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Microwave lenses for high-power phased-array applications

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MICROWAVE LENSES FOR HIGH-POWER PHASED-ARRAY APPLICATIONS
by
Mudar Al-Joumayly
A dissertation submitted in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
(Electrical Engineering)
at the
UNIVERSITY OF WISCONSIN-MADISON
2011
UMI Number: 3471444
All rights reserved
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UMI 3471444
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MICROWAVE LENSES FOR HIGH-POWER
PHASED-ARRAY APPLICATIONS
submitted to the Graduate School of the
University of Wisconsin-Madison
in partial fulfillment of the requirements for the
degree of Doctor of Philosophy
By
Mudar Alaa Al-Joumayly
Date of final oral examination:
May 4, 2011
Month and year degree to be awarded: May 2011
The dissertation is approved by the following members of the Final Oral Committee:
Nader Behdad, Assistant Professor, Electrical and Computer Engineering
Susan C. Hagness, Professor, Electrical and Computer Engineering
Daniel van der Weide, Professor, Electrical and Computer Engineering
Akbar M. Sayeed, Professor, Electrical and Computer Engineering
Francesco A. G. Volpe, Assistant Professor, Engineering Physics
© Copyright by Mudar Al-Joumayly 2011
All Rights Reserved
1
To my wonderful parents Rafah and Alaa
To my lovely wife Ola and my precious son Firas
11
ACKNOWLEDGMENTS
It definitely has been an exciting and a rewarding journey. It all started when I met Professor
Nader Behdad at the University of Central Florida five years ago. I would like to thank him for
giving me the opportunity to work with him and continue my post graduate studies. I am deeply
grateful to him for his mentorship, support, and advise that I won't soon forget. He taught me
many important lessons in academia and life which I believe will always guide me for the rest of
my life.
I would also like to thank the members of my doctoral committee: Professors Daniel Van der
Weide, Akbar Sayeed, Francesco Volpe, and especially Professor Susan C. Hagness. I really enjoyed attending Professor Hagness' computational methods in electromagnetics course and having
had the privilege of working with her.
I would like to thank my colleagues and friends at the Antenna, RF, Microwave, and Integrated
Systems (ARMI) at the University of Central Florida. Special thanks and appreciations go out
to Rajesh C. Pariyani, Yazid Yusuf, and Ajay Subramanian for always being there in tough times
and for their personal friendship outside the lab. In addition, I would like to thank my colleagues
and friends at University of Wisconsin-Madison. I especially would like to thank Suzette Aguilar,
Meng Li, and Bin Yu for their friendship and for making the last two years enjoyable.
I would like to acknowledge my wife Ola for her support and love during the years of our
marriage, the birth of my precious Wisconsinite son Firas, who was born in September of 2010
and who was a great inspiration for me. Also, I would like to acknowledge my wonderful sisters
Sabreen, Nesreen, and Yasamen for their love and support.
Last but not least, I would like to thank my parents Alaa and Rafah. It is your constant love,
endless support, great sacrifices, and guidance that made this accomplishment possible.
iii
TABLE OF CONTENTS
Page
LIST OF TABLES
vii
LIST OF FIGURES
xii
ABSTRACT
1
Introduction
1.1
1.2
1.3
1.4
Motivation
Proposed Approach
Literature review
1.3.1 Microwave Lenses
1.3.2 Single-band frequency selective surfaces
1.3.3 Multi-band frequency selective surfaces
Thesis overview
1.4.1 Chapter 2: Hybrid Frequency Selective Surfaces with Third-Order Responses
1.4.2 Chapter 3: Synthesis of Hybrid Frequency Selective Surfaces with OddOrder Frequency Responses
1.4.3 Chapter 4: Dual-Band Frequency Selective Surfaces Using Hybrid Periodic Structures
1.4.4 Chapter 5: Second-Order Frequency Selective Surfaces Using Non-resonant
Periodic Structures
1.4.5 Chapter 6: Synthesis of Generalized Frequency Selective Surfaces of Arbitrary Order Using Non-resonant Periodic Structures
1.4.6 Chapter 7: Design of Planar Microwave Lenses Using Sub-Wavelength
Spatial Phase Shifters
1.4.7 Chapter 8: Design of Planar True-Time-Delay Lenses Using Sub-Wavelength
Spatial Time-Delay Units
1.4.8 Chapter 9: Study of Power Handling Capability of Miniaturized-Element
Frequency Selective Surfaces
xxiv
1
1
5
7
7
9
13
14
15
16
16
17
17
18
18
19
iv
Page
2
Hybrid Frequency Selective Surfaces with Third-Order Responses
20
2.1
2.2
20
22
22
26
30
33
2.3
2.4
3
Synthesis of Hybrid Frequency Selective Surfaces with Odd-Order Frequency Responses
3.1
3.2
3.3
3.4
4
Introduction
Generalized FSS Topology
3.2.1 Topology and Equivalent Circuit Model
3.2.2 Synthesis Procedure for Third-Order FSSs
3.2.3 Synthesis Procedure for Higher-Order Bandpass FSSs (N > 3)
FSS Implementation and Verification of the Synthesis Procedure
3.3.1 FSS Implementation
3.3.2 Synthesis Procedure Verification
Conclusions
35
35
36
36
39
40
. 42
42
44
48
Dual-Band Frequency Selective Surfaces Using Hybrid Periodic Structures
49
4.1
4.2
49
51
51
54
58
58
65
68
4.3
4.4
5
Introduction
FSS Design
2.2.1 Principles Of Operation
2.2.2 Design Procedure
Experimental Verification and Measurement Results
Conclusions
Introduction
Principles of Operation and the Design Procedure
4.2.1 Principles of Operation
4.2.2 Design Procedure
Theoretical and Experimental Verification
4.3.1 Simulation Results
4.3.2 Experimental Verification
Conclusions
Second-Order Frequency Selective Surfaces Using Non-resonant Periodic Structures 70
5.1
5.2
5.3
Introduction
Principles of Operation and FSS Design Procedure
5.2.1 Principles of operation
5.2.2 Design Procedure
Experimental Verification and Measurement Results
:
70
71
71
73
78
v
Appendix
Page
5.4
6
Synthesis of Generalized Frequency Selective Surfaces of Arbitrary Order Using
Non-resonant Periodic Structures
6.1
6.2
6.3
6.4
7
78
83
88
89
Introduction
89
Generalized Synthesis Procedure
90
6.2.1 Topology and Equivalent Circuit Model
90
6.2.2 General Synthesis Procedure
93
Validation of the Proposed Synthesis Procedure
98
6.3.1 Synthesis of A Third-Order Bandpass FSS: Design and Simulation
98
6.3.2 Synthesis of A Third-Order Bandpass FSS: Fabrication and Measurement . 102
6.3.3 Synthesis of a Fourth-Order Bandpass FSS
105
Conclusions
107
Design of Planar Microwave Lenses Using Sub-Wavelength Spatial Phase Shifters . 109
7.1
7.2
7.3
7.4
8
5.3.1 FSS Prototypes and Simulation Results
5.3.2 Measurement Results
Conclusions
Introduction
109
Lens Design and Principles of Operation
112
7.2.1 Miniaturized-Element Frequency Selective Surfaces as Spatial Phase Shifters 114
7.2.2 Lens Design Procedure
115
Experimental Verification
122
7.3.1 Measurement System
123
7.3.2 Measurement Results
126
Conclusions
135
Design of Planar True-Time-Delay Lenses Using Sub-Wavelength Spatial TimeDelay Units
147
8.1
8.2
147
149
8.3
8.4
Introduction
Lens Design and Principles of Operation
8.2.1 Miniaturized-Element Frequency Selective Surfaces as Spatial Time Delay
Units
8.2.2 Lens Design Procedure
Numerical Verification
Conclusions
150
153
158
160
VI
Appendix
Page
9
Study of Power Handling Capability of Miniaturized-Element Frequency Selective
Surfaces
162
9.1
9.2
9.3
9.4
162
163
166
168
Introduction
Dielectric breakdown caused by the electric
FSS failure due to heat dissipation
Conclusions
APPENDIX
field
Physical dimensions of MEFSSs reported in Chapter 8
LIST OF REFERENCES
170
178
Vll
LIST OF TABLES
Table
Page
1.1
Comparison among different types of phase shifters used in passive phased arrays. . .
4
1.2
Comparison among different MMIC technologies used in active phased arrays
4
2.1
Electrical parameters of the equivalent circuit model of the proposed third-order FSS. . 28
2.2
Physical parameters of the proposed third-order FSS
29
3.1
Normalized quality factors and coupling coefficients for third-order coupled resonator
filters with different filter responses
40
3.2
Physical and electrical parameters of the third-order bandpass FSS of Section 3.3.2. . . 45
3.3
Normalized quality factors and coupling coefficients for fifth-order coupled resonator
filters with different filter response types
46
Equivalent circuit values and physical and geometrical parameters of the fifth-order
bandpass FSS discussed in Section 3.3.2
47
Physical and electrical parameters of the structure discussed in Section 4.3.1. The
equivalent circuit values presented in this table are obtained through an optimization
process in Agilent ADS and refer to the circuit model of Fig. 4.2(a). In this optimization procedure the values predicted by (4.1 )-(4.9) were used as initial values
61
3.4
4.1
4.2
Physical and electrical parameters of the proposed dual-band FSS utilizing the slot
resonator shown in Fig. 4.3(b) and its equivalent circuit model shown in Fig. 4.2(a). . 67
5.1
Normalized quality factors and coupling coefficient for realizing different filter responses
73
5.2
Physical parameters of the first fabricated FSS prototype shown in Fig. 5.6
80
5.3
Physical parameters of the second fabricated FSS prototype shown in Fig. 5.8
84
Page
1
Normalized parameters of third-order bandpass coupled resonator filters. For thirdorder coupled resonator filters, r\ = rN = 1
100
2
Physical and electrical parameters of the third-order bandpass FSS of Section 6.3.1. . . 101
3
Physical and electrical parameters of the third-order bandpass FSS of Section 6.3.2.
4
Normalized parameters of fourth-order coupled-resonator filters with various response
types
106
Equivalent circuit values and physical and geometrical parameters of the fourth-order
bandpass FSS discussed in Section 6.3.3
106
Physical and electrical properties of the spatial phase shifters that populate each zone
of the first prototype. Insertion loss values are in dB and all physical dimensions are in
jiva. For all of these designs, w2 = w4 = 2.4 mm and h1>2 = ^2,3 = ^3,4 = ^4,5 = 0.5
mm
122
5
1
2
. 103
Physical and electrical properties of the spatial phase shifters that populate each zone
of the second lens prototype. Insertion loss values are in dB and all physical dimensions are in /mi. For all of these designs, w2 = w4 = 2.4 mm and hi^ = ^2,3 =
^3,4 = ^4,5 = 0.5 mm
123
1
Field enhancement factors of MEFSSs with different Butterworth responses. The field
enhancement factor is calculated for different response orders and fractional bandwidths. 164
2
Field enhancement factors of FSSs with different Chebyshev responses. The field
enhancement factor is calculated for different response orders and fractional bandwidths. 164
3
Field enhancement factor of MEFSSs as a function of unit cell size. All the MEFSSs
are designed to have a second-order bandpass response operating at 10 GHz with 20%
bandwidth
165
Temperature rise of MEFSSs with different Butterworth responses. The temperature
rise is caused by an average power of 2.2 KW/m2 that is incident on the different
MEFSSs. The temperature rise is calculated for different response orders and fractional bandwidths
167
4
IX
Appendix
Table
9.5
9.6
Temperature rise of MEFSSs with different Chebyshev responses. The temperature
rise is caused by an average power of 2.2 KW/m2 that is incident on the different
MEFSSs. The temperature rise is calculated for different response orders and fractional bandwidths
Page
168
Temperature rise of MEFSSs with two different response types as a function of unit
cell size for an incident power density of 2.2 KW/m2. All the MEFSSs are designed
to have a second-order bandpass response operating at 10 GHz with 20% bandwidth. . 168
A. 1 Physical parameters of a second-order MEFSS with a Butterworth response operating
at 10 GHz with different fractional bandwidths. The unit cell dimensions of all designed prototypes are 6.5 mm x 6.5 mm. The substrates have a dielectric constant of
er= 3.38. wi>2, s1, s2, /iQ{i,2}> andfya{i,2}refer to the geometrical parameters identified
in Fig. 6.1 of Chapter 6
171
A.2
Physical parameters of a second-order MEFSS with a Chevyshev response operating
at 10 GHz with different fractional bandwidths. The unit cell dimensions of all designed prototypes are 6.5 mm x 6.5 mm. The substrates have a dielectric constant of
er= 3.38. wit2, si, s2, /ia{i,2}> andfya{i,2}refer to the geometrical parameters identified
in Fig. 6.1 of Chapter 6
171
A.3
Physical parameters of a second-order MEFSS with Butterworth response operating
at 10 GHz with a fractional bandwidth of 20%. The physical parameters are for different unit cell dimensions. The substrates have a dielectric constant of er= 3.38 and
a thickness of /ia{i,2}=fy3{i,2}:=:0.592 mm. D, w^2, S\, s2, /ia{i,2}> andfyg{i,2}refer to
the geometrical parameters identified in Fig. 6.1 of Chapter 6
171
A.4
Physical parameters of a second-order MEFSS with a Chebyshev response operating
at 10 GHz with a fractional bandwidth of 20%. The physical parameters are for different unit cell dimensions. The substrates have a dielectric constant of er= 3.38 and a
thickness of /ia{i,2}=^/3{i,2}=0.51 mm. D, w\)2, s\, s2, ha{i}2}, andfyg{i,2}refer to the
geometrical parameters identified in Fig. 6.1 of Chapter 6
172
A.5
Physical parameters of a third-order MEFSS with a Butterworth response operating at
10 GHz with a fractional bandwidth of 10%. D, w\j2, u>2,3, s\, s2, s 3 , /ia{i,2}, fys{i,2}j
ha{2,3}, andfyg{2,3}refer to the geometrical parameters identified in Fig. 6.1 of Chapter 6.172
A.6
Physical parameters of a third-order MEFSS with a Butterworth response operating at
10 GHz with a fractional bandwidth of 20%. D, ty li2 , u>2,3, si, s2, s 3 , ha{it2}, fys{i,2}5
ha{2fi}, and ^{2,3} refer to the geometrical parameters identified in Fig. 6.1 ofChapter6.172
X
Appendix
Table
Page
A.7
Physical parameters of a third-order MEFSS with a Butterworth response operating at
10 GHz with a fractional bandwidth of 30%. D, tulj2, w2}3, s1, s2, s3, ha^t2}, fys{i,2}5
ha{2,3}, and ^{2,3} refer to the geometrical parameters identified in Fig. 6.1 of Chapter 6.173
A.8
Physical parameters of a third-order MEFSS with a Chebyshev response operating at
10 GHz with a fractional bandwidth of 10%. D, u>1]2, u>2>3, Si, s2, S3, /ia{i,2}, fys{i,2}5
^a{2,3}5 and hj3{2,3} refer to the geometrical parameters identified in Fig. 6.1 of Chapter 6.173
A.9
Physical parameters of a third-order MEFSS with a Chebyshev response operating at
10 GHz with a fractional bandwidth of 20%. D, wi)2, w2>3, si, s2, s 3 , ha{i,2}, fy3{i,2}>
^a{2,3}, and ^{2,3} refer to the geometrical parameters identified in Fig. 6.1 of Chapter 6.173
A. 10 Physical parameters of a third-order MEFSS with a Chebyshev response operating at
10 GHz with a fractional bandwidth of 30%. D, w\y2, w2,3, si, s 2 , s 3 , /iQ{i,2}, fys{i,2}5
ft-a{2,3}> and /i/3{2,3} refer to the geometrical parameters identified in Fig. 6.1 of Chapter 6.174
A.l 1 Physical parameters of a fourth-order MEFSS with a Butterworth response operating
at 10 GHz with a fractional bandwidth of 10%. D, wij2, u>2;3, tu3)4, s\, s2, s3, s 4 ,
ha{i,2}, hp{i,2], ha{2,3}, ^3(2,3}, ha{3A}, and % 3 ,4} refer to the geometrical parameters
identified in Fig. 6.1 of Chapter 6
174
A. 12 Physical parameters of a fourth-order MEFSS with a Butterworth response operating
at 10 GHz with a fractional bandwidth of 20%. D, w li2 , w2>3, tu3)4, Si, s 2 , s3, s 4 ,
^a{i,2}, ^{1,2}, ^«{2,3}, ^{2,3}, ^a{3,4}, and ^{3,4} refer to the geometrical parameters
identified in Fig. 6.1 of Chapter 6
175
A. 13 Physical parameters of a fourth-order MEFSS with a Butterworth response operating
at 10 GHz with a fractional bandwidth of 30%. D, w\)2, w2j3, w3)4, si, s 2 , s3, s 4 ,
^{i,2},fya{i,2},/ia{2,3}. ^3{2,3}, ^{3,4}, and ^{3,4} refer to the geometrical parameters
identified in Fig. 6.1 of Chapter 6
175
A. 14 Physical parameters of a fourth-order MEFSS with a Chebyshev response operating
at 10 GHz with a fractional bandwidth of 10%. D, u>1)2, u>2)3, u>3)4, si, s 2 , s 3 , s 4 ,
/ia{i,2}, ^/3{i,2}, ^a{2,3}, ^{2,3}, ^Q{3,4}, and ^{3,4} refer to the geometrical parameters
identified in Fig. 6.1 of Chapter 6
176
A. 15 Physical parameters of a fourth-order MEFSS with a Chebyshev response operating
at 10 GHz with a fractional bandwidth of 20%. D, u>1>2, w2>3, u>3j4, si, s2, s3, s 4 ,
ha{i,2}, hp{i,2}, ha{2j3}, /i/s{2,3}, ^a{3,4}, and /i/3{3,4} refer to the geometrical parameters
identified in Fig. 6.1 of Chapter 6
176
XI
Appendix
Table
A. 16 Physical parameters of a fourth-order MEFSS with a Chebyshev response operating
at 10 GHz with a fractional bandwidth of 30%. D, w1>2, w2^, u>3,4, s1, s2, s3, s 4 ,
ha{i,2},fy3{i,2},/ia{2,3}, ^{2,3}, ^a{3,4}, and ^{3,4} refer to the geometrical parameters
identified in Fig. 6.1 of Chapter 6
Page
177
DISCARD THIS PAGE
Xll
LIST OF FIGURES
Figure
Page
2.1
Topology of the third-order bandpass FSS presented in Section 2.2. (a) Three-dimensional
view and (b) Side view
22
2.2
Top view of the unit cell of the third-order bandpass FSS studied in Section 2.2. The
view of the capacitive patch is shown on the left hand side and the topology of the
utilized miniaturized slot antenna is shown on the right hand side
23
2.3
Equivalent circuit model of the third-order bandpass FSS studied in Section 2.2
24
2.4
Expanded equivalent circuit model of the third-order bandpass FSS studied in Section 2.2. The impedance inverters and the slow-wave resonators are clearly indicated.
24
(a) A unit cell of the proposed FSS is placed inside a waveguide with periodic boundary conditions to simulate the frequency response of the infinitely large third-order
FSS studied in Section 2.2. (b) Equivalent circuit model of a simplified version of
the FSS composed of ONLY the dielectric substrates and the miniaturized slot antenna. This equivalent circuit is used to optimize the topology of the miniaturized slot
antenna to match its frequency response to that of this equivalent circuit
26
Transmission and reflection coefficients of a typical third-order bandpass FSS with
equivalent circuit parameters and dimensions shown in Tables 2.1 and 2.2. The results obtained using full-wave FEM simulations as well as those obtained using the
equivalent circuit model are shown here
29
Transmission coefficients of the FSS studied in Section 2.2 with the parameters given
in Tables 2.1 and 2.2 as a function of the angle of incidence. The results are obtained
using full-wave FEM simulations
30
Photograph of the FSS measurement fixture used to characterize the frequency response of the third-order bandpass FSS. The edges of the fixture are covered with RF
absorbers to reduce the effects of scattering on the performance of the measurement.
In the photograph, the FSS test sample is placed in the opening and the photograph of
the FSS test sample is also shown in the inset of the figure
31
2.5
2.6
2.7
2.8
Figure
2.9
Pag
Measured and simulated transmission coefficients of the test sample of the third-order
FSS. Simulations are performed using FEM simulations in Ansoft HFSS
32
2.10 Measured transmission coefficients of the third-order bandpass FSS for various angles
of incidence ranging from 6 = 10° to 50°
33
3.1
3.2
3.3
3.4
4.1
4.2
(a) 3-D topology a frequency selective surface composed of successive arrays of subwavelength capacitive patches and miniaturized slot resonators, (b) Side view of the
FSS and the top views of a constituting unit cell of the structure
37
(a) Equivalent circuit model of the general FSS shown in Fig.3.1(a) for normal angle
of incidence, (b) Transmission lines which represent the dielectric substrates in the
equivalent circuit model of Fig. 3.2(a) can be represented by their simple LC equivalent network. The replacements are shown in dark gray, (c) The general circuit model
representing an Nth order, bandpass coupled resonator filter. The equivalent circuit
model shown in Fig. 3.2(b) can be obtained from that shown in Fig. 3.2(c) if L 1} L 3 ,
..., LN can be designed to have infinite inductance values
38
Reflection and transmission coefficients of the third-order bandpass FSS discussed in
Section 3.3.1. Measurement results as well as those obtained through full-wave EM
simulations in HFSS and the equivalent circuit model (EQC) are presented
45
Transmission and reflection coefficients of the fifth-order bandpass FSS discussed in
Section 3.3.1. Results predicted from the equivalent circuit model of Fig. 3.2(a)
(EQC) are compared with the full-wave EM simulation results obtained in HFSS. The
electrical and geometrical parameters of this FSS are provided in Table 3.4
48
(a) 3D topology of the proposed dual-band FSS. (b) Side view of the structure showing
the dielectric substrates, metal layers, and the prepreg bonding films, (c) Top view of
one unit cell of the structure. The views of the capacitive patch and the resonator are
shown separately for better clarification
50
Equivalent circuit models of the proposed dual-band device, (a) The most general
equivalent circuit model, (b) A simplified equivalent circuit model which ignores the
mutual coupling between the resonator layers and the parasitic inductances associated
with the compact resonators, (c) The circuit model of part (b) is further simplified by
replacing the short pieces of transmission lines with their lumped element equivalents,
(d) A general topology of a fourth-order, coupled resonator filter. It can be shown that
under certain circumstances this circuit can be converted to the one shown in part (c). . 52
XIV
Appendix
Figure
4.3
4.4
4.5
4.6
4.7
4.8
Page
(a) A simplified circuit model of the FSS without the two capacitive patch layers. This
equivalent circuit model is used in conjunction with the full-wave simulations of the
FSS without the exterior capacitive patch layers to fine tune the dimensions of the
slot-line resonators as well as the separation between the two resonator layers, (b) To
increase the value of the parasitic inductance of the hybrid resonators located on the
interior layers of the structure, the unit cell of the two middle layers can be modified
as indicated in the figure
56
Calculated transmission and reflection coefficients of the proposed dual-band device,
(a) Transmission coefficient (b) Reflection coefficient. The results obtained from the
equivalent circuit model of Fig. 4.2(b) with the values predicted directly from the
equations provided in Section 4.2.2 is shown in red (up triangle symbols). The results
predicted from the circuit of Fig. 4.2(b) with tuned values are shown in blue (down
triangle symbols). Full-wave simulations results are shown in black (circle symbols)
and the results predicted by the equivalent circuit model of Fig. 4.2(a) with the values
given in Table 4.1 are shown in green (square symbols)
59
Full-wave EM simulation results of the transmission coefficients of the dual-band FSS
discussed in Section 4.3.1 with parameters shown in Table 4.1 for various oblique
angles of incidence, (a) TE polarization (b) TM polarization
62
Calculated and measured transmission coefficients and the calculated reflection coefficients of the proposed dual-band FSS. Measurement results are compared to the
theoretically predicted ones from the equivalent circuit model of Fig. 4.2(a) and the
full-wave EM simulation results conducted in CST Microwave studio. The structure
exploits the new resonator architecture shown in the right side of Fig. 4.3(b). This increases the values of L s2 and L s3 to create a transmission null at a frequency between
fu and fi
63
Calculated transmission coefficients of the dual-band FSS described in Section 4.3.1,
which uses the slot resonator topology shown in the right hand side of Fig. 4.3(b).
The effects of changing the length of the inductive strips on the frequency of the
transmission null and the frequency of the second band of operation are examined. As
can be seen, by increasing the length of the inductive strip (L), the frequency of the
transmission null decreases as expected from the theory presented in Section 4.3. . . .
65
Calculated transmission coefficients of the dual-band FSS discussed in Section 4.3.1
with parameters shown in Table 4.2 for various oblique angles of incidence; the results are obtained from full-wave EM simulations in CST Microwave Studio, (a) TE
polarization (b) TM polarization
66
XV
Appendix
Figure
4.9
Measured transmission coefficients of a fabricated prototype of the proposed dualband FSS studied in Section 4.3.2 for various oblique angles of incidence, (a) TE
polarization (b) TM polarization
Page
69
5.1
Topology of the low-profile, second-order bandpass FSS presented in Section 5.2. . . . 72
5.2
A simple equivalent circuit model for the low-profile FSS shown in Fig. 5.1 and
discussed in Section 5.2
72
(a) Simplified equivalent circuit model of the FSS presented in Fig. 5.1. (b) By
converting the T-network composed of LT1, L2, and LT2 into a 7r-network composed
of L\, Lm, and L 3 , the simplified equivalent circuit shown in Fig. 5.3(a) is converted
to a classic second-order coupled resonator filter utilizing inductive coupling between
the resonators
74
Calculated transmission and reflection coefficients of the equivalent circuit model of
the second order FSS, shown in Fig. 5.2, and the simplified circuit model, shown in
Fig. 5.3 for the example studied in Section 5.2
77
A unit cell of the proposed FSS is placed inside a waveguide with periodic boundary
conditions to simulate the frequency response of the infinitely large second-order FSS.
79
Topology and photograph of the fabricated FSS prototype utilizing regular patch capacitors. A bonding layer (RO4450B Prepreg from Rogers Corporation) is used to
bond the two substrate layers together as shown
80
Calculated transmission coefficients of the second-order FSS with parameters shown
in Table 5.2. (a) TE polarization (b) TM polarization. Results are obtained using
full-wave EM simulations in HFSS
81
Topology and photograph of the fabricated FSS prototype utilizing interdigital capacitors. A bonding layer (RO4450B Prepreg from Rogers Corporation) is used to bond
the two substrate layers as shown in this
figure
82
Calculated transmission coefficients of the second-order FSS utilizing interdigital capacitors with parameters shown in Table 5.3. (a) TE polarization (b) TM polarization.
Results are obtained using full-wave EM simulations in Ansoft's HFSS
83
5.10 Photograph of the free-space measurement system used to characterize the frequency
response of the fabricated FSSs
85
5.3
5.4
5.5
5.6
5.7
5.8
5.9
XVI
Appendix
Figure
Page
5.11 Comparison between the measured and simulated results obtained from full-wave EM
simulations as well as the equivalent circuit model shown in Fig. 5.2 for the two FSS
prototypes studied in Section 5.3. The figures include the measured responses of
the FSSs with and without range gating. Range gating removes the ripples caused by
multiple reflections between the two antennas. All results are for normal incidence, (a)
The FSS prototype utilizing simple patches (b) The FSS prototype utilizing interdigital
patches
86
5.12 Measured frequency responses of the two fabricated FSS prototypes for the TE polarization and various angles of incidence, (a) The FSS prototype utilizing simple
patches (b) The FSS prototype utilizing interdigital patches
86
5.13 Measured frequency responses of the two fabricated FSS prototypes for the TM polarization and various angles of incidence, (a) The FSS prototype utilizing simple
patches (b) The FSS prototype utilizing interdigital patches
87
6.1
6.2
6.3
6.4
6.5
(a) 3-D topology of a frequency selective surface composed of successive arrays of
sub-wavelength capacitive patches and wire grids with sub-wavelength periodicitties.
(b) Side view of the FSS. (c) Top views of a constituting unit cell of the structure. . . .
90
(a) Equivalent circuit model of the general FSS shown in Fig. 6.1(a) for normal angle
of incidence, (b) Transmission lines which represent the dielectric substrates in the
equivalent circuit model of Fig. 6.2(a) can be represented by their simple LC equivalent network. The replacements are shown in dark gray, (c) The equivalent circuit
model shown in Fig 6.2(b) can be further simplified by ignoring certain parasitic capacitances, (d) Finally a T to TV transformation is used to convert the equivalent circuit
model of Fig. 6.2(c) to that of Fig. 6.2(d)
91
Reflection and transmission coefficients of the 3 r d order bandpass FSSs discussed in
Section 6.3.1. Results directly predicted by the equivalent circuit model shown in
Fig. 6.2(a) and those calculated using full-wave EM simulations in Ansoft HFSS are
presented here
101
Side view of the FSS discussed in Section 6.3.2 showing the different metal, dielectric,
and prepreg layers and their relative positions with respect to each other
101
Reflection and transmission coefficients of the bandpass FSS discussed in Section 6.3.2.
Simulation results in HFSS and theoretically predicted ones are compared with the
measurement results
102
XV11
Appendix
Figure
6.6
6.7
6.8
7.1
7.2
7.3
7.4
Page
Simulated and measured transmission and reflection coefficients of the third-order
bandpass FSS discussed in Section 6.3.2 for oblique incidence angles and Transverse
Electric (TE) polarization, (a) Results obtained from full-wave EM simulations in
Ansoft HFSS (b) Measurement Results
104
Simulated and measured transmission and reflection coefficients of the third-order
bandpass FSS discussed in Section 6.3.2 for oblique incidence angles and Transverse
Magnetic (TM) polarization, (a) Results obtained from full-wave EM simulations in
Ansoft HFSS (b) Measurement
105
Reflection and transmission coefficients of a fourth-order bandpass FSS with a maximally flat response type studied in Section 6.3.3. Theoretically predicted results are
compared with those obtained from full-wave EM simulations in Ansoft HFSS
107
(a) Topology of a conventional double-convex dielectric lens, (b) A planar microwave
lens composed of arrays of transmitting and receiving antennas connected to each
other using transmission lines with variable lengths, (c) Topology of the proposed
MEFSS-based planar lens
Ill
Phase delay of a spherical wave at the input of the lens aperture (referenced to the
phase at the center of the aperture) and the phase delay, which must be provided by
the lens to achieve a planar wavefront at the output aperture of the lens. The results
are calculated for a lens with aperture size of 23.4 cm x 18.6 cm and focal distance
of 30 cm
112
(a) 3D topology of an iV'^-order MEFSS reported in Chapter 6. (b) 2D top view of the
unit cell of a sub-wavelength capacitive patch layer (top) and an inductive wire grid
layer (bottom). The two layers are used a building block for the 7Vt?l-order MEFSS.
(c) Equivalent transmission line circuit model of the Nth-order MEFSS
113
Simulated transmission coefficients (magnitude and phase) of three MEFSSs having
second-, third-, and fourth-order bandpass responses. Simulation results are obtained
in CST Microwave Studio. As the order of the MEFSS response increases, the phase
of the transmission coefficient changes over a wider range within the pass band of the
MEFSS
114
Appendix
Figure
7.5
Top view of the proposed planar MEFSS-based lens. A spherical wave is launched
from a point source located at the focal point of the lens, (x = 0, y = 0, z = —f). To
transform this input spherical wave-front to an output planar one, k0r + $(x, y) must
be a constant for every point on the aperture of the lens. $(x, y) is the phase delay,
which is provided by the lens
Page
116
7.6
Topology of the two lens prototypes described in Section II. The lens aperture is divided into a number of concentric zones populated with identical spatial phase shifters
(a) First prototype with five concentric zones, (b) Second prototype with ten zones. . . 1 1 8
7.7
Layouts of the two lens prototypes designed using the proposed design procedure
discussed in Section II-B. The lens aperture is discretized into small area pixels. Each
pixel is occupied by a spatial phase shifter with physical dimensions of 6.1 mm x 6.1
mm designed using the design procedures discussed in Section II-B. (a) Layout of the
first prototype, where the lens aperture is divided into five discrete zones. The zones
are numbered from 0-4. The specifications of the spatial phase shifters that populate
each zone are listed in Table 7.1. (b) Layout of the second prototype, where the lens
aperture is divided into ten discrete zones. The zones are numbered from 0-9. The
specifications of the spatial phase shifters that populate each zone are listed in Table 7.2.120
7.8
Frequency responses of the spatial phase shifters that populate the lens aperture of (a)
first prototype and (b) second prototype. The spatial phase shifters are designed using
the procedure described in Section II
121
Photograph of the fabricated lens prototype that consists often zones. In this figure,
only one of the five metallic layers that constitute the lens is visible (the first capacitive
layer). Notice that the sizes of the capacitive patches on the layer decrease as we move
away from the center of the lens
124
7.9
7.10 (a) Perspective view of the measurement setup used to experimentally characterize the
performance of the two fabricated lens prototypes, (b) In a set of measurements, the
lens is excited with a plane wave and a receiving probe is swept in a rectangular grid
in the vicinity of its focal point (x = 0, y = 0, z = —f) to characterize its focusing
properties as a function of frequency. The measurement grid's area is 8 cm x 8 cm and
the resolution is 1 cm. (c) In another set of measurements, the lens is illuminated with
plane waves arriving from various incidence angles and the received power pattern on
the focal arc of the lens is measured. Here, a probe is swept over the focal arc with 2°
increments to measure the received power.
