown in Fig. 3.1 follows a similar procedure as the one described in Section 3.3.1. First, we will determine the element
values of the equivalent circuit model of the structure shown in Fig. 3.2(e) from the desired filter response of the structure. Then, these element values will be related to those of
the circuit model shown in Fig. 3.2(b) and finally to the geometrical and physical parameters of the MNG and ENG layers of the structure shown in Fig. 3.1. With regards to the
equivalent circuit model shown in Fig. 3.2(e), the inductance values of the first and last
resonators are determined using:
L1(N ) =
δZ0
ω0 q1(N )
(3.12)
where q1 and qN are the normalized quality factors of the first and last resonators of the
filter, respectively. Then, the inductance value of the second resonator is selected to meet
the following condition:
L2 > L1 (δk1,2 )2
(3.13)
The inductance values of inductors L3 , L4 , ..., LN −2 must be chosen to satisfy the following condition:

Li > 
2
δki−1,i

δk
− √i−2,i−1
√1
Li−1
Li−2
i = 3, 4, · · · , N − 2
, i ̸=
N +1
for Odd N
2
(3.14)
50
where ki,j represents the normalized coupling coefficient between the ith and j th resonators
of the coupled-resonator filter model shown in Fig. 3.2(e). For odd N values, the inductance value of the middle inductor L i+1 can be obtained from:
2
i−1
1
L i+1
2
=
2
∑
i−1
(−1)(k+1)
k=1
1
L( i+1 −k)
+
(−1)(k+1)
k=1
2
i = 3, 5, 7, · · · , N
2
∑
1
L( i+1 +k)
2
N : Odd.
(3.15)
For even N values, in addition to the condition specified by (3.14), the inductance
values chosen for L3 , L4 , ..., LN −2 must satisfy the following condition:
−1
2
∑
i
k=0
−1
2
∑
i
(−1)
k
1
L( i −k)
2
=
(−1)k
k=0
1
L( i +1+k)
2
i = 4, 6, 8, · · · , N
N : Even.
(3.16)
Finally, the value of the inductor of the second to last resonator LN −1 , shown in Fig. 3.2(e),
is chosen according to the following condition:
LN (δkN −1,N )2 < LN −1 < (
1
√
δkN −2,N −1
LN −2
+
)2
(3.17)
δkN −1,N
√
LN
The conditions specified in (3.13)-(3.17) indicate that the inductor values of the coupledresonator filter shown in Fig. 3.2(e) are not unique. In other words, for a given desired
frequency response, more than one choice of inductor values exist that results in the desired, a priori known, frequency response. This is a consequence of the fact that the
coupled-resonator filter shown in Fig. 3.2(e) is fundamentally an under-determined system. This, however, provides a degree of flexibility in choosing the parameters of the ENG
and MNG layers that result in a given response.
51
After determining all the inductance values of coupled-resonator filter, the capacitance
values of the first and last resonators can be obtained from:
δk1,2
√
L1 L2
1
δkN −1,N
CN = 2
− 2√
ω0 L N
ω0 LN −1 LN
C1 =
1
ω02 L1
−
ω02
(3.18)
(3.19)
The capacitance values of the remaining resonators are calculated using:
Ci =
1
ω02 Li
−
ω02
δki,i+1
δki−1,i
√
− 2√
Li−1 Li ω0 Li Li+1
i = 2, 3, · · · , N − 1.
(3.20)
The coupling capacitors can be obtained from:
Ci,i+1 =
ω02
δki,i+1
√
Li Li+1
i = 1, 2, · · · , N − 1.
(3.21)
The values of the capacitors (Ci∗ ) shown in Fig. 3.2(d) can be calculated from:
C1∗ = C1
(3.22)
CN∗ −1 = CN
(3.23)
∗
Ci∗ = Ci − Ci−1
i = 2, · · · , N − 2.
(3.24)
Via a π to T transformation, the series and parallel capacitance values of each T network
shown in Fig. 3.2(c) can be calculated from:
2Ci∗ Ci,i+1 + Ci∗ Ci∗
Ci∗
∗
2Ci Ci,i+1 + Ci∗ Ci∗
i
Cp =
Ci,i+1
Csi =
i = 1, 2, · · · , N − 1.
(3.25)
(3.26)
52
The electric plasma frequency of the ith ENG layer for a given thickness shown in Fig.
3.2(a) can be obtained from the previously calculated inductance values using (3.27):
√
1
1
i
fEN
ω02 +
i = 1, 2, · · · , N.
(3.27)
G =
i
2π
ϵ0 hEN G Li
where hiEN G is the thickness of the ith ENG layer and its specific value can be chosen
arbitrarily5 . The thickness and magnetic plasma frequency of ith MNG layer can be related to the capacitance values calculated in the previous steps as shown in the following
equations:
i
fM
NG
hiM N G
v
(
)
u
Cpi
i
u
ϵ
2
+
i
MNG
Cs
1t 2
=
ω0 +
2 i i
2π
Z0 Cs Cp
(
)
Cpi
−1
cosh
1 + Ci
s
=√
(
)
2
i
2
µ0 ϵ0 ϵiM N G (2πfM
)
−
ω
0
NG
(3.28)
(3.29)
i = 1, 2, · · · , N − 1.
(3.28) and (3.29) show that the magnetic plasma frequency and the thicknesses of the
MNG layers are frequency dependent. Similar to the process followed in Section 3.3.1, ω
is replaced by ω0 in (3.28) and (3.29) to obtain constant values of the thickness and the
magnetic plasma frequency.
5
The chosen value of hiEN G must be as small as possible to ensure that the approximations described in
Section 7.3 remain valid.
53
3.4
Verification of the Synthesis Procedure Using Full-Wave EM Simulations
3.4.1
Transmission Through A Single MNG Layer Sandwiched by Two
ENG Layers
In our first design example, we consider a three-layer structure composed of one MNG
layer sandwiched by two ENG layers on the two sides. Using the synthesis procedure
presented in Section 3.3.1, we will determine the geometrical and physical parameters of
different MNG and ENG layers that result in a flat transmission coefficient (a Butterworth
filter response) with a center frequency of operation of 2.4 GHz and a fractional bandwidth
of δ = 20%. Table 3.1 shows the normalized quality factors and the normalized coupling
coefficient of an ideal coupled-resonator filter with a second-order Butterworth response.
In this synthesis procedure, the relative permeability of the ENG layer, µEN G , and the relative permittivity of the MNG layer, ϵM N G , can be chosen arbitrarily. Close examination
of the procedures presented in Sections 3.3.1 and 3.3.2 reveals that choosing small µEN G
values enhances the agreement between the response of the proposed structure and an ideal
coupled-resonator filter response. This is due to the fact that the values of the series inductors LiS shown in Fig. 3.2(b) and Fig. 3.4(b) are directly proportional to µiEN G . Since the
effect of these series inductors are ignored in the proposed synthesis procedure, choosing
small relative permeability values reduces the impact of this assumption. Therefore, in
this synthesis procedure, we choose µEN G = 1.
Similarly, by examining the synthesis procedures presented in Sections 3.3.1 and 3.3.2,
the effect of choosing specific ϵM N G values on the accuracy of the proposed synthesis procedure can be evaluated. In particular, (3.10) and (3.28) show that ϵEN G can significantly
influence the value of the magnetic plasma frequency. Since the proposed synthesis procedure is based on the assumption that the entire operating frequency band of interest falls
54
Table 3.1 Normalized quality factors and coupling coefficients of the coupled-resonator
bandpass filters examined in this paper.
Filter Type
Order (N )
δ
q1
Butterworth
2
20%
1.4142 1.4142
1
Chebychev (0.5 dB ripple)
5
40%
1.8068 1.8068
1
Filter Type
rN
k1,2
Butterworth
1
0.7071
Chebychev (0.5 dB ripple)
1
0.6519
k2,3
qN
k3,4
r1
k4,5
0.5341 0.5341 0.6519
below the magnetic plasma frequency of the MNG material, the choice of ϵM N G can influence the validity of this procedure. Thus, (3.10) suggests that ϵM N G should be chosen
as large as possible to enhance the accuracy of the proposed synthesis procedure. For
the three-layer structure described above, if the synthesis procedure is carried out with
ϵM N G = 1, (3.10) predicts fM N G to be 2.42 GHz (the values of the remaining physical
and geometrical parameters of this structure are shown in Table 3.2). This value of fM N G
is in the middle of the desired transmission window. Above fM N G , the transmission-line
model shown in Fig. 3.2(a) will no longer be equivalent to that shown in Fig. 3.2(b)
and the entire synthesis procedure falls apart. The effect of ϵM N G on the proposed synthesis procedure is examined by designing the aforementioned tri-layer structure for four
different ϵM N G values. Following the synthesis procedure presented in Section 3.3.1, the
parameters of this structure are obtained for ϵM N G = 1, 10, 100, and 200 and the results
are presented in Table 3.2 (Case 1 - Case 4). The frequency response of these structures
are calculated using full-wave numerical EM simulations in CST studio and the results are
presented in Fig. 3.5. As can be seen, by increasing ϵM N G , the agreement between the
response of the ideal filter and that of the tri-layer structure is improved as expected.
55
R: εMNG = 1
R: εMNG = 10
R: εMNG = 100
R: εMNG = 200
R: Ideal
T: εMNG = 1
T: εMNG = 10
T: εMNG = 100
T: εMNG = 200
T: Ideal
Figure 3.5 The reflection (left) and transmission (right) coefficients of the tri-layer
structure studied in Section 3.4.1 for different ϵM N G values obtained using full-wave EM
simulations in CST Microwave Studio. The desired (ideal) filter response is also shown
as reference. The physical parameters of the structures are provided in Table 3.2 (Case 1 Case 4). These results show the effect of the dielectric constant of the MNG layer of a
tri-layer structure (ENG-MNG-ENG) on the accuracy of the synthesis procedure
described in Section 3.3.1.
The thickness of the (thin) ENG layers constituting the structure, hEN G , is another design parameter that can be chosen freely. To examine the effect of this design parameter
on the frequency response of the tri-layer structure examined here, three different structures with different hEN G values are designed following the synthesis procedure presented
in Section 3.3.1. The physical parameters of these are provided in Table 3.2 (Case 4 Case 6) and their frequency responses are calculated in CST Microwave Studio. Figure
3.6 shows the simulated frequency responses of these structures as well as the frequency
response of the ideal coupled-resonator bandpass filter with a second-order Butterworth
response with f0 = 2.4 GHz and δ = 20%. As can be seen, in general, a good agreement is observed between the ideal filter responses and the transmission coefficients of
56
Table 3.2 Physical parameters and dimensions of the tri-layer structures studied in
Section 3.3.1.
1,2
1
2
hEN G fEN G = fEN
hM N G
fM N G
ϵM N G
G
Case 1
500 µm
40.3 GHz
368 mm
2.42 GHz
1
Case 2
500 µm
40.3 GHz
36.8 mm
2.63 GHz
10
Case 3
500 µm
40.3 GHz
3.68 mm
4.19 GHz
100
Case 4
500 µm
40.3 GHz
1.84 mm
5.41 GHz
200
Case 5
100 µm
90.0 GHz
1.84 mm
5.41 GHz
200
Case 6
10 µm
284.5 GHz
1.84 mm
5.41 GHz
200
the ENG-MNG structures. Additionally, it can be observed that as hEN G decreases, the
agreement between the simulated transmission coefficients and the ideal filter response
improves as described earlier in this sub-section. Based on the results shown in Figs. 3.5
and 3.6, it can be seen that the equivalency between the multi-layer structure shown in Fig.
3.1 and an N th -order coupled-resonator bandpass filter is best when the ENG layers are
extremely thin and the MNG layers have high relative permittivity values.
3.4.2
EM Wave Tunneling Through Four MNG Layers Sandwiched by
Five ENG Slabs
The second structure that we consider in this paper is a multi-layer structure composed
of four MNG layers sandwiched by five ENG layers. Following the synthesis procedure of
Section 3.3.2, we seek to determine the physical and geometrical parameters of the structure that results in an equal-ripple transfer function centered at 2.4 GHz with a fractional
bandwidth of δ = 40%. As described in Section 3.3.2, the multi-layer structure shown
in Fig. 3.1 with N ENG layers is equivalent to an N th -order banspass filter. Therefore,
57
T: Ideal
T: hENG=0.01mm
T: hENG=0.1mm
T: hENG=0.5mm
R: Ideal
R: hENG=0.01mm
R: hENG=0.1mm
R: hENG=0.5mm
Figure 3.6 The transmission and reflection coefficients of the tri-layer structure
composed of one MNG layer sandwiched by two ENG layers discussed in Section 3.4.1
obtained using full-wave EM simulations in CST Studio. The desired (ideal) filter
response is also shown for comparison. The physical parameters of the structures are
given in Table 3.2 (Case 4-Case 6).
in synthesizing the proposed structure, the parameters of a fifth-order coupled-resonator
bandpass filter with an equal-ripple response type, shown in Table 3.1, are used. As described in Section 7.3, such a fifth-order bandpass filter is an under-determined system and
more than one combination of element values exist that result in a given transfer function.
Table 3.3 shows the element values of the equivalent circuit model of this filter determined
from (3.12)-(3.21). Using these values in conjunction with (3.22)-(3.29), the physical parameters of the different ENG and MNG layers constituting the structure are calculated
and the results are presented in Table 3.4. The transmission and reflection coefficients of
this multi-layer structure are also calculated using full-wave numerical EM simulations in
CST Microwave studio and presented in Fig. 3.7. Additionally, Fig. 3.7 shows the trans-
58
T: Ideal
T: Analytical
T: Full Wave
R: Ideal
R: Analytical
R: Full Wave
Figure 3.7 The transmission and reflection coefficients of the multi layer structure
composed of four MNG layers sandwiched by five ENG layers discussed in Section 3.4.2
obtained from full-wave EM simulations in CST Studio. The desired (ideal) filter
response is also shown for comparison. The physical parameters of this structures are
given in Table 3.4.
mission and reflection coefficients of an ideal coupled-resonator filter with a fifth-order
equal-ripple response. As can be observed, in general, a good agreement between the two
results is observed. The discrepancies can be attributed to the finite thickness of the ENG
layers as well as the effect of ϵM N G on the accuracy of the response as described in Section 3.3.1. Close examination of the ideal filter response shown in Fig. 3.7 shows that the
ripples in the transmission coefficient of the filter are not of equal magnitude. This is due
to the fact that the well-known coupled-resonator filter synthesis procedure that we used
in this work is based on a narrow-band approximation [63]. As the bandwidth of the filter
increases, the accuracy of this approximation reduces as can be observed from the transfer
function of the ideal filter shown in Fig. 3.7. Nonetheless, the transmission coefficient
59
of the multi-layer ENG-MNG structure closely follows the response of the ideal filter as
expected.
Table 3.3 Equivalent circuit values of the fifth-order coupled-resonator filter of the type
shown in Fig. 3.2(e), which is studied in Section 3.3.2.
3.4.3
Parameter
C1 = C5
C2 = C4
C3
L1 = L5
Value
0.502 pF
0.957 pF
0.91 pF
5.535 nH
Parameter
L2 = L4
L3
C1,2 = C4,5
C2,3 = C3,4
Value
2.767 nH
2.767 nH
0.29 pF
0.34 pF
The Effect of Loss
Complete transmission of EM waves through the multi-layer structure shown in Fig.
3.1 is only possible when the structure is lossless. In reality, however, both the ENG
and MNG layers will be lossy. In the Drude model used for ϵEN G (µM N G ), this loss can
be modeled by considering the relative permittivity (relative permeability) of the ENG
(MNG) material to be a complex number. This can be done by considering a nonzero
electric (magnetic) collision frequency term, νEN G (νM N G ), in the Drude model used for
the relative permittivity (relative permeability) of the ENG (MNG) materials:
To demonstrate the effect of the material loss on the response of the structure, the total
power loss of several multi-layer structures of the type shown in Fig. 3.1 are calculated
and the results are presented in Fig. 3.8. In obtaining these results, it is assumed that all
of these structures have a Butterworth response type centered at 2.4 GHz and use lossy
ENG and MNG layers with νEN G /ω = 0.005 and νM N G /ω = 0.005. Figure 3.8 shows
the percentage of the incident power lost in the structure as a function of the fractional
60
Total Power Loss [%]
100
2 ENG Layers
80
3 ENG Layers
4 ENG Layers
60
5 ENG Layers
40
20
0
10
15
20
25
30
35
40
Fractional Bandwidth ( ) [%]
Figure 3.8 The percentage of the total power lost in the multi-layer structure shown in
Fig. 3.1 as a function of the number of ENG layers, N , and the fractional bandwidth, δ.
The results are obtained for multi-layer structures exhibiting a maximally flat bandpass
response centered at 2.4 GHz. It is assumed that the ENG and MNG materials are lossy
with νEN G /ω = νM N G /ω = 0.005.
bandwidth of the transmission window as well as the number of ENG layers, N 6 . As can
be observed, the power loss increases with increasing the number of the layers (i.e., the
order of the filter) and reduces with increasing the fractional bandwidth of the transmission
window. This trend is consistent with the behavior of microwave filters, where losses
increase with decreasing the bandwidth of the filter and increasing the order of the filter
response [65].