125
XIX
Appendix
Figure
Page
7.11 Measured focusing gain of the first prototype (with five zones) in a rectangular grid in
the vicinity of its expected focal point (x = 0 cm, y = 0 cm, z = —30 cm). In all of
the figures, the horizontal axis is the x axis with units of [cm] and the vertical axis is
the z axis with units of [cm]. The color bar values are in dB. The x marker in all of
these figures represents the exact coordinate where the focusing gain maxima occur,
(a) 8.0 GHz, (b) 8.5 GHz, (c) 9.0 GHz, (d) 9.5 GHz, (e) 10.0 GHz, (f) 10.5 GHz, (g)
11.0 GHz, (h) 11.5 GHz, and (i) 12.0 GHz
127
7.12 Measured focusing gain of the second prototype (with ten zones) in a rectangular grid
in the vicinity of its expected focal point (x = 0 cm, y = 0 cm, z = —30 cm). In all
of the figures, the horizontal axis is the x axis with units of [cm] and the vertical axis
is the z axis with units of [cm]. The color bar values are in dB. The x marker in all of
these figures represents the exact coordinate where the focusing gain maxima occur,
(a) 8.0 GHz, (b) 8.5 GHz, (c) 9.0 GHz, (d) 9.5 GHz, (e) 10.0 GHz, (f) 10.5 GHz, (g)
11.0 GHz, (h) 11.5 GHz, and (i) 12.0 GHz
128
7.13 Calculated and measured focusing gains of the two lens prototypes at their expected
focal point (x = 0 cm, y = 0 cm, z = —30 cm) as a function of frequency. The 3 dB
gain bandwidths of the two prototypes are respectively 19.2% and 20% for the first
and second prototypes
129
7.14 Calculated focusing gain of the first prototype (with five zones) in a rectangular grid
in the vicinity of its expected focal point (x = 0 cm, y = 0 cm, z = —30 cm). In all
of the figures, the horizontal axis is the x axis with units of [cm] and the vertical axis
is the z axis with units of [cm]. The color bar values are in dB. The x marker in all of
these figures represents the exact coordinate where the focusing gain maxima occur,
(a) 8.0 GHz, (b) 8.5 GHz, (c) 9.0 GHz, (d) 9.5 GHz, (e) 10.0 GHz, (f) 10.5 GHz, (g)
11.0 GHz, (h) 11.5 GHz, and (i) 12.0 GHz
130
7.15 Calculated focusing gain of the second prototype (with ten zones) in a rectangular grid
in the vicinity of its expected focal point (x = 0 cm, y = 0 cm, z = —30 cm). In all
of the figures, the horizontal axis is the x axis with units of [cm] and the vertical axis
is the z axis with units of [cm]. The color bar values are in dB. The x marker in all of
these figures represents the exact coordinate where the focusing gain maxima occur,
(a) 8.0 GHz, (b) 8.5 GHz, (c) 9.0 GHz, (d) 9.5 GHz, (e) 10.0 GHz, (f) 10.5 GHz, (g)
11.0 GHz, (h) 11.5 GHz, and (i) 12.0 GHz
131
7.16 Received power pattern on the focal arc of the first fabricated prototype (with five
zones) when excited with plane waves arriving from various incidence angles, (a)
Calculated, (b) Measured
133
XX
Appendix
Figure
Page
7.17 Received power pattern on the focal arc of the second fabricated prototype (having
ten zones) when excited with plane waves arriving from various incidence angles, (a)
Calculated, (b) Measured
134
7.18 Calculated and measured power patterns on the focal arc of the first fabricated prototype (having five zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz,
(c) 10.5 GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves
excitation with 0° angle of incidence (normal incidence). These power patterns are
normalized to the maximum measured value, which occurs at the focal point when it
is excited with a normally incident plane wave operating at 10 GHz. Therefore, 0 dB
refers to the power level received on the focal point of the lens when it is excited with
a normally incident plane wave operating at 10 GHz
137
7.19 Calculated and measured power patterns on the focal arc of the first fabricated prototype (having five zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz,
(c) 10.5 GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves
excitation with 15° angle of incidence. These power patterns are normalized to the
maximum measured value, which occurs at the focal point when it is excited with
a normally incident plane wave operating at 10 GHz. Therefore, 0 dB refers to the
power level received on the focal point of the lens when it is excited with a normally
incident plane wave operating at 10 GHz
138
7.20 Calculated and measured power patterns on the focal arc of the first fabricated prototype (having five zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz,
(c) 10.5 GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves
excitation with 30° angle of incidence. These power patterns are normalized to the
maximum measured value, which occurs at the focal point when it is excited with
a normally incident plane wave operating at 10 GHz. Therefore, 0 dB refers to the
power level received on the focal point of the lens when it is excited with a normally
incident plane wave operating at 10 GHz
139
7.21 Calculated and measured power patterns on the focal arc of the first fabricated prototype (having five zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz,
(c) 10.5 GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves
excitation with 45° angle of incidence. These power patterns are normalized to the
maximum measured value, which occurs at the focal point when it is excited with
a normally incident plane wave operating at 10 GHz. Therefore, 0 dB refers to the
power level received on the focal point of the lens when it is excited with a normally
incident plane wave operating at 10 GHz
140
XXI
Appendix
Figure
Page
7.22 Calculated and measured power patterns on the focal arc of the first fabricated prototype (having five zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz,
(c) 10.5 GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves
excitation with 60° angle of incidence. These power patterns are normalized to the
maximum measured value, which occurs at the focal point when it is excited with
a normally incident plane wave operating at 10 GHz. Therefore, 0 dB refers to the
power level received on the focal point of the lens when it is excited with a normally
incident plane wave operating at 10 GHz
141
7.23 Calculated and measured power patterns on the focal arc of the second fabricated
prototype (having ten zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz,
(c) 10.5 GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves
excitation with 0° angle of incidence (normal incidence). These power patterns are
normalized to the maximum measured value, which occurs at the focal point when it
is excited with a normally incident plane wave operating at 10 GHz. Therefore, 0 dB
refers to the power level received on the focal point of the lens when it is excited with
a normally incident plane wave operating at 10 GHz
142
7.24 Calculated and measured power patterns on the focal arc of the second fabricated
prototype (having ten zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz,
(c) 10.5 GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves
excitation with 15° angle of incidence. These power patterns are normalized to the
maximum measured value, which occurs at the focal point when it is excited with
a normally incident plane wave operating at 10 GHz. Therefore, 0 dB refers to the
power level received on the focal point of the lens when it is excited with a normally
incident plane wave operating at 10 GHz
143
7.25 Calculated and measured power patterns on the focal arc of the second fabricated
prototype (having ten zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz,
(c) 10.5 GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves
excitation with 30° angle of incidence. These power patterns are normalized to the
maximum measured value, which occurs at the focal point when it is excited with
a normally incident plane wave operating at 10 GHz. Therefore, 0 dB refers to the
power level received on the focal point of the lens when it is excited with a normally
incident plane wave operating at 10 GHz
144
XX11
Appendix
Figure
Page
7.26 Calculated and measured power patterns on the focal arc of the second fabricated
prototype (having ten zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz,
(c) 10.5 GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves
excitation with 45° angle of incidence. These power patterns are normalized to the
maximum measured value, which occurs at the focal point when it is excited with
a normally incident plane wave operating at 10 GHz. Therefore, 0 dB refers to the
power level received on the focal point of the lens when it is excited with a normally
incident plane wave operating at 10 GHz
145
7.27 Calculated and measured power patterns on the focal arc of the second fabricated
prototype (having ten zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz,
(c) 10.5 GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves
excitation with 60° angle of incidence. These power patterns are normalized to the
maximum measured value, which occurs at the focal point when it is excited with
a normally incident plane wave operating at 10 GHz. Therefore, 0 dB refers to the
power level received on the focal point of the lens when it is excited with a normally
incident plane wave operating at 10 GHz
146
8.1
8.2
8.3
8.4
(a) Topology of a conventional double-convex dielectric lens, (b) Topology of the
proposed MEFSS-based planar true-time-delay lens
148
Relative time delay that different rays of a spherical wave experience (with respect to
the ray that passes through the center of the aperture) at the input of the lens aperture
and the time delay profile, which must be provided by the lens to achieve a planar
wavefront at the output aperture of the lens. The time delay is referenced to the time it
takes the wave to propagate from focal point to the center of the aperture. The results
are calculated for a circular aperture lens with aperture diameter of 18.6 cm and focal
distance of 30 cm
149
(a) 3D topology of an Nth-order MEFSS discussed in chapter 6. (b) 2D top view of the
unit cell of a sub-wavelength capacitive patch layer (top) and an inductive wire grid
layer (bottom). The two layers are used a building block for the Nth-order MEFSS.
(c) Equivalent transmission line circuit model of the iV^-order MEFSS
150
Calculated frequency response of three MEFSSs having second-, third-, and fourthorder bandpass responses, (a) Magnitude and phase of transmission coefficients, (b)
Magnitudes of transmission coefficient and the corresponding group delays. As the
order of the MEFSS response increases, the phase of the transmission coefficient
changes over a wider range within the highlighted region of the MEFSS bandpass.
Hence, a larger group is achieved within that frequency range of interest
151
Appendix
Figure
8.5
Calculated frequency responses of three MEFSSs having fourth-order bandpass responses with different fractional bandwidths. (a) Magnitude and phase of transmission
coefficients, (b) Magnitudes of transmission coefficient and the corresponding group
delays. As the bandpass fractional bandwidth of the MEFSS response increases, the
slope of the transmission phase coefficient decreases within the highlighted region of
the MEFSS bandpass. Hence, a smaller group is achieved within that frequency range
of interest
Page
152
8.6
Top view of the proposed MEFSS-based true-time-delay planar lens. A spherical wave
is launched from a point source located at the focal point of the lens, (x = 0, y —
0, z = —f). To transform this input spherical wavefront to an output planar one,
T(x, y) + TD(x, y) must be a constant for every point on the aperture of the lens.
TD(x, y) is the time delay, which is provided by the lens. T(x, y) is the time it takes
for the wave to travel from the focal point of the lens, (x = 0, y = 0, z = — / ) , to a
point at the lens aperture, (x, y, z = 0)
154
8.7
Topology of the MEFSS-based true-time-delay lens prototype with a circular aperture.
The lens aperture is divided into a number of concentric zones populated with identical
spatial time-delay units
157
8.8
Calculated received power in the vicinity of the expected focal point of the circular
true-time-delay lens prototype. The received power is calculated over a rectangular
grid centered at the expected focal point (x = 0 cm, y = 0 cm, z = — 30 cm). In all
of the figures, the horizontal axis is the x axis with units of [cm] and the vertical axis
is the z axis with units of [cm]. The color bar values are in dB. The x marker in all of
these figures represents the exact coordinate where the received power maxima occur.
(a) 8.0 GHz, (b) 8.5 GHz, (c) 9.0 GHz, (d) 9.5 GHz, (e) 10.0 GHz, (f) and 10.5 GHz. . 158
8.9
Calculated received power pattern on the focal arc of the circular true-time-delay lens
prototype when excited with plane waves arriving from various incidence angles, (a)
8.0 GHz, (b) 8.5 GHz, (c) 9.0 GHz, (d) 9.5 GHz, (e) 10.0 GHz, (f) and 10.5 GHz. . . . 159
XXIV
ABSTRACT
Phased-array antennas have received great attention due to their wide use in many military and
civilian systems. Despite the significant developments that have been made over the past decades,
developing a low-cost and high-power phased-array system is still a challenge. A compromise
must always be made amongst system losses, power handling capabilities, cost, and architecture
complexity. In this dissertation, a new approach is presented which is expected to enable the
development of affordable, efficient, high-power, and wideband phased-array antenna systems.
This approach is based on using tunable space-fed planar microwave lenses. The main features
of these lenses are wideband operation, capability of handling high RF power levels, and reduced
system loss. The design of these planar lenses is accomplished by populating the lens aperture with
spatial true-time-delay units. Beam scanning is achieved by controlling the time delay gradient
over the lens aperture. These true-time-delay units are implemented using sub-wavelength periodic
structures.
Non-resonant structures and hybrid structures are two different classes of periodic structures that
have been proposed and studied. Both structures are composed of multiple impedance surfaces
separated from each other using ultra-thin dielectric substrates. In both classes, the constituting
elements of these surfaces have very small dimensions compared to the operating wavelength and
have ultra-thin profiles. Moreover, their responses are stable with respect to the angle and polarization of incidence of the incident electromagnetic wave. An analytical synthesis procedure has
been developed to design these structures with a variety of responses of any order. Investigations
XXV
of the two classes reveal that the non-resonant structures are more suitable for the intended application, due to their elements' simplicity and high power handling capability. On the other hand,
the hybrid periodic structures can be designed to operate as spatial filters with either single- or
dual-band operation. The ultra-thin profile, light weight, and the reduced sensitivity to the angle
of incidence, make these hybrid structures an attractive solution for a multitude of applications.
Examples include radomes to reduce the radar cross section of military targets, and dichroic feeds
for reflector antennas.
1
Chapter 1
Introduction
1.1
Motivation
Since their introduction in the early 1950's, phased-array antennas have been used in a mul-
titude of applications such as radars [1], beam forming [2], imaging [3], and remote sensing [4].
In the recent years, they have also been used in commercial applications such as radio-frequency
identification (RFID), multiple-input multiple-output wireless communication systems (MIMO),
and collision avoidance radars. A phased-array is an array of antennas in which the relative phases
and amplitudes of the respective signals feeding the antennas, or received by the antennas, are controlled such that its effective radiation pattern is directed in the desired direction and suppressed in
all other directions. Depending on the system architecture, phased arrays can be categorized into
two main categories of passive arrays and active arrays. In passive electronically steered arrays
(PESAs), a single transmitter and receiver are connected to all the array elements through a feeding network. The relative phases between the elements are altered by using phase shifters at each
radiating element. In active electronically steered arrays (AESAs), each radiating element has a
dedicated transmitter and receiver (T/R) module, which contains power amplifiers, phase shifters,
low noise amplifier (LNAs), variable gain amplifiers, and other components of a fully functional
transceiver.
Each category (PESA and AESA) has its own advantages and disadvantages. In PESAs, the
signal is generated from a single source, and is split up amongst the radiating elements. Therefore,
the, losses in the system become important. Reducing the system loss, which is caused by phase
shifters and the feeding network, has always been a challenge. A sizable body of research has been
2
conducted on reducing the losses in the passive phased-array systems to increase their efficiency
and sensitivity [5]-[23]. Reducing losses of the feeding network requires using low-loss transmission lines such as a waveguide. However, this may not only increase the size and the weight of the
overall array, but also increase the complexity of the feeding network particularly when the array
is large.
To reduce the phase shifter losses, different types of phase shifters have been investigated.
Prominent among these include phase shifters based on ferrites, PIN diodes, ferroelectric materials,
and micro-electro-mechanical systems (MEMS). The differences among these phase shifters are
PvF insertion loss, switching times, and power handling capability. Waveguide-based ferrite phase
shifters are recognized for their high power handling capabilities, a moderate switching time of
(50 // sec) [5], and a high figure of merit (FOM) (e.g. 257°/dB [5]). The figure of merit of a phase
shifter is defined by the degrees of the phase shift the device provide per dB of its insertion loss.
The main drawback of these phase shifters, however, is that they are biased using magnets, thereby
increasing the weight and size of the phase shifter. Most recently, a team at Northeastern University
has addressed these drawbacks and reported the deposition of hexaferrite thick films [7] to be used
for building miniature phase shifters with FOM approaching 200°/dB [8]. Diode phase shifters
have also been used in passive phase arrays design. They are low cost and have fast switching time
(50-200 nsec) and FOM of 211°/dB [9]-[10]. Ferroelectric materials, which are characterized by
a change in their permittivity when imposed to a DC bias voltage, have also been investigated to
develop affordable phase shifters. Several groups have proposed different phase shifter topologies
that have been investigated and implemented [13]-[15]. This type of phase shifter is low cost, has
a small size, and is useful for applications operating at low microwave frequencies (< 15 GHz)
[16]. However, its main drawback is the relatively small FOM, 93°/dB [17]. RF MEMS is an
emerging technology that has shown a great potential towards lowering the cost of phase shifters
used in PESAs. Phase shifters designed using this technology offer several advantages including
low power consumption, moderate FOM, reduced size, and most importantly lowered cost. In
addition, they can operate at microwave and millimeter wave frequencies with FOM of 100°/dB
[24]. Since the introduction of this technology, MEMS switches have improved their reliability.
3
The state of the art switches can now operate with greater than 900 billion switch cycles [22].
However, the power handling of this type of shifters is about 1 Watt [23], which limits their use
to passive array applications, where the radiated power per element is low. A comparison among
different types of phase shifters are shown in Table 1.1. As can be seen from this table, no one
type of phase shifter has the desirable properties of high figure of merit, very low bias power, and
high power handling capabilities. A compromise has always to be made depending on the desired
requirements of the passive phased array.
In an active array, the single transmitter used in a passive array is replaced by (T/R) modules
distributed over the array elements. Distributing the T/R modules over the AESA aperture offers
several advantages compared to the PESAs. It provides a complete flexibility in determining the
relative phase and amplitude at each array element and, hence, allows more flexibility in beam
scanning and synthesizing very low sidelobe levels. This improvement of the system's performance and capabilities comes at the expense of increased complexity of the active phased array
systems, high fabrication cost, limited RF output power levels, and increased system's generated
heat. Different solid state technologies have been used over the past years in an attempt to reduce the cost and increase the output power of active phased arrays. Gallium Arsenide (GaAs)
monolithic microwave integrated circuits (MMICs) have been successfully applied for designing
T/R modules to build active phased array systems. The RF output power of GaAs power amplifier
approaches 50 Watt at S-band and 1 Watt at Ka band [25]. This RF output power limitation is
primarily due to the low breakdown field of the GaAs [25]. Silicon Germanium (SiGe) technology
has also been investigated due to its low cost and ability to integrate the T/R modules with the rest
of the system circuits on a single chip. However, its RF output power, which is about 50 mW [27],
is much less than that of GaAs technology which limits its use to low power phased-array applications only. In the recent years, Silicon (Si) technology has seen a considerable development in its
speed. Si MMIC now operates at microwave frequencies. It holds the promise of low cost and low
power for the receiver parts of T/R modules. It also has the advantage of allowing the integration
of many functions on a single chip. However, the low operating voltages of Si devices need voltage
converters and also lead to a large device periphery. Thus, the device complexity increases.
4
Shifter Type
Ferrite
Ferrite
PIN diode
Ferroelectric
MEMS
(waveguide)
(thin-film)
Figure of Merit
257°/dB [5]
200°/dB [8]
211°/dB [10]
93°/dB [17]
100°/dB [24]
Switching Time
50 [is[5]
-
50 - 200 ns [10]
30 ns [20]
1 - 20 /is [23]
Biasing
< 1 W [5]
< 100 Oe [8]
< 10mW[ll]
< 10/uW[18]
< 1 nW [16]
Power Handling
< 600 W [6]
-
< HOW [12]
1 W[19]
< 1 W [23]
Phase Shift
Analog
Analog
Digital
Analog
Analog
Table 1.1 Comparison among different types of phase shifters used in passive phased arrays.
In the recent years, active arrays have seen a significant improvement in their RF output power
capabilities due the development of Silicon Carbide (SiC) and Gallium Nitride (GaN) technologies
in fabricating the T/R modules. The improvement in the RF output power is achieved due to the
improvement in the material properties, such as high electric breakdown field of the semiconductor
material, high saturated electron drift velocity, and improved thermal conductivity. However, SiC
technology is primarily used for frequencies below S-band, due to its poor electron transport properties, with power handling capabilities of 60 Watt [28]. Moreover, the high cost, small size, and
low quality of the SiC substrate wafers have limited their use. On the other hand, GaN technology
has shown the possibility of providing 15-120 Watt in a frequency range from L-band to 50 GHz
[29]-[30], since it has the highest power density among other technologies. A comparison amongst
the different MMIC technologies are summarized and listed in Table 1.2.
MMIC Technology
GaAs
SiGe
SiC
GaN
Si
Frequency Range (GHz)
2-80
6-100
1-4
3-100
1 - 11
Power Added Efficiency (PAE)
42% [26]
25% [27]
40% [28]
38% [30]
19% [31]
Average Output Power
11 [26]
1 [27]
100 [28]
58 [30]
0.22 [31]
at X-band
at L-band
Table 1.2 Comparison among different MMIC technologies used in active phased arrays.
5
Despite the development in the MMIC technologies over the past decades that helped to reduce
cost, complexity, and increase power handling of such systems, there are still challenges that need
to be addressed. One of these challenges is developing an efficient heat management technology
that can handle the heat generated by the T/R modules. Unless there is system to dissipate the
heat generated by these modules, they can not be used to their full capacities of generating high RF
power. One possible solution to reduce the generated heat is to fabricate these modules on substrate
with large thermal conductivity, such as SiC. This allows lower temperature rise caused by the self
heating. Thus, higher RF output power and improved efficiency can be achieved [32]. The other
challenge is that despite the fact that the cost of the T/R modules have dramatically decreased by
orders of magnitude, their cost is still a limiting factor that hinders the use of active phased array
in many commercial applications.
In summary, despite the significant developments that have been made over the past decades,
neither PESAs nor ASEAs have been successful for low-cost and high-power phased-array system. A compromise has always to be made amongst system's losses, power handling capabilities,
cost, and architecture complexity. In this thesis, we present a new approach has the potential of
developing affordable, efficient, high-power, wideband phased-array antenna systems.
1.2
Proposed Approach
We introduce a new approach to design affordable low-loss high-power passive phased array
antennas. This approach is based on using tunable space-fed planar microwave lenses. The main
features of these lenses are wideband operation and the capability of handling high RF power
levels. Moreover, the space feeding mechanism used in these lenses offers not only simplicity of
the feeding network but also reduced system loss. The beam forming in these lenses is achieved
by populating the lens aperture with spatial true-time-delay units. Thus, the time delay gradient
over the lens aperture is controlled. These true-time-delay units are implemented using periodic
structures.
Two different classes of periodic structures have been proposed and studied: non-resonant
structures and hybrid structures. Both structures are composed of multiple impedance surfaces
6
separated from each other using ultra-thin dielectric substrates. In the first class, two types of nonresonant reactive impedance surfaces are used. Capacitive surfaces composed of two-dimensional
(2D) periodic arrangement of rectangular patches and inductive surfaces composed of 2D periodic
arrangement of wire grid. In the second class, a combination of non-resonant capacitive impedance
surfaces and resonant impedance surfaces composed of 2D arrangement of miniaturized high-Q
slot resonators are used. In both classes, the constituting elements of these surfaces have very
small dimensions compared to the operating wavelength and have ultra-thin profile.
Our investigation to these two classes reveals that the non-resonant structures are more suitable
for the intended application. They have the advantage of high power handling capability for both
continuous and pulsed microwave signals, in part due to the non-resonant nature of their unit cells.
In addition, they have a more stable response with respect to the angle and polarization of incidence
of the electromagnetic wave impinging upon its surface. One of the most unique characteristics
of these non-resonant periodic structures is the ability to localize the spectral response within
the constituting unit cell itself. This localization is of great importance in simplifying the design
procedure of microwave lenses, allowing each unit cell to be treated and designed independently
to operate as a spatial phase shifter or true-time-delay unit. An analytical synthesis procedure has
been developed to design these elements with a variety of responses of any order. Depending on the
location of the unit cell within the lens, the required time delay (or phase shift), introduced by each
unit cell, is calculated and used in the element design process. The lens aperture is then populated
with these time delay units and is space-fed by a feed antenna. The beam steering is achieved by
controlling the time-delay gradient over the lens aperture. To achieve that, two different techniques
are being investigated by either using liquid metal or optical signal. However, these techniques are
out of the scope of this work and will not be discussed here.
Despite the fact that the hybrid periodic structures are not optimum for designing wideband
high power microwave lenses, they can be designed to operate as frequency selective surfaces
with either single- or dual-band of operation and higher-order filtering characteristics. The ultrathin profile, light weight, and the reduced sensitivity with respect to the angle of incidence, make
these hybrid structures an attractive solution for many applications such as radomes to reduce the
7
radar cross section of military targets or dichroic feeds for reflector antennas. Comprehensive
design procedures for both single- and dual-band selective surfaces have also been developed,
experimentally verified, are presented,in this dissertation.
1.3
Literature review
As mentioned in the previous section, we are proposing a new approach for designing mi-
crowave lenses using a new class of periodic structure. Over the past decades, there have been
a tremendous body of research in the area of microwave lenses and periodic structures. In this
section, we present a summary of different techniques that have been reported for designing planar
microwave lenses, single-, and multi-band frequency selective surfaces.
1.3.1
Microwave Lenses
Microwave and millimeter-wave lenses have been used extensively for a variety of applications
in imaging [150], beam scanning [155], and beam-shaping [156]. Dielectric lenses are one of the
first structures investigated [33]-[34]. The main advantage of this type of lenses is that they are
wide band. The drawback, however, is the lens size which results in a heavy and costly structure.
Moreover, dielectric lenses suffer some losses from the inherent internal reflections caused by the
high refractive index of lens materials. In this regard, the research in this field has focused on
finding alternative ways to design the planar, low profile, and low cost lenses.
The concept of planar lens array has been investigated by a number of researchers [161]-[166].
Traditionally, planar microwave lenses are designed using planar arrays of transmit and receive
antennas connected together using a phase shifting mechanism. Phase shifting is achieved using
various different techniques including using different types of transmission lines with variable
lengths between the arrays of receiving and transmitting antennas. In Rotman's lens [161], microstrip type of transmission lines are used, whereas in Ruze's lens [160], a parallel-plates guiding
structure is used. Another approach to achieve the required phase shift is introduced by McGrath
[162]-[163]. In which the two arrays of transmit and receive patch antennas are connected to each
8
other through different lengths of microstrip lines, which are coupled to each other using feedthrough pins or slot apertures. Engineered artificial materials have also been investigated [167].
Using this approach, structures consisting of stacked parallel metal plates, or arrays of parallel
metal wires, behaving as artificial dielectrics with the permittivity less than one, are used in the
lens design. These lenses, however, suffer from the same impedance matching problem as the
dielectric ones. In addition, their effective refractive index strongly depends on the frequency. In
another approach, Fresnel zone plate (FZP) are employed as a focusing element to design a variety
of microwave lenses [170]-[172]. The classical microwave FZP lens consists of circular concentric
metal rings that lay over the odd or even Fresnel zones [170]-[171]. Recently, it has been reported
that combining the FZP lens and the FSS leads to a new compound diffractive FZP-FSS lens with
enhanced focusing and frequency filtering characteristics [172].
Metamaterials have received a great attemtion due their capability of guiding the electromagnetic waves. In [43]-[168], planar microwave lenses are designed using metamaterial complementary structures. Depending on the geometry of constituting elements used in the complementary
structure, microwave lenses with different operational bandwidths can be obtained. Gradient index
metamaterials (GRIN), introduced by Smith [45], have been used to design flat lenses. Metamaterial GRIN lenses with negative index have shown their focusing abilities [46] and [47]. However,
this type of lenses operate over a narrowband due to the resonant nature their constituting elements.
An alternative approach to achieve metamaterial gradient index broadband lenses is by choosing
the constituting resonant elements to operate in the non-resonant region of the structure [48]. Flat
lenses based on left-handed metamaterial (LHM) have also been reported [49] and [50]. This type
of lenses has shown sub-wavelength resolution capability. However, LHM-based lenses are lossy
and operate over a very narrow band.
Pozar [165] has introduced an alternative approach based on using a modified version of
frequency-selective surfaces (FSS). The resonant unit cells of the FSS are de-tuned from their
natural operating frequency to obtain the required transmission phase. In his approach, Pozar used
first- and second-order frequency selective surfaces in the design of lens structure. However, the
9
frequency response of these FSS orders fail to achieve sufficient phase transition between the input and output wavefronts. This drawback is overcome by using antenna filter antenna approach
[166] in which frequency selective surfaces with higher order frequency response can be realized
and utilized to achieve the required 0°-360° phase shift. Due to the relatively large inter-element
spacing, at least half a wavelength spacing, the structure is more sensitive to the angle of incidence
of the electromagnetic wave impinging upon its surface. Consequently, a moderate scanning performance can be achieved up 30°.
In most of these techniques, however, the microwave lens is designed based controlling the
phase shift gradient over its aperture. This inherently results in a very narrowband lens, regardless
of how wide the operational bandwidth of the elements populating the lens aperture.
1.3.2
Single-band frequency selective surfaces
One of the earliest and easiest approaches to achieve frequency selectivity is to populate the
periodic structure with resonant elements such as antennas. The element type and geometry, interelement spacing, the substrate parameters, and the presence or absence of superstates are important parameters that will determine the overall frequency response of the structure, such as its
bandwidth, transfer function, and the dependence of its frequency response on the incidence angle
and polarization of the incident electromagnetic wave.
When designing FSS to have the desired spectral and spatial performance, the proper choice
of constituting elements type and geometry is of utmost importance [57]. We can divide FSSs
into two categories of resonant- and non-resonant-element based on the type of unit cells they
use. With regards to the non-resonant type of elements, FSSs are also categorized into two main
groups: patch elements and aperture elements. It has been shown that a simple periodic structure
consisting of array of metallic patches has a low-pass characteristic [58]-[60] and it is considered
as a capacitive surface. In other word, this surface transmits the low frequency contents of the
electromagnetic wave impinging upon its surface and reflects the higher frequency contents. The
complementary structure of the metallic patches, array of apertures in a metallic screen, has been
shown to have an inductive response, and is thus acting as a high-pass filter.
10
With regards to the resonant type of elements, FSSs are generally divided into two main
groups: dipole-type elements and slot-type elements. Over the past decades, different geometries of resonant-type elements were introduced to design FSSs with bandpass and band-stop filtering characteristics. These elements include but are not limited to the following: circular shapes
[60]-[62], dipoles [63]-[66], cross dipoles [67]-[69], tripoles, Jerusalem cross [69]-[74], three- or
four-legged dipoles [75], circular rings [76]-[79], square loops [80]-[82], etc. The choice of these
elements depends on the design requirements, such as the dependence on the incidence angle of
the electromagnetic wave, operational bandwidth, and level of cross polarization. For instance, if
a narrow bandwidth is required, the four- and three-legged loaded elements are used. Whereas the
hexagon elements are used to achieve a very wideband of operation. In another example, dipoles
provides a narrow band of operation. However, it has the worst operating frequency stability with
angle of incidence, since the resonant frequency of the dipole depends on the projected length of
the dipole in the direction of the incident wave. On the other hand, rings and square loops are
considered the most stable elements with the angle of incidence.
Another parameter that has a significant effect on the FSS design and performance is the interelement spacing. The larger the inter-element spacing is, the narrower the operational bandwidth,
and vice versa (see Section 2.2, pp. 28 in [57]). However, care should be taken when the interelement spacing is increased. This causes an earlier onset of grating lobes (see Section 5.14, pp.
175-184 in [57]) and also a greater variation in the frequency response of the structure with angle
of incidence and polarization of the incident electromagnetic wave.
In some applications where sharp frequency selectivity is required, FSSs with higher-order
frequency responses are needed. Using a single layer periodic structure composed of one of the elements mentioned earlier does not provide the fast roll-off required to achieve the sharp frequency
selectivity. In such cases, FSSs with higher order bandpass or bandstop frequency response is
achieved by cascading multiple first-order FSS layers with a quarter wavelength spacer, which
acts as an impedance inverter, between each layer. While this quarter-wavelength spacing may be
useful in certain applications for structural rigidity, it presents a problem for other applications especially at low microwave frequencies, as well as applications where conformal FSSs are required.
11
Aside from the mechanical suitability, using quarter-wavelength spacers increases the sensitivity of
the frequency response of such higher-order FSSs to the angle of incidence of the electromagnetic
wave. The remedy for this shortcoming includes using superstrate dielectric stabilizers at outer
layers of the cascaded structure which further increases the overall thickness of the FSS [57] (see
Section 7.5, pp. 240 - 255).
For applications in the millimeter-wave (MMW) region, conventional FSSs composed of metallic materials become very lossy. All dielectric frequency selective surfaces (DFSS) have been
investigated as an alternative to the conventional FSSs, since they offer the advantages of low absorption loss compared to metallic screens [83]-[85]. This concept was presented by Bertoni in
1989 [83]. It has been shown that all dielectric periodic structure will support an infinite set of
modes with different wavenumbers (propagation modes) [117]. At low frequencies, only the fundamental mode will propagate through the structure and it acts approximately as if it has a uniform
dielectric constant equal to the average of that for the period layer. Hence, the DFSS will have
transmission properties similar to those of a uniform layer. At higher frequencies, however, the
first two modes propagate through the DFSS. These modes are excited at the top surface of the
DFSS layer by the incident wave and are multiply reflected within that layer. At a certain frequency, the phases of the two modes in the layer will be such that they add destructively for the
transmitted plane wave and constructively for the reflected plane wave. Thus, a total reflection is
achieved. However, at other frequencies, the phases of these two modes are such that they add
for the transmitted plane wave and cancel for the reflected wave to achieve a total transmission.
However, despite the DFSS advantage of low dielectric loss, these structures are more difficult and
more expensive to fabricate.
Resonant cavities used as constituting unit cells of FSS have also been studied and investigated
to achieve a sharp frequency selectivity [87]-[88]. These cavities are implemented using substrate
integrated waveguide (SIW) technology [89]. The response of this type FSS differs from that of
the conventional FSS in that has two main resonant modes, the resonance generated by periodicity
and the resonance generated by the SIW cavity. The energy in such structures is coupled via two
coupling slots on front and back sides of the FSS, respectively. The combination of using SIW
12
resonant cavities and coupling slots in the FSS design offers not only a low-loss structures but also
a high quality factor. This way highly selective structure with a very narrow operational bandwidth
and a sharp roll-off at the passband edges is obtained [87].
In another approach, the concept of designing FSSs based on microstrip patch antennas has
been investigated and reported in [90] and [91]. In [90], an FSS with a second-order bandpass
response is obtained by using two arrays of patch antennas coupled together using non-resonant
coupling apertures in their common ground planes. These coupling apertures are utilized to control the total transmission within the operational bandwidth which is usually set by the frequency
characteristics of the patch antennas. In [91], a similar design is used, except this time the coupling aperture is a slot resonator (first-order bandpass filter) utilizing coplanar waveguide (CPW)
technology. This approach allows the synthesis of higher-order bandpass FSS with a more general
category of the filtering characteristics, depending on the topology of CPW filters used between
the two arrays of antennas. However, in these two techniques, resonant patch antennas are used
in periodic structures with periodicities in the order of half a wavelength. This large periodicity is
generally undesirable, since it will lead to an early onset of grating lobes. In addition, it can easily
support surface waves at oblique angles of incidence and increase the sensitivity of their responses
to the angle of incidence.
Recently, a new class of frequency selective surfaces called miniaturized-element frequency
selective surfaces was developed [92]-[93]. In [92], the building block of the FSS is a combination
of capacitive and inductive elements. This block is implemented using a two-dimensional array of
capacitive patches and an inductive wire grid, each printed on a side of a very thin substrate, to constitute a parallel L-C circuit that behaves as a first-order bandpass filter. Higher-order structures of
this class of FSS is achieved by cascading multiple layers of these blocks with quarter-wavelength
spacing between them. However, this will increase the sensitivity of the overall structure to the
angle of incidence of the impinging electromagnetic wave. In [93], the FSS building block is a
combination of series resonant and inductive elements. The inductive element is implemented by
an array of wire grid, whereas the series resonant element is implemented by an array of square
loops. These two printed layers are separated by a very thin dielectric substrate. The structure
13
is a parallel combination of highly coupled band-stop and inductive surfaces. By choosing elements dimensions and controlling the alignment between the two surfaces, bandpass or bandstop
characteristics are obtained.
1.3.3
Multi-band frequency selective surfaces
Several categories of techniques have been used to design multi-band frequency selective surfaces in the past. Prominent among these include using perturbation techniques, multi-element
unit cells, multi-resonant unit cells, genetic algorithm design techniques, and using complementary structures. Perturbation was used to convert a single layer, single-band, first-order FSS to a
dual-band one [94]. This was achieved both through perturbing the constituting elements of the
structure as well as the spacing between the elements. In element perturbation, a certain percentage of the elements in the array were de-tuned to change their resonant frequency while in spacing
perturbation, the elements were the same but the periodicity of the array was irregular.
Periodic structures having composite unit cells have also been used to design multi-band FSSs
[79],[95]-[97]. In such designs, the unit cell of the structure is composed of two or more resonant
elements, the resonant frequencies of which determine the operational bands of the FSS. For example, the FSS introduced in [79], which is composed of two concentric circular rings, exhibits
two reflection bands. The lower frequency corresponds to that of the larger ring. Whereas the
higher frequency corresponds to that of the smaller ring. However, using a single screen FSS does
not provide an optimum reflection within the two bands especially at the lower and higher ends of
each band. To improve the reflections at these ends, a second identical screen is placed approximately quarter-wavelength away from the first screen at the frequency of operation of the lower
band [95]. While the second screen serves as a transmission impedance-matching layer to tune out
the unwanted reflections, it increases the overall thickness and weight of structure which makes it
impractical in some applications. Fractal structures are also used as constituting unit cells of frequency selective surfaces and have shown to be capable of providing multi-band operation owing
to their multi-resonant behavior [98]-[100]. In [98], self-similar patch element is utilized to obtain
the dual-band response, whereas a multi-scaled crossed dipole is utilized in [100], which has the
14
advantage of being dual polarized as well. Genetic algorithm (GA) is extensively used in the design of single- or multi-band frequency selective surfaces [101]-[103], because of its robustness,
accuracy, and efficiency. Another approach that has been utilized in multi-band FSSs design is the
use of multilayer complementary structures [104]-[105]. In this approach, two closely spaced FSS
layers are used, one layer composed of conducting elements and the other composed of aperture
elements. This results in a strong field interaction between the two layers. Consequently, FSSs
with two narrow passbands, separated by a transmission null, are obtained. Finally, designs of
dual-band FSSs composed of sub-wavelength non-resonant and resonant elements have been reported [106]-[107]. In [106], a first-order bandpass response is obtained at each band. Whereas in
[107], a second-order bandpass response is obtained.
In most of these multi-band structures, however, the separation between different bands of
operation is generally large. In the case of concentric rings [79], [95] and square loops [96] the
band separation is determined by the clearance between those elements whereas in fractal FSSs
[98]-[99] an octave separation between the bands is generally present. A multiband FSS with
very close band separation has also been reported recently [108]. This is achieved by designing
two neighboring capacitively loaded ring slot resonators of the same dimension but with different
capacitive loadings. Due to the high quality factor of each slot resonator, which is caused by the
capacitive loading, a very narrow bandwidth is obtained in each band. The main drawback of this
approach is that the small separation between the two bands is limited by the values of capacitors
that are commercially available. In addition, large FSS panels will naturally require many discrete
elements, which may be a practical limitation in some applications.
1.4
Thesis overview
The goal of this thesis is to develop wideband, high-power-capable, planar microwave lenses.
The lens is designed using a new class periodic structures that have been been developed through
this research. The motivation behind this work along with the literature review are discussed in
this chapter. Two new classes of periodic structures have been investigated to design the proposed
15
microwave lens. The first class is presented in Chapter 2 and 3. In this class, the structure is composed of multiple layers of two types of impedance surfaces, non-resonant capacitive surface and
slot high-Q resonant surface. It has been found that these hybrid structures have the capability to
operate over a multiple bands. The approach is discussed in Chapter 4. The second class that has
been investigated for the lens design is presented in Chapter 5 and 6. In this class, the structure is
composed of multiple layers of two types of non-resonant reactive impedance surfaces, capacitive
surface and inductive surface. Capacitive surfaces composed of 2D periodic arrangement of rectangular patches and inductive surfaces composed of 2D periodic arrangement of wire grid. It has
been found that this class is more suitable for designing the low-profile planar microwave lenses.
The first approach for designing planar lenses is presented in Chapter 7. The constituting elements
of the frequency selective surface are designed to operate as spatial phase shifters. The lens aperture is populated with these spatial phase shifter to achieve the required phase profile across the
lens aperture. The second approach for designing microwave lenses is presented in Chapter 8. The
constituting elements of the frequency selective surface are designed to operate as true-time-delay
unit. Hence, a true-time-delay lens operating over a wideband is obtained. Chapter 9 studies the
power handling capabilities of the structures presented in Chapter 5 and 6. The power handling
capabilities of these structures are studied when the structures operate under sustained high power
levels and when pulses with high peak power levels are transmitted through these structures.
1.4.1
Chapter 2: Hybrid Frequency Selective Surfaces with Third-Order Responses
In this chapter, the first class of the investigated periodic structures is presented. The proposed
structure is a three layer structure composed of three metallic layers, separated by two electrically very thin dielectric substrates. Each layer is a two-dimensional periodic structure with subwavelength unit cell dimensions and periodicity. The unit cell of the proposed structure is composed of a combination of resonant and non-resonant elements. It is shown that this arrangement
acts as a frequency selective surface (FSS) with a third-order bandpass response. The proposed FSS
has an extremely low profile and an overall thickness of about A0/24, where A0 is the free space
16
wavelength. As a result of the miniaturized size and the extremely thin unit cells, the proposed FSS
has a reduced sensitivity to the angle of incidence of the electromagnetic wave impinging upon its
surface. The principles of operation along with guidelines for designing the proposed structure are
presented in this chapter. A prototype of the proposed third-order bandpass FSS is also fabricated
and tested using a free space measurement system at C-band.
1.4.2
Chapter 3: Synthesis of Hybrid Frequency Selective Surfaces with OddOrder Frequency Responses
In this chapter, the topology of the proposed structure presented in Chapter 2 is expanded to
synthesize FSSs with bandpass responses of odd-order (N = 3, 5, 7,...). An Nth order of the
proposed FSS has N metallic layers. ^-^ of them are non-resonant capacitive layers, whereas the
other layers, ^f^, are resonant layers. The main advantage of the proposed FSSs, compared to the
traditional ones, is that they have a very low-profile and provide sharp frequency selectivity. An
Nth order of the proposed FSS has an electrical thickness in the order of ~ (N — l)A 0 /50 which is
significantly smaller than that of traditional Nth order FSS (~ (N — l)A 0 /4), where A0 is the free
space wavelength. The synthesis procedure is validated for two FSS prototypes having third- and
fifth-order bandpass responses.
1.4.3
Chapter 4: Dual-Band Frequency Selective Surfaces Using Hybrid Periodic Structures
The hybrid combination of non-resonant and resonant elements proposed in Chapters 2 and 3
are also exploited to achieve periodic structure with dual-band frequency selectivity. The proposed
structures has closely spaced bands of operation and a highly-selective frequency response at each
band. Despite the close separation between the two bands, a very high out-of-band rejection is
maintained between them. An approximate analytical design procedure of such structures is also
provided. The validity of the design procedure is verified using full-wave EM simulations and
experimental characterization of a fabricated prototype operating at C-band.
17
1.4.4
Chapter 5: Second-Order Frequency Selective Surfaces Using Non-resonant
Periodic Structures
In this chapter, the second class of the proposed periodic structure is presented. The proposed
structure is a three layer structure composed of three metallic layers, separated by two electrically
very thin dielectric substrates. The first and the third layers are capacitive surfaces composed of
two-dimensional periodic arrangement of rectangular patches. The middle layer is an inductive
surface composed of two-dimensional periodic arrangement of wire grid. It is shown that this
arrangement acts as a second-order bandpass FSS. The unit cell the proposed FSS has dimensions
in the order of 0.15A0 and an overall thickness of A0/30. An equivalent circuit based synthesis
procedure is also presented. Prototypes operating at X-band are designed, fabricated, and tested.
A free-space measurement setup is used to thoroughly characterize the frequency responses of
these prototypes for both the TE and TM polarizations and various angles of incidence.
1.4.5
Chapter 6: Synthesis of Generalized Frequency Selective Surfaces of Arbitrary Order Using Non-resonant Periodic Structures
The topology of the proposed structure presented in Chapter 5 is expanded to synthesize FSSs
with bandpass responses with a variety of transfer functions of any order (N = 2, 3,4,...). A
comprehensive synthesis procedure for designing these FSSs is presented. An Nth order of the
proposed FSS has 2N — 1 metallic layers. TV of them are non-resonant capacitive layers, whereas
the other layers, TV—1, are non-resonant inductive layers. The proposed FSSs have constituting unit
cells with unit cell dimensions and periodicities in the range of 0.10A0-0.15A0. An Nth order FSS
designed using this synthesis procedure typically has an electrical thickness in the order of ~ (N —
l)A 0 /30 which is significantly smaller than the overall thickness of a traditionally designed Nth
order FSS (~ (N — l)A 0 /4). The proposed FSSs offer several advantages. They can be designed
to provide linear transmission phase within the passband, thereby using them as spatial phase
shifters in the designing microwave lenses, as described in the Chapter 7. Moreover, they allow
for the design of low-profile, can provide wideband of operation with sharp frequency selectivity,
and have stable frequency responses as functions of angle and polarization of incidence of the
18
electromagnetic wave. The comprehensive synthesis procedure is validated for two FSS prototypes
having third- and fourth-order bandpass responses. Principles of operation, detailed synthesis
procedure, measurement results of a fabricated prototype, and implementation guidelines for this
type of FSS are presented and discussed in this chapter.
1.4.6
Chapter 7: Design of Planar Microwave Lenses Using Sub-Wavelength
Spatial Phase Shifters
The generalized miniaturized element frequency selective surfaces discussed in Chapter 6 are
used to design low-profile planar microwave lenses. The proposed planar lenses are obtained by
populating the lens aperture with numerous miniature spatial phase shifters. The topology of each
spatial phase shifter (SPS) is based on the FSS presented in Chapter 6. A procedure for designing
the proposed lenses and their constituting spatial phase shiners is presented in details. This design
procedure is validated experimentally by fabricating two lenses prototype operating at X-band.
These prototypes are fabricated and experimentally characterized using a free-space measurement
system. The fabricated prototypes demonstrate relatively wide bandwidths of approximately 20%.
Furthermore, the two lenses demonstrate stable responses when illuminated under oblique angles
of incidence up to 45°. One characteristic of this type of lenses, however, is that the location of the
focal point is frequency dependent. At frequencies lower than the designed operating frequency,
the focal point is located closer to the lens aperture. As the frequency increases, the location of the
focal point is moved farther away from the lens aperture.
1.4.7
Chapter 8: Design of Planar True-Time-Delay Lenses Using Sub-Wavelength
Spatial Time-Delay Units
The generalized miniaturized element frequency selective surfaces discussed in Chapter 6 are
used to design low-profile planar true-time-delay lenses. The proposed planar lenses are obtained
by populating the lens aperture with numerous miniature true-time-delay units. The topology of
each of these true-time-delay units is based on the FSS presented in Chapter 6. The true-time-delay
unit is obtained by designing the elements of the miniaturized frequency selective surface to have
19
a linear transmission phase within the frequency range of interest. This linear transmission phase
corresponds to a time delay that is provided by each element. A procedure for designing this type of
lenses and their constituting true-time-delay units are presented. The proposed design procedure
is validated numerically by designing a circular true-time-delay lens prototype operating at Xband. It is expected that the lens operate over a relatively wide bandwidth of approximately 27%.
Moreover, the designed lens demonstrate a stable response when illuminated under oblique angles
of incidence up to 45° over the entire band of operation. Finally, it is shown that the designed lens
prototype has frequency-independent focal point, in which the location of the focal point remains
stationary over the entire operational bandwidth.
1.4.8
Chapter 9: Study of Power Handling Capability of Miniaturized-EIement
Frequency Selective Surfaces
In this chapter, a numerical study is conducted to investigate the power handling capability of
the frequency selective surfaces presented in Chapters 5 and 6. In general, the power handling
capability of the frequency selective surfaces are limited by the failure of the material used in these
surfaces. This failure is caused by either heat dissipation or dielectric breakdown. The failure due
to heat dissipation occurs when the structure operates under sustained high power levels. Whereas
dielectric breakdown occurs when the structure is under very high peak power levels. This study
is aimed to give an insight into the effect of various design parameters on the peak and average
power handling capability of the FSS discussed in Chapters 5 and 6.
20
Chapter 2
Hybrid Frequency Selective Surfaces with Third-Order Responses
2.1
Introduction
Frequency selective surfaces (FSSs) have been the subject of intensive investigation for their
widespread applications as spatial microwave and optical filters for more than four decades [109][110]. Traditional FSSs are usually constructed from periodically arranged metallic patches of arbitrary geometries or their complimentary geometries having aperture elements similar to patches
within a metallic screen [110]. These surfaces exhibit total reflection or transmission in the neighborhood of the element resonances for the patches and apertures respectively. One common feature
of the traditional FSS design techniques, widely used nowadays, is that they use resonant type constituting unit cells such as a resonant dipole, slot, circular or rectangular rings, etc. [57]. In such
structures, the size of the resonant elements and the inter-element spacing are generally comparable to one-third to half a wavelength at the desired frequency of operation. However, in practical
applications, FSSs are not infinite in extent and have finite dimensions. Therefore, to observe the
desired frequency response, the finite surface must include a large number of the constituting elements and be illuminated by a planar phase front. For some applications, such as low-frequency
antenna radomes or frequency selective EMI shielding, FSSs of relatively small electrical dimensions that are less sensitive to incidence angle and can operate for non-planar phase fronts are
highly desirable.
Recently, a new type of frequency selective surface with miniaturized elements was presented
in [113], [92]. This type of FSS is composed of a periodic array of sub-wavelength patches printed
on one side of a dielectric substrate and an inductive grid printed on the other side of the same
21
substrate. The combination of the capacitance of the patch array and inductance of the wire mesh
results in a parallel LC combination, which acts as a first-order bandpass filter at the resonant frequency of the structure. In [92], it was shown that the sensitivity of the response of this FSS to
the angle of incidence can be reduced without using any dielectric superstate stabilizer. Moreover, it was demonstrated that a relatively small finite panel of this structure, with panel sizes of
about 2A x 2A demonstrates essentially the same frequency behavior as the full-scale panel. One
significant problem of this approach, however, is that a double-layer structure is used to achieve
a first-order bandpass response [92]. The presence of a separating dielectric substrate between
the inductive and capacitive arrays increases the insertion loss of the first-order FSS. Therefore,
using the structure presented in [92], it is not possible to have a first-order bandpass response with
zero (or very low) insertion loss to fully utilize the benefits offered by using the sub-wavelength
periodic structures.
A single two-dimensional periodic array composed of resonant building blocks such as those
presented in [[57], Chap. 2] or the non-resonant building blocks such as those presented in
[92],[113]-[114] acts as a bandpass or band-stop FSS with a first-order bandpass (or band-stop)
frequency response. Traditionally, to achieve a higher-order filter response, several identical firstorder FSS panels are cascaded with quarter-wavelength spacing between each consecutive panel
[57], [92]. Using this approach, to achieve a third-order bandpass frequency response, three firstorder FSS panels should be cascaded with a panel spacing of A/4 between each panel. This will
result in a composite FSS with an overall electrical thickness of roughly A/2. Irrespective of the
type of unit cell and the periodicity of the structure, this relatively large electrical thickness significantly increases the sensitivity of the response of the FSS to the angle of incidence. Furthermore,
it reduces the amenability of such a structure to applications where conformal frequency selective
surfaces are required. Therefore, FSS design techniques that allow for significantly reducing the
profile (overall thickness) of a higher-order FSSs are highly desirable.
In this chapter, a new class of frequency selective surface is presented which provides an easy
method for designing low-profile bandpass spatial filters with higher-order filter responses. This
structure is composed of two arrays of sub-wavelength capacitive patches separated from a periodic
22
(a)
(b)
Figure 2.1 Topology of the third-order bandpass FSS presented in Section 2.2. (a)
Three-dimensional view and (b) Side view.
array of miniaturized slot antennas using two very thin dielectric substrates. The resulting structure
has three layers covered with metal patterns and two very thin dielectric substrates. It is shown
that the proposed structure acts as an FSS with a third-order bandpass filter. As a result of its low
profile and sub-wavelength periodicity, the frequency response of the proposed structure is less
sensitive to the angle of incidence electromagnetic wave compared to other third-order bandpass
FSSs designed using traditional techniques. In this chapter, the principles of operation of the FSS
along with the full-wave simulation results will be presented in Section 2.2. These principles
are experimentally verified in Section 2.3 by fabricating a prototype of the proposed FSS and
measuring its frequency response using a free space measurement system.
2.2
FSS Design
2.2.1
Principles Of Operation
The topology of the proposed structure is shown in Fig. 2.1. Figure 2.1(a) shows a threedimensional topology of different layers of the structure. The structure is composed of three different metal layers separated from one anther by two very (electrically) thin dielectric substrates.
23
Top View of the Unit Cell
Capacitive Patch
Miniaturized Slot
c, Antenna Aperture f•—•*! '•
y
Figure 2.2 Top view of the unit cell of the third-order bandpass FSS studied in Section 2.2. The
view of the capacitive patch is shown on the left hand side and the topology of the utilized
miniaturized slot antenna is shown on the right hand side.
The top and bottom metal layers consist of identical two-dimensional periodic arrangements of
sub-wavelength capacitive patches. The center metal layer consists of a two-dimensional periodic
arrangement of miniaturized slots etched into a ground plane. The capacitive patch layers are identical and the dielectric substrates are also identical resulting in a symmetric structure with respect
to the plane containing the miniaturized slots. Figure 2.1(b) shows the side view of the FSS. The
overall thickness of the FSS, h, is twice the thickness of the dielectric substrates used to fabricate
the structure on. Generally, this thickness can be made as small as A0/60 to as large as A0/10. Figure 2.2 shows the top view of different layers of the unit cell of the proposed FSS. Each unit cell
has maximum physical dimensions of Dx and Dy in the x and y directions, respectively, which are
the same as the period of the structure in the x and y directions. On the left hand side of Fig. 2.2,
the top view of a single capacitive patch is shown. Assuming that the structure has the same period
in x and y directions, each capacitive patch will be in the form of a square metallic patch with side
length of D — s, where s is the separation between the two adjacent capacitive patches. Since the
structure has sub-wavelength periods and dimensions, i.e. Dx,Dy <C A, the capacitive patches are
non-resonant and their 2-D periodic arrangement simply presents a capacitive wave impedance to
an incident EM wave. The right hand side of Fig. 2.2 shows the top view of the miniaturized slot
used in the unit cell of the FSS. Unlike the capacitive patches, this element is a resonant element.
However, it is designed to occupy an overall area significantly smaller than regular dipole or slot
24
Zo
Z„l
Z„j
Zc
Figure 2.3 Equivalent circuit model of the third-order bandpass FSS studied in Section 2.2.
antennas. As can be seen in Fig. 2.2, the aperture of each slot occupies an area ofDap x Dap, where
Dap is only a fraction of the unit cell size (Dap < Dx, Dy). The topology of the miniaturized slot
antenna used in the unit cell of the proposed structure was first introduced in [115] and details of
its operation along with its behavior as a miniaturized slot antenna are extensively discussed in
this reference. The miniaturized slot shown in Fig. 2.2 is single polarized and in the arrangement
depicted in this figure, permits the transmission of an electric field polarized in the y direction.
Therefore, the frequency response of the proposed FSS utilizing this embodiment of the miniaturized slot is polarization sensitive. If polarization discrimination is not required, a dual-polarized
version of the miniaturized slot antenna shown in Fig. 2.2 may be used.
To understand the principles of operation of the proposed FSS, it would helpful to consider its
equivalent circuit. A simple equivalent circuit of the proposed FSS, for a vertically polarized TEM
plane wave, is shown in Fig. 2.3. The circuit is composed of a parallel LC resonator (Li and Ci)
Impedance Inverters
Capacitively Loaded T-Lines
Figure 2.4 Expanded equivalent circuit model of the third-order bandpass FSS studied in
Section 2.2. The impedance inverters and the slow-wave resonators are clearly indicated.
25
with a parasitic series inductor (L 2 ), separated from two parallel capacitors (C2) with two short
sections of transmission lines with characteristic impedance of Z\ and length of I. The parallel
L\C\ resonator represents the miniaturized slot resonator. The series inductor, L2 represents the
parasitic inductance associated with the electric current flowing in the ground plane of the miniaturized slot. The dielectric substrates supporting the structure are modeled with two short pieces
of transmission lines where the length of each line, £, is equal to the thickness of the substrate and
the characteristic impedance of each line is Zx = Z0/y/e^, where Z0 = 377Q is the free space
impedance and er is the dielectric constant of the substrates used. Free space on both sides of
the FSS is modeled with two semi-infinite transmission lines with characteristic impedances of ZQ.
This circuit model is only valid for normal incidence. For oblique incidence, both the characteristic
impedances of the transmission lines and the values of the capacitances and inductances need to be
altered. The equivalent circuit model of the FSS shown in Fig. 2.3 is basically a third-order bandpass microwave filter composed of two slow-wave capacitively loaded transmission line resonators
separated from a parallel LC resonator with two simple impedance inverters. Figure 2.4 shows the
expanded equivalent circuit model of the filter, where the impedance inverters, the slow-wave transmission line resonators, and the parallel LC circuit are clearly indicated. The impedance inverter
is an inductive network with a transmission line with "negative" electrical length. The principles
of operation of this inductive impedance inverter and its other variations are thoroughly studied in
[116]. Two inductive impedance inverters separate two capacitively-loaded transmission line resonators from the parallel LC resonator. Therefore, the combination of these three resonators and
two impedance inverters results in the third-order bandpass filter shown in Fig. 2.3. By comparing
the two equivalent circuits, it is observed that the negative length of the transmission lines used
in the inductive impedance inverter is absorbed in the positive length of the capacitively loaded
transmission line, thereby further reducing the transmission line length. Moreover, the inductors
of the inverter network, L,, are absorbed in the parallel LC resonator.
26
(a)
Figure 2.5 (a) A unit cell of the proposed FSS is placed inside a waveguide with periodic
boundary conditions to simulate the frequency response of the infinitely large third-order FSS
studied in Section 2.2. (b) Equivalent circuit model of a simplified version of the FSS composed
of ONLY the dielectric substrates and the miniaturized slot antenna. This equivalent circuit is
used to optimize the topology of the miniaturized slot antenna to match its frequency response to
that of this equivalent circuit.
2.2.2
Design Procedure
The proposed FSS can be designed using a simple, three-step design procedure described in
this sub-section. The first step in the design of the FSS is to obtain the appropriate element values
for the equivalent circuit model shown in Fig. 2.3. This can be accomplished using a circuit
simulation software such as the Agilent's Advanced Design Systems (ADS). For example, to obtain
27
a bandpass frequency response with center frequency of operation of 4 GHz and the fractional
bandwidth of 20%, the equivalent circuit element values shown in Table 2.1 can be used. In this
case, it is assumed that a 1.524 mm thick dielectric substrate with the dielectric constant of er=3.5
and loss tangent of tan S = 0.0020 (RF-35 from Taconic corporation) is used. The shape of the
FSS transfer function can be optimized using simple and rapid circuit based simulations. In the
second step of the design process, the miniaturized slot resonator must be designed such that it
has an equivalent circuit (parallel LC resonator) with element values that match those obtained
in the previous step (e.g., L\ and C\ shown in Table 2.1 for the FSS under examination here).
The miniaturized slot resonator is essentially a miniaturized slot antenna composed of a straight
slot section connected to two balanced spirals at each end. The effective electrical length of the
antenna, from one end of one balanced spiral to the corresponding end of the other balanced spiral
is about half a wavelength. Therefore this antenna is a resonant structure, which is meandered in
order to occupy a significantly smaller area compared to non-miniaturized antennas operating at
the same frequency. This structure, which was first introduced and thoroughly studied in [115],
acts as a magnetic Herzian dipole. Similar to any other electrically small antenna, the quality
factor (Q) of this structure is inversely proportional to its occupied area [117]-[118]. Therefore,
by reducing the occupied area of the antenna (i.e. by reducing Dap x Dap as seen in Fig. 2.2)
while maintaining the resonant frequency of the antenna, the Q of the structure can be increased.
This way, by appropriately choosing the aperture dimensions of the miniaturized slot antenna, for
a constant resonant frequency, the desired values for L\ and C\ can be synthesized. In practice
this procedure is carried out using full-wave EM simulations in conjunction with circuit based
simulations. An incomplete version of the unit cell of the proposed FSS, which consists of ONLY
the miniaturized slot resonator sandwiched between the two dielectric substrates, is simulated using
full-wave EM simulations in CST Microwave Studio Suite using its Finite Element Method solver.
The simulations are performed according to the topology shown in Fig. 2.5(a), where the unit cell
of the structure is placed in a waveguide with periodic boundary conditions to emulate an infinite
structure. Finite Element Method (FEM) simulations are performed to calculate the transmission
and reflection coefficients of a vertically polarized TEM wave from this structure. These full-wave
28
Parameter
Zo
Zx
£
er
Value
377 fi
201 Q
1.524 mm
3.5
Parameter
Ci
U
c2
L2
Value
43.2 pF
34.75 pH
0.7536 pF
53.5 pH
Table 2.1 Electrical parameters of the equivalent circuit model of the proposed third-order FSS.
simulation results are then matched to those obtained from the equivalent circuit model shown in
Fig. 2.5(b), where only the effect of the miniaturized slot antenna and the dielectric substrates are
taken into account. The dimensions of the slot resonator are then modified as necessary and this
process is then repeated until the frequency response obtained through the full-wave simulations
of the sandwiched resonator are matched to those of the equivalent circuit model of Fig. 2.5(b)
with the element values obtained previously. After successful completion of this step, the design
of the resonator is finalized. One must note that in this step, we are assuming that the addition
of the capacitive patches on both sides of the slot does not affects the resonant frequency of this
structure. This is a valid assumption, since the width of the slot lines in the miniaturized slot
resonator are significantly smaller (more than 10 times smaller) than the thickness of the dielectric
substrates covering the slot on its both sides. Therefore, the effective dielectric constant that the
slot resonator experiences will not be significantly changed after placing the capacitive patches
underneath the slot, since most of the slot's electric fields are already concentrated in the dielectric
substrates covering the aperture on its both sides.
In the third and final step of the design process, the capacitive patches corresponding to capacitors C2 (in the equivalent circuit model shown in Fig. 2.3) are added to the full-wave simulation
model. These capacitors are in the form of simple, sub-wavelength, non-resonant patches with side
lengths of D — s, where D = Dx = Dy is the period of the structure and is equal to the unit cell
dimensions. To provide the desired capacitance value, the initial dimensions of these capacitors
can be approximated using the formulae available in reference [119] for calculating the capacitance
between two printed metallic strips. Finally, full-wave simulations are performed on the complete
29
Parameter
Dx
Dy
h/2
Value
12.6 mm
12.6 mm
1.524 mm
Parameter
er
s
Value
3.5
150 fiva
Uap
3.3 mm
Table 2.2 Physical parameters of the proposed third-order FSS.
0
-10
m -20
s= -30
0)
o -40
.___u_L_yi —-X
-----^A-
\
At-
• ^ ^ - n j/-
-50 +-60
i
Hi
"
Trans Coeff Eq Circuit
Trans Coeff Full Wave
Refl Coeff Eq Circuit
Refl Coeff Full Wave
-70
—
-10
i
\r
„_L____
2.5
3.0
35
i
i
~\
~~~
i
-80
2.0
i
-20
' (I !
[
H
0
i
—
o
«
,
4.0
\
/!
i
I~t
-30
o
03
-40
~]
I
i
4.5
5.0
55
-50
60
Frequency [GHz]
Figure 2.6 Transmission and reflection coefficients of a typical third-order bandpass FSS with
equivalent circuit parameters and dimensions shown in Tables 2.1 and 2.2. The results obtained
using full-wave FEM simulations as well as those obtained using the equivalent circuit model are
shown here.
unit cell of the proposed FSS consisting of the miniaturized resonator, the capacitive patches, and
the dielectric substrates. Based on the results obtained from the final full-wave simulations, the
dimensions of the capacitive patches are tuned to fine tune the frequency response of the structure
and obtain the desired frequency response.
The design procedure outlined above was carried out for the FSS with equivalent circuit parameter values shown in Table 2.1 and the physical and geometrical parameters of this design are
presented in Table 2.2. Figure 2.6 shows the FSS frequency response as obtained from full-wave
EM simulations in CST Microwave Studio as well as those predicted by the equivalent circuit
model presented in Fig. 2.3. It is observed that the equivalent circuit model accurately predicts
the FSS frequency response. The frequency response of the proposed FSS is also calculated for
30
2
3
4
5
6
Frequency [GHz]
Figure 2.7 Transmission coefficients of the FSS studied in Section 2.2 with the parameters given
in Tables 2.1 and 2.2 as a function of the angle of incidence. The results are obtained using
full-wave FEM simulations.
non-normal angles of incidence and the results are presented in Fig. 2.7. Figure 2.7 shows the
transmission coefficient of the FSS for an obliquely incident plane wave (as shown in Fig. 2.1(b))
for various angles of incidence ranging from 6 = 0° to 50° in 10° degree steps. As observed from
this figure, the frequency response of the FSS is not considerably affected1 as the angle of incidence
increases from 9 = 0° to 9 = 40°. As 9 is increased beyond 40°, however, the response starts to
deviate from that of the normal incidence. Nevertheless, the structure demonstrates a rather stable
frequency response as a function of angle of incidence without the aid of any dielectric superstrates
that are commonly used to stabilize the frequency response of FSSs for oblique angles of incidence
[57].
2.3
Experimental Verification and Measurement Results
The performance of the proposed FSS is experimentally demonstrated using a simple free-
space measurement setup. Using this approach, a relatively large panel of the third-order bandpass
FSS operating at 4.0 GHz, with the dimensions of 24" x 18" (or equivalently 61 cm x 46 cm), is
1
This is in comparison with a traditional third-order bandpass FSS obtained by cascading three first order FSS
panels a quarter-wavelength apart.
31
. =. * !
i * »
4
-
*
( £ .'