6
The number of MNG layers is N-1
61
MNG
Layers
ENG
Layers
MNG Layers
2
1
hMNG
hMNG
N-1
N
hMNG
hMNG
x
y
x
z
1
ENG
h
2
ENG
h
2
1
z hENG hENG
N-2
N-1
hENG
hENG
N
N-1
hMNG
hMNG
2
1
hMNG
hMNG
ENG Layers
(a)
L 1 C1
Z0
C2
L1,2
N-2 N-1
hENG
hENG
(b)
L2
C3
L2,3
L3
LN-1 CN-1 CN LN
LN-1,N
Z0
(c)
Figure 3.9 (a) 3D topology of the multi-layer structure that is dual of the one shown in
Fig. 3.1. In this case, the structure is composed of N MNG layers separated from each
other by N − 1 ENG layers. (b) Side view of the dual structure. (c) Following a
procedure similar to that described in Section 7.3, it can be shown that the multi-layer
structure of parts (a)-(b) can be modeled with a coupled-resonator filter composed of
series LC resonators coupled to each other using parallel inductors.
3.5
EM Wave Tunneling Through the Dual Multi-Layer MNG-ENG
Structure
If we apply the duality principle to the structure shown in Fig. 3.1, a new multi-layer
structure can be obtained in which N MNG layers sandwich N − 1 ENG layers. This
transforms the problem shown in Fig. 3.1 to that shown in Fig. 3.9. This change, however,
does not change the response of the structure. Following a similar analysis procedure as
the one presented in Section 7.3, it can be shown that the dual structure shown in Fig.
62
Table 3.4 Physical parameters of the nine-layer ENG-MNG composite structure
examined in Section 3.3.2.
1
5
2
3
4
Parameter
fEN
fEN
G = fEN G
G = fEN G = fEN G
Value
227.3 GHz
321.6 GHz
Parameter
h1M N G = h4M N G
1
4
fM
N G = fM N G
Value
690.7 µm
8.4 GHz
Parameter
hEN G
ϵM N G
Value
10 µm
200
Parameter
Value
h2M N G = h3M N G
606.6 µm
2
3
fM
N G = fM N G
8.7 GHz
3.9 is analogous to a coupled-resonator filter composed of series LC resonators coupled to
each other using parallel inductors as depicted in Fig. 3.9(c). This coupled-resonator filter
is in fact dual of the circuit shown in Fig. 3.2(e). Rather than developing a new synthesis
procedure for this structure, the synthesis procedure presented in Section 7.3.2 can be used
in conjunction with the duality principle to synthesize structures of the type shown in Fig.
3.9. This can be demonstrated by examining the dual of the tri-layer structure described in
Section 3.4.1. The dual structure is composed of an ENG layer sandwiched by two MNG
layers. The parameters of this multi-layer structure are provided in Table 3.2 (Case 4 after
exchanging the ENG subscripts MNG and vice versa). The response of this structure is
also calculated using full-wave EM simulations in CST Microwave Studio. Figure 3.10
shows the reflection and transmission coefficients of this structure and compares them with
those of the tri-layer structure examined in Section 3.3.1 (with two ENG and one MNG
layers). As expected, the results match perfectly.
63
Figure 3.10 The transmission and reflection coefficients of the tri-layer structure
discussed in Section 3.4.1 and its dual structure examined in Section 3.5. These results
are obtained from full-wave EM simulation in CST Studio and show perfect agreement.
The physical parameters of the structures are presented in Table 3.2 (Case 4).
3.6
Conclusions
We examined the close relationship that exists between EM wave tunneling through
stacks of single-negative metamaterial slabs and two classical coupled-resonator microwave
filters. We demonstrated that a structure composed of N thin ENG layers separated from
each other by N −1 MNG layers is equivalent to a capacitively-coupled, coupled-resonator
filter with an N th -order bandpass response. Assuming Drude dispersion models for the
relative permittivity (permeability) of the ENG (MNG) layers, we developed an analytical
synthesis procedure and showed that it can be used to determine the physical parameters
of these multi-layer structures from their a priori known desired transfer functions. Using
this synthesis procedure, we found out that the MNG layers must have high relative dielectric constant values to ensure that their magnetic plasma frequencies will not fall within the
desired transmission window (and hence, invalidate the assumption that the materials have
64
negative permeability values). This condition is a consequence of the specific dispersion
model assumed for the MNG material. Several design examples were used in conjunction
with full-wave EM simulations to demonstrate the validity of the proposed analogy and
examine the accuracy of the proposed synthesis procedure. The conclusions of the main
study were also expanded to the dual problem using the duality principle.
65
Chapter 4
High-power microwave filters and frequency selective surfaces exploiting electromagnetic wave tunneling through ϵnegative layers
4.1
Introduction
Electromagnetic (EM) waves are evanescent and exponentially decay along the direc-
tion of propagation in materials with negative relative permittivity (ϵ-negative or ENG) or
negative permeability (µ-negative or MNG). However, in certain multi-layer arrangements
where one or more of the layers is a material with a negative constitutive parameter, resonant tunneling of EM waves can occur and the combined multi-layer structure can become
completely transparent. This phenomenon, in its various forms, has been investigated by
a number of different research groups [57, 27, 31, 32, 60, 30, 61, 33] and its various applications at the RF [73], microwave [74], millimeter-wave [75], and optical frequency
[76, 77] bands have been examined. Recently, we presented a theoretical examination of
the phenomenon of EM wave tunneling through multi-layer arrangements of ENG layers
sandwiched by double-positive (DPS) layers [78] and demonstrated that this phenomenon
can be explained using a classical microwave filter theory. Furthermore, it was demonstrated that a multi-layer structure composed of N ENG layers that are separated by N + 1
DPS layers acts as a coupled-resonator bandpass filter of order N + 1 and a method for
synthesizing this filter from its a priori known transfer function was presented.
66
In addition to explaining an important physical phenomenon using a classical microwave circuit theory, the theoretical studies reported in Ref. [78] suggest an unconventional method for designing microwave filters and frequency selective surfaces (FSSs)1 .
Namely designing passive microwave devices that use only cascaded plasma and dielectric layers. Depending on the method used to implement or emulate the plasma layers,
these devices can be implemented either without using any metallic structure or with minimal use of metals. Therefore, devices of the type reported in Ref. [78] are expected
to be amenable to operation at significantly higher peak power levels compared to conventional microwave devices. This is practically significant in high-power microwave applications, since many conventional methods used to design microwave filters and FSSs
require the use of metallic structures that create significantly high local electric field intensities within the device. When excited at high peak power levels, the field intensity within
these devices can exceed the breakdown limit and the device can ultimately fail. The
physical mechanisms behind the breakdown phenomenon have been studied theoretically
[79, 80, 81, 82, 55, 83] and experimentally [84, 85] in the past. It has been demonstrated
that the threshold at which breakdown occurs depends on the electrical strength of the dielectric materials (or gas) surrounding the device, humidity, and the ratio of pressure to
temperature [82]. Since many different factors impact the formation of electron avalanche
and the eventual breakdown of the device, predicting breakdown events tends to be difficult. Nonetheless, by minimizing the electric field enhancement within a given device,
its breakdown threshold can be enhanced and its peak power handling capability can be
improved.
In this paper, we revisit the theoretical concepts presented in Ref. [78] and investigate
methods that can be used to implement multi-layer filters of the type proposed there. In
1
An FSS acts as a spatial filter for electromagnetic wave propagating in free space
67
particular, we demonstrate that the ENG (or plasma) layers can be implemented using sections of rectangular waveguides, which operate below their cutoff frequencies. Using this,
we design, fabricate, and experimentally characterize a three-layer structure composed of
two DPS and one ENG layers. Using measurements conducted at microwave frequencies,
we demonstrate that this multi-layer structure does indeed act as a second-order bandpass
filter as predicted in Ref. [78]. We also examine the power handling capability of this
structure both theoretically and experimentally. Experimental high-power measurements
are conducted using a high-power magnetron source operating at 9.382 GHz and having
a pulse duration of 1 µs and a peak power of 25 kW. We demonstrate that the vertical
dimensions of the cutoff waveguide used to implement the ENG layer is the main factor
that limits the peak power handling capability of this structure. We examine an alternative
method for implementing the ENG layer, which is more suitable for frequency selective
surface applications. Moreover, this implementation method still lends itself to operation
at extremely high power levels. In particular, we demonstrate that ENG layers capable of
handling extremely high power levels can be implemented by using a thin sheet of metal
perforated with very small sub-wavelength holes. Using this concept, a second prototype is
designed, fabricated, and experimentally characterized both at low and high power levels.
It is demonstrated that the proposed structure can handle extremely high peak power levels
without breaking down. Therefore, this technique presents a viable method for designing
microwave filters and frequency selective surfaces that are capable of handling very high
peak power levels.
68
ε-Negative Layer
ε-Negative Layer
hENG
hENG
x
x
z
y
z
1
hDPS
2
DPS
h
High-εr Dielectric
Layers
(a)
2
1
hDPS
hDPS
High-εr Dielectric
Layers
(b)
Figure 4.1 (a) 3D topology of a tri-layer structure composed of an ϵ-negative layer
sandwiched by two high-permittivity dielectric layers. Each layer extends to infinity in x
and y directions. (b) Side view of the structure.
4.2
A tri-layer spatial filter composed of an ENG layer sandwiched by
two DPS layers
Figure 4.1(a) shows the 3D topology of a tri-layer structure composed of an ENG layer
sandwiched by two high-permittivity dielectric layers. The three layers are assumed to be
infinite in x and y directions and have finite thicknesses along z direction. The side view of
this arrangement is shown in Fig. 4.1(b) and it is assumed that this structure is illuminated
by a vertically incident transverse electromagnetic (TEM) wave.
In Ref. [78], it was demonstrated that such a structure can become completely transparent to the incident EM wave provided that the constitutive parameters of the DPS and
ENG layers are chosen appropriately. In particular, a synthesis formula was provided that
could be used to determine the physical and electrical parameters of the three layers from
its a priori known transfer function. Following this synthesis procedure, we design the
69
Transmission Coefficient
Reflection Coefficient
Figure 4.2 The reflection and transmission coefficients of the tri-layer structure shown in
Fig. 4.1 with physical and geometrical parameters provided in Table 4.1. The
transparency window is centered at 9.382 GHz and it has a factional bandwidth of 50%.
Table 4.1 Physical parameters of the tri-layer structure composed of two DPS and one
ENG layers examined in Section 7.3.
Parameter
ϵDP S
h1,2
DP S
hEN G
fEN G
Value
10
1.81 mm
900 µm
60 GHz
three-layer structure shown in Figs. 4.1(a)-4.1(b) to have a transparency window centered
at 9.382 GHz with a fractional bandwidth of 50%.
2
Following the synthesis procedure provided in Ref. [78], the physical parameters of
the tri-layer structure that results in this frequency response are determined. Since the
dielectric constant of the DPS layers is a free design parameter [78], its value can be chosen
2
The fractional bandwidth is defined as the ratio of the bandwidth of the transmission window to its center
frequency. Bandwidth is defined as the range of frequencies over which the power transmission coefficient
of the device is larger than 1/2 of the value of the transmission coefficient at the center frequency.
70
DPS Layer ENG Layer DPS Layer
(a)
DPS Layer ENG Layer DPS Layer
(b)
Figure 4.3 The electric and magnetic fields within the tri-layer DPS-ENG-DPS device
studied in Section 7.3. (a) Normalized magnitude. (b) Phase. The results are obtained
assuming that a TEM wave, propagating along the +z axis, is normally incident on the
structure. The magnitudes of the total |E| and |H| are normalized respectively to the
magnitudes of the incident electric and magnetic fields. The reference phase plane (zero
phase) is at z = -2 mm.
arbitrarily. As described in Ref. [78], the accuracy of the synthesis procedure is better for
large values of ϵDP S . In this design example, however, we chose DPS layers with relative
permittivity values of ϵDP S =10, since such materials are readily available and have lower
costs compared to materials with higher dielectric constant values. Table 4.1 provides the
final electrical and physical parameters of this optimized tri-layer structure. This structure
is then simulated using full-wave EM simulations in CST Microwave Studio. The ENG
layer is modeled as a dispersive lossless plasma with a Drude model that characterizes its
frequency dependent dielectric constant. The transmission and reflection coefficients of
this structure are calculated and the results are presented in Fig. 4.2. Observe that the
tri-layer structure shown in Fig. 4.1 shows a transparency window spanning the frequency
range of 7.05 GHz - 11.76 GHz, centered at 9.382 GHz, as expected.
To examine the physical process that results in the observed tunneling phenomenon,
the structure shown in Fig. 4.1 is examined analytically. In particular, analytical expressions for the electric and magnetic fields within the DPS and ENG layers are obtained.
71
1
0.9
1.4
0.8
0.7
1.3
0.6
1.2
0.5
0.4
1.1
0.3
0.2
1
0.1
0.9
5 6 7 8 9 10 11 12 13 14 15 0
1
0.9
0.8
0.7
2.8
0.6
2.6
0.5
0.4
2.4
0.3
0.2
2.2
0.1
2
5 6 7 8 9 10 11 12 13 14 15 0
3
hDPS [mm]
fENG [GHz]
Frequency [GHz]
(a)
1.5
hENG [mm]
1
0.9
0.8
0.7
85
0.6
0.5
80
0.4
75
0.3
70
0.2
65
0.1
60
5 6 7 8 9 10 11 12 13 14 15 0
100
95
90
Frequency [GHz]
(b)
Frequency [GHz]
(c)
Figure 4.4 The power transmission coefficients ,|T |2 , of the tri-layer structure studied in
Section 7.3 as a function of frequency and (a) Plasma frequency of the ENG layer, fEN G ,
(b) the thickness of the ENG layer, hEN G , and (c) the thickness of DPS layers, hDP S .
Fig. 4.3(a) shows the magnitudes of the total electric and magnetic fields within this trilayer structure at 9.382 GHz and Fig. 4.3(b) shows the phase of the electric and magnetic
fields. The results are obtained based on the assumption that a TEM wave propagating
in the +z direction is normally incident on the tri-layer structure. Additionally, the magnitudes of the total electric and magnetic fields are normalized to the magnitudes of the
incident electric and magnetic fields respectively and the plane z = -2 mm is assumed to
be the reference plane for phase calculations (i.e., phase of zero). As can be observed, the
EM wave completely tunnels through the structure and a transmission coefficient of unity
is achieved. More importantly, the field distribution within the tri-layer structure shows a
mode pattern, which is characteristic of a resonant structure. This indicates that the complete transparency observed in this DPS-ENG-DPS structure is due to resonant tunneling
as demonstrated in Ref. [78] using the classical circuit theory.
Another important feature of this resonant mode is that the magnitude of the electric
field within this structure decreases compared to that of the incident EM wave. In particular, within the ENG layer, the magnitude of the electric field is reduced to about one half
of that of the incident EM wave. This is beneficial in high-power applications where air or
72
dielectric breakdown can occur if the electric field intensity within the structure exceeds
the breakdown level. Since the electric field intensity within the structure does not increase
compared to that of the incident EM wave, the DPS-ENG-DPS structure is more amenable
to high-power operation.
Using the same analytical method, the effects of the important physical parameters of
this structure on its transmission coefficient are also examined. Fig. 4.4(a) shows the effect of the plasma frequency of the ENG layer on the power transmission coefficient of
the structure. As can be seen, when all the other physical parameters are fixed, increasing fEN G increases the frequency of the transmission window. Fig. 4.4(b) shows that the
thickness of the ENG layer primarily determines the spectral width of the transmission
window and does not significantly affect the center frequency where maximum transmission coefficient occurs. Finally, Fig. 4.4(c) shows that the thickness of the DPS layer
primarily determines the center frequency where peak transmission occurs. In particular,
when all other parameters are fixed, increasing hDP S significantly reduces the center frequency of operation. The results shown in Fig. 4.4 further confirm the resonant nature
of the tunneling phenomenon observed in this structure. As is seen from Fig. 4.4, the
frequency at which resonance occurs is primarily determined by the plasma frequency of
the ENG layer and the thickness of the DPS layer, for a fixed ϵDP S . This observation is
also consistent with the circuit-based explanation of the resonant tunneling phenomenon
provided for this structure in Ref. [78].
4.3
Practical implementation of the DPS-ENG-DPS filter
4.3.1
Emulating Plasma
To practically realize a structure of the type reported in the previous section, a relatively
thick ENG layer with a high plasma frequency should be used, as can be observed from
73
Vector Network
Analyzer (VNA)
High-εr dielectric layers
bc
WR90
Waveguide
x
y
aDPS
ac
b
Cutoff Waveguide DPS
(Emulating ENG Layer)
Figure 4.5 The tri-layer structure shown in Fig. 4.1 is implemented using a section of
cutoff waveguide sandwiched by two high-ϵr dielectric slabs. The waveguide that
emulates the ENG layer operates below its cutoff frequency and has an opening of ac ×
bc and a thickness of hc . Both dielectric slabs have the dimensions of aDP S × bDP S and
thickness of hDP S . This structure is placed inside a standard WR-90 rectangular
waveguide and its frequency response is measured using a calibrated vector network
analyzer.
Table 4.1. Such a plasma layer can be emulated conveniently using a waveguide operating
below its cutoff frequency [86]. The propagation constant and mode impedance for a
rectangular waveguide operating in its dominant Transverse Electric (T E10 ) mode are:
√
β = 2πf µ0 ϵ0
√
1−(
Z0
ZT E = √
1 − (fc /f )2
fc 2
)
f
(4.1)
(4.2)
where f is frequency, µ0 = 4π × 10−7 (H/m), ϵ0 = 8.854 × 10−12 (F/m), Z0 = 377
Ω, and fc is the cutoff frequency of the T E10 mode. As can be seen from (1)-(2), when
f < fc , the propagation constant and the mode impedance both become imaginary. Moreover, using (1)-(2), it can be shown that a section of such a cutoff waveguide acts as a
homonegeous material with a permittivity of ϵ0 (1 − (fc /f )2 ) and a permeability of µ0 .
74
This way, such a cutoff waveguide section can effectively emulate a layer of plasma with
a plasma frequency of fc .
Emulation of an ENG layer with a cutoff waveguide is most suitable for test and experiments conducted in a waveguide environment. In a waveguide, however, the dominant
propagating mode is not TEM and hence, the physical and geometrical values reported in
Table 4.1 cannot be directly used. Nevertheless, the physical phenomenon of EM wave
tunneling does not change (i.e., transparency can still be achieved), since the dominant
T E10 mode of a rectangular waveguide is simply a superposition of two TEM plane waves
propagating with different incidence angles within the waveguide. Thus, the same tunneling phenomenon will be observed when a waveguide version of the tri-layer structure
shown in Fig. 4.1 is used.
A waveguide version of the tri-layer structure shown in Fig. 4.1 was designed using
full-wave EM simulations in CST Microwave Studio. In doing so, WR-90 waveguides
with inner dimensions of 22.86 mm × 10.16 mm and operating in the 8-12 GHz frequency range are used. The two DPS layers are implemented using dielectric slabs with
dimensions of 22.86 mm × 10.16 mm and thicknesses of 1.27 mm. The ENG layer is
implemented with a section of a rectangular waveguide with inner dimensions of 9.5 mm
× 4.28 mm and a length of 6.33 mm. At the desired center frequency of operation, this
waveguide is cutoff and acts as an ENG layer as previously explained.
The waveguide version of the tri-layer structure shown in Fig. 4.1 was fabricated and
a photograph of its different constituting layers is shown in Fig. 4.5. The physical parameters of this prototype are listed in Table 4.2. The transmission and reflection coefficients
of this structure are measured using a vector network analyzer and the results are shown in
Fig. 4.6. The simulated results obtained using full-wave EM simulations are also shown
for comparison. In general, a good agreement is observed between the two results. The
75
Simulation
Measurement
Figure 4.6 The measured and simulated transmission coefficients of the tri-layer
structure shown in Fig. 4.5. Measurement results are obtained using a calibrated vector
network analyzer and the simulation results are obtained using full-wave EM simulations
in CST Microwave Studio.
discrepancies observed between the simulation and measurement results are primarily attributed to the small differences between the physical parameters of the fabricated device
and those of the modeled one. In particular, the exact dimensions of the cutoff waveguide
that emulates the ENG layer are determined by the accuracy of the machining process
used to fabricate it. Additionally, there is a 2.5% uncertainty in the precise value of the
dielectric constant of the substrates used to implement the DPS layers. Nevertheless, the
measurement results show a transparency window which covers the operating frequency
of 9.382 GHz as expected.
4.3.2
High Power Measurements
The measurement results shown in Fig. 4.6 are conducted when the tri-layer structure
is illuminated with a relatively low power level of 0.3 mW. However, the operation of this
76
E-H Tuner
Reflection
Detector
Transmission
Detector
DUT
Magnetron
Match Load
Figure 4.7 Experimental test-bed used to measure the transmission and reflection
coefficients of the DPS-ENG-DPS filters under high-power excitation levels. The device
under test (DUT) is illuminated at various power levels to detect the level at which air
breakdown occurs. The magnetron source generates a short pulse with a single frequency
of 9.382 GHz, a duration of 1 µs and a peak power level of 25 kW.
device under high-power excitation conditions is also of interest. To investigate this, we
examined the peak power handling capability of this tri-layer filter using the experimental
test-bed shown in Fig. 4.7. The tri-layer structure shown in Fig. 4.5 is placed inside a standard WR-90 waveguide. A high power magnetron source is used to illuminate the device
with a single-frequency signal (9.328 GHz) that lasts for approximately 1 µs. The source
is capable of generating a maximum output power level of 25 kW and the actual output
power level can be controlled by using the combination of a circulator and an E-H tuner
that are also shown in Fig. 4.7. Two directional couplers connected to crystal detectors are
used in series with the waveguide setup to sample the amplitude of the transmitted and the
reflected signals. The outputs of the crystal detectors are connected to an oscilloscope and
are measured as a function of time.
77
|T|2: 4.2 kW
(a)
|R|2: 4.2 kW
|T|2: 4.4 kW
|R|2: 4.4 kW
(b)
Figure 4.8 Normalized reflection and transmission coefficients of the DPS-ENG-DPS
structure examined in Section 7.3.2. The measurements are conducted in time domain
using the experimental setup shown in Fig. 4.7. (a) Incident power level of 4.2 kW. (b)
Incident power level of 4.4 kW. As can be seen, air breakdown occurs at about 200 ns
after the start of the high-power pulse.
The power handling capability of the device is determined by gradually increasing
the input power level and monitoring the transmission and reflection coefficients of the
device to identify when breakdown occurs. When the incident power level is below the
breakdown level, the power transmission coefficient of the device is close to 1 and the
power reflection coefficient is close to zero. When the device breaks down in air, a streaming plasma is created between the top and bottom walls of the waveguide that emulates
the ENG layer and effectively short circuits this waveguide. This short circuit results in
increasing the reflection coefficient of the device and decreasing its transmission coefficient. Thus, by monitoring the time-domain reflection and transmission coefficients of the
device and gradually increasing the input power level, the breakdown threshold can be
experimentally determined.
78
Table 4.2 Physical and geometrical parameters of the DPS-ENG-DPS filter examined in
Section 7.3.2. The photograph of the device is shown in Fig. 4.5.
Parameter
ϵDP S
h1,2
DP S
a1,2
DP S
b1,2
DP S
Value
10.2
1.27 mm
22.86 mm
10.16mm
Parameter
hc
ac
bc
Value
6.33 mm
9.5 mm
4.28 mm
Table 4.3 Physical parameters of DPS-ENG-DPS filter examined in Section 4.4-A. The
photograph of the device is shown in Fig. 4.10.
Parameter
ϵDP S
h1,2
DP S
a1,2
DP S
b1,2
DP S
Value
10.2
1.4 mm
22.86 mm
10.16mm
Parameter
hc
ac
bc
Value
4 mm
7.8 mm
10.16 mm
Table 4.4 Physical parameters of DPS-ENG-DPS structure examined in Section 4.4-B.
The photograph of the device is shown in Fig. 4.14.
Parameter
ϵDP S
1,2
hDP
S
a1,2
DP S
b1,2
DP S
Value
10.2
1.9 mm
22.86 mm
10.16mm
Parameter
hwg
awg
bwg
wwg
Value
0.1016 mm
2.34 mm
0.816 mm
0.2 mm
Figures 4.8(a) and 4.8(b) show the normalized power reflection and transmission coefficients in time domain for the incident power levels of 4.2 kW and 4.4 kW, respectively.
The normalized power reflection coefficient ,|R|2 , is defined as the ratio of the detected
reflected signal when the DUT is placed in the waveguide environment to the detected
79
V/m
2000
x
4
x 10 V/m
5
4. 5
4
3. 5
3
2. 5
2
1. 5
1
0. 5
x
1500
y
1000
y
500
0
(a) WR90 Waveguide
Cutoff Waveguide
(b) WR90 Waveguide
Figure 4.9 The magnitude of electric field distribution over the cross section of the
waveguide version of the tri-layer structure shown in Fig. 4.5 at (a) input port of the
standard WR-90 waveguide (b) the interface between the dielectric substrate and the
waveguide operating below the cutoff frequency. In both cases, the incident power level
is 1 Watt.
reflected signal when the DUT is replaced by a PEC termination. Similarly, the normalized power transmission coefficient ,|T |2 , is defined as the ratio of the detected transmitted
signal when the DUT is placed in the waveguide environment to the detected transmitted
signal when the DUT is removed. As can be seen from Fig. 4.8(a), when the input power
level is 4.2 kW, the transmission coefficient is close to 1 and the reflection coefficient is
close to zero. This is expected, since the frequency of the incident signal is well within
the transmission window of the device. As the incident power is increased to 4.4 kW, air
breakdown occurs shortly after the start of the illumination. Figure 4.8(b) shows that the
power reflection coefficient of the device starts to increase approximately 200 ns after the
start of the incident pulse. Similarly, the transmission coefficient of the device decreases
as expected.
80
Dielectric Layers
x
aDPS
am
y
bm
bDPS
Modified Cutoff Waveguide
Figure 4.10 A modified waveguide version of the tri-layer structure shown in Fig. 4.5 is
composed of a section of cutoff waveguide with rectangular cross section sandwiched by
two high-ϵr dielectric substrates. This rectangular waveguide operates below its cutoff
frequency and has an opening of am × bm and a thickness of hm . The height of this cutoff
waveguide (bm ) is increased compared to the height of the cutoff waveguide of Fig.
4.5(b) to increase the power handling capability of the ENG layer. Both dielectric slabs
have the dimensions of aDP S × bDP S and a thickness of hDP S .
4.4
Enhancing the Power Handling Capability of the DPS-ENG-DPS
filter
4.4.1
High-power capable DPS-ENG-DPS filter using cutoff waveguides
In Section 7.3.2, we showed that the waveguide version of the tri-layer structure shown
in Fig. 4.5 has a transient peak power handling capability of 4.2 kW. The peak power
handling capability of this structure is limited by the maximum local electric field intensity
within the structure. Figure 4.9 shows the magnitude of the electric field distribution at two
locations over the cross section of the waveguide version of the tri-layer structure shown
in Fig. 4.5. Figure 4.9(a) shows the E-field distribution in the cross section of the input
waveguide (outside of the tri-layer device) for an incident power density of 1 Watts. As
can be observed, the maximum electric field intensity associated with this incident wave
is approximately 2000 V/m. Figure 4.9(b) shows the electric field distribution in the cross
81
V/m
2000
x
x
1500
y
1000
V/m
3500
3000
2500
y
2000
1500
1000
500
500
0
(a) WR90 Waveguide
0
Modified
WR90 Waveguide
Cutoff Waveguide (b)
Figure 4.11 Magnitude of the electric field distribution in the cross section of the
DPS-ENG-DPS structure shown in Fig. 4.10. (a) E-field distribution in the cross section
of the input waveguide (outside of the device) for an incident power level of 1 W. (b)
E-field distribution in the cross section at the boundary between the DPS layer and the
cutoff waveguide emulating the ENG layer (the boundary closer to the input) for an
incident power level of 1 W.
section of the waveguide at the boundary of the DPS and the ENG layer (the boundary
closer to the input). As can be observed, the maximum electric field intensity is now
increased to approximately 50,000 V/m. This corresponds to a local field enhancement
factor of approximately 25. This local electric field enhancement causes air breakdown
when the input power exceeds the breakdown threshold level of air. This limits the power
handling capability of the tri-layer structure shown in Fig. 4.5. To increase the power
handling capability of this structure, the local electric field intensity in the ENG layer
should not increase significantly compared to the electric field intensity of the incident
EM wave.
Examination of the field distribution plots shown in Fig. 4.9 reveals that the large electric field intensity observed in the ENG layer of the structure shown in Fig. 4.5 is due to
the small vertical dimension of the cutoff waveguide used to implement it. While the horizontal dimension of this cutoff waveguide is determined by the plasma frequency of the
ENG layer, its vertical dimension is not restricted in this design. Therefore, to maximize
82
Simulation
Measurement
Figure 4.12 The simulated and measured transmission coefficients of the DPS-ENG-DPS
filter shown in Fig. 4.10.
the power handling capability of the device, the vertical dimension of the cutoff waveguide
must be selected to be as large as possible. In reality, the vertical dimension (the height)
of the cutoff waveguide can be chosen to be the same as the height of a standard WR-90
waveguide. This modified waveguide version of the DPS-ENG-DPS filter was fabricated
and its photograph is shown in Fig. 4.10. The physical and geometrical parameters for this
tri-layer device are shown in Table 4.3. Figure 4.11(a) shows the amplitude of the electric
field distribution in the cross section of the input waveguide for this device for an incident
power level of 1 W. Fig. 4.11(b) shows the amplitude of the electric field distribution at the
boundary of the DPS-ENG layer (close to the input). The maximum electric field intensity
within the ENG layer occurs in this cross section of the device. As can be seen from Figs.
4.11(a) and 4.11(b), the local electric field intensity is only enhanced by a factor of 1.5.
Therefore, this device is expected to handle significantly higher power levels compared to
the one shown in Fig. 4.5.
83
|T|2: 25 kW
|R|2: 25 kW
Figure 4.13 Time-domain transmission and reflection coefficients of the DPS-ENG-DPS
filter shown in Fig. 4.10. The measurements are conducted using the setup shown in Fig.
4.7 at a power level of 25 kW. Observe that the tri-layer structure maintains its
transparency when excited with a high power signal and does not break down.
This modified waveguide version of the DPS-ENG-DPS filter is experimentally characterized both at low and high power levels. Figure 4.12 shows the measured and calculated transmission coefficients of this device. As can be seen, the transparency window
covers the operating frequency of the high-power magnetron source (9.382 GHz). The
time-domain transmission and reflection coefficients of this device are also measured using the high-power measurement setup shown in Fig. 4.7. The results of this measurement
are shown in Fig. 4.13. As can be observed, the filter operates without breaking down.
Indeed, since the local electric field intensity within this device is only slightly increased
compared to that of the incident E-field intensity, the power handling capability of this
structure is expected to be significantly higher than 25 kW. However, due to the power
limitations of our current measurement setup, we are not able to conduct experiments at
higher power levels.
84
Dielectric Layers
bwg
awg
aDPS
wwg
Wire Grid
bDPS
Figure 4.14 An alternative method for implementation of the tri-layer structure shown in
Fig. 4.1. In this case, the structure is composed of a metallic sheet perforated with small
sub-wavelength holes sandwiched by two dielectric substrates. The sub-wavelength holes
have an opening of awg × bwg and are separated from each other by a distance of wwg .
The perforated sheet has a thickness of hwg and it emulates the ENG layer. Both
dielectric slabs have the dimensions of aDP S × bDP S and a thickness of hDP S .
4.4.2
High-power capable DPS-ENG-DPS filter using a perforated metal
layer with sub-wavelength holes
The power handling capability of a DPS-ENG-DPS filter can be further improved by
arranging several cutoff waveguides side by side to obtain a two-dimensional periodic
structure. To maintain the same plasma response, the dimensions of each cutoff waveguide must be reduced and the overall depth of the structure must also be reduced. In the
limit, this structure simplifies to a single thin sheet of metal perforated with many rectangular sub-wavelength holes. A modified version of the tri-layer DPS-ENG-DPS structure
utilizing this type of an ENG structure is also designed and implemented as shown in Fig.
4.14. Figure 4.15(a) shows the electric field distribution in the cross section of the input
waveguide for this device when the structure is illuminated with an incident power level
of 1 W. In this device, the maximum electric field intensity also occurs at the boundary of
85
V/m
2000
x
V/m
2000
x
1500
y
1000
1500
y
1000
500
500
0
(a) WR90 Waveguide
Wire Grid
(b)
WR90 Waveguide
Figure 4.15 Magnitude of the electric field distribution in the cross section of the
DPS-ENG-DPS structure shown in Fig. 4.14. (a) E-field distribution in the cross section
of the input waveguide (outside of the device). (b) E-field distribution in the cross section
at the boundary between the DPS layer and the perforated metallic sheet that emulates the
ENG layer (the boundary closer to the input). Both figures are obtained for an incident
power level of 1 W.
Simulation
Measurement
Figure 4.16 The measured and simulated transmission coefficients of the DPS-ENG-DPS
structure shown in Fig. 4.14.
the DPS and ENG layers. Fig. 4.15(b) shows the electric field distribution in this plane.