^
' ^ . ^ II l
^ki <:'
t ' i
* *• ,
I ' > '.
I l l :
I I I
i I :
i i !
* I !
11 .-
f' i
fit
i-f*
if f
>^|k^
Figure 2.8 Photograph of the FSS measurement fixture used to characterize the frequency
response of the third-order bandpass FSS. The edges of the fixture are covered with RF absorbers
to reduce the effects of scattering on the performance of the measurement. In the photograph, the
FSS test sample is placed in the opening and the photograph of the FSS test sample is also shown
in the inset of the figure.
designed and fabricated. A wooden fixture with dimensions of 6 feet x 4 feet, with an opening with
dimensions of 24" x 18" at its center is also built. The wooden fixture is covered with 0.005" thick
copper plates on its both sides everywhere except on the aperture opening. This fixture is placed
between two double-ridge horn antennas connected to the two ports of a vector network analyzer.
The horn antennas are not identical but their operating bandwidths overlaps in the 3 GHz - 5 GHz
region, which entirely covers the operational bandwidth of the proposed FSS. To ensure that the
FSS is excited with plane waves, the fixture is placed in the far field of the two horn antennas.
The transmission coefficient of this structure without the presence of the FSS (open aperture) is
measured and this result is used for calibration purposes. In the next step, the finite FSS panel is
placed in the opening of the fixture and the transmission results are measured once again. Using
these two measurement results, the transmission response of the FSS is obtained.
32
r"
CD
•o
/
\
'
/
V
-y—\-^—-ir
•10
/J
O
o
_: -20
/
\V,>
.
.
•
i
\..
\i
_
1
x
*
i
/^S
0)
w
i
-30
t -=.—-' = •
-40
™ =
=,
= -
_ _ _
3.0
L
i
Trans. Coeff. Meas.
Trans. Coeff. Sirn.
Refl. Coeff. Sim.
Refl. Coeff. Meas.
3.5
•'/
f.
V
4.0
1
j
1
4.5
V
5.0
Frequency [GHz]
Figure 2.9 Measured and simulated transmission coefficients of the test sample of the third-order
FSS. Simulations are performed using FEM simulations in Ansoft HFSS.
Figure 2.8 shows a photograph of the FSS panel and the measurement fixture. Figure 2.9
shows the measured frequency response (transmission coefficient) of the FSS along with the fullwave EM simulation results obtained in Ansoft's High Frequency Structure Simulator (HFSS). As
can be seen from this figure, a relatively good agreement between the measured and simulated
results is observed. The main differences observed between the simulated and measured results
are mainly observed in the stop band of the FSS. These discrepancies are attributed to the leakage
from the gaps that exist between the edges of the FSS panel and the metal covered wooden fixture,
scattering from the edges of the fixture, as well as the finite size of the FSS panel. By covering the
opening of the fixture shown in Fig. 2.8 with a metal screen (instead of FSS panel) and performing
a calibrated transmission measurement, the shielding effectiveness of this particular fixture was
measured to be about 25 dB average over the 3.0 GHz - 5.0 GHz frequency range. Nevertheless,
the good general agreement observed between the measurement results and the simulation results
confirms the fact that the proposed structure acts as a third-order bandpass FSS and the asserts
validity of the proposed design procedure. At its center frequency of operation, the fabricated FSS
demonstrates a measured insertion loss of about 0.4 dB (for normal incidence), which is mainly
attributed to the Ohmic and the dielectric losses of the structure. The transmission coefficients of
the FSS is also measured for oblique angles of incidence and the results are presented in Fig. 2.10.
33
CO
•4—'
c
(U
"o
!t=
Q
O
O
c
o
V)
en
to
3.0
3.5
4.0
4.5
5.0
Frequency [GHz]
Figure 2.10 Measured transmission coefficients of the third-order bandpass FSS for various
angles of incidence ranging from 6 = 10° to 50°.
As observed from this figure, the frequency response of the FSS changes with the angle of incidence. However, compared to third-order bandpass FSSs designed using traditional techniques,
the structure demonstrates a more stable frequency response. Therefore, it may be possible to further enhance the angular stability of the frequency response of this structure using the techniques
presented in [57]. Prominent among these technique is the utilization of dielectric superstrates on
both sides of the FSS used to achieve angular stabilities up to 85° as described in [57] and its cited
references.
2.4
Conclusions
A new technique for designing low-profile frequency selective surfaces with third-order band-
pass responses was presented in this chapter. Unlike traditional FSS design techniques that achieve
higher-order bandpass responses by cascading multiple bandpass FSSs of first order, the proposed
technique allows for designing third-order bandpass FSSs with an extremely small overall profile.
It was demonstrated that using this technique, third-order FSSs with an overall thickness of A0/24
or smaller can easily be designed. The principles of operation of the proposed FSS along with a
simple, three-step procedure for designing the proposed structure was also presented. Finally, the
34
validity of the proposed concept was experimentally demonstrated by fabricating an FSS test sample and measuring its frequency response using simple free space measurements. The frequency
response of this test sample was measured both for normal incidence and for oblique angles of incidence. The measurement results for oblique angles of incidence demonstrate that the frequency
response of this structure is not as sensitive to the angle of incidence of the EM wave as those of
traditional third-order bandpass FSSs.
35
Chapter 3
Synthesis of Hybrid Frequency Selective Surfaces with Odd-Order
Frequency Responses
3.1
Introduction
In the previous chapter, we presented a new technique for designing a low-profile, third-order
bandpass FSS using a combination of resonant and non-resonant elements. The FSS consists of
three metal layers and two dielectric substrates where the first and the third metal layers are in
the form of two-dimensional (2-D) periodic arrangement of sub-wavelength capacitive patches,
while the middle metal layer is a periodic structure composed of miniaturized slot resonators. A
qualitative explanation was provided describing why this particular structure operates as a thirdorder bandpass FSS.
In this chapter, we extend the work presented in Chapter 2 and generalize that FSS topology
to allow for the design of low-profile FSSs with any higher-order bandpass responses of odd-order
(N = 3, 5, 7, ...). Furthermore, we provide a procedure for synthesizing the proposed general,
odd-order bandpass FSSs. This procedure is based on a generalized equivalent circuit model of
the structure that allows for synthesizing the FSS from system level performance indicators such
as as center frequency of operation (/ 0 ), fractional bandwidth (5 = BW/f0),
response type (e.g.
Chebyshev, Butterworth, etc.), and response order (JV = 3, 5,...). In what follows, the design
procedure and the generalized synthesis procedure of the proposed structures are presented. The
proposed synthesis procedure is validated with two design examples of FSSs having third- and
fifth-order bandpass responses.
36
3.2
Generalized FSS Topology
3.2.1
Topology and Equivalent Circuit Model
Fig. 3.1(a) shows the three-dimensional (3-D) topology of the proposed FSS. The structure is
composed of several metal layers separated from one another by very thin dielectric substrates.
Each metallic layer is either in the form of 2-D periodic arrangement of sub-wavelength capacitive
patches or a two-dimensional periodic arrangement of miniaturized slot resonators. Figure 3.1(b)
(top portion) shows the side view of the structure. As can be seen, the first and last layer of the
structure are always composed of sub-wavelength capacitive patches, while the layers in between
are composed of miniaturized slot resonators and capacitive patches repeated sequentially. Figure 3.1(b) (bottom portion) depicts the top view of one unit cell of the sub-wavelength capacitive
patches and the miniaturized slot resonators. The different capacitive patch layers (or different
miniaturized slot layers) are not necessarily identical to one another. Additionally, the structure is
always composed of an odd number of metal layers and an even number of dielectric substrates that
separate them from each other. An FSS of this topology, which is composed of TV metallic layer
acts a Nth order bandpass FSS, where TV is always an odd number (i.e., iV = 3, 5, 7, ...). In this
particular Nth order bandpass FSS, ^f^ of the metallic layers are composed of miniaturized resonators and ^ p of the layers are composed of sub-wavelength capacitive patches. To understand
the principles of operation of this general odd-order bandpass FSS, we use a simple equivalent
circuit model of the FSS as shown in Fig. 3.2(a). Here, the sub-wavelength capacitive patches are
modeled by parallel capacitors (Cn, C^,..., CLN) in a transmission line circuit. The miniaturized slot resonators are modeled using parallel LC resonators
(L2CL2,
L^CL^
...,
LN-iCL(N^i))
and the ultrathin dielectric substrates separating the metallic layers from one another are modeled by transmission lines with lengths of /i li2 , /i2,3, •••, ^./v-i,./v and characteristic impedances of
^i,2, Z 2 ,3,..., ZN-itN.
Free space on each side of the FSS is modeled with two semi-infinite trans-
mission lines with the characteristics impedances of Z0 = 377fL
This equivalent circuit (Fig. 3.2(a)) can be converted to the one shown in Fig. 3.2(b) by converting the transmission lines to their equivalent LC network using the Telegrapher's equations
37
Miniaturized Slot Resonators
Figure 3.1 (a) 3-D topology a frequency selective surface composed of successive arrays of
sub-wavelength capacitive patches and miniaturized slot resonators, (b) Side view of the FSS and
the top views of a constituting unit cell of the structure.
[125]. With the assumption that the electrical length of a transmission line is small (i.e., t < A/12
or fit < 30°), a short piece of a transmission line with a length Az can be modeled with a series inductor and a parallel capacitor with inductance and capacitance values of LAz and CAz, where L
(H/m) and C (F/m) are inductance and capacitance per unit length of the line, respectively. Based
on this, Lj )t+ i = ^o^r,t,t+iK,i+i, where iir,i,i+i and hlyl+i are the relative permeability and the
thickness of the substrate which separates the ith and (i + l)th metallic layers from one another.
Similarly, C M + i = e0er.ij!l+i/iJjj+i/2, where er^l+\ is the relative permittivity of the same substrate.
In this equivalent circuit, all the parallel capacitors at each node can be combined and lumped into
a single capacitor. Our synthesis procedure is based on the fact that the circuit model shown in Fig.
3.2(b) is a special case of the coupled resonator filter shown in Fig. 3.2(c), where the inductors
of resonators with odd-order, N = 1,3,5,..., are infinite (i.e., Li, L 3 , ..., LN —> oo). However,
this can only be done for symmetric coupled resonator networks (i.e., networks with symmetric
LC values with respect to the center resonator). Consequently, FSSs of the type shown in Fig.
3.1 can only be used to synthesize symmetric response types (such as a Chebyshev or a Butterworth response), whereas the coupled resonator filter topology shown in Fig. 3.2(c) can be used to
38
•CL1
N-2.N-1
"2,3
1,2
^C, U
cL2
L2:
T^L(N-2)L
N-l.N
N-lr
c'
c
^L(N-1)^LN
h 1,2
2,3
L,2
L
2,3
/YY\.
rrrx.
•
C
Cn
•
^Ll
: :
T C
^2
c2,3
^L2
L
c3.4
cL3
L N-l.N
rrr\
fYY\
k- *
* 4
- -' T c cN-l.N
N lN
^L(N-2)
(b)
L N-2.N-1
2,3
_TYY\_
CjL;
L N-2.N-1
c
L
l,2
rm__
C
2L2
CI
^3
C I
^3
N-l, N
N-2.N-1
(a)
^N-2^N-2
^N.j
c
^L(N-l)
^LN
L N-l.N
rrr\_
CN-1 LN-1
C L
„„
(c)
Figure 3.2 (a) Equivalent circuit model of the general FSS shown in Fig.3.1(a) for normal angle
of incidence, (b) Transmission lines which represent the dielectric substrates in the equivalent
circuit model of Fig. 3.2(a) can be represented by their simple LC equivalent network. The
replacements are shown in dark gray, (c) The general circuit model representing an Nth order,
bandpass coupled resonator filter. The equivalent circuit model shown in Fig. 3.2(b) can be
obtained from that shown in Fig. 3.2(c) if L1} L 3 , ..., LN can be designed to have infinite
inductance values.
synthesize filters with non-symmetric response types (e.g., maximally flat group delay) as well as
symmetric ones.
In the coupled resonator filter topology shown in Fig. 3.2(c), the shunt blocks in the circuit
model represent parallel LC resonators, while the series blocks represent the coupling networks,
where series inductors are used to couple the resonators to each other. The order of the filter is
determined by the number of the shunt resonators (TV) and the response type of the filter is determined by the normalized quality factor of the first and Nth resonators (gi and qN), the normalized
39
load and source impedances (rx and rN)1, and the normalized coupling coefficients between the
resonators (/cii2, k2>3, —, kN-i>N) [126]. Therefore, by specifying the frequency of operation, f0,
FSS bandwidth, BW, and the response type and order, all of the element values of the equivalent
circuit shown in Fig. 3.2(c) can be determined. The synthesis procedure ensures that, for symmetric networks, the inductance values of L l9 L 3 , L 5 ,..., LN (in Fig. 3.2(c)) will become infinite.
This will automatically convert the equivalent circuit model shown in Fig. 3.2(c) to that of our
proposed FSS (shown in Fig. 3.2(b) and Fig. 3.2(a)). Once the parameter values of the equivalent
circuit model shown in Fig. 3.2(a) are obtained, they can be mapped to the geometrical parameters
of the proposed FSS shown in Fig. 3.1. The generalized synthesis procedure developed for this
type of FSS results in slightly different formulae for FSSs with third-order bandpass response and
any higher-order bandpass responses other than the third order (N = 5, 7,...). Therefore, first
in Section 3.2.2, the design procedure of the third order FSS is presented and the general design
procedure for FSSs with orders of N > 3 is presented in Section 3.2.3.
3.2.2
Synthesis Procedure for Third-Order FSSs
A third-order bandpass FSS of the topology shown in Fig. 3.1(a) is composed of three metallic
layers (two capacitor layers and one resonator layer) separated from one anther by two very thin
dielectric substrates. The operation of this FSS can be described using the equivalent circuit model
shown in Fig. 3.2(c) with N = 3. For this circuit, the capacitors of the first and the third resonators
(Ci and C3) are determined from:
tfi = ^?h
LOQZQO
c
* = -h
(3 1}
-
U)0Z0d
where Z0 = 377f2, 5 = BW/f0 is the fractional bandwidth, and qi and q3 are the normalized
quality factor of the first and third resonators. Values of gi and q3 can be found from filter synthesis
handbooks and are provided in Table 3.1 for third-order coupled resonator filters with various
response types. Since q1 = q3 (due to symmetry of the network), C\ and C3 are equal to one
another. The capacitance values of the resonator in the middle layer can then be calculated from
1
For odd-order bandpass filters,
7"I=TJV=1
40
Filter Type
0.1
Q3
h2
&23
Butterworth
1
1
0.7071
0.7071
Chebsshev (0.0 ldB ripple)
1.1811
1.1811 0.6818
0.6818
Chebyshev (0.1 dB ripple)
1.4328
1.4328 0.6618
0.6618
Chebyshev (0.5dB ripple)
1.8636
1.8636 0.6474
0.6474
Table 3.1 Normalized quality factors and coupling coefficients for third-order coupled resonator
filters with different filter responses.
(3.2):
Co
a
Ci
(3.2)
where ki<2 and k2j3 are the normalized coupling coefficients between the first and second, and
second and third resonators, respectively and their values are provided in Table 3.1. For symmetric
networks, k1>2 = k2j3. These particular choices of capacitor values C\, C2, and C 3 causes the
inductance of the first and third resonators (Li and L 3 ) of Fig. 3.2(c) to become infinite. This
transforms the equivalent circuit of Fig. 3.2(c) to that of Fig. 3.2(b) (for TV = 3). By determining
the value of C 2 , L2 can be determined from:
L,=
1
u (C2 - kly25^C^C~2 - k2t36VUZ%)
2
(3.3)
Finally, the values of the coupling inductors can be calculated from:
Ll}2
1
UJ;\k\2,oyC\C2
.
L2,3 —
1
%k2,3svc&
(3.4)
UJ,
Because of symmetry, L 1 2 = 1/2,3- Synthesis formulae provided in (3.1)-(3.4) can be used to
synthesize a third-order bandpass FSS of the type shown in Fig. 3.1(a) (for TV = 3).
3.2.3
Synthesis Procedure for Higher-Order Bandpass FSSs (N > 3)
For bandpass FSSs with higher-order responses (i.e., TV = 5, 7,...), a similar approach is followed to obtain the equivalent circuit model parameters. First, all the capacitor values are determined. From the calculated capacitance values, the values of the coupling inductors and the
41
inductances of the resonators are calculated. Since conversion of the circuit shown in Fig. 3.2(c)
to that of Fig. 3.2(b) is only possible for symmetric networks, symmetry is used to greatly simplify the calculation of the equivalent circuit values. To ensure that this communication remains
concise, details of the derivation procedure are not presented and only the main design equations
are given. For the generalized filter topology of Fig. 3.2(c), the procedure given below can be used
to determine the values of all capacitors:
1. Calculate the capacitance values of the first and the second layers using (3.5):
Cl =
7T^
C*2 = , o xo
0-5)
2. Choose Ci (for i =3, 5,...) according to the following condition:
G:< • V C i - l ~
ki-2,i-lfiyCi-2,2
h-i,i8
N+l
i = 3, 5,...; i < —-—
(3.6)
3. Calculate Cj (for j = 4, 6,...) capacitor values using (3.7):
n — ( VCi-1 ~
Ci
(
~
k
w
j-2,j-i^\/Cj-2.2
}
J=4,6,...;j<^ii
(3.7)
4. Repeat steps 2 and 3 until all capacitor values are calculated. At this point, the only undetermined capacitor values are those of the center elements for iV" = 5,9,13, ....
5. For N=5, 9, 13,..., calculate the value of the remaining capacitor from:
CN+1
2
=
kN+i N+SJCN+S)2
2 '
2
V
O (fcjV-l N+l
2 ' 2
A
y
CN-l
2
+
N = 5,9,13,....
(3.8)
2
6. Since the equivalent circuit of the FSS is symmetric, Q =
CN-(I-I)
for i = 1, 2,...,
JV+1
2 •
42
The procedure described above can be used to calculate the capacitance values of all the capacitors
in the equivalent circuit model of Fig. 3.2(c). It can be mathematically proven that these particular
choices of capacitance values cause L\, L3,..., and LN to become infinite (see Fig. 3.2(c)). Also,
(3.6) implies that for any given filter response with N > 7, more than one circuit exists that satisfies
the design equations. This is indeed the case and stems from the fact that the equivalent circuit
of the proposed FSS (shown in Fig. 3.2(b)) is an underdetermined system (for iV > 7). Having
determined all the capacitor values, the series coupling inductor values can be calculated from:
Li,i+i = —^jFinally, the inductor values L2, L4,...,
LJV-I
,
(3.9)
are determined from:
—— = UJ (Q+i — kiti+i8-\JCiCi+i
-ki+1,i+2Sy/Ci+1Ci+2)
i = l,3,...,N-2.
(3.10)
These design formulae allow for the synthesis of the FSS's equivalent circuit shown in Fig. 3.2(b)
for N > 5. In the next section, we will demonstrate how this equivalent circuit parameter values
can be converted to the geometrical and physical parameters of the proposed generalized FSS.
3.3
FSS Implementation and Verification of the Synthesis Procedure
3.3.1
FSS Implementation
The values of the inductances obtained in (3.4) and (3.9) can be used to obtain the thicknesses
of the magneto-dielectric substrates used in the FSS topology of Fig. 3.1. This can be done using
Telegrapher's equivalent circuit model for a short section of a transmission line with the aid of
(3.11):
V n =
Li i+1
'
i = l,2,...,N-l.
(3.11)
where /ir,i,i+i is the relative pearmeability of the magnetodielectric substrates separating the ith
metallic layer from the (i + l)th layer. The effective value of the capacitances of the parallel LC
43
resonators and the sub-wavelength capacitive patches of Fig. 3.2(a) can be calculated using:
e
Ch\ — C\
^LN
CLi
~Ci
=
0 e r,l,2^1,2
2
<
~
2
2
i = 2,3,...,iV-l.
^N
(A 1 A)
where e^i+i is the dielectric constant of the magneto-dielectric substrate separating the ith and
(i + l)th metallic layers from each other. In (3.12), the capacitances are obtained by subtracting
the capacitance contributions from the transmission lines from the values of Ci,C 2 , ...CN. In
this procedure, we assume that the constitutive parameters (er and yur) of the magneto-dielectric
substrates separating the metallic layers of the FSS from one another are known a priori.
Mapping the capacitor values of each capacitive layer to the geometrical parameters of the
periodic structures can be performed in a manner similar to that described in Chapter 2. The
effective capacitance of a 2-D periodic arrangement of sub-wavelength capacitive patches having
a square shape with periodicity of D and separation of 2s between adjacent capacitive patches (as
shown in Fig. 3.1(b)) can be calculated using (3.13) [127]:
2D
1
C = e0ee//—ln(-^)
TV
sm^j
(3.13)
where e0 is the free space permittivity, e e // is the effective permittivity of the medium in which the
capacitive patches are located, D is the period of the structure, and 2s is the gap spacing between
the adjacent patches. This closed form formula provides an approximation of the capacitance
value of the capacitive layer. (3.13) can be used to determine the spacing between the patches,
2s, once the unit cell size and the capacitance values, CLI, CL3, •••, CLN of the capacitive layers
are determined. The particular choice of the unit cell size is arbitrary. However, choosing small
values of D results in very small values of s, which may not be practical to fabricate. The next
step is to map the parallel LC resonator to the geometrical parameters of the periodic structure
of the miniaturized slot resonator. The values of inductor Li and capacitor Cu in each resonator
layer are determined from (3.2), (3.3), and (3.12) for third-order FSS and (3.7), (3.10), and (3.12)
44
for the generalized FSS with N > 3. In general, each resonator layer can have slightly different
resonant frequency and a different quality factor. The resonant frequency of the slot resonator is
determined by its electrical length, whereas the quality factor (Q) of the miniaturized slot resonator
is inversely proportional to its occupied area [115]. Consequently, by reducing the occupied area
of the slot resonator while maintaining its electrical length, the quality factor of the structure is
increased. While closed form formulae do not exist that can map the geometrical dimensions of
these miniaturized resonators to their equivalent circuit parameters, a detailed empirical procedure
for doing this is provided in Chapter 2 and will not be repeated here.
3.3.2
Synthesis Procedure Verification
The FSS synthesis procedure developed in the previous sections is verified by designing two
FSS prototypes using the equations provided in the previous sections. In the first example, a
bandpass FSS with a third order Chebyshev response (with 0.1 dB ripple), a center frequency of
operation of 4.0 GHz, and a fractional bandwidth of 8 = BW/f0 = 20% is designed. Using the
design equations (3.1)-(3.4) in conjunction with the parameters of a coupled resonator Chebyshev
filter given in Table 3.1, the initial values of the equivalent circuit parameters shown in Fig. 3.2(a)
and (b) are derived. In this derivation, it is assumed that a material with a dielectric constant of er =
3.5 and relative permeability of jir = 1 (RF-35 from Taconic Corp. [128]) is used as the dielectric
substrates. Using (3.1)-(3.4) the parameters of the equivalent circuit model shown in Fig. 3.2(a)
are calculated to be CL1 = CL3 = 730.3 fF, CL2 = 43.11 pF, L2 = 38.01 pH, and hli2 = h2,3 =
1.666 mm. In practical applications, it may not be possible to obtain dielectric substrates with
the exact thicknesses that (3.11) predicts. Therefore, in this verification procedure, we selected
hi,2 = ^2,3 = 1.524 mm, which is the closest commercially available substrate thickness to the
one predicted by (3.11). The response of the equivalent circuit model shown in Fig. 3.2(a) was then
optimized in a circuit simulation software, Agilent's Advanced Design System (ADS), to account
for the change in h\^ and /i2,3 values and the equivalent circuit parameter values were modified
slightly to compensate for this change. The new values of this equivalent circuit model are shown
in Table 3.2. Using the mapping procedure presented in Section 3.3.1, these values are used to
45
Parameter
CLI
Cl2
C*L3
L2
Value
769.74 fF
46.915 pF
769.74 fF
32.98 pH
Parameter
h\2
^23
fr-12
£r23
Value
1.524 mm
1.524 mm
3.5
3.5
Parameter
2s
Dx
Dy
•L^ap
Value
150 lira
12.5 mm
12.5 mm
3.3 mm
Table 3.2 Physical and electrical parameters of the third-order bandpass FSS of Section 3.3.2.
Frequency [GHz]
Figure 3.3 Reflection and transmission coefficients of the third-order bandpass FSS discussed in
Section 3.3.1. Measurement results as well as those obtained through full-wave EM simulations
in HFSS and the equivalent circuit model (EQC) are presented.
obtain the geometrical parameters of the constituting unit cells of the FSS shown in Fig. 3.1(b).
Table 3.2 shows the values of the parameters of the equivalent circuit model of this FSS (shown in
Fig. 3.2(a)) as well as the physical and geometrical parameters of its constituting unit cells (shown
in Fig. 3.1).
The third-order FSS with the parameters shown in Table 3.2 is simulated using full-wave EM
simulations in Ansoft High Frequency Structure Simulator (HFSS) and its transmission and reflection coefficients are calculated. The structure is then fabricated using the procedure described
in Chapter 2 and its transmission and reflection coefficients are measured using a free-space measurement setup. Figure 3.3 shows the comparison between the equivalent circuit model response
46
Filter Type
Qi
95
&12
&23
&34
/?45
Butterworth
0.618
0.618
1
0.5559
0.5559
Chebyshev (0.0ldB ripple)
0.9766
0.9766
0.7796
0.5398
Chebyshev (0.1 dB ripple)
1.3031
1.3031 0.7028
Chebyshev (0.5dB ripple)
1.8068
1.8068 0.6519
ri
r5
1
1
1
0.5398
0.7796
1
1
0.5355
0.5355
0.7028
1
1
0.5341
0.5341
0.6519
1
1
Table 3.3 Normalized quality factors and coupling coefficients for fifth-order coupled resonator
filters with different filter response types.
predicted by the proposed synthesis procedure, the results obtained using full-wave EM simulations, and the measurement results of the third-order FSS whose parameters are given in Table 3.2.
Details of the measurement procedure and the experimental setup used to obtain these results are
presented in Chapter 2 as well. As is observed, the filter-based synthesis procedure proposed for
the third-order FSS provides a simple and accurate method of synthesizing FSSs of the type shown
in Fig. 3.1 and the results predicted by the procedure are in good agreement with the measurement
results as well as those obtained in HFSS.
To validate the accuracy of the general synthesis procedure presented in Section 3.2.3, equations (3.5)-(3.10) are used in conjunction with the parameters of a fifth-order inductively coupled,
coupled resonator bandpass filter provided in Table 3.3 to synthesize a fifth-order bandpass FSS
with a Chebyshev response with 0.1 dB ripple, a center frequency of/0 = 4.05 GHz, and fractional
bandwidth of 5 = 19%. The mapping procedure described in Section 3.3.1 is then used to determine the physical parameters of the FSS unit cells. The parameters of the equivalent circuit model
of the FSS as well as the parameters of the constituting unit cell are presented in Table 3.4. The
fifth-order FSS is also simulated using full-wave EM simulations in HFSS and its transmission and
reflection coefficients are calculated and presented in Fig. 3.4 along with the results obtained from
the equivalent circuit model of Fig. 3.2(a). In these full-wave simulations, the actual substrate
thicknesses predicted by (3.11) are used, even though these exact substrate thicknesses are not
commercially available. This is done to demonstrate the agreement between the results obtained
using the proposed circuit-based FSS synthesis procedure and those obtained using full-wave EM
47
Parameter
CLI,
CL5
CL2,
CL4
CL3
^ 2 , -^4
40.12 pH
Value
678.3 fF
39.9 pF
1.612 pF
Parameter
ft-1,2, ^4,5
^•2,3, ^-3,4
Mr,i,j
^r,i,j
Value
1.721 mm
1.482 mm
1.0
3.5
J-^ap2t J-^ap4:
2si,2s5
2s 3
3.38 mm
220 /im
130 jLtm
Parameter
Value
^ i )
J-Jy
12.7 mm
Table 3.4 Equivalent circuit values and physical and geometrical parameters of the fifth-order
bandpass FSS discussed in Section 3.3.2.
simulations in HFSS. As can be observed, a good agreement is obtained between the circuit-based
synthesis procedure and the full-wave HFSS simulations.
The main discrepancy observed between the results obtained using the equivalent circuit model,
the full-wave simulations, and the measurement is the presence of an out of band transmission null
at
/ > fo, which is not predicted by the equivalent circuit model. This out of band null happens
at 5.27 GHz for the third-order FSS and at 5.22 GHz for the fifth-order one. The specific physical
mechanisms that give rise to this null are studied in Section 2.2. It is demonstrated that this null
is caused by the particular method used to implement the miniaturized slot resonators and can be
predicted in the equivalent circuit model of Fig. 3.2(a) by considering the equivalent circuit model
of the miniaturized slot resonators to have an inductor in series with the parallel LC resonator (See
Fig. 2.3). This model, however, is not used in the current manuscript for two main reasons: 1)
Using this model significantly complicates the proposed synthesis procedure to the point where
derivation of closed form synthesis formulae is not possible, and 2) For a given miniaturized slot
resonator and periodicity, the location of that transmission null is uniquely determined and cannot
be changed as a design parameter. In spite of this discrepancy, the proposed synthesis procedure
does a very good job of predicting the actual response of the FSS in its pass-band.
48
Frequency [GHz]
Figure 3.4 Transmission and reflection coefficients of the fifth-order bandpass FSS discussed in
Section 3.3.1. Results predicted from the equivalent circuit model of Fig. 3.2(a) (EQC) are
compared with the full-wave EM simulation results obtained in HFSS. The electrical and
geometrical parameters of this FSS are provided in Table 3.4.
3.4
Conclusions
A comprehensive method of synthesizing frequency selective surfaces with bandpass responses
of odd-order (N = 3, 5,...) is presented. The presented procedure is verified by full-wave EM
simulations and the results are corroborated by free-space measurements of an FSS prototype.
This general synthesis procedure allows for the development of ultra-thin, low-profile frequency
selective surfaces with higher-order bandpass responses and very high out-of-band rejection.
49
Chapter 4
Dual-Band Frequency Selective Surfaces Using Hybrid Periodic
Structures
4.1
Introduction
In modern complex military systems (such as aircrafts, missiles, and naval vessels), a mul-
titude of different electromagnetic and optical sensors are used. These sensors operate all over
the electromagnetic (EM) wave spectrum ranging from microwave and millimeter-wave (MMW)
frequency bands to sub-MMW, far infrared, and optical bands. In such applications, frequency
selective surfaces with multiple independent transmission bands are required [73].
Several categories of techniques have been used to design multi-band frequency selective
surfaces in the past. Prominent among these include using perturbation techniques [94], multielement unit cells [79]-[97], multi-resonant unit cells [98]-[100], genetic algorithm design techniques [101]-[103], and using complementary structures [104]-[105]. In most of these multi-band
structures, however, the separation between different bands of operation is generally large. In the
case of concentric rings [79], [78] and square loops [96] the band separation is determined by the
clearance between those elements whereas in fractal FSSs [98]-[99] an octave separation between
the bands is generally present. A multi-band FSS with very close band separation has also been reported recently [108]. This is achieved by designing two neighboring capacitively loaded ring slot
resonators of the same dimension but with different capacitive loadings. Due to the high quality
factor of each slot resonator, which is caused by the capacitive loading, a very narrow bandwidth
is obtained in each band. In addition, the separation between the bands is limited by the values of
50
capacitors that are commercially available and large FSS panels will naturally require many discrete elements, which may be a practical limitation in some applications. Recently, several studies
have reported the use of non-resonant sub-wavelength periodic structures in designing frequency
selective surfaces [92], [106]. It was later demonstrated that combining non-resonant elements
with miniaturized resonant structures in a multi-layer structure can lead to the design of frequency
selective surfaces with more complex transfer functions [107], in which a dual-band FSS with
independent bands of operation and second-order bandpass response at each band.
In this chapter, we present a new design for a low-profile, dual-band FSS with closely spaced
bands of operation. The FSS demonstrates a second-order response at each band of operation and
achieves very high out of band rejection. The proposed design is based on a fourth-order coupled
resonator filter, where the four poles of the transfer function are split into two groups of two
closely spaced poles. The proposed FSS is a four-layered structure with a thin overall thickness
of 0.06A;, where A; is the wavelength at the lower band of operation and unit cell dimensions of
(a)
(c)
Figure 4.1 (a) 3D topology of the proposed dual-band FSS. (b) Side view of the structure
showing the dielectric substrates, metal layers, and the prepreg bonding films, (c) Top view of one
unit cell of the structure. The views of the capacitive patch and the resonator are shown separately
for better clarification.
51
0.125A; x 0.125A;. The combination of small cell size and the extremely thin profile results in
a dual-band second-order bandpass FSS with a frequency response, which is not very sensitive
to the angle of incidence of the incident wave. In what follows the principles of operation, the
design procedure, and the measurement results of a fabricated prototype of the proposed device
are presented and discussed.