As can be observed from Figs. 4.15(a) and 4.15(b), the maximum electric field intensity
in the device is practically the same as the maximum electric field intensity of the incident
86
|T|2: 25 kW
|R|2: 25 kW
Figure 4.17 Time-domain transmission and reflection coefficients of the DPS-ENG-DPS
tri-layer structure shown in Fig. 4.14. The measurements are conducted using the setup
shown in Fig. 4.7 at a power level of 25 kW. Observe that the tri-layer structure maintains
its transparency when excited with a high power signal and does not break down.
EM wave. More specifically, the maximum E-field intensity within the device is only 1.08
times that of the incident EM wave. Therefore, this device is expected to be capable of
handling extremely high power levels. Assuming that air breakdown occurs for an electric
field intensity of 3 MV/m, the peak power handling capability of this device can be approximated to be around 1.78 MW in the WR-90 waveguide environment. This is close to
the air breakdown threshold. The response of this type of DPS-ENG-DPS filter is also experimentally characterized at both low and high power levels in a manner similar to what
was described earlier in this paper and the results are presented in Figs. 4.16 and 4.17. Fig.
4.16 shows the simulated and measured transmission coefficients of this device. As can be
seen, the device demonstrates a transparency window that covers the operating frequency
of the high-power magnetron source. Fig. 4.17 shows the time-domain transmission and
reflection coefficients of this device measured at 9.382 GHz and at a peak power level of
87
25 kW. As expected, the filter maintains its transparency for the incident power level of 25
kW. Indeed, based on the earlier discussions, the power handling capability of this device
is expected to be significantly higher than 25 kW.
4.5
Conclusions
In this paper, we experimentally examined the phenomenon of electromagnetic wave
tunneling through an ϵ-negative layer and its applications in designing filters and frequency
selective surfaces capable of handling extremely high power levels. Two methods for implementing such multi-layer structures at microwave frequencies was presented. The first
method utilizes a cutoff waveguide to emulate the ENG layer. In the second implementation method, a thin metal sheet perforated with numerous sub-wavelength rectangular
holes is exploited to emulate the ENG layer. It was demonstrated that both methods can
be used to design waveguide filters with extremely high power handling capabilities. In
particular, by judicious design of the ENG layer, the electric field enhancement within
the device can be minimized. This results in DPS-ENG-DPS filters that are capable of
handling extremely high peak power levels. Two such filters were designed and experimentally characterized at high power levels. It was demonstrated that these filters do not
break down under high-power excitations with power levels as high as 25 kW, although
the actual peak power handling capability of these devices is expected to be significantly
higher than this.
88
Chapter 5
Impact of Microwave Induced Breakdown on the Responses
of High-Power Microwave Metamaterials
5.1
Introduction
Periodic structures have a wide range of applications at RF, microwave, and millimeter-
wave frequency bands [87]. In recent years, with the emergence of the field of metamaterials, they are receiving even more attention, since most metamaterials are implemented
using periodic structures with sub-wavelength periods. Metamaterials, in various different
implementations, have been used for a wide range of applications at RF [88], microwave
[24],[89]– [90], mm-wave [91], and optical [92] frequency bands. The applications of
metamaterials in high-power microwave (HPM) systems have also been investigated in
recent years. Metamaterials have been used in the design of novel HPM sources [37] –
[39],[93]. The interaction mechanisms between metamaterials and electron beams have
also been investigated for various applications [1] [94]–[95]. Finally, metamaterials have
also been used to design passive microwave devices such as antennas [96] and frequency
selective surfaces [56],[97] for HPM applications in recent years.
At high-power levels, however, metamaterials are susceptible to breakdown. Usually
breakdown events in metamaterials and periodic structures are accompanied by the creation of plasma within the unit cell of the device. The nature of this plasma determines
how the response of the device changes in the event of a breakdown. In many high-power
89
microwave applications, it is important to be able to evaluate how the response of the
structure changes in the event of a breakdown. An example that illustrates this point is
a bandstop spatial filter commonly used to protect sensitive electronics from strong jamming or interfering signals. In such an application, if the spatial filter is illuminated with
a strong enough signal that can induce breakdown within the structure (e.g. an electromagnetic pulse), the bandstop filter can be made partially or completely transparent. This
allows a significant portion of the energy of the EM pulse to pass through the spatial filter
and damage sensitive electronics that are supposed to be protected by it. This example,
among many others that can be envisioned, illustrates the importance of predicting the
responses of RF/microwave metamaterials in the event of a breakdown.
In this work, we examine breakdown events in single-layer metasurfaces that operate
in air and at atmospheric pressure levels. Since many metamaterials and periodic structures are used in devices that normally operate in air and at atmospheric pressure levels
(such as antenna apertures, filters, etc.), examination of this operational scenario is of practical interest. The metasurfaces examined in this work are in the form of two-dimensional
periodic structures where the unit cell of the structure is composed of miniature LC resonators. Three different metasurfaces with different constituting unit cells are examined.
When these metasurfaces are illuminated with a low-power electromagnetic wave, they
reflect the incident EM wave and are completely opaque. On the other hand, these metasurfaces are designed to become transparent when a localized low-impedance connection
(e.g. a short circuit) is created within the capacitors used in each unit cell. In principle,
such a localized connection can be created by a localized discharge within the capacitive
gaps of the resonators used in each unit cell. On the other hand, transparency will not be
achieved if the discharge is not localized, is not confined to the capacitor region, or is very
lossy. Therefore, by measuring the responses of these metasurfaces under high power excitation (before and after breakdown) the impact of the generated plasma on the frequency
90
responses of these devices can be determined. To facilitate the process of high-power
measurements, the experimental part of this work is conducted on unit cells of these metasurfaces, which are placed in a WR-90 waveguide environment. The waveguides used in
these experiments are pressure sealed and have gas inlets and outlets which allow the experiments to be conducted for different gas compositions and pressure levels. An X-band
magnetron with a pulse duration of 1 µs, a frequency of 9.382 GHz, and a peak power
level of 25 kW is the HPM source used to induce breakdown in these metasurfaces. Two
sets of high-power measurements are conducted. The first set of experiments is conducted
in air and at atmospheric pressure levels. The second set of experiments is conducted in
Argon at different pressure levels. We demonstrate that, in air at atmospheric pressure
levels, the responses of the metasurfaces can be predicted by modeling the discharge with
a localized resistive-inductive impedance in parallel with the capacitors employed within
each unit cell. HPM measurement results obtained in this case agree well with the model
in all three metasurfaces. On the other hand, when these metasurfaces are placed in a
high-pressure Argon environment, the discharge is no longer localized and predicting the
exact frequency responses of the metasurfaces becomes challenging.
5.2
Metasurface Design and Principles of Operation
5.2.1
Design Procedure
Figure 5.1(a) shows the three-dimensional (3D) topology of a metasurface of the type
examined in this work. The structure is in the form of a two-dimensional (2D) periodic
structure with sub-wavelength periods. Structures similar to this are used in the construction of various types of metamaterials [98], reactive impedance surfaces [99], and frequency selective surfaces [62],[100]. A constitutive unit cell of the metasurface is also
shown in Fig. 5.1(b). As can be seen, the unit cell of the structure is composed of two
91
E
k
H
^
z
y^
x^
(a)
(b)
(c)
(d)
Figure 5.1 (a) 3D topology of a single-layer metasurface illuminated by a transverse
electromagnetic (TEM) wave. (b) One possible unit cell of the metasurface composed of
a series LC resonator. (c) A different unit cell of the metasurface composed of two
meandered inductors and a gap capacitor. For the same resonant frequency, this LC
resonator has a higher quality factor compared to the one shown in part (b). (d) Another
possible unit cell of the metasurface composed of two series LC resonators in parallel
with each other. The two resonators have the same resonant frequency but they have
different loaded quality factors. The resonator on the left has a higher Q than the one on
the right.
straight wire sections connected to two capacitive plates. If a ŷ-polarized EM wave illuminates this unit cell, the straight wire sections interact with the magnetic field of the
incident EM wave and act as inductors. On the other hand, the ŷ polarized electric field
92
creates positive and negative charge buildups on the lower and upper edges of the two parallel strips and the parallel strips act as a capacitor. Consequently, the combination of the
vertical and horizontal wire strips act as a series LC circuit. At the resonant frequency of
this LC circuit, the EM wave is completely reflected by the metasurface and the structure
remains completely opaque. When the power of the incident EM wave increases, however, breakdown can occur within this structure. In this event, based on the nature of the
plasma created during breakdown, the response of the metasurface changes. Assuming
that breakdown creates a localized discharge in the capacitive gap region, which creates a
low-impedance1 path between the two plates of the capacitor, the unit cell of the structure
will be converted from a series LC circuit to a series RL circuit. Depending on the value
of the inductance and the period of the structure, the metasurface can be made transparent
at the operating frequency. On the other hand, if the discharge is not localized, predicting
the response of the metasurface after breakdown becomes challenging and transparency
will most likely not be achieved.
Other unit cells can also be envisioned for this type of metasurface. Fig. 5.1(c) shows
another implementation of the unit cell shown in Fig. 5.1(b) where the inductive wire is
meandered to increase its inductance. In this structure, the capacitance is formed between
the open (pointed) ends of the inductive wire strips. For the same resonant frequency, the
unit cel shown in Fig. 5.1(c) has a higher loaded quality factor compared to the one shown
in Fig. 5.1(b). This results in a higher field intensity within the gap region of the capacitor
of this unit cell. Fig. 5.1(d) shows a third potential implementation for the unit cell where
two parallel LC resonators are embedded within one unit cell of the metasurface. These
two resonators have the same resonant frequency but different quality factors.
1
This low-impedance path is expected to have an impedance, which is inductive resistive.
93
10.16 mm
w
t
d
h
l
22.86 mm
(a)
5
x 10 V/m
3
2.5
2
1.5
1
0.5
V/m
15000
10000
5000
0
(b)
(c)
Figure 5.2 (a) 3D topology of the waveguide version of the unit cell of a metasurafce
composed of two inductive strips separated from each other by a capacitive gap of 0.2
mm. The physical dimensions are shown in Table 5.1. (b) The magnitude of electric field
intensity in the cross section of the device before breakdown. (c) The magnitude of the
electric field intensity in the cross section of the device obtained when the capacitive gap
is loaded using a resistive-inductive impedance with a value of 32 + j0.6 Ω. Assuming
that the discharge is localized to the capacitive gap, this field distribution demonstrates
the field distribution within the device after breakdown occurs.
The structure shown in Fig. 5.1 is a two-dimensional metasurface composed of more
than one unit cell. However, to facilitate high-power measurements and experimental characterizations in this work, we examine only a unit cell of this structure within a waveguide
environment. This allows us to conduct accurate high-power measurements in a controlled
environment. Figures 5.2(a), 5.3(a), and 5.4(a) show the waveguide versions of the unit
cells shown in Figs. 5.1(b), 5.1(c), and 5.1(d) respectively. These unit cells have transmission nulls at the resonant frequencies determined by the capacitance and inductance values
of their series LC resonators.
94
10.16 mm
w
d2
d3
h
t
d1
22.86 mm
(a)
5
x 10 V/m
2.5
4
x 10 V/m
2
2
1.5
1.5
1
0.5
1
0.5
0
(b)
(c)
Figure 5.3 (a) 3D topology of the waveguide version of the unit cell of a metasurafce
composed of two meander inductive strips separated from each other by a capacitive gap
of 0.2 mm. The physical dimensions are shown in Table 5.2. (b) The magnitude of
electric field intensity in the cross section of this device before breaking down. (c) The
magnitude of the electric field intensity in the cross section of the device obtained when
the capacitive gap loaded with a resistive-inductive impedance of 32 + j0.6 Ω. Assuming
that the discharge is localized to the capacitive gap, this figure demonstrates the field
distribution within the device after breakdown occurs.
For the unit cell shown in Fig. 5.2(a), the capacitance value is determined by the
separation of the parallel strips, h, the lengths of the strips, l, and the widths of the strips,
d. Increasing l and d and decreasing h results in increasing the capacitance value. For the
structure shown in Fig. 5.3(a), the inductance value is determined by the overall length
and width of the meandered strip and the capacitance value is determined by the width of
the strip and the gap separation, h. The unit cells shown in Figs. 5.2(a), 5.3(a), and 5.4(a)
are placed in a WR-90 waveguide to emulate a two-dimensional infinite periodic structure
95
Table 5.1 Physical parameters of the waveguide version of the unit cell of the
metasurface shown in Fig. 5.2.
Parameter
w
d
h
l
t
Value
0.2 mm
0.3 mm
0.2 mm
1.45 mm
100 µm
[101]. In a WR-90 waveguide, however, the dominant mode is a TE10 mode and not a
TEM. Therefore, the physical dimensions of the waveguide versions of the unit cells of
these metasurfaces will be different from those of the free-space versions. Nonetheless,
because the TE10 mode is the superposition of two plane waves propagating with different
incidence angles within the waveguide, the physics of the problem does not change and the
same opacity to transparency transition is expected to be observed from these structures
as well. The high-power Magnetron source used in these experiments has a frequency of
9.382 GHz. Therefore, all three metasurfaces are designed to have transmission nulls at
that frequency. The design optimization of the dimensions of these structures is performed
via full-wave EM simulations in CST Microwave Studio and the physical dimensions of
the device shown in Fig. 5.2(a) are shown in Table 5.1. For an incident peak power level
of 1 Watt in an empty section of a WR-90 waveguide, the maximum electric field intensity
is 2127 V/m. For the same peak power level, when the unit cell shown in Fig. 5.2(a) is
placed within the WR-90 waveguide, the peak E-field intensity increases significantly as
shown in Fig. 5.2(b).
When the incident power level exceeds the threshold breakdown power level, air breakdown is expected to occur at the location of maximum electric field intensity, (i.e., between
the capacitive gaps). This discharge is expected to create a low-impedance connection between the two opposite sides of this capacitive gap [102]. In this case, the device’s unit
cell will be reconfigured from an LC resonator to an inductor in series with a small resistor
96
w1
10.16 mm
w2
h2
t
22.86 mm
(a)
d1
d2 h1
l2
High-Q
Resonator
l1
Low-Q
Resonator
4
x 10 V/m
4
x 10 V/m
15
10
5
(b)
0
4
x 10 V/m
10
(c)
7
6
5
4
3
2
1
0
4
x 10 V/m
2
8
1.5
6
(d)
4
1
2
0.5
(e)
0
Figure 5.4 (a) 3D topology of the waveguide version of the unit cell of a metasurface
composed of two parallel LC resonators. Both LC resonators have similar topology
shown in Fig. 5.2(a) and maintain the same capacitive gap of 0.2 mm. The physical
dimensions are shown in Table 5.3. (b) The magnitude of electric field intensity in the
cross section of this device before breaking down. (c) The magnitude of the electric field
intensity in the cross section of the device obtained when the capacitive gap of the high-Q
LC resonator (on the left) is loaded with a resistive-inductive impedance of 32 + j0.6 Ω.
(d) The magnitude of the electric field intensity in the cross section of the device obtained
when the capacitive gap of the high-Q LC resonator (on the right) is loaded with a
resistive-inductive impedance of 32 + j0.6 Ω. (e) The magnitude of the electric field
intensity in the cross section of the device obtained when the capacitive gaps of both LC
resonators are loaded with the resistive-inductive impedances of 32 + j0.6 Ω.
(modeling the discharge). To model the frequency response of this device after breakdown
occurs, a resistive-inductive impedance is used to bridge the capacitive gap to emulate the
97
Table 5.2 Physical parameters of the waveguide version of the unit cell of the
metasurface shown in Fig. 5.3.
Parameter
d1
d2
d3
Value
1.6 mm
2.25 mm
2.58 mm
Parameter
w
h
t
Value
0.2 mm
0.2 mm
100 µm
discharge. The value of this resistive-inductive impedance is assumed to be 32 + j0.6 Ω.
This value is experimentally determined from a series of high-power breakdown experiments conducted in our group for similar structures 2 . Figure 5.2(c) shows the magnitude
of the electric field intensity in the cross section of this device (with the capacitive gap
loaded with the LR impedance emulating the discharge) for an incident power level of 1
Watt. As can be observed, the field distribution is reminiscent of an inductively loaded
waveguide. The simulated transmission coefficient of this device before breakdown is
shown in Fig. 5.6(a). The expected transmission coefficient of the device after breakdown
(assuming a localized discharge) is also simulated using full-wave EM simulations in CST
Microwave Studio and the result is shown in Fig. 5.6(b). As can be observed from this
figure, the transmission coefficient of the device increases significantly. These results will
be discussed further in Section 5.3.1.
The unit cell shown in Fig. 5.3(a) is designed to breakdown at a lower incident power
level compared to the one shown in Fig. 5.2(a). In this structure, the capacitance is formed
between the two edges of the meandered inductive wire grids. A small triangle is added
2
A unit cell with similar topology shown in Fig. 5(a) is examined in the experimental test-bed shown in
Fig. 7. To obtain the correct impedance value of localized discharges, we start with a value of 0 Ω in fullwave EM simulations in CST Microwave Studio that the simulated responses are deviated from the measured
ones. The discrepancy between the simulated and measured results decreases by gradually increasing the
impedance value until simulated power transmission coefficient matches with the measured one.
98
at the edge on each side of the gap to increase the local field enhancement factor and
facilitate breakdown. The vertex to vertex distance is maintained at 0.2 mm. Since the
capacitance value of this device is substantially reduced due to the reduction of the area
of the capacitive surface, the LC resonant frequency moves to a higher frequency if the
inductance value is not changed. To maintain the same resonant frequency of 9.382 GHz
(the frequency of operation of the magnetron), the value of the inductance is increased
by meandering the wire as can be seen from Fig. 5.3(a). The physical parameters of this
device are provided in Table 5.2. Figure 5.3(b) shows the magnitude of the electric field
intensity in the cross section of this device when illumined with an incident power level
of 1 Watt. As expected, the maximum electric field intensity occurs at the capacitive gap.
Similar to the previous case, the response of the device after breakdown is predicted by
loading the gap with a resistive-inductive impedance. The field distribution in the cross
section of the device in this case is shown in Fig. 5.3(b) for an incident power level of
1 Watt. The simulated frequency response of the device before breakdown is presented
in Fig. 5.6(a). The anticipated frequency response of the device after breakdown is also
simulated and the result is presented in Fig. 5.6(b). As can be seen, at 9.382 GHz, this
device is expected to transition from total opacity before breakdown to near complete
transparency after breakdown.
Figure 5.4(a) shows a waveguide version of the unit cell of a metasurface composed
of two series LC resonators in parallel with each other. Each resonator has a similar
topology to the device shown in Fig. 5.2(a). Both resonators are designed to have the
same resonant frequency. However, the loaded quality factors (Q) of the two resonators
are different 3 . Even though the capacitive gap width is maintained at 0.2 mm in both
3
In particular, one of the resonators has a larger inductor and smaller capacitor value than the other
one. In other words, the product of the values of L and C in each resonator is the same (the same resonant
frequency) but their L/C ratios are different (i.e., one has a higher loaded Q than the other).
99
Table 5.3 Physical parameters of the waveguide version of the metasurafce shown in Fig.
5.4.
Parameter
d1
w1
h1
l1
t
Value
0.3 mm
0.8 mm
0.2 mm
4 mm
Parameter
d2
w2
h2
l2
Value
0.3 mm
0.2 mm
0.2 mm
1.83 mm
100 µm
resonators, the difference between their Qs yields different local field enhancement factors
at their respective capacitive gap locations. This unit cell is used to investigate the effect
of the resonator Q and the coupling between the two resonators on the breakdown event.
The physical parameters of this device are shown in Table 5.3.
Figure 5.4(b) shows the magnitude of the electric field distribution in this device for an
incident power level of 1 Watt. Figures 5.4(c), 5.4(d), and 5.4(e) show the E-field distribution in the cross section of the device when the high-Q resonator (left), low-Q resonator
(right), and both resonators break down respectively. The frequency response of this device before breakdown is shown in Fig. 5.6(c). The expected transmission coefficients of
the device after breakdown are shown in Fig. 5.6(d). Here, the results for two different
breakdown cases (when the high-Q resonator breaks down only and when both resonators
breakdown together) are shown4 . As can be seen, at 9.382 GHz, the transmission coefficients of the device in all three states are significantly different from one another. This
allows for using the measured transmission coefficient of the device to determine which
resonators (if any) breaks down under high-power excitation.
4
Notice that when the incident power level is high enough to cause breakdown in the low-Q resonator,
the high-Q resonator is also expected to breakdown.
100
Table 5.4 Comparison of the measured power reflection and transmission coefficients of
the metasurface shown in Fig. 5.2(a) with the simulated ones.
Sim.(before breakdown)
Sim.(after breakdown)
Meas.(before breakdown)
Meas.(after breakdown)
|R|2
|T |2
1
0
0.26 0.66
0.91
0
0.21 0.66
Table 5.5 Comparison of the measured power reflection and transmission coefficients of
the metasurface shown in Fig. 5.3(a) with the simulated ones.
Sim.(before breakdown)
Sim.(after breakdown)
5.3
|R|2
|T |2
1
0
0.09 0.88
Meas.(before breakdown)
0.92
0
Meas.(after breakdown)
0.09
0.6
Fabrication and Experimental Characterization
5.3.1
Fabrication and Low-Power Measurements
The unit cells of the three metasurfaces discussed in the previous section are fabricated
using chemical etching of stainless steel sheets. The photographs of the three fabricated
unit cells are shown in Fig. 5.5. Each metal sheet is 0.1 mm thick. The devices are
placed inside a standard WR-90 rectangular waveguide and their frequency responses are
measured using a calibrated vector network analyzer at an incident power level of 0.3 mW
over the frequency range of 7.5 GHz to 11.5 GHz.
101
Table 5.6 Comparison of the measured power reflection and transmission coefficients of
the metasurface shown in Fig. 5.4(a) with the simulated ones.
Sim.(before breakdown)
Sim.(after breakdown)
Meas.(before breakdown)
Meas.(after breakdown)
|R|2
|T |2
1
0
0.64 0.24
0.92
0
0.61 0.18
Figures 5.6(a) and 5.6(c) show the measured transmission coefficients of these metasurfaces. As can be seen from these figures, a good agreement between the simulated and
measured results is observed. A transmission null is observed near 9.382 GHz for these
devices, which indicates that they are opaque at this operating frequency when illuminated
with a low power level.
5.3.2
High Power Measurements In Atmospheric Air Environment
The fabricated devices are designed to show partial transmissions when breakdown
occurs and creates a low-impedance connection between the two plates of the capacitors.
To investigate this experimentally, the transmission and reflection coefficients of these devices are measured, at a single frequency, under high-power excitation. The experimental
test-bed used to accomplish this is shown in Fig. 5.7. This experimental test-bed is composed of a 25 kW magnetron source, WR-90 rectangular waveguides, a pressurized sealed
section of a WR-90 waveguide, directional couplers, crystal detectors, an oscilloscope, and
a vacuum system. The device under test is placed within the pressurized section composed
of WR-90 rectangular waveguide with gas inlet and outlet. Two mica pressure windows
(with VSWR of 1.15:1) are used at both ends of this pressurized waveguide section to seal
102
this section off. This way, the experiments can be conducted at different pressure levels
and for different gas compositions. All waveguide connections are sealed by O-rings and
the pressure of this chamber is controlled by a mass flow controller, a turbo pump, and a
scroll pump where the pressure is recorded by the pressure detector. The magnetron source
generates a short pulse with a single frequency of 9.382 GHz, a duration of 1 µs, and a
peak power level of 25 kW. The power level of the incident pulse can be adjusted from 1
W to 25 kW using the combination of a circulator and an E-H tuner. When the devices are
illuminated by the incident pulse, the reflected and transmitted pulses are detected by the
crystal detectors connected to the outputs of two waveguide directional couplers that are
used to sample the transmitted and reflected signals. The nonlinear crystal detectors detect
the amplitudes of the reflected and transmitted waves.
Figure 5.8 shows the normalized power reflection and transmission coefficients, in
time domain, for the three fabricated unit cells. The normalized time-domain power reflection coefficient is defined as the ratio of the electric signal detected by the reflection
detector when the device is placed inside the pressurized chamber to the electric signal detected by the reflection detector when the device under test (DUT) is removed and replaced
by a short circuit. Similarly, the normalized time-domain power transmission coefficient
is defined as the ratio of the electric signal detected by the transmission detector when
the DUT is placed inside the pressurized chamber to the electric signal detected by the
transmission detector when the DUT is removed. As can be observed from Figs. 5.8(a),
5.8(c), and 5.8(e), before breakdown occurs, all devices are reflective as indicated by the
small transmission and large reflection coefficients observed from these figures. On the
other hand, as the incident power level is increased, breakdown occurs in these devices
approximately halfway through the pulse and an abrupt change in the reflection and transmission coefficients of the devices can be observed. Figures 5.8(b) and 5.8(d) show the
transmission coefficients of the structures shown respectively in Fig. 5.2(a) and 5.3(a).
103
Vector Network Analyzer
WR90
Waveguide
(a)
(b)
(c)
Figure 5.5 Photographs of the fabricated unit cells of the metasurfaces shown in (a) Fig.
5.2, (b) Fig 5.3, and (c) Fig. 5.4.
As can be observed in this case, the transmission coefficient after breakdown agrees very
well with the predicted values obtained from full-wave EM simulation results (e.g. see
the transmission coefficient of the device at 9.382 GHz shown in Fig. 5.6(b)). In this
case, a two-stage operation (opaque or semi-transparent) is observed. Additionally, after
breakdown, until the end of the pulse, the transmission coefficient of the device remains
stable and does not change with time. Figure 5.8(f) shows the transmission coefficient
of the metasurface shown in Fig. 5.4(a) when the incident power level is high enough to
cause breakdown in the device. The measured transmission coefficient of this device (after
breakdown) is consistent with the predicted results shown in Fig. 5.6(d) when both resonators break down. In theory, one can expect to observe three different operational states
104
for this device at different incident power levels. These correspond to situations where
no breakdown occurs, when the high-Q resonator breaks down, and when both resonators
break down5 . In practice, however, whenever breakdown was observed to occur in this
device, both resonators appeared to breakdown together. This is evident from Fig. 5.8(f)
where the transmission coefficient of the device shows two major states corresponding to
|T |2 = 0 and |T |2 = 0.2. The fact that both resonators in this structure break down is
further confirmed visually by taking photographs of unit cell of the metasurface during the
breakdown process and observing two distinct discharges.
Because of the higher local field enhancement factor of the high-Q resonator, this resonator is expected to breakdown at a significantly lower power level than the low-Q resonator. Therefore, one would expect to observe a breakdown event in this structure where
only the high-Q resonator breaks down and the low-Q one does not. This is naturally expected to occur at an incident power level which is high enough to cause breakdown in the
high-Q resonator but not high enough to cause break down in the low-Q resonator. Despite
this expectation, however, this mode of operation was not observed for this unit cell and
whenever breakdown occurred, it was observed in both resonators. This phenomenon most
likely indicates that the breakdown of the low-Q resonator is facilitated by the localized
discharge created in the high-Q resonator. Currently, the exact physical mechanisms that
result in this behavior are under investigation and experiments are being devised to test this
hypothesis. Nonetheless, this observation itself is important in predicting the performance
of HPM metamaterials and periodic structures that use complex unit cells in the event of a
breakdown.
5
Notice that when the power is high enough to cause breakdown in the low-Q resonator, breakdown in
the high-Q resonator is also expected. Therefore a situation where only the low-Q resonator breaks down is
not expected to occur in practice.
|T|2
|T|2
105
(a)
(b)
Simple LC Resonator Fig. 2 (Meas.)
Meandered LC Resonator Fig. 3 (Meas.)
Simple LC Resonator Fig. 2 (Sim.)
Meandered LC Resonator Fig. 3 (Sim.)
|T|2
|T|2
Simple LC Resonator Fig. 2 (Sim.)
Meandered LC Resonator Fig. 3 (Sim.)
(c)
Measurement
Simulation
(d)
High Q LC Resonator Break Down
Both LC Resonators Break Down
Figure 5.6 (a) The measured and simulated transmission coefficients of the metasurfaces
shown in Fig. 5.2 and 5.3 when no breakdown occurs. (b) Simulated transmission
coefficient of the metasurfaces shown in Fig. 5.2 and 5.3 when breakdown occurs. These
results are obtained based on the assumption that breakdown event in these devices
creates a localized discharge confined to the region of the capacitive gap. (c) Measured
and simulated transmission coefficients of the metasurface shown in Fig. 5.4. (d)
Simulated transmission coefficient of the metasurface shown in Fig. 5.4 for two different
possible breakdown scenarios: the high-Q resonator (on the left) breaks down only and
both resonators break down together.
The comparison between the measured and simulated transmission and reflection coefficients of these three devices, before and after breakdown, are shown in Tables 5.4,
5.5, and 5.6. As can be observed, a good agreement between the measured and simulated
106
Pressure
Detector
Reflection
Detector
Transmission Mass Flow
Detector
Controller
Argon
Gas
DUT
E-H Tuner
Magnetron
Pressurized
Waveguide
Match Load
Figure 5.7 Experimental test-bed used in the high-power measurements. The magnetron
source generates a short pulse with a single frequency of 9.382 GHz, a duration of 1 µs,
and a peak power level of 25 kW.
Table 5.7 Breakdown power level for the three devices discussed in this paper when they
operate in Air at atmospheric pressure level and in Argon at 600 torr.
Air
Argon
Simple LC Resonator (Fig. 5.2)
18.3 kW
15.9 kW
Meandered LC Resonator (Fig. 5.3)
7.6 kW
6.1 kW
Double LC Resonator (Fig. 5.4)
20.2 kW
16.5 kW
results is observed in general. This indicates that, when breakdown occurs in Air at atmospheric pressure levels, the response of the structure can be predicted with a reasonable
degree of accuracy by modeling the discharge as a low-impedance connection path within
the unit cell of the metamaterial.
107
(a)
(c)
(e)
(b)
(d)
(f)
Figure 5.8 Normalized, time-domain power transmission and reflection coefficients in
measured in air and at atmospheric pressure level for the device shown in Fig. 5.2
illuminated with a peak power of (a) 18.3 kW and (b) 20.2 kW, the device shown in Fig.
5.3 illuminated with a power of (c) 7.6 kW and (d) 10.1 kW, and the device shown in Fig.
5.4 illuminated with a peak power of (e) 20.2 kW and (f) 21 kW.
5.3.3
High Power Measurements In Argon
The responses of these three metasurfaces are also measured at high-power levels when
they operate in an Argon environment. These three devices are placed within the pressurized rectangular waveguide chamber one at a time. First, the air is pumped out of the
pressurized chamber down to a low pressure of 11 mTorr and then the chamber is filled
with Argon with a purity of 99.999% (i.e. 1 ppm O2 , 1 ppm H2 O, 0.5 ppm THC, 0.5 ppm
CO and 0.5 ppm CO2 ). The pressure of this chamber is controlled using the mass flow
controller, turbo pump, and scroll pump. The breakdown threshold level of the devices
108
(a)
(c)
(f)
(b)
(d)
(e)
Figure 5.9 Normalized, time-domain power transmission and reflection coefficients in
measured in Argon and at pressure of 600 torr for the device shown in Fig. 5.2
illuminated with a peak power of (a) 15.9 kW and (b) 16.5 kW, the device shown in Fig.
5.3 illuminated with a power of (c) 6.1 kW and (d) 7.6 kW, and the device shown in Fig.
5.4 illuminated with a peak power of (e) 16.5 kW and (f) 18.3 kW.
are determined by gradually increasing the incident power level and monitoring the transmission and reflection coefficients of the device. Figure 5.9(a) and Fig. 5.9(b) show the
normalized power reflection and transmission coefficients of the device shown in Fig. 5.2,
at a pressure level of 600 Torr, when the device is illuminated with a peak power level
of 15.9 kW and 16.5 kW, respectively. For the incident power of 16.5 kW, shortly after
start of the breakdown, the transmission coefficient increases to about 0.7 and rapidly falls
down afterwards. Subsequently, the transmission coefficient fluctuates between 0 and 0.5
109
in the remainder of the pulse 6 . This fluctuations in the measured transmission coefficient
are accompanied by similar fluctuations in the measured reflection coefficient albeit on a
smaller scale. Examining the measured power transmission and reflection coefficients of
the device reveals that breakdown in this situation is accompanied by significant absorption of the energy of the pulse. Notice that the total sum of the transmitted energy and
reflected energy (|T |2 + |R|2 in Fig. 5.9(b)) has decreased significantly after breakdown
compared to before breakdown. Thus, breakdown in this operational condition is accompanied by absorption of a significant part of the energy of the pulse. Additionally, the
rapid fluctuations of the transmission coefficient as a function of time indicate significant
temporal variation in the nature of the created plasma.
A similar phenomenon is observed for the unit cell of the metasurface shown in Fig
5.3(a). Fig. 5.9(c) and Fig. 5.9(d) show the measured transmission and reflection coefficients of this device when it is illuminated with a power level of 6.1 kW and 7.6 kW,
respectively. The response of this device, after breakdown, is similar to the previous one
where a significant fluctuation in the transmission coefficient of the device is observed.
Figures 5.9(e) and 5.9(f) show the measured reflection and transmission coefficients of the
metasurface shown in Fig. 5.4(a). The behavior of this device in Argon at 600 Torr is also
similar to the previous two cases examined. The breakdown threshold for these devices in
provided in Table 5.7.
5.4
Conclusions
We examined the impact of RF breakdown on the performance of metamaterials and
periodic structures operating at high-power microwave levels. We demonstrated that, when
6
In Argon environment, it is easier to create electron avalanche because electrons hardly lose energy
due to elastic collisions with neutral argon. However, in air environment, oxygen molecules require more
threshold energy for ionizations [103] –[105].