4.2
Principles of Operation and the Design Procedure
4.2.1
Principles of Operation
The topology of the proposed dual-band FSS is shown in Fig. 4.1. The device's side view and
the top view of the different layers of the unit cell of the structure are also respectively shown in
Fig. 4.1(b) and (c). The structure consists of three thin dielectric substrates that separate four metal
layers from each other. Two of the metal layers, located on the exterior surfaces of this device,
are in the form of two-dimensional periodic structures composed of sub-wavelength capacitive
patches. Each capacitive patch is in the form of a square metallic patch with a side length of
D — s, where s and D are the separation between the two adjacent patches and the period of the
structure, respectively. The dimensions of each capacitive patch are significantly smaller than a
wavelength (i.e. D «
X) such that at the desired frequencies of operation, the capacitive patches
are not resonant and their periodic arrangement simply presents a capacitive surface impedance to
an incident electromagnetic wave. The interior metallic layers of the structure are composed of
two-dimensional arrangements of compact slot resonators all etched in the same common metal
plane. The top view of the unit cell of the miniaturized slot resonator is also shown in Fig. 4.1(c).
Each slot resonator is composed of a straight slot section at the middle connected to two arms at
each end and occupies an area ofDap x Dap. The overall electrical dimension of each slot resonator,
as measured from one end of one arm to the end of the other arm located in the opposite side of
the straight slot section, is approximately 0.5Ag, where Xg is the slot line's guided wavelength.
Therefore, unlike the capacitive patches, these slot layers are resonant structures.
The operation of this device can best be understood by examining its equivalent circuit model,
shown in Fig. 4.2(a), and drawing an analogy between this model and that of a regular microwave
52
TC/
Z
TCr2?Lr2
12
Z
2,3
TCr3?Lr3
Z
:3,4
TC4
(b)
i Cl,2
C
C
3,4
l,2
C
3,4
" "(c)"
L
L
l,2
^3,4
_nnn
2,3
YYYV_
±i
C;L;
C;L;
C;L;
zn
C4L;
(d)
Figure 4.2 Equivalent circuit models of the proposed dual-band device, (a) The most general
equivalent circuit model, (b) A simplified equivalent circuit model which ignores the mutual
coupling between the resonator layers and the parasitic inductances associated with the compact
resonators, (c) The circuit model of part (b) is further simplified by replacing the short pieces of
transmission lines with their lumped element equivalents, (d) A general topology of a
fourth-order, coupled resonator filter. It can be shown that under certain circumstances this circuit
can be converted to the one shown in part (c).
filter. In this equivalent circuit model, which is valid for a vertically polarized (y directed) normally incident EM wave, the two exterior metallic layers are modeled with two capacitors, Ci
53
and C4, placed in parallel with a transmission line. The interior metallic layers are modeled with
two hybrid (series-parallel) resonators consisting of Lr2, Cr2, and Ls2 and L r3 , Cr3, and L s3 . The
parallel combination of the Lr and Cr represent the resonant behavior of the compact slot line
resonators, while the series inductor Ls represents a parasitic inductance associated with the flow
of electric current in the common ground plane of the compact slot line resonators. The dielectric substrates, sandwiched between the ith and (i + l)th metallic layers, are modeled with short
pieces of transmission lines with lengths and characteristic impedances of hi}i+i and Ziti+1. Here
Zi<i+i = ZQ^J'/i^j+i/'e^j+i,
where ZQ = 377 fl is the free space impedance and ^r,i,i+i and tr,i,i+i
respectively are the relative permeability and the relative permittivity of the magneto-dielectric
substrate separating the ith metal layer from the (i + l)th layer. Since the two compact slot layers
are located very close to each other, they will be mutually coupled to each other inevitably. This
mutual coupling is taken into account in the equivalent circuit of Fig. 4.2(a) by allowing Lr2 and
L r3 to be mutually coupled to each other with a coupling coefficient of K2. Similarly, the two
closely spaced inductors Ls2 and L s3 can couple to each other and their mutual coupling is taken
into account using a coupling coefficient of K\. Finally, in the equivalent circuit model of Fig.
4.2(a), two semi-infinite transmission lines with the characteristics impedances of 377Q are used
to model the free space on both sides of the FSS.
To better demonstrate the principles of operation of the proposed dual-band FSS, we can simplify the equivalent circuit of Fig. 4.2(a) to that shown in Fig. 4.2(b). In this simplified circuit
model, direct mutual coupling between the resonators and the series inductors of the interior layers, Ls2 and Ls3, are neglected. This circuit is further simplified by replacing the short sections of
the transmission lines with their equivalent circuit models, composed of a series inductor and two
shunt capacitors, as shown in Fig. 4.2(c). This circuit is a special case of a fourth-order coupledresonator bandpass filter shown in Fig. 4.2(d). One in which the inductor values of the first and
fourth resonator (L[ and L;4 in Fig. 4.2(d)) are infinite. While we don't present the proof here, it
can be theoretically proven that a specific choice of capacitor values of the four parallel resonators
of Fig. 4.2(d) will result in the values of L[ and L'4 to become infinite. This way, the equivalent
circuit model of Fig. 4.2(d) can be converted to that shown in Fig. 4.2(c). Alternatively, this means
54
that the simplified equivalent circuit model of the proposed FSS shown in Fig. 4.2(b) is that of a
fourth-order bandpass coupled resonator filter. Our goal, however, is to utilize the proposed structure to achieve a dual-band operation. This will be achieved in two steps. First, we can separate
the four poles of the transmission function of this fourth-order bandpass FSS into two groups, with
two poles in each group. Doing this results in achieving a basic dual-band response with a moderate out-of-band rejection in the frequency range between the two bands. Then, we will introduce
a transmission null in the frequency range between the two bands of operation to significantly enhance the out-of-band isolation between the two bands of operation. This way, a dual-band FSS
with closely spaced bands of operation, 2nd order bandpass response in each band, and considerable out-of-band rejection may be obtained. In what follows, a detailed design procedure will be
presented and this process will be explained in a greater detail.
4.2.2
Design Procedure
The design procedure of the proposed device is based on synthesizing the desired filter response from the equivalent circuit models presented in Fig. 4.2 and mapping these equivalent
circuit parameter values to the physical and geometrical parameters of the proposed FSS. While
a comprehensive theoretical procedure for synthesizing a dual-band transfer function from the
equivalent circuit models of Fig. 4.2 is not available, we have developed an analytical procedure
for approximating the values of the equivalent circuit parameters of these structures for a desired
dual-band operation. This procedure will be used to obtain the values for parameters of the equivalent circuit models shown in Fig. 4.2. In this approximate procedure, we assume that the proposed
FSS has two bands of operation with center frequencies of fi and fu, which refer respectively to
center frequencies of operation of the lower and upper bands and an operational bandwidth of BW
at both bands of operation. With this assumption, the values of the first and the fourth capacitors
of the equivalent circuit model shown in Fig. 4.2(d) are determined from:
1/
1
r~il
~
4
^
~ 7.8b4:BWZ<
(4.1)
55
Where Z0 = 377H is the free space impedance. Since the conversion of the equivalent circuit
model shown in Fig. 4.2(d) to that of Fig. 4.2(c) is only possible if the structure is symmetric,
C[ = C4. This symmetry also applies to the capacitors of the second and third resonators, and
their values are determined from:
Here, f0 — ia±h. a n j F is the spectral separation factor defined as:
F =
-" + 1
(4 3)
K
BW
}
The particular choice of C'2 and C'z obtained from equation (4.2) ensures that inductors values of
L[ and L\ are very large, such that they can be removed without altering the filter response. After
determining all the capacitor values, the inductor values of the second and third resonators are
determined from:
jj — j_j —
^___
2
3
cu2(C^-^(0.8409Fv / C i (C^ + 0.5412C^)
(4 4-)
where ui0 = 27rf0. The coupling inductors are calculated from:
Lt
2
= L3 4 =
L2
3
'
=
.
l
,
3Acu0BWVF + 0.5C2
(4.5)
(4.6)
These values of coupling inductors are then used to determine the lengths of the short sections of
transmission lines, shown in Fig. 4.2(b) from:
hiii+1 = — ^ ± -
i = l,2,3
(4.7)
where /i 0 = Air x 10~7(H/m) is the permeability of the free space and /J,r,i,i+i is the relative
permeability of the magneto-dielectric substrate separating the ith metal layer from the (i + l)th
layer. The net capacitance values of the capacitors in the capacitive patch layers and the capacitors
in the shunt resonators can be calculated using:
d = C, = C[ -
e
^
h
^
(4.8)
56
ur2 — o2
(4.yj
where e^j+i represents the dielectric constant of the substrate that separates the ith metal layer
from the (i + l)th layer. The second term in equations (4.8)-(4.9) represents the shunt capacitors
of the simplified circuit model of the short transmission line. Because of symmetry, C 4 = C\ and
Cr2 = L>r3-
Using equations (4.1)-(4.9), the initial values of the simplified equivalent circuit model of the
FSS, shown in Fig. 4.2(b), can be derived.
These values can then be mapped to the geometrical and physical parameters of the FSS. This
mapping procedure is accomplished in two steps. In the first step, the slot resonators embedded in
the interior layers of the structure must be designed such that their equivalent circuit model resembles those of the middle hybrid resonators (composed of Lr, Cr) in Fig. 4.2(b). The slot resonators
are essentially slot transmission lines with an overall length of about half a guided wavelength
Figure 4.3 (a) A simplified circuit model of the FSS without the two capacitive patch layers. This
equivalent circuit model is used in conjunction with the full-wave simulations of the FSS without
the exterior capacitive patch layers to fine tune the dimensions of the slot-line resonators as well
as the separation between the two resonator layers, (b) To increase the value of the parasitic
inductance of the hybrid resonators located on the interior layers of the structure, the unit cell of
the two middle layers can be modified as indicated in the figure.
57
(A5/2 as shown in Fig. 4.1). Similar to a slot antenna, the quality factor of these resonators are
inversely proportional to the area they occupy (i.e., Dap x Dap) [117]. Therefore, from an equivalent circuit point of view the overall electrical length of each slot line resonator, Xg/2, or its first
resonant frequency, fg, determines the product of LrCr as fg ;=s \j\/LrCr.
On the other hand,
the occupied area of each resonator determines its loaded quality factor (QL) which in turn determines the ratio of Cr/Lr as QL oc \ j ^ - Therefore, the equivalent circuit parameters of the
resonators can be mapped to their topologies by determining appropriate values for the occupied
dimensions and the resonant frequencies of the resonators. It is also important to note that other
resonator topologies could be used instead of the specific slot resonator used here as long as these
resonators demonstrate a parallel resonant behavior with the quality factors determined using this
design procedure. The mapping between the topology of the slot resonators and the equivalent
circuit parameter values is accomplished by using a combination of circuit-based simulations and
full-wave EM simulations using an iterative procedure similar to the one described in Chapters 2
and 3. In this iterative procedure, a simplified model of the structure composed of only the two
resonators and the three dielectric substrates (excluding the exterior capacitive patches) is placed
in a waveguide with periodic boundary conditions to emulate the infinite array of the slot resonator
structures. The structure is then simulated using full-wave EM simulations to calculate its reflection and transmission coefficients. The calculated frequency response is then compared to that of
the simplified equivalent circuit model of this topology, shown in Fig. 4.3(a). For a given slotline
width, the overall length of the slot resonators and the occupied area are changed in each iteration
to match the frequency response obtained from full-wave simulation of the simplified structure
to that obtained from the equivalent circuit model shown in Fig. 4.3(a). In addition to these parameters, however, the thickness of the middle substrate, h2^, must also be changed. This is due
to the fact that the mutual coupling effects were ignored in our analytical procedure to simplify
the synthesis of the structure. However, the mutual coupling between these resonator layers (as
indicated with K\ and K2 in Fig. 4.2(a)) affects the response of the circuit. To compensate for
the effect of these mutual couplings, the value of the series inductor L2}3 in Fig. 4.2(d) must be
reduced, which is accomplished by reducing the thickness of the middle substrate (h2j3). Since
58
this equivalent circuit model of the FSS and the FSS itself are symmetric, the two inner resonator
layers are identical. Therefore, only three parameters (resonator length, resonator occupied area,
and the separation between the two resonator layers) should be changed in this procedure. The
second step of the design procedure involves determining the dimensions of the sub-wavelength
capacitive patches. The initial dimensions of the capacitive patches can be calculated using the
following formula [131]:
c =t
*-"lT^*t&)
Here, e0 = 8.85 x 10~12 (F/m) is the permittivity of free space and e e // ss
<4J0)
+< r 1,2
2'
is the relative
permittivity of the medium in which the capacitive patches are embedded. Once the initial dimensions of the capacitive patches are determined, the response of the FSS can be calculated using
full-wave EM simulations and an iterative procedure can be used to fine tune the dimensions of the
sub-wavelength capacitive patches.
4.3
Theoretical and Experimental Verification
4.3.1
Simulation Results
The design procedure described in the previous section is used to design a dual-band FSS with
a second-order bandpass response at each band of operation. In this example, the assumption is
that the substrates are non-magnetic and all of them have a relative dielectric constant of er = 3.5.
The center frequencies of operation of the first and the second bands are chosen to be respectively
ft = 4.0 GHz and fu = 7.0 GHz and the bandwidth at each band is BW = 0.8 GHz. Using
(4.1)-(4.9), the values of the simplified equivalent circuit parameters shown in Fig. 4.2(b) are
calculated to be C= 397.7 fF, Cr2=Cr3= 1.205 pF, Lr2 = Lr3= 1.147 nH, hli2=h3A= 1.578 mm,
and /i2,3= 2.95 mm. These values are used in the equivalent circuit model of Fig. 4.2(b) to calculate
the transmission and reflection coefficient of the FSS for normal incidence angles and the results
are presented in Fig. 4.4. As can be observed, the approximate synthesis procedure results in a
dual-band operation. However, as the analytical synthesis procedure presented in Section 4.2.2
is an approximate one, the predicted values result in a dual-band frequency response with center
59
-30
EQC: Fig. 4.2(a)
-CSTMWS
-EQC: Fig. 4.2(b)-Synthesis Procedure
EQC: Fig.4.2(b)-Optimized Values
-40
-50
Frequency [GHz]
0- ! » » » »
-10
s . -20
•»-••
+ » » • » » «
(b)
8 -so
o
-n—EQC: Fig. 4.2(a)
-•— CST MWS
EQC: Fig. 4.2(b)-Synthesis Procedure
EQC: Fig.4.2(b)-Optimized Values
=d -40
an
-50
-60
2
3
4
5
6
7
8
Frequency [GHz]
Figure 4.4 Calculated transmission and reflection coefficients of the proposed dual-band device.
(a) Transmission coefficient (b) Reflection coefficient. The results obtained from the equivalent
circuit model of Fig. 4.2(b) with the values predicted directly from the equations provided in
Section 4.2.2 is shown in red (up triangle symbols). The results predicted from the circuit of Fig.
4.2(b) with tuned values are shown in blue (down triangle symbols). Full-wave simulations
results are shown in black (circle symbols) and the results predicted by the equivalent circuit
model of Fig. 4.2(a) with the values given in Table 4.1 are shown in green (square symbols).
frequencies of operation of ft = 3.7 GHz and fu = 7.0 GHz. To obtain the desired frequency
response from the circuit shown in Fig. 4.2(b), we fine tuned the values obtained from the synthesis
procedure. This was done by conducting circuit simulations in Advanced Design Systems (ADS)
software package from Agilent Technologies, Inc. [132]. We were able to fine tune the response
of this circuit by changing the values of Lr2 = Lr3 from 1.147 nH to 746 pH and the values of
C r2 = C r 3 from 1.205 pF to 1.424 pF without changing the other parameters. The calculated
60
frequency response of the equivalent circuit of Fig. 4.2(b) with these new values is also shown in
Fig. 4.4. As can be seen, a dual-band response with transmission windows at the desired frequency
bands is now achieved.
Next, the calculated values of the shunt resonators, Lr2 and L r 3 and C r2 and C r3 , are used
to design the slot resonator following a procedure that was described in detail in the previous
section. As indicated in the previous section, in the full-wave simulations, it is necessary to reduce
the thickness of /i2>3 in addition to determining the optimum dimensions of the slot resonators.
This is done to compensate for the effects of mutual couplings, which are not taken in to account
during the analytical synthesis procedure. In the present case, this procedure resulted in reduction
of the value of /i2,3 from 2.95 mm to 1.5 mm. Once the dimensions of the slot resonator are
optimized, the two exterior patch layers are introduced to the circuit and the overall structure of the
proposed FSS, shown in Fig. 4.1, is optimized using full-wave EM simulations in CST Microwave
Studio [133]. The calculated frequency response of the structure obtained from the full-wave EM
simulations is also shown in Fig. 4.4. As is observed, a dual-band response with ft = 4.0 GHz
and /„ = 7.0 GHz is now obtained. Furthermore, the full-wave EM simulation results predict the
existence of a transmission null at a frequency slightly above fu. This transmission null is caused
by the series inductances L s2 and L s3 in the equivalent circuit model shown in Fig. 4.2(a) as well
as the mutual coupling effects between the two resonators. To demonstrate that the equivalent
circuit model of Fig. 4.2(a) does indeed represent the proposed FSS accurately, we conducted
an optimization procedure in ADS to determine the values of the equivalent circuit parameters
of the circuit shown in Fig. 4.2(a) that result in the best match with full-wave EM simulation
results. In this optimization procedure, however, only the values of the effective inductances and
capacitances are allowed to change. The results of this optimization procedure are given in Table
4.1. The frequency response of the structure, as predicted from the equivalent circuit model of Fig.
4.2(a) with the values given in Table 4.1, is also presented in Fig. 4.4. As can be observed, an
excellent agreement between the two is achieved.
The sensitivity of the frequency response of the proposed FSS to the angle of incidence of
the EM wave for both the transverse electric (TE) and the transverse magnetic (TM) polarizations
61
Parameter
D
s
J-^ap
^ 1 , 2 , /l3,4
Value
9.2 mm
0.11 mm
5.18 mm
1.578 mm
Parameter
^2,3
er
C
Oy-
Value
1.5 mm
3.5
460 fF
1.424 pF
Parameter
J-Jy
LB
Kx
K2
Value
746 pH
85 pH
-0.86
-0.08
Table 4.1 Physical and electrical parameters of the structure discussed in Section 4.3.1. The
equivalent circuit values presented in this table are obtained through an optimization process in
Agilent ADS and refer to the circuit model of Fig. 4.2(a). In this optimization procedure the
values predicted by (4.1)-(4.9) were used as initial values.
is also studied using full-wave EM simulations in CST Microwave Studio. Figure 4.5 shows the
transmission coefficients of the FSS at various oblique angles of incidence for both TE and TM
polarizations. As observed, the frequency response of the structure, for oblique angles of incidence
up to 45°, is relatively stable for both polarizations. It is observed that for TE polarization, the outof-band rejection level is increased as the angle of incidence increases. This occurs due to the
increase in the loaded quality factor of the shunt resonators, which is caused by the increase in the
wave impedance for the TE polarization as the incidence angle increases (i.e., ZTE — Z0/ cos(9)
where 6 is the incidence angle with respect to the normal to the surface of the FSS). This increase
in the loaded quality factor also causes the bandwidth of each resonator to decrease, and hence the
ripple level increases in the pass band of both bands for the TE polarization. The opposite behavior
is observed for the TM polarization, where the loaded quality factor of the shunt resonators are
decreased as the incidence angle increases. This is due to the decreasing wave impedance of the
TM mode as the angle of incidence is increasing (i.e., ZTM
= Z0 cos(6>)). Consequently, the
bandwidth of each resonator is increased, and the out-of-band rejection is decreased.
As can be seen from Fig. 4.4, the FSS provides a very sharp response and the out-of-band rejection at frequencies above fu and below ft is very high (better than typical second-order responses).
This is due to the fact that this structure is actually providing a fourth-order response and pole
separation is used to achieve a dual-band operation. However, for exactly the same reason, only
62
n
•w
-in
CO
•9flCD
O
^n
O
to
c
4D
•
8=0°
W
U—l J
CD
^D60-
—™— 0-30
- T — e=45°
i
•
Frequency [GHz]
•0
*=
CD
O
O
to
c
co
4
5
6
Frequency [GHz]
Figure 4.5 Full-wave EM simulation results of the transmission coefficients of the dual-band FSS
discussed in Section 4.3.1 with parameters shown in Table 4.1 for various oblique angles of
incidence, (a) TE polarization (b) TM polarization.
a moderate out of band rejection of about 15 dB is achieved between the two bands of operation.
This may not be enough in certain applications. Therefore, to improve the out-of-band rejection in
the frequency range between ft and fu, a transmission null can be placed in this frequency band.
In the proposed structure, this can be achieved by increasing the values of the series inductors L s3
and L s4 . In addition to having a parallel resonance, the two hybrid resonators of Fig. 4.2(a) also
have a series resonance. The frequency of this series resonance can be effectively controlled by
changing the value of Ls. At this series resonant frequency, the FSS demonstrates a transmission
null and acts as a totally reflective structure. Normally, the value of Ls is small enough that this
63
CO
c
CO
•T: EQC
•T: CST MWS
•T: Meas
R: CST MWS
R:EQC
Frequency [GHz]
Figure 4.6 Calculated and measured transmission coefficients and the calculated reflection
coefficients of the proposed dual-band FSS. Measurement results are compared to the
theoretically predicted ones from the equivalent circuit model of Fig. 4.2(a) and the full-wave EM
simulation results conducted in CST Microwave studio. The structure exploits the new resonator
architecture shown in the right side of Fig. 4.3(b). This increases the values of Ls2 and L s3 to
create a transmission null at a frequency between fu and ft.
transmission null occurs at a frequency higher than fu. This can be seen in Fig. 4.4, where the
transmission null occurs just slightly above the frequency of operation of the second band, fu.
Therefore, by increasing the values of L s2 and L s3 , the frequency of this null can be reduced and
a very sharp transmission null can be placed between the two pass-bands of the structure. In practice, Ls can be altered by appropriately modifying the unit cell of the resonator layer, as shown in
Fig. 4.3(b). Here, the slot resonators in the two interior layers are patterned on finite ground planes
that are connected to each other through inductive strips with a length of L and width of W. The
inductance value of the strips can be controlled by changing the values of L and W. In particular,
Ls can be increased by increasing L and decreasing W in Fig. 4.3(b).
This new modification is applied to the unit cell of the structure whose frequency response
is depicted in Fig. 4.4. The frequency responses of the structure is calculated using full-wave
64
EM simulations in CST Microwave Studio and the results are presented in Fig. 4.6. Moreover, the
equivalent circuit model of this new design is also modified to account for this increase in the value
of the series inductors and its response is also depicted in Fig. 4.6. A relatively good agreement
is observed between the frequency response of the modified FSS and its equivalent circuit model.
The figure also shows clearly the improvement in the out-of-band rejection, where the transmission
null is created at 4.9 GHz. As is observed from this figure, the introduction of this null will reduce
the frequency of operation of the second band (fu) from 7.0 GHz down to 5.5 GHz. This is due
to the fact that increasing the values of these inductors affects both the series and the parallel
resonant frequencies of the hybrid resonators of Fig. 4.2(a). This effect is further illustrated in
Fig. 4.7, where the values of these inductors, Ls2 = Ls3, are changed by changing the length of the
inductive strip, L. As can be seen, increasing the length of this inductive strip increases the value
of L s , which will in turn reduce both the frequency of the transmission null and the value of fu.
Choosing small values of L allows for keeping fu close to the original values used in the synthesis
procedure. However, this comes at the expense of having a relatively high insertion loss at the
second band of operation as seen from Fig. 4.7 and it may increase the fabrication complexity.
Alternatively, one can use the design procedure discussed in the previous section in a different way
to achieve dual-band operation with given final bands of operation and a good isolation between
the two bands. For example, if the end goal is to have a dual-band operation with two bands of
operation at 4.0 GHz and 7.0 GHz and a transmission null in between them, the design procedure
presented in the previous section can be used with initial values of fi = 4.0 GHz and a higher
fu value (e.g. fu = 9.0 GHz). The design procedure can then be used to achieve a dual-band
operation without a transmission null in between. Then, by adding a transmission null between the
two bands, the operating frequency of the second band can be reduced from 9.0 GHz to 7.0 GHz.
The transmission coefficients of the modified FSS at various oblique angles of incidence for
both TE and TM polarizations are also calculated using full-wave EM simulations in CST Microwave Studio and the results are presented in Fig. 4.8. As observed, the structure demonstrates a
stable frequency response as the angle of incidence is increased for both TM and TE polarizations
of incidence.
65
Figure 4.7 Calculated transmission coefficients of the dual-band FSS described in Section 4.3.1,
which uses the slot resonator topology shown in the right hand side of Fig. 4.3(b). The effects of
changing the length of the inductive strips on the frequency of the transmission null and the
frequency of the second band of operation are examined. As can be seen, by increasing the length
of the inductive strip (L), the frequency of the transmission null decreases as expected from the
theory presented in Section 4.3.
4.3.2
Experimental Verification
To validate the proposed design procedure, a prototype of the dual-band FSS is also fabricated
and characterized using a free space measurement system. The two interior metal layers are fabricated on two sides of a 60 mil thick RF-35 dielectric substrate [134] from Taconic Corporation.
Each of the exterior metal layers (capacitive patches) is fabricated on one side of a 60 mil thick
RF-35 substrate and the other side of the substrate is completely etched. The dielectric substrates
are bonded together with the aid of two 4.5 mil thick bonding films (TPG-30 from Taconic Corp.).
The bonding film has a dielectric constant of 3.5 and a loss tangent of tan(<5) = 0.0018. The
relative position of various metal layers, the bonding films and the dielectric substrates are shown
in Fig. 4.1(b). As can be seen, the bonding films and the metallic layers are located strategically to
ensure that the symmetry of the structure is maintained. The physical and geometrical parameters
66
of this fabricated FSS prototype as well as the parameters of its equivalent circuit model shown
in Fig. 4.2(b) are provided in Table 4.2. The size of the fabricated panel is 24" x 18". The overall thickness of the panel, inclusive of all the metal, dielectric, and bonding film layers, is 4.79
mm which corresponds to an electrical thickness of 0.06A; for the first band (A; is the free space
wavelength at 3.95 GHz) and to an electrical thickness of 0.08AU for the second band (Au is the
free space wavelength at 5.4 GHz). A 6' x 4' fixture with a center opening is built and is covered
with 0.005" thick copper sheet except on the aperture opening. Two dipole antennas are placed at
-10
2,
it
CD
O
-20
o
w
c
to
5
-30
•
U—u
—•—0=15°
(a)
-40
™ ^ — 0=45°
3.0
™
U—JU
i
•
3.5
4.0
4.5
5.0
5.5
6.0
5.0
5.5
6.0
Frequency [GHz]
J?*
-10
m
T3
1 -20
o
c/i
—•—9=0°
2 -30
H
•(b)
-40
3.0
— • — 9=15
— • — 9=30°
— 9=45°
3.5
4.0
4.5
Frequency [GHz]
Figure 4.8 Calculated transmission coefficients of the dual-band FSS discussed in Section 4.3.1
with parameters shown in Table 4.2 for various oblique angles of incidence; the results are
obtained from full-wave EM simulations in CST Microwave Studio, (a) TE polarization (b) TM
polarization.
67
Parameter
D
s
-L^ap
Value
9.5 mm
0.15 mm
5.4 mm
Parameter
hp
er
£rp
Value
0.11mm
3.5
3
Parameter
h
L
W
Value
1.524 mm
1 mm
1.25 mm
Parameter
C
Value
444 IF
2.37 pF
537.36 pH
Parameter
L8
Ki
K2
Value
1.01 nH
-0.19
-0.08
\ j
r
±Jy
Table 4.2 Physical and electrical parameters of the proposed dual-band FSS utilizing the slot
resonator shown in Fig. 4.3(b) and its equivalent circuit model shown in Fig. 4.2(a).
distance of 2' from each side of the fixture to ensure that the FSS is excited with plane waves. The
transmission coefficient of this fixture without the presence of the FSS is measured and is used for
FSS calibration. The FSS panel is then placed in the opening of the fixture and the transmission
coefficient is measured and calibrated using the first transmission measurement. The measured
transmission response of the FSS along with simulated response is presented in Fig. 4.6. A relatively good agreement between the measured results, the equivalent circuit predicted ones, and the
results obtained from the full-wave EM simulations is observed. The fabricated FSS demonstrates
a 0.5 dB measured insertion loss at the center frequency of operation of the first band, and a 0.7
dB measured insertion loss at the center frequency of operation of the second band. This insertion
loss is attributed to the ohmic and the dielectric losses in the structure.
The sensitivity of the frequency response of the fabricated structure to the angle of incidence is
also examined by measuring the transmission coefficient of the structure for various oblique angles
of incidence in the 6 = 0 — 45° for both the TE and TM polarizations and the results are presented
in Fig. 4.9. Figure 4.9(a) shows the measured frequency response for the TE polarization and
Fig. 4.9(b) shows the response of the structure for the TM polarization. As is observed from this
68
figure, the frequency response of the structure is relatively stable to variations of incidence angle
up to 45°. Factors that help achieve this stability include the small periodicity of the structure, its
extremely thin profile, as well as the presence of a sharp transmission null between the two bands
of operation, the frequency of which does not significantly change as the angle of incidence is
changed [57].
The fact that the two pass-band's center frequencies are almost independent of the polarization
and angle of incidence can be explained if we consider the variations of the equivalent circuit parameter values for the TE and TM modes separately. It can be shown that for the TM polarization,
as the angle of incidence changes, the element values of the equivalent circuit model shown in Fig.
4.2(a)-(c) do not change significantly. Therefore, the poles and zeros of the transfer function of
this equivalent circuit model will not change with incidence angle for the TM polarization and the
two operating frequencies do not change significantly. For the TE polarization, on the other hand,
as the angle of incidence increases, the effective capacitance of the capacitive layers decreases (see
[131]) whereas the values of the coupling inductors {L\i, L23, and L34) increase. Other parameters
of these equivalent circuit models will not significantly change with incidence angle for the TE
polarization. Decreasing capacitance values results in increasing the bandwidth of the FSS in each
band and the separation between the two bands of operation. On the other hand, the increasing of
L12, L23, and L34 results in decreasing the bandwidth of the FSS at each band of operation and the
reduction of the separation between the two bands. These two effects tend to cancel each other for
the TE polarization and hence, the frequencies of operation of the two bands do not significantly
change as the incidence angle is increased. However, the bandwidth at each band of operation
changes for both the TE and TM polarizations as described in the previous section.
4.4
Conclusions
We presented a method for designing highly-selective, dual-band frequency selective surfaces
with closely spaced bands of operation. It was demonstrated that the proposed FSS can be designed
using a multi-stage design procedure, which takes advantage of an approximate analytical synthesis
procedure. Implementation of this design procedure was demonstrated by presenting a design
69
-10
2. -20
a;
o
O
CO
-30
c
CD
•
-40
-•—9=0°
-•—9=15°
- « * — 9=30°
- T — 9=45°
(a) :
-50
3.0
1
3.5
4.0
4.5
5.0
5.5
6.0
5.0
5.5
6.0
Frequency [GHz]
J^
-10
•
CD
2- -20
CD
O
o -30
<n
-Hi—9=0°
-•—9=15°
CO
-40
-+-
0=30°
•(b) - 1
-^r— 9=45°
-50
3.0
3.5
4.0
4.5
Frequency [GHz]
Figure 4.9 Measured transmission coefficients of a fabricated prototype of the proposed
dual-band FSS studied in Section 4.3.2 for various oblique angles of incidence, (a) TE
polarization (b) TM polarization.
example, which followed the presented design procedure step by step. The design procedure was
validated through the use of equivalent circuit model simulations, full-wave EM simulations, and
measurement of a fabricated prototype operating at 4.0 and 5.4 GHz. Furthermore, experimental
characterization of the fabricated prototype demonstrate a stable frequency response as a function
of angle of incidence of the EM wave for both the TE and the TM polarizations.
70
Chapter 5
Second-Order Frequency Selective Surfaces Using Non-resonant
Periodic Structures
5.1
Introduction
A new class of frequency selective surfaces (FSSs) are introduced in this chapter. In this tech-
nique, the constituting elements of the FSS are non-resonant structures that are combined to create
a second-order bandpass spatial filter. Both the unit cell dimensions and the periodicity of the
structure are considerably smaller than the wavelength. This helps reduce the sensitivity of the
response of this FSS to the angle of incidence and assures that no grating lobes can be excited
for any real angle of incidence [57]. Furthermore, the overall profiles (thicknesses) of the secondorder FSSs are extremely small (equal to A0/30). This thickness is considerably smaller than
the overall thickness of a traditional second-order FSS designed by cascading two first-order FSS
panels a quarter-wavelength apart, which is at least a quarter wavelength. The combination of
sub-wavelength unit cell dimensions and small period as well as the small overall thickness of the
proposed structure results in a second-order FSS with a frequency response that is less sensitive to
the angle of incidence of the electromagnetic wave compared to that of a traditional second-order
bandpass FSS obtained by cascading two first-order FSS panels a quarter-wavelength apart. Such
low-profile FSSs can be especially useful at lower frequencies where the wavelengths are long and
traditional second-order FSSs will be too thick, bulky, and heavy to use.
The design procedure and the principles of operation of the proposed FSS are first presented.
For a normally incident plane wave, a simple equivalent circuit model is developed for the proposed
FSS and it is shown that the structure is equivalent to a second-order coupled resonator spatial filter.
71
A synthesis procedure along with closed form formulas are also developed, presented, and verified
experimentally by fabricating two FSS prototypes operating at X-band.
5.2
Principles of Operation and FSS Design Procedure
5.2.1
Principles of operation
Figure 5.1 shows the three-dimensional view of different layers of the proposed FSS. The
structure is composed of three different metal layers separated from one another by two very thin
dielectric substrates. The top and bottom metal layers consist of two two-dimensional (2-D) periodic arrangements of sub-wavelength capacitive patches. The center metal layer consists of a 2-D
periodic arrangement of metallic strips in the form of a wire grid. In its basic and simplest form, the
capacitive patch layers are identical and so are the dielectric substrates. This results in a structure
that is symmetric with respect to the plane containing the wire grids. The overall thickness of the
FSS is twice the thickness of the dielectric substrates, h, used to fabricate the structure on. Figure
5.1 also shows the top view of different layers of the unit cell of the proposed FSS. Each unit cell
has maximum physical dimensions of Dx and Dy in the x and y directions, respectively, which
are also the same as the period of the structure in the x and y directions. Each capacitive patch in
the capacitive layer is in the form of a square metallic patch with side length of D — s, where s
is the separation between the two adjacent capacitive patches. The top view of the inductive strips
used in the middle layer of the unit cell of the FSS is also shown in Fig. 5.1. As is observed from
this figure, the structure maintains its original shape as it is rotated by 90°. This ensures that the
frequency response of the structure is polarization insensitive for normal incidence. Assuming that
the structure has the same period in x and y directions, the inductive layer will be in the form of
two metallic strips perpendicular to each other with a length of Dx=Dy=D, and width of w.
To better understand the principles of operation of this structure, it is helpful to consider its
simple equivalent circuit shown in Fig. 5.2, which is valid for normal incidence. The patches
in the first and third metallic layers are modeled with parallel capacitors (Ci and C3), while the
wire grid layer is modeled with parallel inductor (L2). The substrates separating these metal layers
are represented by two short pieces of transmission lines with characteristic impedances of Z\ =
72
Capacitive Patph--
Capacitive Layers
- Inductive Layer
Wire Grid Inductor
---Dx.,,--'
Figure 5.1 Topology of the low-profile, second-order bandpass FSS presented in Section 5.2.
Zo/y/cn and Z2 = Zo/\Ar2 and lengths of h\ and h2, where erl and er2 are the dielectric constants
of the substrates and Z0 = 377S1 is the free space impedance. The half-spaces on the two sides
of the FSS are modeled with semi-infinite transmission lines with characteristic impedances of
ZQI = ZQT\ and ZQ2 = Z0r2, where r\ and r 2 are normalized source and load impedances and
ri = r2 = 1 for free space. This equivalent circuit can be further simplified to obtain the one shown
in Fig. 5.3(a). In this circuit, the short transmission line sections are replaced with their equivalent
circuit model composed of a series inductor and shunt capacitor. This circuit is a second-order
ZQI
z2,h2
Z1,h1
Z0r]
;C7
u
Z02
Z0r2
r^Ci
Figure 5.2 A simple equivalent circuit model for the low-profile FSS shown in Fig. 5.1 and
discussed in Section 5.2.
73
coupled resonator bandpass filter. The second-order nature of this filter can be clearly observed
if one converts the T-network composed of inductors LT1, L2, and Ly2 in Fig. 5.3(a) to a TTnetwork composed of the inductors L\, Lm, and L 3 as shown in Fig. 5.3(b) [126]. As can be
observed, the circuit shown in Fig. 5.3(b) is composed of two parallel LC resonators coupled
to one another using a single inductor, Lm. This circuit is a classic example of a second-order
coupled-resonator bandpass filter as described in [126]. The two circuits shown in Fig. 5.3(a) and
5.3(b) are both second-order coupled resonator structures and are completely equivalent to each
other. The equations relating the values of the inductors LT1, L2, and LT2 to those of L l5 Lm,
and L 3 are also provided in Fig. 5.3(b). Since the equivalent circuit shown in Fig. 5.3(a) is an
approximated version of the one shown in Fig. 5.2, the proposed FSS will act in a manner similar
to that of a second-order, coupled-resonator bandpass filter.
5.2.2
Design Procedure
The proposed FSS can be designed using a simple and systematic approach. The design procedure starts with the equivalent circuit model shown in Fig. 5.3(a). The fractional bandwidth of the
FSS, S, its center frequency of operation, f0, and the response type are generally known a priori.
Filter Type
<7i
«?2
ki2
Butterworth
1.4142
1.4142
0.70711
1
Chebyshev (0.0ldB ripple)
1.4829
1.4829
0.7075
1.1007
Chebyshev (0.1 dB ripple)
1.6382
1.6382
0.7106
1.3554
Chebyshev (0.5dB ripple)
1.9497
1.9497
0.7225
1.9841
Maximally flat delay
0.5755
2.1478
0.8995
Linear phase equiripple (E=0.05)
0.648
2.1085
0.8555
Linear phase equiripple (E=0.5)
0.8245
1.98
0.7827
Gaussian
0.4738
2.185
0.9828
r\
T2
Table 5.1 Normalized quality factors and coupling coefficient for realizing different filter
responses.
74
Short Transmission Lines
^01
^0ri
LT1
cr
^ 7 7
LT2
i
n-v~,
Z r
^02
02
^^^^^^1
I
\
}L2
^Cr2|
<•
-,c3
i_
(a)
Coupling Inductor
^01
Resonators
L.
^ori
2
02
m
Cr
-^33
LrLT+L2(l+LT/LT2)
Z r
02
<~nCT2
T7 n~-C
L3=LrL2{\+LTJLT)
krLT1+LT+(LT1LT2)/L2
(b)
Figure 5.3 (a) Simplified equivalent circuit model of the FSS presented in Fig. 5.1. (b) By
converting the T-network composed of L T1 , L2, and LT2 into a 7r-network composed of L\, Lm,
and L3, the simplified equivalent circuit shown in Fig. 5.3(a) is converted to a classic
second-order coupled resonator filter utilizing inductive coupling between the resonators.
In the coupled resonator filter topology presented in Fig. 5.3, the normalized loaded quality factor
of the resonators, qi and q2, the normalized coupling coefficient between the two resonators, ki2,
and the normalized source and load impedances, 7*1 and r2, are determined by the desired response
type (e.g. Butterworth, etc.). The values of these parameters are provided in [126] and also given
in Table 5.1 for a number of common second-order responses. The classical equations used for the
design of second-order coupled-resonator filters of the type shown in Fig. 5.3(b) can be found in
[126]. In this case, we have used these equations and after some simple algebraic manipulations
75
and with the aid of the reverse transformation equations provided in Fig. 5.3(b), we have obtained
the element values of the equivalent circuit model shown in Fig. 5.3(a). This way, from the values
of 8, /o, <7i, q2, ku, ri, and r 2 , the values of inductors L T1 , LT2, and L2 can be determined using
the following equations:
Z0n
uoQiku
LT2 =
k125
1 - [ki2oY
q1 r2
V ?2 J*i
Z0r2
k12S
.
q2 n
7—x— - — x (1 - k12SJ
uj0q2k12
1 - {k125y
V <?i r 2
1 - (Kl20) 2
W0Kl2
(5.2)
V 91 ^2
Where 5 = BW/fo is the fractional bandwidth of the structure and BH^ is the 3 dB transmission
bandwidth of the filter. As mentioned in the previous sub-section, LT\ and LT2 are the series
inductors representing the short transmission lines on both sides of the inductive layer, as shown
in Fig. 5.2. The value of this inductance can be calculated from the value of the inductance per
unit length of the short transmission line. Using the Telegrapher's model for TEM transmission
lines, LT is simply equal to fiofirh, where fi0 is the permeability of free space, jj,r is the relative
permeability of the dielectric substrate used and h is the length of the transmission line (equal to
the thickness of the dielectric substrate). Therefore, the thickness of the substrates, h\ and h2 can
be calculated from:
h1
^ -
(5.4)
h2 = ± ^
(5.5)
=
M0/V2
Using a procedure similar to the one used to obtain the values of LT\, L2, and LT2, the values
of the capacitances C\ and C3 can be calculated from:
C =
Ql
- e°€rlhl
u0Z0riS
2
(5 6)
76
n
—
g2
€
o£r2h2
u0Z0r2d
rf.
„.
2
Where the second terms in (5.6) and (5.7) represent the capacitance values of the shunt capacitors,
CT\ and CT2, which are part of the simplified model used for the short transmission line section.
Also, note that jjr and er in equations (5.4)-(5.7) are design parameters that can be chosen freely
and are determined by the type of the dielectric substrate used (although generally jir = 1, since
most dielectric substrates are non-magnetic). As an example, the above formulas are used to
design a second-order Butterworth filter with center frequency of f0 = 10 GHz and a fractional
bandwidth of S = 20% (i.e. BW = 2 GHz) using a dielectric substrate with jir — 1 and er = 3.4.
The desired values for the capacitor, inductor, and substrate thickness are found using (5.1)-(5.7)
to be C\ — C 3 = 289.5 fF, L2 = 122.4 pH, and h\ = h2 = 0.59 mm, respectively. Using these
values, the frequency responses of the two filters shown in Fig. 5.2 and 5.3 are calculated and
presented in Fig. 5.4. As can be seen, a very good agreement between the two results is observed.
The minor discrepancies observed between the two is attributed to the approximate model used for
short lengths of transmission lines. This model is generally valid so long as the electrical length of
the line is less than 30° or h < A 0 /(3 v / e r ), where A0 is the free space wavelength.
The next step in the design procedure is to map the desired inductor, L2, and capacitors, C\ and
C 3 , values obtained from the above formulas to geometrical parameters of the periodic structures.
Due to the close proximity of the three metal layers, the presence of the inductive layer will affect the capacitance of the capacitive layer and vice versa. Therefore, the exact dimensions of the
capacitive patches and inductive wire grid should be optimized using numerical EM simulations.
Nevertheless, a first order approximation in a closed form formula can be used as a starting optimization point for the full-wave EM simulations. The effective capacitance value of a 2-D periodic
arrangement of square metallic patches with side lengths of D — s and the gap spacing of s can be
calculated using [144]:
2D
1
C = e0eeff— l n ( — — )
7T
sin^
(5.8)
77
CD
•o
CO
-I—«
c
o -20
"o
CD
O
O
ID
-40
DC
o3
CO
£Z
CO
-60
—
Simplified Cir. Trans. Coef.
Equivalent Circ. Trans. Coef.
— Simplified Cir. Ref. Coef.
Equivalent Cir. Ref. Coef.
-80
8
9
10
11
12
13
14
15
Frequency [GHz]
Figure 5.4 Calculated transmission and reflection coefficients of the equivalent circuit model of
the second order FSS, shown in Fig. 5.2, and the simplified circuit model, shown in Fig. 5.3 for
the example studied in Section 5.2.
Where e0 is the free space permittivity, e e // is the effective permittivity of the medium in which
the capacitive patches are located, D is the unit cell size, and s is the spacing between two adjacent
capacitive patches. This formula provides a first order approximation for the capacitance values
and can be used to determine the period of the structure, D, once the spacing between the patches,
s, and the capacitance values, C\ and C 3 , are determined. The gap spacing, s, is mainly determined
by the minimum feature that can be fabricated using the fabrication technology of choice. In lowcost lithography fabrication techniques, minimum features of 150 jiva can be easily fabricated on
high-frequency microwave laminates. As a general guideline, reducing the gap spacing will result
in reduction of the period of the structure, D, which is desirable. Once the D is determined, the
initial width of the inductive strips can be approximated using the following formula [144]:
T
D
W
X
27r--smf|'
(5.9)
78
Where D is the period of the structure, /io is the free space permeability, neff is the effective
permeability, and w is the strip width. Similar to the previous case, this formula is also valid only
when the inductive wire grid is placed in a homogeneous medium away from any metallic objects
or scatterers. Therefore, the value of the strip width, w, obtained from (5.9) must only be used as
the first-order approximation and a starting point for the full-wave optimization of the structure.
Starting with the initial dimensions of the structure obtained using the procedure mentioned
above, the frequency response of the structure can then be obtained using full-wave EM simulation. The simulations are carried out using the Finite Element Method (FEM) method in Ansoft's
High Frequency Structure Simulator (HFSS). Figure 5.5 shows the setup used for simulations. A
unit cell of the structure is placed inside a waveguide with Periodic Boundary Conditions (PBC)
walls. The transmission and reflection coefficients of this structure are then simulated and the frequency response of the infinitely large structure is obtained. The geometrical parameters of the
FSS unit cell are then manually tuned, by successive full-wave EM simulations and changing the
geometrical parameters of the structure accordingly, to match the full-wave frequency response to
the desired response predicted by the equivalent circuit model and equations (5.1)-(5.7).
5.3
Experimental Verification and Measurement Results
5.3.1
FSS Prototypes and Simulation Results
The procedure presented in Section 5.2 was followed to design an X-band FSS prototype operating with a with a center frequency of f0 — 10 GHz. This frequency band was chosen mainly for
measurement simplicity as many waveguide components and standard antennas are available that
operate in this band. The equivalent circuit parameters of the structure are determined using equations (5.1)-(5.7) assuming that the dielectric substrate has an er = 3.4 and a jir = 1 (RO4003 from
Rogers Corp.). The geometrical parameters of the FSS (D, s, and w) are obtained using the procedure mentioned in Section 5.2. The unit cell dimensions of the FSS are Dx = Dy = D = 5.8 mm,
which is slightly smaller than A 0 /5, where A0 is the free space wavelength at 10 GHz. The FSS
shown in Fig. 5.1 is a multi layer structure which is fabricated on two dielectric substrates that
are bonded together using a thin layer of bonding material. To accurately model the frequency
79
Figure 5.5 A unit cell of the proposed FSS is placed inside a waveguide with periodic boundary
conditions to simulate the frequency response of the infinitely large second-order FSS.
response of the structure, the effect of this thin bonding film must also be taken into account.
Therefore, the design is modified to take into account the presence of the 0.091 mm thick bonding
material, as shown in Fig. 5.6. The final design parameters of the structure and its unit cell are
listed in Table 5.2. The structure presented in this figure is not symmetric anymore because of
the existence of the bonding film which adds electrical length to one of the transmission lines and
not to the other. As indicated by (5.1)-(5.7), this added electrical length can be compensated by
reducing the capacitance of the capacitive layer on the side with a thicker effective substrate.
The sensitivity of the frequency response of the first FSS prototype to the angle of incidence
is examined using full wave simulation in HFSS. Figure 5.7(a) shows the calculated frequency
response of the first FSS prototype for the TE polarization and various angles of incidence. For
the TE polarization, as the angle of incidence increases, the bandwidth decreases and the bandpass
ripple level increases. However, the structure's frequency response does not considerably change
80
SI
J---
Top view Photo
Capacitive Layers
Figure 5.6 Topology and photograph of the fabricated FSS prototype utilizing regular patch
capacitors. A bonding layer (RO4450B Prepreg from Rogers Corporation) is used to bond the two
substrate layers together as shown.
for incidence angles in the range of 0° < 9 < 45°. Figure 5.7(b) shows the computed frequency
response of the same structure for the TM incidence and various angles of incidence. Similar to
the previous cases, it is predicted that the frequency response of the structure will not considerably
change for 0° < 9 < 45°. The variations observed in the bandwidth of the structure as the angle of
Parameter
D
w
•si
Value
5.8 mm
2.5 mm
0.15 mm
Parameter
•S2
h
t
Value
0.18 mm
0.5 mm
0.091 mm
Table 5.2 Physical parameters of the first fabricated FSS prototype shown in Fig. 5.6.
81
o
0
-10
-10
^
-20
^
^
'
i
-20
sV
i
m
2,-30
S"
^-40
^-40
o
H-50
t/i
o
^-50
9-0
i/l
9-1 5
C
2-60
h-70
-80 5
9 = 30°
-
-
-
6
i
9 = 60°
7
8
—4,
— - ' !
9=45°
9
10
11
r
12
13
14
15
i
1
Xx
30
c
2-60
H
-70
-80
1
I
X.
i
''
„
--
9-0
9-15
9 = 30°
—\A/
!!
9 = 45°
!
9 = 60°
9
10
* in
|
1
-1
!
11
Frequi sncy [GHz]
Frequency [GHz]
(a)
(b)
12
13
|
r I ',
14
15
Figure 5.7 Calculated transmission coefficients of the second-order FSS with parameters shown
in Table 5.2. (a) TE polarization (b) TM polarization. Results are obtained using full-wave EM
simulations in HFSS.
incidence varies, can be attributed to the change of wave impedance, which in turn will change the
loaded quality factor of the resonators of the coupled resonator FSS. For the TE incidence, as the
angle of incidence changes, the wave impedance changes as Z0/ cos(9) [145]. Therefore, for large
incidence angles, the loaded quality factor (QL) of the parallel resonators of Fig. 5.3(b) increases or
equivalently the bandwidth of each resonator decreases. This is evident from Fig. 5.7(a), where the
FSS bandwidth corresponding to each transmission pole is reduced and consequently the pass-band
ripple is increased. The wave impedance for the TM polarization, however, varies as Z 0 cos(#).
Therefore, for large incidence angles, QL decreases for the TM mode resulting in the broadening
of the FSS bandwidth as observed from 5.7(b).
To further reduce the sensitivity of the response of the FSS to the angle of incidence, the unit
cell size of the FSS can further be miniaturized. This is accomplished by reducing the unit cell
dimensions while maintaining the effective capacitance and inductance values. If the unit cell size,
D, is decreased, the capacitance value can be maintained by decreasing the spaces between the
patches, s, or increasing the effective side length of the capacitor. Decreasing the spacing is feasible in theory but is limited by fabrication techniques used. Therefore, achieving very small s
82
Top view Photo
Figure 5.8 Topology and photograph of the fabricated FSS prototype utilizing interdigital
capacitors. A bonding layer (RO4450B Prepreg from Rogers Corporation) is used to bond the two
substrate layers as shown in this figure.
values is rather difficult and requires special fabrication conditions. Thus, to increase the capacitance within the same occupied area, interdigital capacitors can be used. Figure 5.8 shows the
three-dimensional view and photographs of different layers of a second FSS prototype that utilizes
interdigital capacitive patches. The interdigital capacitive patches are designed to be symmetric
for horizontal and vertical polarizations. As shown in Fig. 5.8, six fingers are added in each unit
cell. Using the interdigital capacitor increases the effective capacitance value of the unit cell for
the same period and unit cell dimensions. This allows for reducing the unit cell dimension, D,
to achieve the same capacitance value and operating frequency at 10 GHz. Using this structure
with a gap spacing of s = 0.15 mm, the unit cell dimension is reduced to 4.5 mm (or equivalently
0.15Ao). The values of the inductor width, w, as well as the length of the fingers of the interdigital
capacitors, L\, L 2 , L 3 , and L 4 , are manually fine tuned using successive full-wave EM simulations
83
o
^-P
_ \
-10
yy
'
_,
\
1
:::k::
I
i
-:
1
I
1
I
^j-20
o
u
8-30
U
„
0-0
C-40
0-1 5
0 = 30°
-50
-60
0=45°
-
-
-
0 = 60°
10
rAA
1 ^
!
'
i
W \
^ '
-30
c
ro
12
13
14
v
\ \ _>\ ^ y
Y\ \-\ >\ '
.-^Z-^'-L.— V ^\ .A^ xL\ .
V \
1
:
\V\
~i"
r~
j ^ * * ^
i
^
i
„
0=0
.
-
15
t
^/y-'
s / ^
-40
!
-50
11
1
-10
^T"'\ " T s -
/ //' "
/ /<*'
> __
|
-
-
0-1 5
0 = 30°
i
9 = 45"
!
0 = 60°
i
i
1
1
i
i
10
•
i
l
,
'
l
X > 'N v
\
Xi
!
i
1
!
1
1
1
i
1
\
11
12
13
14
15
Frequency [GHz]
Frequency [GHz]
(a)
(b)
Figure 5.9 Calculated transmission coefficients of the second-order FSS utilizing interdigital
capacitors with parameters shown in Table 5.3. (a) TE polarization (b) TM polarization. Results
are obtained using full-wave EM simulations in Ansoft's HFSS.
and changing the parameter values accordingly. The physical and geometrical parameters of the
unit cell of the second FSS prototype are listed in Table 5.3. The frequency response of the second FSS prototype is also calculated using full-wave simulations in HFSS. Figure 5.9(a) shows
the frequency response of the structure for the TE polarization and various angles of incidence.
Figure 5.9(b) shows the same result for the TM polarization. As is observed from these figures,
the frequency response of the structure does not considerably change for incidence angles in the
range of 0° < 9 < 45° for the TE and TM polarizations. A comparison of Fig. 5.7(a) and 5.9(a)
also reveals that the frequency response of the second FSS prototype is more stable as a function
of incidence angle compared to that of the first one. This is expected since the latter prototype has
a smaller period than the former.
5.3.2
Measurement Results
The principles of operation of the proposed FSS are experimentally verified by fabricating prototypes of the two FSSs presented in Section 5.3.1 and measuring their frequency responses using
a free space measurement setup. The fabrication is performed by first patterning the capacitive
84
and inductive layers on two sides of a 0.5 mm RO4003 substrate (from Rogers corporation). The
other capacitive layer is patterned on one side of another substrate while etching away the copper
cladding on the other side. The two substrates are then thermally compressed together with a bonding film (RO4450B) with the thickness of 0.091 mm in between them. The physical dimensions
of the fabricated prototypes are 20.3 cm x 25.4 cm, which correspond to electrical dimensions
of 7A0 x 8A0. This ensures that the FSSs are excited with a uniform plane wave in free space
measurements. The diffractions from the edges of the FSS panel is another problem that affect the
measurement accuracy. To minimize this effect, the fixture shown in Fig. 5.10 is built. The fixture
is made out of copper with very large dimensions 120 cm x 90 cm with an opening of 20.3 cm
x 25.4 cm in the center to accommodate the FSS panel.
The FSS measurement system shown in Fig. 5.10 consists of two standard horn antennas
operating at the X-band with the fixture described above placed in between. The line of sight
between the two antennas passes through the center of the fixture and the antennas are located
about 90 cm apart from each side of the fixture to ensure the formation of a uniform plane wave
impinging upon the FSS surface. The measurement of the fabricated FSS is performed in two
simple steps. First, the transmission response of the system without the FSS is measured. This
measurement is used to calibrate the FSS response. Then, the frequency response of the structure
with the presence of the FSS is measured. By normalizing this measured response to that of the
calibration data, the frequency response of the FSS can be obtained. This is performed for both FSS
prototypes and the results are presented in Fig. 5.11. As observed from this figure, the frequency
Parameter
D
w
s
Value
4.5 mm
2 mm
0.15 mm
Parameter
h
t
Li
Value
0.5 mm
0.091 mm
1.195 mm
Parameter
L2
L3
LA
Value
0.675 mm
0.975 mm
0.6 mm
Table 5.3 Physical parameters of the second fabricated FSS prototype shown in Fig. 5.8.
85
Figure 5.10 Photograph of the free-space measurement system used to characterize the frequency
response of the fabricated FSSs.
responses obtained using this technique have ripples over the measured frequency range. These
ripples are caused by the multiple reflections between the two antennas. To remove these ripples,
the normalized response is transformed to the time domain. The FSS response is then range gated
in the time domain and only the main transmission is retained. Then, the gated data is transformed
back to the frequency domain and the ripples are removed to obtain the clear frequency response
shown in Fig. 5.11. The figure also includes the full-wave EM simulation results obtained in
Ansoft HFSS as well as the frequency response predicted by the equivalent circuit model shown in
Fig. 5.2. As can be seen from this figure, a good agreement between the measured and simulated
results is observed within the X-band. At the center frequency of operation, the fabricated FSSs
demonstrate a measured insertion loss of about 0.3 dB, which is mainly attributed to the Ohmic
and the dielectric losses of the structure.
The sensitivity of the response of the structure to the angle of incidence is also examined for
both fabricated FSS prototypes. Figure 5.12(a), shows the measured frequency response of the
first FSS prototype for the TE polarization and various angles of incidence. A relatively good
agreement between the measured and simulated response is observed. As expected, as the angle
of incidence of the TE mode increases, the ripple level increases and bandwidth decreases. The
same behavior is observed for the interdigital FSS, as shown in Fig. 5.12(b). However, as observed
86
9
10
11
Frequency [GHz]
(a)
9
10
11
Frequency [GHz]
(b)
Figure 5.11 Comparison between the measured and simulated results obtained from full-wave
EM simulations as well as the equivalent circuit model shown in Fig. 5.2 for the two FSS
prototypes studied in Section 5.3. The figures include the measured responses of the FSSs with
and without range gating. Range gating removes the ripples caused by multiple reflections
between the two antennas. All results are for normal incidence, (a) The FSS prototype utilizing
simple patches (b) The FSS prototype utilizing interdigital patches.
Frequency [GHz]
Frequency [GHz]
(a)
(b)
Figure 5.12 Measured frequency responses of the two fabricated FSS prototypes for the TE
polarization and various angles of incidence, (a) The FSS prototype utilizing simple patches (b)
The FSS prototype utilizing interdigital patches.
87
from this figure, the frequency response of the second FSS prototype (with smaller period) is less
sensitive to variations of incidence angle than that of the first one. This is expected, since the unit
cell of the periodic structure is reduced. The discrepancy between the measurement and simulation
results is attributed to working out of the operating bands of the standard X-band horn antennas
and the coax to waveguide transition, the numerical errors in simulations, tolerances and general
inaccuracies involved in fabrication process, inaccuracies in the exact values of the parameters of
the dielectric substrates used (±1.49% as specified by the manufacturer), inaccuracies involved in
the measurement process, as well as the finite size of the FSS. The frequency responses of the two
FSS prototypes for the TM mode and various angles of incidence are shown in Fig. 5.13. The
measured frequency responses of both FSSs are found to be more sensitive to the variations of the
angle of incidence for the TM polarization. Nevertheless, both structures demonstrate a relatively
consistent response for the TM polarization for incidence angles in the range of 0° < 6 < 30°.
7
8
9
10
11
Frequency [GHz]
(a)
12
13
7
8
9
10
11
Frequency [GHz]
12
13
(b)
Figure 5.13 Measured frequency responses of the two fabricated FSS prototypes for the TM
polarization and various angles of incidence, (a) The FSS prototype utilizing simple patches (b)
The FSS prototype utilizing interdigital patches.
88
5.4
Conclusions
A new technique for design and synthesis of frequency selective surfaces with second-order
bandpass responses was presented. The proposed technique is based on a coupled resonator filter
topology and its implementation using sub-wavelength periodic structures composed of capacitive
patches and inductive grids. The technique allows for designing second-order bandpass FSSs
with an extremely small overall profile in the order of 0.03A0. The presented structure can be
especially useful for designing second-order bandpass FSSs operating at lower frequencies (with
longer wavelengths), where traditional second-order FSSs will be too thick, bulky, and heavy to be
effectively used.
89
Chapter 6
Synthesis of Generalized Frequency Selective Surfaces of Arbitrary Order Using Non-resonant Periodic Structures
6.1
Introduction
In the previous chapter, we have shown that non-resonant periodic structures can be used to
design ultra-thin and low-profile bandpass FSSs with second-order bandpass responses. This is in
sharp contrast with traditional FSS design techniques which require cascading two first order FSS
panels a quarter wavelength apart to achieve a second-order filtering responses [57].
In this chapter, we present a comprehensive procedure for synthesizing bandpass FSSs with
non-resonant constituting elements and transfer functions of any arbitrary order. The constituting
elements of this family of FSSs are sub-wavelength arrangements of capacitive patches and wire
grids. The synthesis procedure is based on a generalized equivalent circuit model of the FSS, whose
values can be determined from system level performance indicators such as center frequency of
operation, f0, operational bandwidth, BW, desired out of band rejection determined by the order
of the response, N, and response type (e.g., Chebyshev, maximally flat group delay, etc.). These
equivalent circuit parameter values are then related to the physical, geometrical, and electrical
parameters of the FSS to complete the synthesis procedure. The main advantage of this type
of FSS is its high-power capabilities, in part due to the non-resonant nature of its constituting
elements. The proposed FSSs have extremely thin overall thickness. An Nth order bandpass FSS
of this generalized architecture has a typical thickness of (N — l)A 0 /30, where A0 is the freespace wavelength, whereas a traditional FSS with an Nth order bandpass response has a minimum
thickness of (N — l)A 0 /4. Moreover, due to the reduced overall FSS thickness and the small
90
Sub-A, Capacitive
Patches
Sub-A
Inductive x
Grid
Sub-wavelength Inductive Grid
PlN-l N)
afN-l N}
Sub-wavelength Capacitive Patches
(b)
Dielectric
Substrates \
-<
* — > •
s/2
Capacitive Patch
Inductive Grid
y
Li
(a)
(c)
Figure 6.1 (a) 3-D topology of a frequency selective surface composed of successive arrays of
sub-wavelength capacitive patches and wire grids with sub-wavelength periodicitties. (b) Side
view of the FSS. (c) Top views of a constituting unit cell of the structure.
periodicity of the constituting non-resonant unit cells of the structure, the frequency responses of
these structures are significantly more stable with respect to the angle of incidence of the EM wave
compared to those of traditional FSSs. Achieving similar stability factors from a traditional FSS is
only possible when it is enclosed by relatively thick superstrate dielectric layers on each side (e.g.
see Fig. 7.12 of [57]). This will considerably add to the thickness and weight of these structures
to a point that their use will become impractical below a certain frequency threshold (e.g. at lower
RF/micowave frequencies).
6.2
Generalized Synthesis Procedure
6.2.1
Topology and Equivalent Circuit Model
Figure 6.1(a) shows the three-dimensional (3-D) topology of the proposed FSS. The FSS consists of a multi-layered structure, where a number of metal layers are separated from one another by very thin dielectric substrates. Each metallic layer is in the form of a two-dimensional
91
(2-D) periodic arrangement of sub-wavelength capacitive patches or a 2-D wire grid with subwavelength periodicity. Figure 6.1(b) shows the side view of the structure. In this particular
FSS topology, the first and last layers of the assembly are always composed of sub-wavelength
capacitive patches while the layers in between are composed of sub-wavelength inductive grids
and capacitive patches repeated sequentially. Figure 6.1(c) depicts the top view of a unit cell of
the sub-wavelength capacitive patches and the inductive wire grid. The different capacitive patch
ML2}
a{1.2}
L
L
a/1.2}
'i
T S t
a{N-l,N)
fi(N-l,N}
PJN-l.N)
a/M-l.m
iHJ,2}
T~LI
~L ir~£ c
2 cN.!T Tfc/T^^-f
jcr
c,^1.2}
Ca,'l.21
0N
B/N-IM
La{N-l,N}
Pi 1.2}
Z r
r
ca{N-l,N}
(b)
Lajl,2)
J\f[c>
pjN-l,Nf,
._rrp
ZgrJ
—.C,
L,
T
C
2
C,
L,
C
N T
Z
f,
(c)
c(N-l,Nj
0(1.2}
ny\
ZSJ_JC;
First Resonator
X,
T
C
Second Resonator
2
C
(d)
,v-;T
C
NT
Z r
0N
Nlh Resonator
Figure 6.2 (a) Equivalent circuit model of the general FSS shown in Fig. 6.1(a) for normal angle
of incidence, (b) Transmission lines which represent the dielectric substrates in the equivalent
circuit model of Fig. 6.2(a) can be represented by their simple LC equivalent network. The
replacements are shown in dark gray, (c) The equivalent circuit model shown in Fig 6.2(b) can be
further simplified by ignoring certain parasitic capacitances, (d) Finally a T to TV transformation is
used to convert the equivalent circuit model of Fig. 6.2(c) to that of Fig. 6.2(d) .
92
layers (or wire grids) are not necessarily identical to one another. Moreover, in its current embodiment, the FSS is always composed of an odd number of metal layers and an even number
of dielectric substrates isolating them from each other. An FSS of this topology, which is composed of M metallic layers acts an ^ ^
order bandpass FSS, where M is always an odd number
(i.e., M = 3, 5, 7, ...). Alternatively, to achieve a bandpass response of order N, the FSS must
be composed of 2N — 1 metal layers and 2N — 2 dielectric substrates separating them. In such
an Nth order bandpass FSS, N of the metallic layers are capacitive and N — 1 are inductive. To
understand the principles of operation of this general bandpass FSS, we use a simple equivalent
circuit model for the FSS as shown in Fig. 6.2(a). Here, the sub-wavelength capacitive patches are
modeled by parallel capacitors (Ci, C 2 ,..., CV) in a transmission line circuit. The sub-wavelength
inductive wire grids are modeled using parallel inductors (Li)2,1/2,3, •••>
LN-I,N)
and the ultrathin
dielectric substrates separating the metallic layers from one another are modeled by transmission lines with lengths of /ia{i,2},fys{i,2},•••, ha{N-i,N},
^/3{JV-I,JV>
and characteristic impedances
of Za{i,2}5 -Z/?{i,2}) •••) ^a{iv-i,7V}, Zp{N-i,N}- The semi-infinite spaces on each side of the FSS are
modeled by semi-infinite transmission lines with characteristics impedances of ZQT\ and Znr^f,
where Z0 = 377S7 is the free space characteristics impedance and r\ and rN are normalized source
and load impedances. If the FSS is surrounded by free space r\ = rN = 1. Otherwise, appropriate
values of r\ and r^ must be chosen.