110
operated in air and at atmospheric pressure level, RF breakdown creates localized discharge(s) within the unit cell of a metasurface. We demonstrated that the effect of this
discharge on the frequency response of the periodic structure can be estimated using fullwave EM simulations by assuming that the localized discharge creates a low impedance
connection path within the unit cell of the structure. This was tested experimentally by
examining three different single-layer metasurfaces that were designed to be opaque at
low power levels but could become partially transparent when they break down. We also
demonstrated that in metamaterials where the unit cell is composed of two resonators with
different loaded quality factors (which result in different local field enhancement factors),
breakdown is always observed in both resonators. In such structures, it is likely that breakdown is first initiated in the high-Q resonator and the discharge in the low-Q resonator is
mediated by the initial discharge. This hypothesis, however, is yet to be tested and experiments are currently being devised to test this. Finally, it was demonstrated that the
nature of the generated plasma will significantly affect the response of the periodic structure. This was experimentally demonstrated by creating breakdown in these metasurfaces
in a high-pressure Argon environment.
111
Chapter 6
Investigating the Physics of Simultaneous Breakdown Events
in High-Power-Microwave (HPM) Metamaterials with MultiResonant Unit Cells and Discrete Nonlinear Responses
6.1
Introduction
Periodic structures are widely used at RF, microwave, and millimeter-wave (MMW)
frequencies for various applications [11] –[18]. Recently, with the emergence of the field
of metamaterials, sub-wavelength periodic structures and their applications have received
significant attention at frequencies ranging from microwave to visible radiation [106] –
[113]. For example, in the RF to MMW regimes, sub-wavelength periodic structures have
been used to develop high-impedance surfaces for antenna applications [19], effective materials with properties not found in nature [20] –[22], perfect lenses [114], miniaturizedelement frequency selective surfaces [64], and wide scan angle microwave lenses [115]–
[24]. The application of metamaterials in high-power microwave (HPM) systems has also
been investigated recently. Examples include studies of the interaction of electron beams
with metamaterials [1], new microwave sources [2] –[39] and cathodes [93]. Passive HPM
device application studies include the design of HPM antennas [96], filters [97], and frequency selective surfaces [56]. One of the challenges for periodic structures and metamaterials designed for HPM applications is the potential for breakdown and arcing within
unit cells of the structure. Internal breakdown within the structure can alter performance or
112
cause permanent damage. Figure 6.1 shows several representative metamaterial structures
that may be used for low- or high-power microwave applications. While these structures
look different and are designed for different applications, they all have several common
features. Specifically, their unit cells have sub-wavelength sizes and incorporate miniature
or sharp metallic features and narrow gaps. These attributes often result in local electric field intensity levels exceeding the intensity of the incident field by several orders of
magnitude. Thus, RF-induced breakdown in such structures is a real possibility, even at
incident wave intensities well below breakdown thresholds.
Recently, we examined the impact of RF-induced breakdown on the microwave response of HPM metamaterials [4]–[116]. One of the interesting findings was the observation of simultaneous breakdown discharges at two separate sites within a multi-resonator
unit cell of a metasurface [4]. This simultaneous discharge occurred despite the fact that
the electric field intensity at one of the resonator sites was well below the threshold intensity required for breakdown under normal circumstances. This suggests that in such
structures, breakdown at one site facilitates the discharge at the other site having a lower
local field intensity. A similar phenomenon has also been recently reported on metasurfaces with single resonator unit cells where breakdown at one unit cell was observed to
facilitate breakdown at neighboring unit cells and eventually over the entire surface [3].
This was also despite the fact that the field intensities at the neighboring unit cells were less
than the normal breakdown threshold level. This suggests that the phenomenon reported
in [4] is also applicable to metamaterials with simple unit cell topologies as well those
using more complicated ones such as that reported in [4]. In this paper, we investigate
this simultaneous breakdown using a modified version of the metasurface reported in [4].
The unit cell of our metasurface is composed of two series LC resonators with different
resonant frequencies and slightly different quality factors (Q’s). Using this metasurface,
113
we examine three candidate mechanisms for the simultaneous breakdown discharges: energetic electrons, ultraviolet (UV) radiation, and vacuum ultraviolet (VUV) radiation. To
determine the main coupling mechanism between the two resonators during the discharge,
a non-metallic barrier opaque to UV or VUV photons and energetic electrons was placed
between the resonators within the unit cell during HPM measurements. When this physical barrier was placed between the two resonators, simultaneous breakdown no longer
occurred. In this case, each resonator broke down at the power level corresponding to
that creating the local field intensity necessary for breakdown at that resonator’s location.
By repeating these high-power microwave experiments with different barriers between the
two resonators, it was established that VUV radiation from the discharge at the resonator
with a lower electric field breakdown threshold causes simultaneous breakdown at the
other resonator where the field intensities are otherwise too low. These results suggest
that in HPM metamaterials, creation of localized and isolated discharges may trigger large
scale breakdown at other locations within the structure where the electric field intensities
are not high enough to cause a discharge under normal circumstances. This phenomenon
needs to be taken into account when designing a metamaterial or any periodic structure
that is intended to be used in an HPM system. In what follows, the details of the design
and experiments conducted in this work are presented and discussed.
6.2
A Metasurface with a High-Contrast, Three-Stage Nonlinear Response
Figure 6.2(a) shows the photograph and detailed topology of the unit cell of a meta-
surface composed of two different resonators reported in [4]. In this structure, the two
resonators were designed to have the same resonant frequency but different loaded quality
114
Complementary Split Ring
Resonator Metamaterial
Split Ring Resonator
Metamaterial
(a)
Electron Beam
(b)
Metallic Metamaterial
-Based Mask
Electron Beam
H
E
k
Metallic Ring
Background Gas
Electron Beam
(c)
(d)
Figure 6.1 Different metamaterial topologies and applications. (a) 3D topology of arrays
of split ring resonators cascaded sequentially to achieve negative permeability. (b)
Topology of an active negative index metamaterial powered by an electron beam [1]. (c)
A metamaterial based Cerenkov maser composed of metallic rings used to obtain a
desired engineered dielectric constant [2]. (d) 3-D topology of metamaterial-based
structure used to facilitate plasma generation within a distributed discharge limiter [3].
factors and hence, different threshold breakdown power levels. Specifically, the breakdown power level of the high-Q resonator was shown to be significantly lower than that of
the low-Q resonator. However, the high-power experiments reported in [4] demonstrated
115
0.24
0.08
Before breakdown
Single resonator breaks down
Both resonators break down
(a)
(b)
Figure 6.2 (a) Topology of the unit cell of a metasurface, which was examined in [4].
The unit cell is composed of two resonators with different quality factors but the same
resonant frequencies. The resonator on the left has a higher Q and lower breakdown
threshold power level than the one on the right. (b) Simulated time-domain transmission
coefficients of this device under three operational conditions (no breakdown, high-Q
resonator breaks down, and both resonators breakdown simultaneously).
that, whenever breakdown occurred in this metasurface, both resonators were observed
to breakdown together. Moreover, this breakdown occurred at a power level that corresponded to the lower breakdown threshold of the high-Q resonator. In other words, the
breakdown in the high-Q resonator seemed to trigger the breakdown in the low-Q resonator and significantly reduce its breakdown threshold level. To identify the cause of
this phenomenon, in the present work, we devised a set of experiments and carried them
out on a modified version of the structure reported in [116]. The modification was selected to facilitate the diagnosis of the breakdown effects. Figure 6.2(b) shows the simulated transmission coefficients of the device reported in [4] for all three scenarios (i.e., no
breakdown, high-Q resonator breakdown, and both resonators breakdown together). Our
diagnostic process involves measuring the transmission coefficient of the device to determine its operating state and identify whether any of the resonators has broken down [4].
116
0.63
0.39
Before breakdown
Single resonator breaks down
Both resonators break down
(a)
(b)
Figure 6.3 (a) Topology of the unit cell of the modified metasurface examined in this
work. The unit cell is composed of two resonators with considerably different resonant
frequencies. The resonator on the left has a resonant frequency of 9.112 GHz and the one
on the right has a resonant frequency of 7.516 GHz. Under high-power excitation at 9.382
GHz, the resonator on the left is expected to breakdown at a lower threshold power level
due to its higher local field enhancement factor. (b) Simulated time-domain transmission
coefficients of this device under three operational conditions (no breakdown, high-Q
resonator breaks down, and both resonators breakdown simultaneously).
Unfortunately, the small contrast between the transmission coefficients of two of the operating states of the metasurface reported in [4] (i.e., one resonator breaking down versus
both resonators breaking down) made reliable diagnosis very difficult. To circumvent this
challenge, a modified version of this metasurface shown in Fig. 6.3(a) was used for the
study described in this paper. In this new metasurface, rather than using two resonators
with the same resonant frequency but different Q’s, two resonators with different resonant frequencies are used. Both resonators used in this device have high Q’s and they are
designed to have the same capacitive gap width of 0.2 mm. The left resonator (see Fig.
6.3(a)) is composed of two meandered inductive strips and a narrow capacitive strip width
with a resonant frequency of 9.112 GHz. The right resonator has longer meandered inductive strips and a slightly wider capacitive strip width than the left resonator and hence,
117
9.382 GHz
9.382 GHz
|T|2
|T|2
0.63
0.39
(b)
(a)
Frequency [GHz]
Simulation
Measurement
Frequency [GHz]
Single resonator breaks down
Both resonators break down
Figure 6.4 (a) The simulated and measured transmission coefficients of the device shown
in Fig. 6.3 under low-power excitation. (b) The simulated transmission coefficients of the
device as a function of frequency for situations where either one (the high-frequency) or
both resonators break down.
it has a lower resonant frequency of 7.516 GHz. The composite resonator (left and right
resonators) has a resonant frequency of 9.382 GHz which is the operating frequency of the
high-power source used in our experiments. At the operating frequency of 9.382 GHz, the
field enhancement factor in the capacitive gap region of the left resonator is higher than that
of the right resonator for equal illumination intensity. Consequently, the high frequency
resonator is expected to breakdown first when the incident power level is increased from
a sub-breakdown threshold intensity. Under this scenario (i.e., only the high-frequency
resonator breaking down), the device becomes semi-transparent as shown in Fig. 6.3(b)
and it is expected to exhibit a power transmission coefficient of |T |2 = 0.39. As the power
level increases, both resonators are expected to break down making the structure even
more transparent with a transmission coefficient of |T |2 = 0.63 as shown in Fig. 3(b).
Figure 3(b) shows that this new design offers a significantly larger contrast between the
transmission coefficients of the metasurface under different breakdown scenarios thereby
making the task of diagnosis more reliable.
118
Table 6.1 Comparison of the power reflection and transmission coefficients of the device
shown in Fig. 6.3(a) derived from full-wave EM simulations and HPM measurements
before breakdown occurs and when both resonators breaks down simultaneously.
|R|2
|T |2
Simulated (before breakdown)
1
0
Simulated (both resonators break down)
0.33
0.63
Measured (before breakdown)
0.91
0
Measured (both resonators break down)
0.3
0.59
(a)
(b)
Figure 6.5 (a) Normalized time-domain power reflection and transmission coefficients of
the metasurface shown in Fig. 6.3 when the metasurface is illuminated with a peak power
level of 4.2 kW. No breakdown is observed at this power level. (b) As the power level is
increased to 4.4 kW, breakdown occurs in both resonators making the metasurface
transparent.
6.3
Device Fabrications and Measurement
6.3.1
Fabrication and Low-Power Characterization
Figure 6.3(a) shows a photograph of the waveguide version of the unit cell of the
metasurface considered here. The device is fabricated out of a 0.1 mm thick stainless steel
119
(a)
μ = 31
σ =3.9
(b)
μ = 511.3
σ = 140.2
Figure 6.6 (a) Statistical distribution of the rise time of the breakdown at the power level
of 4.4 kW for a total of 100 pulses. The calculated mean, µ, and standard deviation, σ, of
the distribution are 31 nsec and 4 nsec, respectively. (b) Statistical distribution of the
delay time of the breakdown at the power level of 4.4 kW for a total of 100 pulses. The
calculated mean, µ, and standard deviation, σ, of the distribution are 511 nsec and 140
nsec, respectively. The dashed lines represent the corresponding normal distribution
fitting curves for these statistical distributions.
sheet using chemical etching. This device is placed inside a standard WR90 rectangular
waveguide and its transmission coefficient is measured using a calibrated vector network
analyzer at a low power level of 0.3 mW over the frequency range of 8-12 GHz. At this
power level, a transmission null is observed at 9.382 GHz as desired. Figure 6.4(a) shows
the comparison between the measured and simulated transmission coefficients of this device under low power conditions with no breakdown. Figure 6.4(b) shows the simulated
transmission coefficients versus frequency under different breakdown conditions by modeling the breakdown discharges as electromagnetic short circuits in the capacitive gaps.
120
6.3.2
High Power Measurements
The device is also characterized under high-power excitation conditions using the
setup1 shown in Fig. 7 of [4]. The magnetron source used in these high-power experiments generates a 1 µs pulse at 9.382 GHz and a peak power level of 25 kW. We vary
the incident power level between 0.001 and 25 kW to determine the breakdown power
threshold for the metasurface. Since we did not have access to the required high-speed
ICCD cameras, we were not able to analyze the light emitted by the discharge. Therefore,
we relied on changes in the measured time-domain transmission and reflection coefficient
waveforms to identify whether breakdown had occurred and, if so, which resonators broke
down.
Figure 6.5(a) shows the normalized time-domain power reflection and transmission
coefficients measured in atmospheric pressure air when illuminated with a transient peak
power of 4.2 kW. At this power level, the metasurface was completely opaque indicating
that no breakdown had occurred. As seen in Fig. 6.5(b), as the power level was increased
to 4.4 kW, the abrupt changes of the power transmission and reflection coefficients indicate
that breakdown had occurred halfway through the pulse.
The total time of a complete breakdown process includes the initial time (i.e. delay
time) for developing an electron seed and the forming time (i.e. rise time) for the electron
seed to create electron avalanche [117]. The delay time is defined from the beginning of
the incident microwave pulse to the onset of breakdown. The rise time is defined from
the 5% to 95% of the final value of the pulse. Both the delay time and the rise time depend on pressure, gas composition, humidity, electric field intensity distribution [118], and
electrode geometry. Figure 6.6(a) shows the statistical distribution of the rise time for a
total of 100 pulses. Each pulse is separated by at least 5 seconds to prevent interference
1
Note that in this particular setup, there is no physical barrier placed between the two resonators of the
unit cell of the metasurface.
121
Transmission [%]
10.16 mm
UV Opaque
Substrate
UV Opaque
Kynar® Film
20 mm
1 mm
22.86 mm
(a)
50
45
40
35
30
25
20
15
10
5
0
200
(b)
Thickness: 50 μm
250
300
350
Wavelength [nm]
(c)
Figure 6.7 (a) The setup used to physically isolate the two resonators of the unit cell of
the metasurface under investigation in this paper during the high-power microwave
conditions. A physical barrier with the dimension of 20 mm × 10 mm × 1 mm is placed
between two resonators to minimize the chances of having any physical interaction
between the two resonators during the discharge process. (b) Photograph of the setup
(with one waveguide removed) showing the physical barrier composed of 20 layers of
UV opaque Kynarr films placed between the two resonators. (c) The transmission
spectrum of a single UV opaque Kynarr film with a thickness of 50 µm [5] (Reproduced
with permission from Arkema Inc.). Although it is not shown in this spectrum, this
material also blocks VUV photons with wavelengths below 200nm [6].
between neighboring pulses. The calculated mean, µ, and standard deviation, σ, of the
distribution are 31 nsec and 4 nsec, respectively. This statistical result shows that an electron seed requires 31 nsec to create electron avalanche under external microwave electric
field excitations with a gap width of 0.2 mm in air at atmospheric pressure. The statistical
122
(a)
(b)
(c)
Figure 6.8 (a) Normalized time-domain power reflection and transmission coefficients of
the metasurface measured using the setup shown in Fig. 6.7(a) when it is illuminated with
a power level of 4.2 kW. At this power level, no breakdown is observed. (b) As the power
level is increased to 4.4 kW, breakdown occurs. The values of the power transmission and
reflection coefficients indicate that breakdown occurs at only single (left) resonator in this
case. (c) As the power level is increased to 7.6 kW, the transmission and reflection
coefficients change and the metasurface becomes more transparent. This scenario and the
higher measured transparency are consistent with both resonators breaking down
simultaneously.
distribution of the delay time for the 100 pulses is shown in Fig. 6.6(b). The calculated
mean, µ, and standard deviation, σ, of the distribution are 511 nsec and 140 nsec, respectively. This statistical result demonstrates that 511 nsec on average is needed to develop
an electron seed. These values of µ and σ agree with our previous experimental results
described in [4]. The dashed lines in Figs. 6.6(a)-(b) represent the corresponding normal
distribution curves fitted to these data.