This equivalent circuit (Fig. 6.2(a)) can be converted to the one shown in Fig. 6.2(b) by converting the short transmission lines to their equivalent LC network using the Telegrapher's equations [125]. With the assumption that the electrical length of a transmission line is small (i.e.,
t < A/12 or j3£ < 30°), a short piece of a transmission line with a length Az can be modeled with a series inductor and a parallel capacitor with inductance and capacitance values of
LAz and CAz, where L (H/m) and C (F/m) are inductance and capacitance per unit length of
the line, respectively. This way, L a(/3){iji+1} = /Uo/A-aG9){i,i+i}^a(^){i,t+i}, where /ira(/3){;,i+i} and
ha(/3){i,i+i} are the relative permeability and the thickness of the first (second) substrates (from
the left) which separates the ith and (i + l)th capacitive layers from one another. Similarly,
Ca(p){i,i+i} = eoera(/3){l,i+i}^a(/3){i,i+i}/2, where era(/3){M+i} is the relative permittivity of the first
93
(second) magneto-dielectric substrate separating the ith capacitive layer from the (i + l)th capacitive layer. In this equivalent circuit, the parallel capacitors at the capacitive nodes can be combined
and lumped into a single capacitor. The portions of the capacitors Ca{it2}, Cp^^}, •••, that are parallel with the inductive nodes are simply ignored, since the positive susceptance offered by these
capacitors are considerably smaller than the negative susceptance of the inductors at the inductive
nodes. With this simplifying assumption, the equivalent circuit model of Fig. 6.2(b) is simplified
to the one shown in Fig. 6.2(c). Finally, the T inductive networks separating the capacitors from
one another can be converted to their equivalent TX network as shown in Fig. 6.2(d). The equivalent circuit of the FSS shown in Fig. 6.2(d) represents a bandpass coupled resonator network of
order N. Using this circuit, bandpass FSSs of the type shown in Fig. 6.1 can be synthesized to
demonstrate a variety of different desired transfer functions (e.g., a 4th order maximally flat group
delay response). The response type of the FSS is determined by the normalized quality factor of
the first and Nth resonators (qi and qN), the normalized load and source impedances (r1 and r^v),
and the normalized coupling coefficients between the resonators (fci,2, ^2,3, ••-, kN-ijN).
Therefore,
by specifying the frequency of operation, / 0 , FSS's operational bandwidth, BW, and the response
type and order, all of the element values of the FSS circuit shown in Fig. 6.2(d) can be determined.
This allows us to obtain the equivalent circuit values of the circuit shown in Fig. 6.2(a) and map
these values to the geometrical parameters of the proposed FSS shown in Fig. 6.1.
6.2.2
General Synthesis Procedure
The first step in the proposed synthesis procedure is to obtain the parameters of the equivalent circuit models presented in Fig. 6.2. In the proposed synthesis procedure, we will first
determine all of the capacitor values and use them to obtain the values of the coupling inductors
Lc{i,2}, Lc{2,3}i •••• F° r the generalized FSS equivalent circuit shown in Fig. 6.2(d), the capacitor
values of all the resonators in the FSS can be determined using a step by step procedure. First, we
calculate the values of the first and the last capacitors of the resonators. From the definition of the
normalized quality factor in a couple resonator filter architecture [126], q^.
Qi = uJoC'iZi = qt-
(6.1)
94
where Qi is the loaded quality factor of the first and last resonators, r^ is the normalized impedance
Ti =
ZI/ZQ,
Z0 = 377 £1 is the free space impedance, 5 = BW/f0 is the fractional bandwidth
of the FSS, and o;0 = 2nf0. For i = 1 and N, the capacitors of the first and last resonators are
obtained using (6.1):
C[ = - ^ - =
C'N = - ^ —
(6.2)
In the Nth order coupled resonator filter shown in Fig. 6.2(d), the resonators are inductively
coupled together. In such an inductively coupled filter, the values of the interior coupling inductors
are determined from [126]:
Lc{l,i+i}=
V
>
i = 3,4,...,iV-2.
(6.3)
OKi,i+lI
where L\ is the total shunt inductance of the ith node when all other nodes are shorted to ground
and ki:i+1 is the coupling coefficient between the ith and (i + l)th resonators [126]. In this case,
the total shunt inductance is the parallel equivalent inductor of Li, Lc{i}i+1j, and
J^i
^c{i-l,i}
i-'i
Lc^i^y.
lj
c{i,i+l}
Additionally, in the coupled resonator filter shown in Fig. 6.2(d), the net shunt inductance at each
node must resonate with the net shunt capacitance of the node. Since there is no series capacitor
between the nodes of this filter, the net shunt capacitance of each node is simply the parallel
capacitors shown in Fig. 6.2(d). Therefore, L- = -^y- To make sure that values of Li are positive
and realizable, we must have:
i-'i
^c{i-l,i}
-^c{i,i+l}
or equivalently:
u?C[ > uHki^QC'^
+ uHki-^yJCUC'i
(6.6)
By rearranging (6.6) and solving for C'i+1, we arrive at the condition specified by:
2 = 2,4, 5..., T V - 3 .
(6.7)
95
or equivalently:
n> ^ t V^-i
°*
<
k
~
^
i~2,i-l^\/Cl_2,2
7
X
>
i = 3,4,5...,N-2.
(6.8)
In this equation, /qj represents normalized coupling coefficients between the ith and j t h resonators. This equation is used to determine rest of the capacitors values of C 3 up to CN-2Similarly, the capacitor value of C2 can be determined by enforcing the value of the shunt
inductance L\ to be positive. At this first node, the total shunt inductance of the node, L\, is
obtained from:
1
1
JT = T
1
+7
,m
(-)
6 9
hence, L\ can be realized if:
-^1
L
c{\,2}
or equivalently:
> oj25kh2y/C[C2
UJ2C[
(6.11)
By rearranging (6.11) and solving for C2, the value of the second capacitor is determined:
c
(6 12)
* < TATSI
-
where ki>2 is the normalized coupling coefficient between the first and second resonator of the
equivalent circuit shown in Fig. 6.2(d). Values of the normalized coupling coefficients are provided
in most filter design handbooks (e.g. see [126]).
In a similar manner, by enforcing the values of the shunt inductance, LN, to be positive, the
upper bound for the capacitor value of C^-i can be obtained. The lower bound is obtained by
using (6.6) when i = N — 1 and solving for C ^ ^ :
8 (kN-2,N-i\/C'N_2
+ kN^ltN\/C'N)
<CN_X
< <*iC'N v
{dkN-iNY
(6 13)
-
96
The conditions specified by (6.8), (6.12)-(6.13) indicate that the choices for the capacitors in the
middle layers of the FSS are not unique. This is indeed the case and stems from the fact that the
equivalent circuit of the FSS shown in Fig. 6.2 is an inherently under-determined one. Consequently, for a given set of desired specifications (e.g., / 0 , BW, etc.), more than one solution exists.
This offers the designer the flexibility of choosing values that are more amenable to practical implementations.
After determining the values of all the capacitors in the equivalent circuit model of Fig. 6.2(d),
the values of the shunt inductors can be determined. The inductor value of the first resonator is
obtained by using (6.9), substituting the inductor values of Lc{i,2} using (6.3,) and the inductor
value of L\ by
Jc,:
1
Lx
1
1
L'i
£ C {1,2}
=
CJ2C[
- u2Sk1}2VC[Cl
(6.14)
or equivalently:
Li =
;
kh2Sy/c[a2)
2
u (C[ -
(6.15)
Similarly, the inductor value of the last resonator is obtained by solving for LN using the same
procedure:
1
LN = —^,
;
f r^r-^r,
(C'
—
kj^-i^S
y'C'N_1C'N)I,
CO
N
(6-16)
Finally, the values of the inductors in between are obtained by solving for Li in (6.4):
1
L%
1
Li
1
Lc{i_ij}
1
Lc{iji+1}
(6.17)
and substituting the inductor values of I/c{i-i,«} a n d A:{;,i+i} using (6.3) and the inductor value of
^ by Jc--
l
2
u-=u: {C' -k . 8^CUC'
l
-fci>i+15Jqq+1)
lu
t
i = 2,3,..., N - 1.
(6.18)
97
Finally, the coupling inductors of the equivalent circuit model shown in Fig. 6.2(d) can be calculated by using (6.3) and substituting the inductor value of L^ by -^y'Atf.i+1} - - 3 — - y - = = = , i = 1,2,...,7V- 1
(6.19)
Now that all of the parameter values of the equivalent circuit model shown in Fig. 6.2(d) are
obtained, we can use simple conversion formulas to obtain the parameters of the equivalent circuit
models of the FSS shown in Figs. 6.2(a)-(c). We start this by calculating the effective inductance
values of the inductive layers of the FSS using:
- L +lLlL2+2L
L
l + Lc{l,2} + ALii
2LjyLjv_i
^
T
*-LiN-l
+ ^c{N-l,N}
(6 20
' >
+ L>N
^LjLi+i
2Lj + Lc{ij+i} + 2Lj + i
T
z = 2, 3, . . . , 7 V - 2 .
(6.22)
Using these values, the values of the series inductors representing the substrates of the FSS can be
calculated from (6.23)-(6.28):
L
«{i,2} -
T
2
^{1,2} =
I>lL c{1|2 }
—77fT
(6.23)
^2Lc{1^
(6.24)
2lvjV—l-^c{7V—1 N}
= W~r
—7
T
- TTr
LNLc{N_1}N}
—7
—7—
(6.26)
i = 2,3,...,N-2.
(6.27)
L/3{N-I,N}
L 0{i,i+i} —
, j
(6-25)
La{N-l,N}
2Z/i_i_i
L,
i+l-^c{i,i+l}
2Lj + Lcj^j+i} + 2Li+i
z = 2,3, . . . , 7 V - 2 .
(6.28)
98
Using these equations, the values of the equivalent circuit parameters of the circuit shown in Fig.
6.2(c) are completely determined. Using these values, the parameters of the equivalent circuit
model of Fig. 6.2(a) can be determined. For i = 1,2, ...,N — l,we have:
,
«a(/3){i,i+l} =
Lg(p){i,i+1}
(°./y)
V
Za{j3){i,i+i} =
=
(6.30)
Finally, the values of the capacitors of the capacitive layers can be calculated using:
d = C[ -
e € 1
° - >^1-2>
<^N = <^N
(~>. _ fit _
z
£
(6.31)
{0.32)
Q £ r-(^/3{i-l,i} + ^a{i,i+l})
2
i = 2,3,...,N-l.
(6.33)
Equations (6.2)-(6.33) can be used to calculate the values of the equivalent circuit model of the
FSS shown in Fig. 6.2(a). In the next sub-section, we will demonstrate how this procedure can be
used in practice to design FSSs of the type shown in Fig. 6.1(a).
6.3
Validation of the Proposed Synthesis Procedure
6.3.1
Synthesis of A Third-Order Bandpass FSS: Design and Simulation
The procedure described in Section 6.2 is followed to design a third-order bandpass FSS with
a 0.1 dB ripple Chebyshev bandpass response, center frequency of operation of 10.82 GHz, and
a fractional bandwidth of 5 = 21.7%. Using the parameters of a third-order bandpass coupledresonator response provided in Table 6.1, the values of the capacitors C[ and C'N of the equivalent
circuit shown in Fig. 6.2(d) can be calculated from (6.2). Then, the values of the capacitors in the
middle are chosen using (6.12)-(6.13). As indicated in (6.12)-(6.13), the choice of these capacitors
are arbitrary as long as they meet the condition specified in (6.12)-(6.13). Once these capacitors
values are determined, the values of the inductors L\, L2, and L 3 shown in the equivalent circuit
model of Fig. 6.2(d) are calculated from (6.15)-(6.18). Finally, the values of the two coupling
99
inductors, £c{i,2} and £c{2,3} in Fig. 6.2(d), are calculated from (6.19). Using this procedure, all of
the values of the equivalent circuit model shown in Fig. 6.2(d) are determined. The values of the
equivalent circuit of Fig. 6.2(c) are obtained from these values using the conversion formulas given
in (6.20)-(6.26). The next step of the design procedure is to use the element values of the circuit
shown in Fig. 6.2(c) in conjunction with (6.29)-(6.33) to obtain the values of the equivalent circuit
elements of the circuit shown in Fig. 6.2(a). This, however, can only be done if the constituting
parameters of the magnetodielectric substrates used in the construction of the FSS are known.
In this design procedure, the choice of these parameters are arbitrary and it is assumed that these
values are known a priori. In the case of the third-order bandpass FSS studied here, we have chosen
the Cer-10 material from Taconic Corporation. This non-magnetic dielectric material has a typical
relative permittivity of er = 10 and a loss tangent of tan 8 = 0.0035 at 10 GHz [149]. Using these
values and the parameters of the equivalent circuit model of Fig. 6.2(c), the element values of the
equivalent circuit model of the FSS shown in Fig. 6.2(a) are obtained from (6.29)-(6.33) and the
results are presented in Table 6.2.
The next step of the design procedure is to map the values of the equivalent circuit model of the
FSS to the physical and geometrical parameters of the FSS itself. As a starting point, the values
of C\, C2, and C 3 can be related to the geometrical parameters of the sub-wavelength capacitive
patch layers of the FSS shown in Fig. 6.1 using [127]:
2D
1
Q = e0ee//—ln(-—^)
TV
bin
(6.34)
2D
Here, Q is the effective capacitance of the sub-wavelength capacitive layer, D is the period of
the periodic structure, Sj is the gap between two adjacent capacitive patches on the ith layer, and
eeff is the effective dielectric constant of the medium in which the capacitive layers are embedded.
From this equation, si, s 2 , and s3 can be determined knowing the values of C1-C3 and D as well
as the constitutive parameters of the dielectric substrates used. In this design procedure, D can
be chosen arbitrarily. However, choosing very small values of D results in achieving very small
values of s, which may not be practical to fabricate. The values of L\2
an
d £2,3 can be related to
100
Filter Type
<?i
<?3
ku
&23
Butterworth
1.000
1.000
0.7071
0.7071
Maximally Flat Delay
0.3374
2.2034
1.7475
0.6838
Linear Phase
0.4328
2.2542
1.4886
0.6523
Gaussian
.2624
2.2262
2.1603
0.7416
Chebyshev(0.01 dB)
1.1811
1.1811
0.6818
0.6818
Chebyshev(O.lOdB)
1.4328
1.4328
0.6618
0.6618
Table 6.1 Normalized parameters of third-order bandpass coupled resonator filters. For
third-order coupled resonator filters, ri = rN = 1
the geometrical parameters of the two wire grid structures used through the use of (6.35) [127]:
T
D
,
,
1
(6.35)
2D
Assuming that the period of the structure is fixed in each layer, (6.35) can be used to obtain the
values of wi$ and u>2,3, which are the widths of the two inductive wire grids used in the design
of this third-order bandpass FSS. It is important to note that (6.34) and (6.35) are obtained for
isolated periodic structures. In the present case, however, the capacitive and inductive layers are
placed in close proximity to each other. Therefore, we will use the values of s\, s 2 , s 3 , u>i;2, and
u>2,3 obtained from this procedure as a starting point in a parameter tuning process carried out using
full-wave EM simulations. In this case, we have used full-wave EM simulations in Ansoft High
Frequency Structure Simulator (HFSS) to fine tune these values and achieve the desired response.
Details of this fine tuning procedure are reported elsewhere in Chapters 2-5. The final dimensions
of the FSS are presented in Table 6.2.
Fig. 6.3 shows the reflection and transmission coefficients of this third-order bandpass FSS
predicted using the equivalent circuit model shown in Fig. 6.2(a) as well as those calculated using the full-wave EM simulations conducted in Ansoft HFSS. As can be seen a good agreement
between the two results is observed.
101
Parameter
Ci
c2
c3
Ll,2
Value
232.9 fF
471.4 fF
232.9 fF
177.0 pH
Parameter
-^2,3
a{l,2}
^3(1,2}
^Q{2,3}
Value
177.0 pH
0.556 mm
0.549 mm
0.549 mm
Parameter
^3(2,3}
er
si
S2
Value
0.556 mm
10.0
0.41 mm
0.374 mm
Parameter
S3
Wl>2
^2,3
D
Value
0.41 mm
1.265 mm
1.265 mm
3.5 mm
h
Table 6.2 Physical and electrical parameters of the third-order bandpass FSS of Section 6.3.1.
40
9
10
11
»
12
Frequency [GHz]
Figure 6.3 Reflection and transmission coefficients of the 3 r d order bandpass FSSs discussed in
Section 6.3.1. Results directly predicted by the equivalent circuit model shown in Fig. 6.2(a) and
those calculated using full-wave EM simulations in Ansoft HFSS are presented here.
c.
L 12
C2
L.23
c
V
Metal Layers
ppg
a{l,2}
P{1,2}
a {2,3}
h P{2,3}
Prepreg Bonding yLayers ^ ~~ (~
Figure 6.4 Side view of the FSS discussed in Section 6.3.2 showing the different metal,
dielectric, and prepreg layers and their relative positions with respect to each other.
102
7
8
9
10
11
12
13
Frequency [GHz]
Figure 6.5 Reflection and transmission coefficients of the bandpass FSS discussed in
Section 6.3.2. Simulation results in HFSS and theoretically predicted ones are compared with the
measurement results.
6.3.2
Synthesis of A Third-Order Bandpass FSS: Fabrication and Measurement
To experimentally verify the design procedure described in Section 6.3.1, a prototype of this
third-order bandpass FSS is also fabricated and tested. However, before the structure could be
fabricated, several changes must be made to the values shown in Table 6.2. Since the exact substrate
thicknesses calculated using (6.29) may not necessarily be available commercially, the closest
commercially available substrate thicknesses are used instead. Additionally, the proposed structure
is composed of multiple dielectric substrates that need to be bonded together. This bonding is
achieved by using TPG-30 prepreg material from Taconic Corporation. This bonding film has
a thickness of 4.5 mils and a relative dielectric constant of 3.0 with a loss tangent of tan5 =
0.0038. The presence of this bonding film will also further perturb the response of the FSS from
the one shown in Fig. 6.3. Figure 6.4 shows a cross sectional view of this structure including
the different capacitive and inductive layers and their relative positions with respect to each other,
the dielectric substrates, and the bonding films. These two perturbations will naturally affect the
response of the FSS. To account for these variations, the equivalent circuit model of the FSS,
shown in Fig. 6.2(a) is modified to take into account the presence of the bonding film. Then
using a circuit simulation software, Agilent Advanced Design Systems (ADS), the values of the
equivalent circuit model of the structure are slightly tuned to tune the response of the structure back
103
Parameter
Cx
c2
c3
-^1,2
Value
169.0 fF
367.0 fF
163.0 fF
192.0 pH
Parameter
£2,3
a{l,2}
fys{l,2}
^a{2,3}
Value
226.0 pH
0.64 mm
0.64 mm
0.64 mm
Parameter
fys{2,3}
e r , €rp
s\
•S2
Value
0.64 mm
10.0,3.0
0.48 mm
0.26 mm
Parameter
•S3
™1,2
^2,3
D
Value
0.51 mm
1.35 mm
1.35 mm
3.5 mm
h
Table 6.3 Physical and electrical parameters of the third-order bandpass FSS of Section 6.3.2.
to the desired response. Using a procedure similar to the one described in Section 6.3.1, these new
equivalent circuit values are then related to the physical and geometrical parameters of the thirdorder bandpass FSS. Table 6.3 shows the new values of the equivalent circuit model parameters of
the FSS shown in Fig. 6.4 as well as its geometrical and physical parameter values.
This third-order bandpass FSS is then fabricated and its frequency response is measured using
a free-space measurement setup. The free-space measurement setup uses two standard gain horn
antennas operating at X-band to characterize the response of the system. The measurement bandwidth is constrained by the operational bandwidth of the antennas and the waveguide components
used in the free-space measurement setup (7.0 GHz to 13.0 GHz). The details of the measurement
procedure as well as the measurement setup are provided in Chapter 5 and will not be repeated
here. Figure 6.5 shows the measured transmission and reflection coefficients of the FSS for normal
angle of incidence as well as the the results obtained using full-wave simulations in HFSS and
those obtained from the equivalent circuit model with the values given in Table 6.3. As can be
- observed, a very good agreement between the three transmission coefficient results is observed.
The transmission and reflection coefficients of the FSS are also measured for the TE and TM polarizations for various incidence angles in the 0° to 45° range and the results are presented in Figs.
6.6 and 6.7. Figure 6.6(a) and (b) show the simulated and measured transmission and reflection
coefficients of the third-order bandpass FSS for the TE polarization respectively. As can be seen,
104
9
10
11
Frequency [GHz]
9
10
11
Frequency [GHz]
Figure 6.6 Simulated and measured transmission and reflection coefficients of the third-order
bandpass FSS discussed in Section 6.3.2 for oblique incidence angles and Transverse Electric
(TE) polarization, (a) Results obtained from full-wave EM simulations in Ansoft HFSS (b)
Measurement Results.
a good agreement between the simulated and measured results is observed. Figure 6.7(a) and (b)
show the simulated and measured transmission and reflection coefficients of the same FSS for
oblique incidence angles for the TM polarization. Similar to the previous case, a good agreement
is observed between the measurement and simulated results. Furthermore, as seen from Figs. 6.6
and 6.7, the structure's frequency response does not significantly change as a function of angle
of incidence and polarization of the incident EM wave. This is due to the extremely small overall
profile of the structure (A0/10 at 11.0 GHz) as well as its small unit cell dimensions (0.12A0 at 11.0
GHz). The changes in the reflection coefficients curves shown in Figs. 6.6 and 6.7 as a function of
angle of incidence are naturally more pronounced that those of the transmission coefficients, since
within the FSS pass-band, the reflection coefficient values are small.
105
7
8
9
10
11
12
13
11
12
1
Frequency [GHz]
7
8
9
10
Frequency [GHz]
Figure 6.7 Simulated and measured transmission and reflection coefficients of the third-order
bandpass FSS discussed in Section 6.3.2 for oblique incidence angles and Transverse Magnetic
(TM) polarization, (a) Results obtained from full-wave EM simulations in Ansoft HFSS (b)
Measurement.
6.3.3
Synthesis of a Fourth-Order Bandpass FSS
In Sections 6.3.1 and 6.3.2, the proposed design procedure, was theoretically and experimentally verified for a third-order bandpass FSS. The design procedure presented in this chapter allows
for the synthesis of bandpass FSSs with a variety of different transfer functions (e.g., Chebyshev,
maximally flat, etc.) and response orders (N = 2, 3, 4,...). This is further demonstrated in this subsection by another design example focusing on the development of a fourth-order bandpass FSS
with a maximally flat (Butterworth) response, operating at 10.0 GHz, with a fractional bandwidth
of 5 = 20%. Using the appropriate ktj, q\, and qN values of a fourth-order bandpass coupled
resonator filter provided in Table 6.4 and following a design procedure similar to what is described
in Section 6.3.1, the values of the equivalent circuit model parameters of the FSS (shown in Fig.
6.2(a) for 7V=4) are obtained. These values are then mapped to the physical and geometrical parameters of the FSS with the assumption that a dielectric substrate with'a dielectric constant of
106
Filter Type
<?i
94
&12
&23
&34
Butterworth
0.7654
0.7654
0.8409
0.5412
0.8409
1
Maximally Flat Delay
0.2334
2.2404
2.5239
1.1725
0.6424
1
Linear Phase Equiripple Error (E=0.05)
0.3363
2.2459
1.9325
1.0483
0.6424
1
Gaussian
0.1772
2.2450
3.2627
1.4225
0.6913
1
Chebyshev (0.01 dB ripple)
1.0457
1.0457
0.7369
0.5413
0.7369
1.1007
Chebyshev (0.10 dB ripple)
1.3451
1.3451
0.6850
0.5421
0.6850
1.3554
I'l
r±
Table 6.4 Normalized parameters of fourth-order coupled-resonator niters with various response
types.
er = 3.38 is used. Table 6.5 shows the equivalent circuit parameter values of the FSS along with
its physical and geometrical parameters.
The frequency response of this FSS is also calculated using full-wave EM simulations in Ansoft
HFSS and presented in Fig. 6.8 along with those predicted by the theoretical synthesis procedure.
As can be seen, a good agreement between the two results is observed within the pass-band of
the structure. The major discrepancy observed between the two results happens outside of the
pass-band of the FSS. In particular, the full-wave EM simulation results of the FSS predict the
Parameter
C\,C±
C2,C3
£1,2, £34
£2,3
Value
145.5 fF
481.0 fF
322.1 pH
224.1 pH
Parameter
^a{l,2}
^8(1,2}
^a{2,3}
fys{2,3}
Value
1.067 mm
0.61 mm
0.656 mm
0.656 mm
Parameter
^a{3,4}
fy8{3,4}
er
S\, s 4
Value
0.61 mm
1.067 mm
3.38
0.725 mm
™12
™23
^34
2.05 mm
2.35 mm
2.05 mm
Parameter
Value
S2,S3
0.21 mm
Table 6.5 Equivalent circuit values and physical and geometrical parameters of the fourth-order
bandpass FSS discussed in Section 6.3.3.
107
Frequency [GHz]
Figure 6.8 Reflection and transmission coefficients of a fourth-order bandpass FSS with a
maximally flat response type studied in Section 6.3.3. Theoretically predicted results are
compared with those obtained from full-wave EM simulations in Ansoft HFSS.
presence of an out of band transmission null that occurs at 13.5 GHz, which is not predicted by
the equivalent circuit results. This null is caused by the coupling between the subsequent inductive
and capacitive layers, which are not taken into account in the simplified equivalent circuit models
of Fig. 6.2. These inter layer couplings can be taken into account by modifying the equivalent
circuit of the FSS and allowing for existence of a capacitive coupling between the capacitive and
inductive layers. By doing this alteration, the agreement between the equivalent circuit model
and the full-wave EM simulation can be substantially improved. This, however, complicates the
equivalent circuit model of the FSS to a point, which makes it impossible to derive closed form
synthesis formulas.
6.4
Conclusions
A comprehensive method of synthesizing frequency selective surfaces composed of non-resonant,
sub-wavelength periodic structures was presented. The use of this procedure for synthesizing a
third-order bandpass FSS was demonstrated both theoretically and experimentally. Through a
comparison between measurement, simulation, and theoretically predicted results, the validity of
the proposed design procedure was established. A simple synthesis example demonstrating the
utility of this procedure in synthesizing a fourth-order bandpass FSS was also demonstrated using
full-wave EM simulations. The proposed synthesis technique allows for designing low-profile and
108
thin frequency selective surfaces with frequency responses that are not very sensitive to the angle
of incidence and the polarization of incidence of EM waves. In addition, it allows for designing
the FSS elements to operate as spatial phase shifters. These phase shifters will be used in the next
chapter to design high-power low-cost microwave lenses.
109
Chapter 7
Design of Planar Microwave Lenses Using Sub-Wavelength Spatial Phase Shifters
7.1
Introduction
Lenses are used extensively throughout the microwave and millimeter-wave frequency bands
in applications such as imaging [150], radar systems [151], quasi-optical power combining [152][153], quasi-optical measurement systems [154], and high-gain beam-steerable antenna arrays
[155]-[156]. Dielectric lenses were among the first structures to be investigated and are still used
in various antenna and radar applications [157]-[159]. Dielectric lenses, similar to the one shown
in Fig. 7.1(a), tend to operate over relatively wide bandwidths. However, they suffer from reflection losses caused by mismatch between the refractive index of the lens material and that of its
surrounding environment. Moreover, dielectric lenses that operate at low microwave frequencies
are generally bulky, heavy, and expensive to manufacture. This has limited their use primarily to
millimeter-wave and sub-millimeter-wave frequency bands.
In addition to dielectric lenses, a wide range of other techniques have also been used to design
lenses that operate throughout the RF and microwave frequencies [160]-[172]. In particular, planar microwave lenses have received considerable attention and have been investigated by several
groups of researchers over the years [152], [155], [162]-[166]. Typically, these types of microwave
lenses are designed using planar arrays of transmit and receive antennas connected together using
a phase shifting mechanism as depicted in Fig. 7.1(b). Different techniques have been used to
achieve the required transmission phase between the arrays of receiving and transmitting antennas. Examples include using transmission lines with variable lengths between the two arrays of
110
antennas [164] or using an array of coupling apertures between them [162]-[166]. In [165], two
microstrip patch antennas coupled together using an aperture embedded in their common ground
plane form the basic building block of a flat lens. A similar structure is studied in [166], where
a resonant aperture (filter) is used to couple the arrays of receiving and transmitting microstrip
patch antennas together. However, the relatively large inter-element spacing between the antenna
elements used in such planar arrays is usually a factor that limits their performance under oblique
angles of incidence.
Recently, artificially-engineered materials have also been used to design microwave lenses
[167]-[169]. In [167], a lens consisting of stacked parallel metal plates or arrays of parallel metal
wires is reported. This structure behaves as an artificial dielectric with a permittivity less than
one. Other artificially-engineered materials, such as complementary split ring resonators [168] or
negative-refractive-index transmission lines [169] have also been used to design three-dimensional
microwave lenses operating at microwave frequencies. In another approach, Fresnel zone plates
(FZP) are employed as focusing elements to design a variety of microwave lenses. The classical microwave FZP lens consists of circular concentric metal rings that lay over the odd or even
Fresnel zones [170]-[171].
Frequency selective surfaces (FSSs) have also been used in the design of planar lenses. In [172],
a combination of a Fresnel zone plate lens and an FSS is employed to achieve a new compound
diffractive FZP-FSS lens with enhanced focusing and frequency filtering characteristics. In FSSbased lenses, the phase shift required for beam collimation is achieved from the phase response of
the FSS's transfer function. The phase shift that can be achieved from a bandpass FSS is directly
related to the type and order of its frequency response. Since many planar lenses require phase
shifts in the range of 0° to 360°, FSSs with higher-order responses are generally required. To
achieve a higher-order filter response from a conventional FSS, multiple first-order FSSs must be
cascaded with a spacing of a quarter-wavelength between each panel [173]. This increases the
overall thickness of the FSS and enhances the sensitivity of its frequency response to the angle and
polarization of incidence of the EM wave.
Ill
In this chapter, the generalized MEFSS discussed in Chapter 6 will be used to design lowprofile, planar microwave lenses. In doing this, we treat the unit cells of a typical MEFSS as spatial
phase shifters (SPSs). By populating a planar surface with SPSs that provide different phase shifts,
planar lenses with a desired phase shift profile can be designed as depicted in Fig. 7.1(c). Due
to the sub-wavelength dimensions of their SPSs and their extremely thin profiles, the response of
these planar lenses remain stable over a relatively wide angular range. Moreover, these planar
lenses can operate over relatively wide bandwidths. In what follows, the principles of operation
and the design procedure for the proposed planar lenses is first presented in Section 7.2. In Section
7.3, we discuss the measurement techniques used to characterize the proposed planar lenses and
present the measurement results of two fabricated prototypes. Finally, in Section 7.4, we present a
few concluding remarks.
Double-Convex
Dielectric Lens
1
/
(a)
Conventional
Planar Lens
Rx-^M^Tx
Antenna u Antenna
(b)
Proposed MEFSSBased Planar Lens
(c)
Sub-wavelenth
Spatial
Phase Shifter
Figure 7.1 (a) Topology of a conventional double-convex dielectric lens, (b) A planar microwave
lens composed of arrays of transmitting and receiving antennas connected to each other using
transmission lines with variable lengths, (c) Topology of the proposed MEFSS-based planar lens.
112
7.2
Lens Design and Principles of Operation
Figure 7.1 shows a comparison between a double-convex dielectric lens, a traditional planar
lens composed of arrays of receiving antennas connected to arrays of transmitting antennas using
transmission lines with variable lengths, and the proposed MEFSS-based planar lens. Irrespective
of the implementation technique, the planar microwave lenses shown in Figs. 7.1(b) and 7.1(c)
attempt to mimic the transmission phase profile of the double-convex dielectric lens shown in
Fig. 7.1(a). When illuminated by a point source located at its focal point, the double convex lens
converts an incident spherical wavefront at its input surface to a planar wavefront at its output.
For a radiating point source located at the lens' focal point, the radiated spherical wave arrives at
different points on the lens' aperture with different phase delays. In this case, the ray that passes
through the center of the lens experiences the smallest phase delay in free space. Using this phase
delay as a reference, the excess free-space phase delay that a ray arriving at an arbitrary point
on the aperture acquires can be calculated. Figure 7.2 shows this calculated excess phase delay
profile for a lens with an aperture size of A = 23.4 cm having a focal length of / = 30 cm. From
this curve, the phase delay profile that the lens must provide to achieve beam collimation can be
-12 -10
-8
-6
-4
-2
0
2
4
6
8
10
12
Distance from center of the aperture [cm]
Figure 7.2 Phase delay of a spherical wave at the input of the lens aperture (referenced to the
phase at the center of the aperture) and the phase delay, which must be provided by the lens to
achieve a planar wavefront at the output aperture of the lens. The results are calculated for a lens
with aperture size of 23.4 cm x 18.6 cm and focal distance of 30 cm.