When breakdown occurred, this metasurface showed the maximum transparency indicating that both resonators broke down simultaneously. This agrees well with the results
reported previously in [4]. Additionally, the measured transmission and reflection coefficients of the metasurface agree reasonably well with the simulated values reported in
Table 6.1. However, a transmission coefficient close to |T |2 = 0.39, indicating a singleresonator breakdown event, was never observed in these experiments as the incident power
123
level was varied between 0 and 25 kW. This observation is also consistent with the results
presented in [4] and further suggests that the breakdown event in the resonator with a lower
breakdown threshold facilitates the breakdown of the resonator with a higher breakdown
power threshold. We hypothesized that one of three mechanisms was responsible for this
phenomenon. Specifically, energetic electron transport from the discharge created at the
first resonator to the second one, UV radiation, or VUV radiation. In the following section,
we describe the experiments that were carried out to identify the physical basis underlying
this phenomenon.
6.4
High-Power Measurements With a Physical Barrier Between the
Resonators
To identify the physical mechanism of interaction between the two resonators of this
metasurface, we conducted three sets of experiments in which the two resonators of the
unit cell were physically separated from each other. First, a barrier which was opaque
to electrons, UV, and VUV was placed between the two resonators and a high-power
experiment was conducted. This experiment was intended to confirm the expected three
state operation of the metasurface and rule out any design or fabrication error possibility.
Subsequently, the barrier was replaced with a UV-transparent but VUV-opaque barrier to
examine the effect of UV radiation. Finally, a third VUV-transparent barrier was placed
between the two resonators and high-power measurements were repeated.
Figure 6.7(a) shows the setup used in the first set of experiments. A 20 mm × 10
mm × 1 mm physical barrier was placed between the two series LC resonators to prohibit
any physical interactions between the two resonators. Figure 6.7(b) shows a photograph
of the physical barrier, which consisted of 20 layers of UV opaque Kynarr film2 . The
2
Kynarr film is a registered trademark of Arkema Inc.
124
transmission spectrum of a single UV opaque Kynarr film with a thickness of 50 µm
is shown in Fig. 6.7(c) [5]. Although it is not shown in this spectrum, this multi-layer
substrate also blocks VUV photons in the wavelength range of 100 nm to 200 nm.
Figure 6.8(a) shows the normalized time-domain power reflection and transmission
coefficients of the metasurface, in the presence of the barrier, when illuminated with a
power level of 4.2 kW. At such a low power level, no breakdown was observed, similar to the experiments conducted with no barrier (see Section 6.3-B). As the power level
was increased to 4.4 kW, breakdown occurred halfway through the pulse as shown in Fig
6.8(b). The values of the power transmission and reflection coefficients measured in this
case indicate that breakdown occurred at only one single resonator (the resonator on the
left in Fig. 6.3(a)). Breakdown at only the LC resonator with lower threshold breakdown
power level was confirmed visually by images obtained with a conventional video camera.
As the power level was increased to 7.6 kW, the measured power transmission and reflection coefficients of the device further changed and the device became more transparent
(see Fig. 6.8(c)). The measured response revealed that both resonators broke down simultaneously in this case. Table 6.2 shows the measured and simulated power transmission
and reflection coefficients before breakdown occurred, when only the left resonator broke
down, and when both resonators broke down. The threshold breakdown power levels of
the metasurface for the single and double resonator discharge conditions were 4.4 kW
and 7.6 kW, respectively. The results of this experiment demonstrated that when the two
resonators are isolated, the metasurface functions with impedance states that change discretely with power, as predicted. It also demonstrated that the simultaneous breakdown in
both resonators observed in the absence of a barrier was not due to a change in the electromagnetic field distribution within the metasurface caused by breakdown in the resonator
with a lower breakdown power threshold.
125
6.4.1
Effect of UV Radiation
To investigate whether UV radiation (200-400 nm wavelengths) from a discharge in the
lower breakdown threshold resonator was able to cause simultaneous breakdown of both
resonators, an UV-transparent barrier was placed between the two resonators, as shown
in Fig. 6.9(a)-(b). The 20 mm × 10 mm × 1 mm barrier prohibited energetic electrons
from one resonator to reach the other resonator but allowed the passage of UV photons
between the two resonators. The barrier was made of fused silica. Figure 6.9(c) shows the
transmission spectrum and filtering characteristics of a 1 mm-thick slab of this material.
As can be observed, the material is transparent in the 200nm to 400 nm wavelength range.
While not shown in this figure, fused silica is opaque at wavelengths below 200 nm with a
very high attenuation level below 156 nm [33].
Similar to the previous case, the metasurface (with the barrier between the two resonators) was illuminated with a high-power pulse. The power of the incident pulse was
varied between 0.001 and 25 kW and the transmission and reflection coefficients of the
device were measured. Similar to the previous experiment, three distinct responses were
observed. These corresponded to no breakdown, a single breakdown in just the lefthand
resonator, or simultaneous breakdown of both resonators. The breakdown thresholds were
observed to be identical to the previous case. Therefore, UV radiation was also ruled out as
the physical interaction mechanism causing simultaneous breakdown in the two resonators
of the metasurface.
6.4.2
Effect of VUV Radiation
Photoemission in the VUV range from 100 nm to 130 nm can be significant with high
pressure air discharges [119]. Meanwhile, oxygen gas has low photoabsorption in this
126
UV Transparent
Substrate
10.16 mm
UV Transparent
Fused Silica
20 mm
1 mm
22.86 mm
(a)
Thickness: 1 mm
100
Transmission [%]
(b)
99
99.14% 185 nm
98
97
96
95
0
500
1000
1500
2000
2500
Wavelength [nm]
(c)
Figure 6.9 (a) A UV transparent window with the dimensions of 20 mm × 10 mm × 1
mm is placed between two resonators to allow the propagation of any potential UV
photons that may be generated from the discharge in one resonator to the the location of
the second resonator. (b) Photograph of the setup showing the Fused Silica window
placed between the two resonators. (c) The estimated transmission spectrum of Fused
Silica with a thickness of 1 mm extracted from the transmission spectrum of the sample
with a thickness of 10 mm [7]. Although it is not shown in this spectrum, Fused Silica is
opaque below 156 nm [8].
frequency range [120]. Furthermore, VUV radiation in atmospheric air has been demonstrated to be responsible for development of streamer discharges [121]. To test the hypothesis that VUV radiation from breakdown in the lower threshold resonator induced simultaneous breakdown in the higher threshold resonator, a VUV-transparent physical barrier
127
Table 6.2 Comparison between the simulated and measured power transmission and
reflection coefficients of the metasurface
|R|2 |T |2
Simulated (before breakdown)
1
0
Measured (before breakdown)
0.91
0
Simulated (single resonator breaks down)
0.58
0.39
Measured (single resonator breaks down)
0.56
0.35
Simulated (both resonators break down)
0.33
0.63
Measured (both resonators break down)
0.3
0.57
was placed between the two resonators, as shown in Fig. 6.10(a). The barrier was composed of a 20 mm × 10 mm × 0.5 mm fixture of VUV- and UV-opaque material [9] having
a 10 mm diameter circular opening. A 10 mm diameter, 0.5-mm thick, VUV-transparent
MgF2 window was placed within the circular opening of the fixture. This composite barrier was then placed between the two resonators as shown in Figs. 6.10(a)-(b). Figure
6.10(c) shows the approximate transmission spectrum of a MgF2 window with a thickness
of 0.5 mm [10], verifying high transmissivity to VUV photons with wavelengths greater
than 115 nm.
Using the setup shown in Fig. 6.10, the high-power experiments were repeated. Again,
the power of the incident pulse was increased gradually from 1 W to 25 kW and the
time-domain transmission and reflection coefficients of the device were measured. Figure
6.11(a) shows the normalized time-domain power reflection and transmission coefficients
of the metasurface with the VUV-transparent MgF2 window inserted between the resonators when illuminated with a 4.2 kW incident pulse. At and below this power level,
no breakdown is observed. When the power level was increased to 4.4 kW, breakdown
128
10.16 mm
VUV Transparent
MgF2 Window
VUV Transparent
MgF2 Window
20 mm
0.5 mm
22.86 mm
(a)
100
Thickness: 0.5 mm
(b)
90% 122 nm
Transmission [%]
80
85% 115 nm
60
40
20
0
100
1000
10000
Wavelength [nm]
(c)
Figure 6.10 A physical barrier embedded with a circular VUV transparent MgF2 window
is placed between two resonators to physically forbid electron diffusion and allow the
penetration of VUV emission. (b) Photograph of a dielectric substrate, Rogers 6010.LM
[9], embedded with a circular VUV transparent MgF2 window with the diameter of 10
mm and a thickness of 0.5 mm. (c) The estimated transmission spectrum of a MgF2
window with a thickness of 0.5 mm extracted from the transmission spectrum of the
sample with a thickness of 2 mm [10]. This spectrum demonstrates good transmission for
VUV emission in the range from 115 nm to 130 nm.
occurred halfway through the pulse as shown in Fig. 6.11(b). The measured values of
the transmission and reflection coefficients of the metasurface in this case reveal that both
resonators broke down simultaneously. As the power level was increased above 4.4 kW,
129
(a)
(b)
Figure 6.11 (a) Normalized time-domain power reflection and transmission coefficients
of the device embedded with a VUV transparent MgF2 window when the device is
illuminated with a peak power of 4.2 kW. At this power level, no breakdown is observed.
(b) As the power level is increased to 4.4 kW, breakdown occurs in both resonators
simultaneously as indicated by the measured transmission and reflection coefficient
values.
no further changes were observed in the measured transmission and reflection coefficients
of the device. The measured power transmission and reflection coefficients show good
agreement with the simulated results shown in Table 6.1.
Figure 6.12(a)-(b) show the statistical distributions of the rise time and delay time of
the breakdown when VUV-transparent MgF2 window is inserted between the two resonators. The calculated mean and standard deviation of the distribution of the rise time
are 32 nsec and 5 nsec, respectively. For the delay time, the calculated mean and standard deviation of the distribution are 516 nsec and 151 nsec, respectively. These statistical
distributions are similar to the case in which there is no physical barrier as shown in Fig.
6.6.
Based on these results, it can be concluded that VUV radiation is responsible for simultaneous breakdown in both resonators in such periodic structures.
130
(a)
(b)
μ = 32
σ=5
μ = 516
σ = 151
Figure 6.12 (a) Statistical distribution of the rise time of the breakdown for the case with
VUV physical barrier at the power level of 4.4 kW for a total of 100 pulses. The
calculated mean, µ, and standard deviation, σ, of the distribution are 32 nsec and 5 nsec,
respectively. (b) Statistical distribution of the delay time of the breakdown for the case
with VUV physical barrier at the power level of 4.4 kW for a total of 100 pulses. The
calculated mean, µ, and standard deviation, σ, of the distribution are 516 nsec and 151
nsec, respectively. The dashed lines represent the corresponding normal distribution
fitting curves for these statistical distributions.
6.5
Conclusions
In this paper, we experimentally investigated the cause of simultaneous breakdown at
multiple locations within the unit cell of a metamaterial composed of two LC resonators.
131
When illuminated by high power microwave radiation, the metasurface demonstrated three
distinct responses. It was opaque at low power levels, became partially transparent if
only one of the resonators within the metamaterial unit cell broke down and became more
transparent if both resonators broke down simultaneously. Using high-power microwave
measurements with various dielectric barriers between the two resonators, the cause of
simultaneous breakdown in the two resonators was identified to be VUV radiation from
the lower breakdown threshold resonator causing a reduction in the breakdown threshold
of the other resonator. A similar phenomenon has also been observed in periodic structures
that use a simple unit cell type (e.g., a single resonator) where the breakdown in one unit
cell can cause breakdown in neighbouring unit cells and create a large area discharge over
a rather large surface despite having strong field intensity only over a localized area of that
surface [3]. Understanding this phenomenon and its implications is important in designing
HPM devices that use metamaterials or periodic structures with small, field-enhancing
dimensions and features.
132
Chapter 7
Investigating the Effective Range of VUV-Mediated Breakdown in High-Power Microwave Metamaterials
7.1
Introduction
Periodic structures have long been used at RF and microwave frequencies for a wide
range of applications[11, 13, 14, 122]. Recent examples include design of antenna arrays[123,
124], frequency selective surfaces[35, 125], artificial dielectrics[126, 127], phase shifters[128],
and polarizers[129]. With the emergence of the field of metamaterials, interest in periodic
structures has grown significantly, since metamaterials are implemented using periodic
structures. Metamaterials provide effective medium responses, which cannot be obtained
using naturally occurring materials. They have been used to obtain negative permittivity,
negative permeability, negative index of refraction, invisibility cloaks [54], and superlenses
[114, 112]. Metamaterials have also received attention in the field of high-power microwaves including high-power microwave (HPM) frequency selective surfaces [56, 97],
new HPM sources [40, 41] and amplifiers [37, 38, 2, 1, 42, 43, 44]. Metamaterial structures
use periodic arrangements of metallic and/or dielectric structures with sub-wavelength periods. In such structures the local field intensities within the unit cell of the structure can be
larger than the incident electric field intensity by several orders of magnitude. Therefore,
such structures are highly susceptible to breakdown even when illuminated by moderate
power levels. When breakdown occurs in these structures, the response of the structure
133
HPM radiator
y^
z ^
x
^
Single-layer
metasurface
(a)
z^
HPM radiator
x^
y^
Single-layer metasurface
(b)
Figure 7.1 (a) 3D topology of a single-layer frequency selective surface with a bandstop
response in close proximity to a high-power source. (b) Side view of the structure under
illumination.
will drastically change. Recently, we demonstrated that one can predict how the response
of metamaterials and periodic structures will change when breakdown results in a localized
discharge within the unit cell of the structure. We also demonstrated that if a discharge
is created in a unit cell of a metamaterial, it can cause discharge at other locations within
that unit cell where the localized field intensities may not be enough to cause breakdown
under normal circumstances [4]. This phenomenon results in a distributed discharge that
was exploited to design a distributed discharge limiter as an anti-HPM device [3]. In
Ref. [130], we demonstrated that this phenomenon is due to the generation of vacuum
ultraviolet (VUV) radiation created at the location of the initial discharge. This radiation
propagates to other locations within the structure’s unit cell or other neighboring unit cells
134
and ionizes the air. This reduces the threshold level required to cause breakdown at these
locations and results in nearly simultaneous discharge over a relatively large area in the
periodic structure.
In this paper, we report the results of an experimental investigation of the physical
distances over which VUV generated from one breakdown can mediate others and cause
breakdown at electric field intensity levels below the normal breakdown threshold levels.
A number of different studies have investigated the VUV photoemission in different environments [131, 121, 132, 133, 134, 135, 136]. However, to the best of our knowledge, no
previous study has determined the distance over which VUV photon flux generated from
one discharge remains sufficiently intense to cause a discharge at other locations within a
periodic structure. This study is significant because many periodic structures need to be
placed in close proximity to sources of high-power microwave radiation. Fig. 7.1 shows
such a scenario where a frequency selective surface (FSS) is placed in close proximity to
an HPM antenna. Because of the physical separation between the antenna and the FSS
and the radiation patterns of the antenna, the field intensity over the FSS will not be uniform and the central regions of the FSS are illuminated with higher electric field intensities
than the edges of the structure. In such a situation, the field intensity at the center of the
FSS may be high enough to cause breakdown. The VUV radiation created as a result
of this discharge can then mediate the breakdown at other locations within this periodic
structure resulting in a distributed discharge covering the entire FSS. Knowing the physical distances over which this phenomenon acts can help design the periodic structure in a
manner to ensure that a single discharge does not result in a catastrophic failure. Alternatively in certain applications where the generation of periodic plasmas or large distributed
discharges are required, knowing this interaction distance will aid in designing the structure to ensure that a large scale distributed surface breakdown occurs.
22.86 mm
High-Q
resonator
t
10.16 mm
135
Low-Q
resonator
k
E
H
t
E
H
(a)
k
VUV Photons
(b)
Figure 7.2 (a) 3D topology of the two cascaded unit cells described in Figs. 7.3(a)-(b)
with variable distances. (b) Side view of (a) showing that initial VUV photons generated
at one unit cell propagate and arrive the other unit cell to induce breakdown.
7.2
Experimental Approach
In our experiments, we used a periodic structure composed of two single layer meta-
surfaces cascaded back to back. To simplify the task of high-power measurements, we
conducted our experiments in a waveguide environment. Therefore, only one unit cell
of each metasurface is needed. This configuration is shown in Fig. 7.2, where the two
unit cells of the metasurfaces are placed inside a rectangular waveguide. The unit cells are
both composed of meandered line inductors separated by capacitive gaps as shown in Figs.
7.3(a) and 7.3(b). At the resonant frequency of the inductors and capacitors, each metasurface behaves as a bandstop filter completely reflecting the incident electromagnetic wave.
136
However, the two unit cells are not identical and they have different resonant frequencies
and quality factors (Q). The resonant frequencies and quality factors of each resonator
are chosen to ensure that the cascaded arrangement shown in Fig. 7.2 provides distinct
responses for three different operating states. These include: 1) low incident power level
where no resonator was expected to breakdown; 2) intermediate power levels where the
high Q resonator was expected to breakdown but under normal circumstances the low Q
resonator was not expected to breakdown; and 3) high-power situations where both resonators were expected to breakdown under normal circumstances. When no breakdown
occurred, the device was completely opaque at the operating frequency of 9.382 GHz. For
the intermediate power levels, breakdown in the high-Q resonator short circuited the capacitive gap and converted the LC resonator to a simple resistive-inductive circuit. This
caused partial transparency in the structure and increased the transmission coefficient of
the structure. Finally, as the power level was increased to create breakdown in both resonators, the transparency of the structure further increased and the transmission coefficient
increased as well. These three different scenarios are shown in Fig. 7.3(c) where the simulated transmission coefficient of the device, for a spacing of 6.5 mm between the two
resonators, is shown for these three different conditions. We emphasize that these simulation results were obtained based on the assumption that the resonators did not interact
with each other through VUV radiation. In the experiments, the structure was illuminated
with high-power microwave pulses with durations of 0.8µs and a frequency of 9.382 GHz.
The time-domain power transmission and reflection coefficients of each device were measured for various incident power levels to determine whether or not these three different
responses were observed. If only two of these possible three states were observed (i.e,
complete opacity and maximum transparency), we concluded that VUV radiation facilitates creation of breakdown at the location of the low-Q resonator. This was based on
137
w2
w1
a2
a1 b
1
h2
h1
b2
m2
l2
d1
d2
(a)
(b)
0.65
0.32
Before breakdown
Single resonator breaks down
Both resonators break down
(c)
Figure 7.3 (a)-(b) Unit cells of the two resonators used in a bandstop metasurface
composed of two lumped resonators cascaded with variable distances. (c) The
time-domain power transmission coefficients for three different cases (i.e. no breakdown,
one breakdown, and two breakdowns) for the case where the spacing between the
resonators is 0.2 mm.
the results of previous research published in Ref. [3]. If all three distinct states were observed, however, we concluded that the separation between the two resonators was large
enough so that VUV radiation had decayed substantially. This process was then repeated
for a number of different spacings between the two resonators to determine the physical
distance over which the two resonators could interact through VUV radiation.
138
7.3
Results and Analysis
7.3.1
Device design and simulation results
Figures 7.3(a) and 7.3(b) show the topology of the unit cells of the metasurfaces used
in this experimental study. The unit cells are designed to yield three distinct responses for
resonator separation distances in the range of 6.5 mm to 326 mm. This is done by simulating the response of the structure in CST Microwave Studio for different separations
between the two resonators under different operational conditions (i.e., no breakdown and
one or two breakdowns). The dimensions of the unit cells of these metasurfaces are shown
in Table 7.1. Notice that a total of 8 different pairs of high-Q and low-Q resonators were
designed to conduct these experiments. This was necessary because as the separation between the two metasurfaces was increased, the overall response of the structure changed.
Since we were considering structures in which the separation between the two resonators
increased from very small values (6.5 mm) to very large values (260 mm), a single pair of
high-Q and low-Q resonators would not be able to provide the necessary contrast between
the three different states for all these separation distances. Figure 7.4 shows the transmission coefficient of these devices, as a function of frequency, for the different separations
between the two resonators. As can be observed, at the frequency of operation of 9.382
GHz, in all cases the simulations of the devices predicted three distinct transmissivities.
This satisfied the design conditions required for determining if breakdown has occurred in
this device and if so, in which resonators.
7.3.2
Measurement Results
The unit cells described in the previous section were fabricated using chemical etching
of 0.005”-thick stainless steel sheets. The fabricated structures were placed inside a WR90 rectangular waveguide and their transmission and reflection coefficients were measured
139
Table 7.1 The dimensions of the unit cells of the metasurfaces used in our experiments
with different separation distances.
Distance (mm)
w1
a1
b1
h1
d1
w2
6.5
0.2
1.6
2.58
0.2
1.2
0.2
13.6, 15.16, 15.78, 16.2, and 16.74
0.2
1.6
2.58
0.2
1.2
0.2
16.92, 17.42, 18.58, 20.1, and 51
0.2
1.9
2.58
0.2
1.2
0.2
61.21, 81.31, and 97
0.2
1.85
2.58
0.2
1.2
0.2
110.6
0.2
1.85
2.58
0.2
1.2
0.2
117.1
0.2
1.9
2.58
0.2
1.2
0.2
260
0.2
1.85
2.58
0.2
1.2
0.2
326
0.2
1.85
2.58
0.2
1.2
0.2
Distance (mm)
b2
a2
h2
m2
l2
d2
6.5
2.58 2.75
0.2
0.3
0.4
1.2
13.6, 15.16, 15.78, 16.2, and 16.74
2.58
1.65
0.2
0.3
1
1.2
16.92, 17.42, 18.58, 20.1, and 51
2.58
1.7
0.2
0.3
1.8
1.2
61.21, 81.31, and 97
2.58
2
0.2
0.3
1.4
1.2
110.6
2.58
2.2
0.2
0.3
1.2
1.2
117.1
2.58 2.45
0.2
0.3
1.2
1.2
260
2.58
1.1
0.2
0.3
1.4
1.2
326
2.58
1.6
0.2
0.3
1.3
1.2
140
Table 7.2 Experimental results for the two cases of one breakdown and two breakdowns
when illuminated with 100 pulses.
Distance (mm)1
<16.74
16.74
16.92
17.42
18.58
Three Different States
0
10
24
29
32
Two Different States
100
90
76
71
68
Distance (mm)
20.1
51
71.21
81.31
97
One breakdown
35
44
59
81
83
Two breakdown
65
56
41
19
17
Distance (mm)
110.6
117.1
260
326
One breakdown
95
98
100
100
Two breakdown
5
2
0
0
Table 7.3 Experimental results in nitrogen gas and at atmospheric pressure level for the
two cases of one breakdown and two breakdowns when illuminated with 100 pulses.
Distance (mm)2
<16.92
16.92
17.42
18.58
20.1
Three Different States
0
2
5
12
22
Two Different States
100
98
95
88
78
Distance (mm)
51
71.21
81.31
97
110.6
One breakdown
33
42
52
68
78
Two breakdown
67
58
48
32
22
Distance (mm)
117.1
260
326
One breakdown
84
100
100
Two breakdown
16
0
0
141
9.382 GHz
9.382 GHz
9.382 GHz
0.82
9.382 GHz
0.85
0.89
0.65
0.75
0.68
0.72
0.32
0
0
(a)
No breakdown
Single resonator breaks down
Both resonators break down
(b)
No breakdown
Single resonator breaks down
Both resonators break down
9.382 GHz
9.382 GHz
0.65
0.78
0.71
0
(c)
No breakdown
Single resonator breaks down
Both resonators break down
(d)
No breakdown
Single resonator breaks down
Both resonators break down
9.382 GHz
9.382 GHz
0.7
0.6
0.67
0.56
0
0
(e)
No breakdown
Single resonator breaks down
Both resonators break down
0
(f )
No breakdown
Single resonator breaks down
Both resonators break down
0
0.32
(g)
No breakdown
Single resonator breaks down
Both resonators break down
0
(h)
No breakdown
Single resonator breaks down
Both resonators break down
Figure 7.4 Simulated transmission coefficients of the devices as functions of frequency
for different separation distances betweens the two resonators: (a) 6.5 mm, (b) 13.6 mm,
(c) 16.2mm, (d) 16.74mm, (e) 16.92mm, (e) 17.42mm, (f) 18.58mm, and (g) 97 mm.
using the setup reported in Ref. [4]. The measurements were performed for different power
levels and different separation distances between the two resonators. Each measurement
was performed at least 100 times to obtain statistically relevant information. Additionally,
all the experiments were conducted in air and at atmospheric pressure levels. Based on
the measured time-domain power transmission coefficients, we could determine whether
only one resonator broke down or both resonators broke down. Figure 7.5 shows several
representative samples of the measured time domain transmission coefficients for these
devices at different separation distances. As can be seen, for smaller separation distances
only two distinct transmission coefficients are measured whereas for larger separation distances three distinct transmission coefficients are observed. Table 7.2 shows a summary of
142
0.83
0.83
0.77
0.66
0.63
(a)
(b)
No breakdown
Both resonators break down
(c)
No breakdown
Both resonators break down
0.78
0.61
(d)
No breakdown
Both resonators break down
0.74
0.61
No breakdown
Single resonator breaks down
Both resonators break down
0.66
0.53
0.57
0.28
(e)
No breakdown
Single resonator breaks down
Both resonators break down
(f )
No breakdown
Single resonator breaks down
Both resonators break down
(g)
No breakdown
Single resonator breaks down
Both resonators break down
(h)
No breakdown
Single resonator breaks down
Both resonators break down
Figure 7.5 Measured time-domain transmission coefficients of the devices for different
separation distances between the two resonators: (a) 6.5 mm, (b) 13.6 mm, (c) 16.2mm,
(d) 16.74mm, (e) 16.92mm, (e) 17.42mm, (f) 18.58mm, and (g) 97 mm.
the measurement results for each distance. Specifically, out of the 100 individual measurements conducted for each case, the number of times where either three distinct responses
or only two distinct responses are observed are given. It can be seen that when the separation between the two resonators was shorter than 17 mm, we only observed two distinct
responses (no breakdown and simultaneous breakdown in both resonators). This is similar
to the phenomenon of simultaneous breakdown described in Ref. [137] where the VUV
emission dominates. However, if the distance was larger than this value, we started to detect three distinct responses corresponding to the cases for no breakdown, one breakdown,
and two breakdowns indicating that the intensity of the VUV radiation decayed along the
propagation direction enough so that the VUV mediated breakdown did not occur in every
experiment. Meanwhile, in some experiments, we still observed only two states. The statical results shown in Table 7.2 demonstrate that one breakdown consistently occurred with
143
Data points
Fitting Curve (Polynomial Function)
Figure 7.6 Percentage of the time where two separate breakdown states were observed
for the experiment shown in Fig. 7.2 when the experiments were conducted in air and at
atmospheric pressure levels. The inset shows data points in the range from 0 mm to 20
mm.
100% probability if the separation distance was at least 260 mm. In contrast, two simultaneous breakdowns consistently occurred with 100% probability if the spacing between the
two resonators was less than 17 mm. Therefore, from these experimental results we can
identify three different separation ranges. For distances smaller than 17 mm, simultaneous
breakdown was observed 100% of the time. For intermediate distances in the range of 17
mm to 260 mm, simultaneous breakdown was observed less than 100% of the time and
this percentage decreased as the separation distance increased. Finally, for distances larger
than 260 mm, simultaneous breakdown was never observed in any of the experiments. Fig.
7.6 shows the percentage of times where the three distinct response types were observed
in these devices as a function of the separation distance between the two metasurfaces.
144
Data points
Fitting Curve (Polynomial Function)
Figure 7.7 Percentage of the time where two separate breakdown states were observed
for the experiment shown in Fig. 7.2 when the experiments were conducted in pure
nitrogen gas and at atmospheric pressure levels. The inset shows data points in the range
from 0 mm to 20 mm.
It is known that water vapor and humidity can significantly influence VUV photon
absorption and thus affect the results shown in Fig. 7.6[138, 139]. To suppress the effect
of humidity, the previous experiments were repeated in pure nitrogen gas at atmospheric
pressure level, which is used to emulate dry air due to the fact that air is mostly composed
of nitrogen gas. The measured results are shown in Fig. 7.7 and Table 7.3. Again, we
observed simultaneous breakdown 100% of the time for distances smaller than 17 mm,
less than 100% of the time for intermediate distances in the range of 17 mm to 260 mm,
and none for distances larger than 260 mm. The only difference between Fig. 7.6 and Fig.
7.7 is that the percentage of simultaneous breakdown observations in laboratory air with
uncontrolled, nonzero humidity was higher than that in pure nitrogen gas.
145
7.4
Conclusions
In this work, we experimentally investigated the effective range over which vacuum
ultraviolet (VUV) radiation generated by a localized discharge in a unit cell of a highpower-microwave metamaterial can mediate discharge at other locations within the same
periodic structure. We conducted these measurements in air and at atmospheric pressure
levels, since most periodic structures and metamaterials are operated in such conditions.
In these experiments, we were able to identify three different interaction regions. In the
first region (distances less than 17 mm in our experiments), VUV radiation created by the
initial discharge always mediated breakdown at the location of the other unit cell where
the electric field intensity was not high enough to cause breakdown under normal circumstances. In the second region (distances larger than 260 mm), creation of an initial
discharge was never observed to mediate the creation of breakdown at the location of the
second unit cell. For the intermediate distances, the initial discharge was observed to
mediate discharge at the location of the second resonator with a probability greater than
zero and less than 100 percent. The percentage of the times where this phenomenon was
observed was seen to decrease as the separation between the two resonators increased.
The specific boundaries identified in this investigation (17 mm and 260 mm) are naturally
subject to the physical parameters of the unit cells used (e.g. their maximum field enhancement factor values) as well as environmental conditions over which the experiments
are performed (e.g. humidity, pressure levels, etc.) Nonetheless, they can still shed some
light on critical issues involved in the design of high-power microwave metamaterials.
Specifically, many metamaterials operating in the microwave and millimeter-wave bands
are composed of sub-wavelength periodic structures. At frequencies higher than a few
GHz range, the periodicities of these structures are frequently below 1 cm and their localized field enhancement factors are close to those of the structures reported in this work.
146
The results of this study suggest that in such structures if breakdown occurs in one unit cell
(e.g. due to high localized field intensity), the VUV radiation generated can result in the
creation of a distributed discharge over a large surface area. In other words, a single point
of failure can result in failure of the whole structure. As the use of metamaterials is becoming more widespread in high-power microwave systems, this phenomenon presents an
important design issue that must be taken into account. Alternatively, the VUV mediated
breakdown in periodic structures can be used as a method to generate periodic plasmas
over large surface areas relatively easily (e.g. by creating localized discharges in suitably
designed periodic structures). This can be beneficial in applications where periodic plasma
structures are needed. Finally, this concept may be used to design plasma switchable (or
reconfigurable) periodic structures that may be used for high-power beam steering and
phased array applications.
147
Chapter 8
Future Work
In the above chapters, we have investigated metamaterials for high-power applications
including implementing high-power frequency selective surfaces by using the electromagnetic wave tunneling through ϵ-negative multi-layer structures, experimentally examining
the impact of microwave induced breakdown on the responses of high-power metamaterials, studying the simultaneous breakdown due to VUV photoemission within multiresonator unit cells of metamaterials with discrete nonlinear responses. These studies are
indeed essential in our goal of designing plasma tunable high-power metamaterials. Future work based upon this thesis are still needed to continue the existing efforts toward our
final goal.
8.1
Developing a plasma reconfigurable high-power metasurface
Among versatile tuning techniques, plasma plays an important role in the applications
of tunable antennas and tunable phase shifters because its effective permittivity can be
altered easily by changing the electron densities and it is suitable for operating in highpower systems. The ultimate goal of my research is to explore a plasma reconfigurable
high-power metasurface. The principle mechanism is that the frequency response of the
high-power metasurface can be tuned continuously when microwave-induced breakdown
148
Grating lobe
Incident
waves
(W-band)
Reflected
waves
Reflected
waves
Incident
waves
(W-band)
Metallic Metamaterial
-Based Mask
Polycarbonate
Normal incidnet waves
(X-band)
Before Breakdown (off mode)
After Breakdown (on mode)
(a)
(b)
Figure 8.1 3D topology of a high-power metasurface illuminated with an oblique
incident wave in W-band for the cases (a) before no microwave-induced breakdown
occurs and, (b) after microwave-induced breakdown occurs on the surface.
occurs at different locations or unit cells. For this purpose, I currently conduct a preliminary studying of such a high-power metasurface shown in Fig. 8.1 acting as a switch. The
device is designed being opaque when illuminated with an oblique incident wave in Wband shown in Fig 8.1.(a) where the incident power levels are not high enough to induce
breakdown on the surface of the device. In addition to the reflected wave in the opposite
direction following the Snells law, a grating lobe (higher-order mode) is detected in the
broadside direction due to the contribution of each unit cell. This case is referred as off
mode of the device. The on mode is referred to that when the device is illuminated with a
high-power incident wave in X-band on the other side, the resulted electric field intensities
on the surface are high enough to generate breakdown over entire surface and eliminate the
grating lobe show in Fig. 8.1(b). My current plan is to design this metasurface with fullwave EM simulation in CST Microwave Studio and experimentally examine this device
before and after breaking down. This research is expected to show that the characteristics
149
of high-power metasurface can be switched by plasma and it provides beneficial information for plasma tunable periodic structures and metamaterials.
150
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