113
Sub-wavelength
Capacitive Patches
Dielectric
Substrate
Sub-wavelength
Inductive Grid
(c)
Figure 7.3 (a) 3D topology of an 7V*h-order MEFSS reported in Chapter 6. (b) 2D top view of the
unit cell of a sub-wavelength capacitive patch layer (top) and an inductive wire grid layer
(bottom). The two layers are used a building block for the Nth-order MEFSS. (c) Equivalent
transmission line circuit model of the Nth-order MEFSS.
calculated. This phase delay changes from zero at the edge of the aperture to k0(y/(A/2)2
+ f2 —
f) at its center, where k0 = 27r/A0 is the free space wavenumber, A0 is the free space wavelength,
/ is the focal length, and A is the aperture size of the lens. Figure 7.2 also shows the phase
delay that the lens must provide at every point on its aperture to collimate the input spherical
wavefront. In a conventional convex lens (Fig.7.1(a)), this phase shift is provided in a continuous
fashion by the lens. In particular, the surface profiles of the two curved surfaces determine the
lens' phase transmission function. In planar lenses, however, there is no flexibility in choosing the
surface profiles of the input and output apertures of the lens. Therefore, alternative phase shining
mechanisms are usually used. In the next two sub-sections, we discuss a method for achieving the
desired phase shift by synthesizing sub-wavelength spatial phase shifters that accomplish this task.
114
7.2.1
Miniaturized-Element Frequency Selective Surfaces as Spatial Phase Shifters
Figure 7.3(a) and (b) shows the three-dimensional (3-D) topology and the unit cells of a generalized MEFSS with an 7Vt/l-order bandpass response discussed in Chapter 6. Figure 7.3(c) shows
the equivalent circuit model of this structure. A comprehensive analytical design and synthesis
procedure for this structure is reported in Chapter 6. Using this method, by specifying the center
frequency of operation of the MEFSS, / 0 , its operational bandwidth, BW, and its response type
and order, all of the element values of the equivalent circuit model shown in Fig. 7.3(c) can be determined and mapped the geometrical parameters of the MEFSS shown in Fig. 7.3(a). Compared
to regular FSSs with the same response type and order, MEFSSs have significantly lower overall
thicknesses and smaller periodicities. Moreover, they demonstrate very stable frequency responses
as functions of the incidence angle and the polarization of the incident wave as shown in Chapter
6.
Within its pass band, a bandpass MEFSS allows the signal to pass with little attenuation. However, the transmitted signal experiences a phase shift, which can be determined by the phase of
the MEFSS's transmission coefficient. Therefore, within its pass band, a bandpass MEFSS acts
as a phase shifting surface (PSS). The unit cell of such an MEFSS is the smallest building block
7
8
9
10
11
12
13
Frequency [GHz]
Figure 7.4 Simulated transmission coefficients (magnitude and phase) of three MEFSSs having
second-, third-, and fourth-order bandpass responses. Simulation results are obtained in CST
Microwave Studio. As the order of the MEFSS response increases, the phase of the transmission
coefficient changes over a wider range within the pass band of the MEFSS.
115
constituting the PSS. This way, each unit cell acts as a spatial spatial phase shifter. SPSs providing
different phase shifts can be used to populate the aperture of a lens and provide the phase shift
profile shown in Fig. 7.1(c). In an MEFSS, different phase shifts can be achieved at a single frequency by de-tuning the center frequency of operation of the MEFSS to control the transmission
phase. This can be seen by examining the magnitudes and phases of the transmission coefficients
of several MEFSSs shown in Fig. 7.4. Within their pass bands, the transmission phases of all three
MEFSSs change with frequency. The third-order MEFSS, for example, provides a phase shift of
—40° at 9 GHz and a phase shift of —310° at 11 GHz. If the center frequency of operation of
this MEFSS is tuned, both the magnitude and the phase responses shift with frequency. Therefore,
for a fixed frequency that falls within the pass band of both structures, only a change in phase is
observed. This method can be used to achieve MEFSSs that provide different transmission phase
shifts at a single frequency. The maximum phase shift that can be achieved from an MEFSS depends on the order of its filter response. As seen from Fig. 7.4, the higher the-order of the response
is, the larger the phase shift range will be. For example, the second-order MEFSS whose response
is shown in Fig. 7.4 provides a maximum phase variation of 180° within its pass band. This phase
shift range increases to 270° and 360° for the third-order and fourth-order MEFSSs, respectively.
Therefore, the desired phase shift profile of a planar lens determined from its aperture size, A,
and focal distance, / , determines the order of the MEFSSs that need to be used to synthesize the
required spatial phase shifters.
7.2.2
Lens Design Procedure
Let's assume that the planar microwave lens, shown in Fig. 7.5, is located in the x — y plane
and has a rectangular aperture with aperture dimensions of Ax and Ay. A point source located at
the lens' focal point at (x = 0,y = 0,z = —f) radiates a spherical wavefront that impinges upon
the surface of the lens and is transformed to a planar wavefront at the output aperture. At the lens'
input surface, the input spherical wave can be expressed as:
E?n(x,y,z
= 0) = eAm(x,y)e-^r
(7.1)
116
rjkTM-J^ M
r
M;^j
/ >
/
'
V
•jkfe -J*
^
rjlrfj -jty
.
.r . .
/
/
1 ;
i
LX
o-
y
/
z=0 z=h
Figure 7.5 Top view of the proposed planar MEFSS-based lens. A spherical wave is launched
from a point source located at the focal point of the lens, (x = 0, y = 0, z — —/). To transform
this input spherical wave-front to an output planar one, kor + <&(x, y) must be a constant for every
point on the aperture of the lens. $(x, y) is the phase delay, which is provided by the lens.
where Am(x, y) is the amplitude of the electric field of the incident spherical wave on the (x, y, 0)
plane and r = \Jx2 + y2 + f2 is the distance between an arbitrary point on the lens' aperture
specified by its coordinates (x, y, z = 0) and the focal point of the lens (x = 0, y = 0, z = — / ) .
The electric field distribution at the output aperture of the lens can be expressed as:
Eout(x, y,z = h) = eAout(x,
where Aout(x,y)
r x
y)e-^ e-^ ^
(7.2)
is the amplitude of the electric field over the output aperture of the lens and
<£(x, y) is the phase delay provided by the spatial phase shifters of the lens at point (x, y). To
ensure that the output aperture of the lens represents an equiphase surface, the term k0r + $(x, y)
117
must be a constant. Consequently, $(x, y) can be calculated as:
$(x, y) = -kQ[r - /] + fc0(V(A/2)2 + / 2 - / ) + <50
(7.3)
where $ 0 is a positive constant that represents a constant phase delay added to the response of every
SPS on the aperture of the lens. Let's assume that the lens' aperture is divided into M concentric
zones populated with spatial phase shifters of the same type. If the coordinates of a point located
at the center of zone m are given by (xm, ym, z = 0) (where m = 0,1,..., M — 1), then the desired
transmission phase required from SPSs that populate this zone can be calculated from:
$(xm,yrn)
= -k0[rm -f]
+ fc0(^(A/2)2 + / 2 - / ) + $ 0
(7.4)
Having determined the transmission phase for each zone, the lens design procedure can be summarized as follows:
1. Determine the center frequency of operation of the lens, fiens2. Determine the size of the lens aperture, Ax, Ay, and the focal distance, / .
3. Divide the lens aperture into M concentric discrete regions or zones, where M is an arbitrary
positive integer.
4. Determine the transmission phase delay profile of the lens by calculating the phase delay
required from SPSs populating each zone using (7.4).
5. Depending on the maximum variation of $(x, y) over the lens' aperture, determine the order
of the MEFSS needed to implement the spatial phase shifters. For example, Fig. 7.4 shows
that a fourth-order frequency response would be sufficient if a phase variation of more than
270° but less than 360° is needed.
6. Use the synthesis procedure described in Chapter 6 to design the spatial phase shifter that
populates the central zone of the lens. This SPS should provide a phase delay larger than
k0 ( Y / ( ^ 4 / 2 ) 2 + f2 — f) at the desired frequency of operation, fiens. From the phase response
of this SPS at fiens, the value of $ 0 can be determined. $ 0 is the difference between the actual
phase delay provided by the SPS and k0(y/(A/2)2
+ f2 -
f).
118
23.4 cm
23.4 cm
.& <n po
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Figure 7.6 Topology of the two lens prototypes described in Section II. The lens aperture is
divided into a number of concentric zones populated with identical spatial phase shifters (a) First
prototype with five concentric zones, (b) Second prototype with ten zones.
7. Use the synthesis procedure described in Chapter 6 to design the spatial phase shifters that
populate zone m = 1,2,..., M — 1. These SPSs should provide the required phase shift
®(xm, Vm) calculated from (7.4). In doing this, the MEFSS designed for zone 0 can be used
as a starting point and its frequency response can be shifted towards higher frequencies. This
decreases the phase shift provided by the structure at fiens.
We applied the aforementioned procedure to design two planar lens prototypes shown in Fig.
7.6. The lenses are designed to operate at 10 GHz and both have focal lengths of / = 30 cm and
aperture dimensions of Ax = 23.4 cm and Ay = 18.6 cm. With these specifications, the maximum
phase variation required over the aperture of the lens is 169° along the y direction and 264° along
the x direction. The desired phase profile of the lens, along the x axis is shown in Fig. 7.2. As can
be seen from Fig. 7.4, this phase shift range can be achieved from a third-order MEFSS. The unit
cell of a third-order MEFSS consists of three capacitive and two inductive layers separated from
one another by very thin dielectric substrates.
119
The spatial phase shifters used in this lens are designed based on the assumption that they will
operate in an infinite two-dimensional periodic structure. However, the proposed lens is inherently
non-periodic, since it uses a number of different SPSs over its aperture. Therefore, the frequency
responses of the SPSs used in the design of the lens are expected to change when placed in this
operational environment. In our first proof-of-concept experiments, we considered two lens prototypes with different number of zones. In the first prototype, the lens' aperture is divided into five
discrete zones, as shown in Fig. 7.6(a) and in the second prototype the aperture is divided into
10 zones as seen in Fig. 7.6(b). Figure 7.7 shows the detailed layout of the two lens prototypes
and the arrangement of the spatial phase shifters that populate different zones of each lens. Each
square cell in these figures represents one SPS with physical dimensions of 6.1 mm x 6.1 mm.
The numbers on each rectangular cell represent the zone in which the SPS is located. For a lens
with only a few discrete zones, the structure can be considered to be locally periodic in each zone.
Therefore, the phase responses of SPSs that populate each zone are expected to be closer to the
ideal case (infinite periodic structure). Such a lens, however, does not provide a smooth phase delay profile over its aperture. On the other hand, a lens with a higher number of zones can provide
a smooth phase variation over its aperture. However, since fewer SPSs of the same type are used
within each zone, the uncertainty in their actual frequency response could be higher. The two lens
prototypes shown in Fig. 7.7(a) and 7.7(b) are representative examples of these two scenarios and
were chosen to experimentally study these potential tradeoffs.
For each lens, following the aforementioned design procedure, the required transmission phase
shift is first calculated for each zone. Then, the spatial phase shifter (MEFSS unit cell) that populates Zone 0 of the lens (ZO-SPS) is designed. The structure is then simulated using full-wave
EM simulations in CST Microwave Studio. In doing this, the MEFSS unit cell is simulated as part
of an infinite periodic structure by placing it inside a waveguide with periodic boundary condition
(PBC) walls. The structure is then excited by a plane wave and the magnitude and phase of its
transmission coefficient are calculated. The physical and geometrical dimensions of the ZO-SPS
are used as a reference for designing the SPSs that populate other zones of the lens. Since the SPSs
120
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(b)
Figure 7.7 Layouts of the two lens prototypes designed using the proposed design procedure
discussed in Section II-B. The lens aperture is discretized into small area pixels. Each pixel is
occupied by a spatial phase shifter with physical dimensions of 6.1 mm x 6.1 mm designed using
the design procedures discussed in Section II-B. (a) Layout of the first prototype, where the lens
aperture is divided into five discrete zones. The zones are numbered from 0-4. The specifications
of the spatial phase shifters that populate each zone are listed in Table 7.1. (b) Layout of the
second prototype, where the lens aperture is divided into ten discrete zones. The zones are
numbered from 0-9. The specifications of the spatial phase shifters that populate each zone are
listed in Table 7.2.
of each zone must have a different relative transmission phase, they are designed by slightly detuning the frequency response of the ZO-SPS. Doing this shifts both the magnitude and the phase of
the transfer function of the MEFSS response in frequency. Therefore, for a given fixed frequency,
a relative phase shift can be achieved. In a third-order MEFSS, this de-tuning can be achieved by
changing the size of the three capacitive patch layers without changing the inductive layers. This
facilitates the design of the lens by reducing the number of variables that must be changed from
one SPS to the other. Having achieved the desired relative transmission phases, the spatial phase
shifters are then populated over their corresponding zones.
In general, the frequency and phase responses of each SPS are functions of angle and the
polarization of incidence of the EM wave. In designing SPSs that populate each zone of the lens,
these effects are taken into account. The SPSs designed for the two lens prototypes shown in Fig.
121
(b)
Figure 7.8 Frequency responses of the spatial phase shifters that populate the lens aperture of (a)
first prototype and (b) second prototype. The spatial phase shifters are designed using the
procedure described in Section II.
7.7 provide almost identical phase shifts (with less than 2° difference) under oblique incidence
angles for the transverse electric (TE) and transverse magnetic (TM) polarizations. This is mostly
due to the fact that all SPSs of these lenses operate over relatively small incidence angles (20° or
less). Tables 7.1 and 7.2 show the physical and geometrical parameters of SPSs, which populate
different zones of each lens. In all of these designs, the desired response is achieved by only
changing the dimensions of the three capacitive patches of a third-order MEFSS (Du _D3, and D 5
in Fig. 7.3) while keeping the dimensions of the two inductive grids the same (u>2 and iu4 in Fig.
122
Desired
Simulated
Simulated
Phase
Phase
Ins. Loss
0
-198.6°
-198.6°
0.50
5950
5890
1
-182.6°
-182.5°
0.57
5934
5871
2
-155.6°
-152.8°
0.31
5910
5825
3
-114.6°
-113.4°
0.3
5870
5760
4
-61.6°
-65.3°
0.8
5820
5665
Zone
Patch Size
D!,D5
Patch Size
D3
Table 7.1 Physical and electrical properties of the spatial phase shifters that populate each zone
of the first prototype. Insertion loss values are in dB and all physical dimensions are in /j,m. For
all of these designs, u>2 = WA = 2.4 mm and hit2 = /&2,3 = ^3,4 = ^4,5 — 0.5 mm.
7.3). Figure 7.8(a) and 7.8(b) show the magnitude and phase of the transmission coefficients of
the spatial phase shifters that populate different zones of the two lens prototypes. The frequency
ranges where the magnitudes of the responses overlap are highlighted in both figures. The overlap
region spans a fractional bandwidth of 10% and 8% for the first and second prototypes respectively.
This overlap range can be considered as a lower bound for the bandwidth of the lens.
7.3
Experimental Verification
The two lens prototypes discussed in the previous section were fabricated using standard PCB
fabrication techniques. Both prototypes use four 0.020" -thick dielectric substrates (RO4003C from
Rogers Corporation). Different metal layers are patterned on one or both sides of each substrate
and the substrates are bonded together using three 0.004" thick bonding films (RO4450B from
Rogers Corporation). The overall thickness of each lens, inclusive of the bonding films, is 2.3 mm
or equivalently 0.08A0, where A0 is the free space wavelength at 10 GHz. The dimensions of the
spatial phase shifters are 6.1 mmx6.1 mm or equivalently 0.2A0 x 0.2A0. Figure 7.9 shows the
photograph of the fabricated prototype whose layout is shown in Fig. 7.7(b).
123
Zone
Patch Size
Patch Size
Desired
Simulated
Simulated
Phase
Phase
Ins. Loss
0
-198.6°
-198.6°
0.5
5950
5890
1
-193.6°
-193.5°
0.6
5945
5880
2
-185.6°
-184.3°
0.54
5934
5873
3
-172.6°
-174.5°
0.45
5930
5860
4
-155.6°
-156.7°
0.38
5910
5850
5
-134.9°
-139.2°
0.30
5900
5808
6
-109.6°
-114.3°
0.30
5880
5775
7
-80.6°
-85.5°
0.37
5850
5720
8
-47.6°
-49°
1.37
5810
5650
9
-11.6°
-34.9°
2.1
5800
5635
Di,D5
D3
Table 7.2 Physical and electrical properties of the spatial phase shifters that populate each zone
of the second lens prototype. Insertion loss values are in dB and all physical dimensions are in
fira. For all of these designs, w2 = w4 = 2.4 mm and /ii>2 = /i2,3 = ^3,4 = ^4,5 = 0.5 mm.
7.3.1
Measurement System
To characterize the two fabricated prototypes, two different types of measurements are carried
out using the measurement setup shown in Fig. 7.10. The measurement setup consists of a large
metallic screen with dimensions of 4' x 3'. The screen has an opening with dimensions of 19 cm
x 24 cm at its center to accommodate the fabricated lens prototypes. An X-band horn antenna with
aperture dimensions of 7.5 cm x 9 cm is placed 120 cm away from the test fixture and radiates a
vertically polarized EM wave to illuminate the lens. At the other side of the lens, a probe is used to
measure the received signal. The probe is also vertically polarized and is in the form of an openended, semi-rigid coaxial cable with its center conductor extended by 10 mm. Both the transmitting
antenna and the receiving probe are connected to a vector network analyzer. Figure 7.10(b) shows
a top view of the measurement system and the scenario used for the first series of measurements.
In this case, the fixture that holds the receiving probe is swept over a measurement grid with
124
;.. spatial phase
•'L. / shifter
23.4 cm (7.8^ @ 10 GHz)
Figure 7.9 Photograph of the fabricated lens prototype that consists often zones. In this figure,
only one of the five metallic layers that constitute the lens is visible (the first capacitive layer).
Notice that the sizes of the capacitive patches on the layer decrease as we move away from the
center of the lens.
dimensions of 8 cm x 8 cm in the x — z plane with 1 cm increments in the x and z directions.
The measurement grid is centered at the expected focal point of the lens. The measurement of
the fabricated lens prototypes are conducted in two steps. For each grid point, the transmission
response of the fixture without the lens is measured first. Then, the transmission response of the
system with the presence of the lens is measured. By normalizing the latter measured values to the
former one, the focusing gain of the lens at each grid point can be calculated. These measurements
are then repeated for every point in the 8 cm x 8 cm grid to obtain a two-dimensional plot of the
focusing gain of the lens in the vicinity of its expected focal point (x = 0, y = 0, z = —30 cm).
This measurement is carried out for both fabricated prototypes at different frequencies across the
X-band.
In the second series of measurements, we examined the performance of the fabricated prototypes under oblique incidence angles. This is done by illuminating the lens with a plane wave from
various incidence angles ranging from normal to 60° and measuring the field distribution over the
125
focal arc of the lens using the measurement set-up shown in Fig. 7.10(c). Here, the receiving probe
is mounted on a rotating arm with its axis of rotation at the center of the lens. The length of the
arm can be adjusted as desired. The receiving probe is swept over the focal arc with 2° increments
and the received power is measured.
PEC wall
Probe
Horn
VNA
/V
(a)
_ „ PEC wall - _ _
Probe
^ .Adjustable
arm
>x
V Focal point
\
\
Focal arc
Focal point
Measurement
Grid
(b)
(c)
Figure 7.10 (a) Perspective view of the measurement setup used to experimentally characterize
the performance of the two fabricated lens prototypes, (b) In a set of measurements, the lens is
excited with a plane wave and a receiving probe is swept in a rectangular grid in the vicinity of its
focal point (x = 0, y = 0, z = — / ) to characterize its focusing properties as a function of
frequency. The measurement grid's area is 8 cm x 8 cm and the resolution is 1 cm. (c) In another
set of measurements, the lens is illuminated with plane waves arriving from various incidence
angles and the received power pattern on the focal arc of the lens is measured. Here, a probe is
swept over the focal arc with 2° increments to measure the received power.
126
7.3.2
Measurement Results
Figure 7.11 shows the focusing gain of the first fabricated prototype measured in the frequency
range of 8-12 GHz with 0.5 GHz increments. At each frequency, the location where the maximum
value of the focusing gain is achieved is indicated with a cross (x) symbol. At 10 GHz, a maximum
focusing gain of 9.6 dB is achieved at (x = 0,y = 0,2 = —28 cm), which is 2 cm away from
the expected focal point of the lens. This can be attributed to the difference between the actual
and simulated responses of the SPSs, numerical errors in full-wave simulations, and fabrication
tolerances. Further examination of Fig. 7.11 reveals that at frequencies below 10 GHz, the focal
point1 is located closer to the lens and it moves away from the lens as frequency is increased. This
frequency-dependent movement of the focal point can be explained using the Fermat's principle.
It states that the path taken between two points by a ray of light is the path that can be traversed
in the least time. In the context of a focusing system, such as a reflectarray or a microwave lens,
Fermat's principle requires that the net time delay acquired by any ray propagating from the focal
point to the aperture must be the same. Therefore, to achieve a frequency-independent focal point,
the spatial phase shifters of a planar lens (or a reflectarray) must act as true time delay (TTD)
units that provide different time delay values based on their position of the lens' aperture. In
frequency domain, a constant time delay corresponds to a phase response that changes linearly
with frequency. Therefore, the phase responses of different time-delay units that populate the
aperture will be linear functions of frequency with different slopes. As can be seen from Figs.
7.8(a) and 7.8(b), the phase responses of different SPSs used in the two lens prototypes can roughly
be approximated with linear functions of frequency. However, all of these linear functions have
approximately the same slope. This frequency-dependent focal point movement is not unique
to this lens and is observed in any non-TTD microwave lens or reflectarray. Nonetheless, as is
observed from Fig. 7.11, the focusing properties of the lens remain within acceptable margins in
the vicinity of the desired frequency of operation.
The same measurement is also conducted to characterize the response of the second prototype.
Figure 7.12 shows the measured focusing gain of the second lens prototype over the same 8 cm x
1
Deflned as the location where the focusing gain attains its maximum value.
127
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Figure 7.11 Measured focusing gain of the first prototype (with five zones) in a rectangular grid
in the vicinity of its expected focal point (x = 0 cm, y = 0 cm, z = —30 cm). In all of the figures,
the horizontal axis is the x axis with units of [cm] and the vertical axis is the z axis with units of
[cm]. The color bar values are in dB. The x marker in all of these figures represents the exact
coordinate where the focusing gain maxima occur, (a) 8.0 GHz, (b) 8.5 GHz, (c) 9.0 GHz, (d) 9.5
GHz, (e) 10.0 GHz, (f) 10.5 GHz, (g) 11.0 GHz, (h) 11.5 GHz, and (i) 12.0 GHz.
8 cm rectangular grid in the frequency range of 8 GHz to 12 GHz with 0.5 GHz increments. Figure
7.12(e) shows that a maximum focusing gain of 10 dB is achieved at (x = 0,y = 0, z = —29
cm) at 10 GHz, which is 1 cm away from the expected focal point of the lens. The slight increase
in the focusing gain is attributed to the better approximation of the desired phase delay profile
128
0
2
4
6
(i)
Figure 7.12 Measured focusing gain of the second prototype (with ten zones) in a rectangular
grid in the vicinity of its expected focal point (x = 0 cm, y = 0 cm, z = — 30 cm). In all of the
figures, the horizontal axis is the x axis with units of [cm] and the vertical axis is the z axis with
units of [cm]. The color bar values are in dB. The x marker in all of these figures represents the
exact coordinate where the focusing gain maxima occur, (a) 8.0 GHz, (b) 8.5 GHz, (c) 9.0 GHz,
(d) 9.5 GHz, (e) 10.0 GHz, (f) 10.5 GHz, (g) 11.0 GHz, (h) 11.5 GHz, and (i) 12.0 GHz.
across the lens aperture compared to that of the first prototype. Similar to the previous case, a
frequency-dependent movement of the focal point is also observed in this prototype.
Figure 7.13 shows the measured focusing gains of the two prototypes, at the expected focal
point of the lens (x = 0,y = 0,z = — 30 cm), as a function of frequency. As can be seen, the gain
129
9
10
11
Frequency [GHz]
Figure 7.13 Calculated and measured focusing gains of the two lens prototypes at their expected
focal point (x = 0 cm, y = 0 cm, z = —30 cm) as a function of frequency. The 3 dB gain
bandwidths of the two prototypes are respectively 19.2% and 20% for the first and second
prototypes.
of the first prototype does not vary by more than 3 dB in the frequency range of 9.4-11.4 GHz,
which is equivalent to a fractional bandwidth of 19.2%. For the second prototype, this bandwidth
is extended to 20% (9.4-11.5 GHz).
The electrical dimensions of the fabricated lens prototypes are relatively large and the structures have very small minimum features that gradually change over their apertures. For example,
the gaps between the capacitive patches of the lens shown in Fig. 7.9 have dimensions of 150 jj,m
at the center of the aperture and gradually increase to 300 /im at its edges. Modeling this structure
using full-wave numerical simulations requires using a very dense grid system that can resolve
the small variations in the physical dimensions of the structure from one SPS to the other (see
the dimensions reported in Tables 7.1 and 7.2). Moreover, the overall electrical dimensions of the
structure are significantly larger than the minimum feature of a given SPS. Consequently, conducting a full-wave EM simulation of these prototypes requires significant computational resources
and will be very time-consuming. A simple alternative method to model the proposed lens is to
consider it as a two-dimensional antenna array with an element spacing of P x P, where P is the
130
(g)
(h)
(i)
Figure 7.14 Calculated focusing gain of the first prototype (with five zones) in a rectangular grid
in the vicinity of its expected focal point (x = 0 cm, y = 0 cm, z = —30 cm). In all of the figures,
the horizontal axis is the x axis with units of [cm] and the vertical axis is the z axis with units of
[cm]. The color bar values are in dB. The x marker in all of these figures represents the exact
coordinate where the focusing gain maxima occur, (a) 8.0 GHz, (b) 8.5 GHz, (c) 9.0 GHz, (d) 9.5
GHz, (e) 10.0 GHz, (f) 10.5 GHz, (g) 11.0 GHz, (h) 11.5 GHz, and (i) 12.0 GHz.
physical size of each spatial phase shifter of the lens (6.1 mm). Considering each element of this
array to be an electrically-small Hertizan dipole with a given amplitude and phase, one can easily
calculate the overall array gain of this structure and evaluate it in the vicinity of its expected focal
point. The amplitude and phase of each radiating element of this array can be determined from the
131
(g)
(h)
(i)
Figure 7.15 Calculated focusing gain of the second prototype (with ten zones) in a rectangular
grid in the vicinity of its expected focal point (x = 0 cm, y = 0 cm, z = —30 cm). In all of the
figures, the horizontal axis is the x axis with units of [cm] and the vertical axis is the z axis with
units of [cm]. The color bar values are in dB. The x marker in all of these figures represents the
exact coordinate where the focusing gain maxima occur, (a) 8.0 GHz, (b) 8.5 GHz, (c) 9.0 GHz,
(d) 9.5 GHz, (e) 10.0 GHz, (f) 10.5 GHz, (g) 11.0 GHz, (h) 11.5 GHz, and (i) 12.0 GHz.
simulated values of insertion loss and transmission phase of each SPS provided in Tables 7.1 and
7.2. This process is carried out for both lens prototypes. Figures 7.14 and 7.15 show the calculated
focusing gains of the first and second prototypes, respectively, over the same rectangular grid and
frequency range. A similar frequency-dependent movement of the focal point is also observed in
132
both prototypes. However, the locations of the calculated focal points are slightly different from
the corresponding measured ones. This can be attributed to the tolerances in fabricating the multiple layers of the lens prototypes, along with the numerical errors in the full-wave simulations of
the SPSs. Additionally, the fact that these SPSs are designed in a periodic environment, but operate
in a non-periodic environment contributes to this discrepancy. This simplified modeling process is
also used to calculate the focusing gain at the focal point as a function of frequency. The calculated
values of the focusing gain are found to be very close to the measured ones for both prototypes.
The calculated focusing gains, at (x = 0, y = 0, z = —30 cm), are shown in Fig. 7.13 along with
the measured ones. As can be seen the calculated results match very well with the measured ones
over the entire frequency band of operation of the lens.
The performance of the two fabricated prototypes are also characterized under oblique incidence angles. Each lens is illuminated with vertically-polarized plane waves with incidence angles
of 0°, 15°, 30°, 45°, and 60° in the x — z plane and the received power pattern on the focal arc
is measured. Figures 7.16(a) and 7.16(b) show respectively the simulated and measured power
patterns on the focal arc of the first lens prototype. Figures 7.17(a) and 7.17(b) show the same
results for the second prototype. For each lens, the incident power density is maintained for all
incidence angles. All measured power values are normalized to the maximum measured value,
which occurs at the focal point for normal incidence. Therefore, in these figures, 0 dB refers to the
power level received on the focal point of the lens when it is excited with a normally incident plane
wave. These simulated and measured results show the expected and actual performance of the lens
response under oblique incidence angles. As the angle of incidence changes, the power pattern on
the focal arc is steered by an angle equal to that of the incident angle and its peak value decreases.
For incidence angles up to 45°, however, this decrease is relatively small (1.3 dB for first prototype
and 1.6 dB for second prototype at 45°). Beyond this angle, however, the response of the lens starts
to deteriorate quite rapidly and at 60° the maximum value of the measured power pattern of the
first and second prototypes are reduced respectively by 7.8 dB and 7.1 dB compared to the peak
values measured for normal incidence. Nonetheless, the measured results shown in Figs. 7.16(b)
and 7.17(b) demonstrate a good performance for incidence angles up to 45° for both prototypes.
133
Angle [deg]
(a)
Angle [deg]
(b)
Figure 7.16 Received power pattern on the focal arc of the first fabricated prototype (with five
zones) when excited with plane waves arriving from various incidence angles, (a) Calculated, (b)
Measured.
Additionally, comparison of the calculated results with the measured ones shows a relatively good
agreement between the two despite the simplified method used to model the structure. Comparing
the measurements shown in Figs. 7.16 and 7.17 reveals that the beam-width of the power pattern
of the second prototype is narrower than that of the first one. Additionally, the performance of the
second prototype at oblique angles of incidence is slightly better than that of the first one. These
effects may be attributed to the smoother phase delay profile across the aperture of this prototype.
134
This indicates that the performance of this type of lens can potentially be further enhanced by increasing the number of zones. In the limiting case, the width of each zone can be made as narrow
as that of one spatial phase shifter (6.1 mm in this case).
The power patterns of the two fabricated prototypes are measured at the focal arc and under oblique angles of incidence. This measurement is conducted over the operational bandwidth.
For both prototypes, the power patterns on the focal arc are measured in the frequency range 9.5
o
- 0 U -|
"-
1
-75
1
1
-60
1
1
-45
1
1
-30
\
1
1
1
|J—i
-15
0
15
Angle [deg]
1—i
30
1
1—|—i—
45
60
75
(a)
0 -ou
"-
-)
-75
1
1
-60
1
1
-45
1
1
-30
1
>]
1
1
1
J
r
-15
0
15
Angle [deg]
1
1
30
1
1
45
.
1
60
•
1
75
(b)
Figure 7.17 Received power pattern on the focal arc of the second fabricated prototype (having
ten zones) when excited with plane waves arriving from various incidence angles, (a) Calculated.
(b) Measured.
135
GHz-11.5 GHz with 0.5 GHz increments. These power patterns are measured when the lens is illuminated with plane waves with incidence angles of 0°- 60° with 15° increments. Figures 7.18-7.22
show the simulated and the measured power patterns of the first prototype on the focal arc over the
operational bandwidth and for angles of incidence 0°- 60°. Figures 7.23-7.27 show similar power
patterns of the second prototypes. All the power patterns are normalized to the measured peak
value at 10 GHz under normal EM wave illumination. At all angles of incidence, a relatively good
agreement is observed between the simulated and measured patterns over the operational bandwidth, especially within the main lobe. Comparing the power patterns in (a)-(e) of all the figures
shows that the beam-width of the power pattern on the focal arc becomes narrower as frequency is
increased. This is due to the fact that the lens aperture is electrically larger at higher frequencies.
These results indicate that the focusing performance of this type of lenses does not deteriorate over
the operational bandwidth of the lens.
7.4
Conclusions
A new technique for designing low-profile planar microwave lenses is introduced in this chap-
ter. The proposed lenses use the unit cells of suitably-designed miniaturized-element frequency
selective surfaces as their spatial phase shifters. A procedure for designing such planar lenses was
also presented. The lens design procedure is based on designing each spatial phase shifter of the
lens by treating it as part of an infinite 2-D periodic structure. Two prototypes that approximate
the phase profile of a conventional convex dielectric lens were designed following this procedure.
The first prototype accomplishes this by dividing the aperture of the lens into five phase-shifting
zones and thus providing only a rough approximation of the ideal phase profile. In the second prototype, the lens aperture is divided into ten zones and a smoother phase variation over the aperture
is obtained. Both prototypes were fabricated and experimentally characterized using a free-space
measurement setup. The two prototypes operate over a relatively wide band of operation with
fractional bandwidths of respectively 19.2% and 20%. Additionally, under oblique angles of incidence, both prototypes demonstrate a good performance up to 45°. As a simple analysis technique,
136
the two lenses were treated as two-dimensional antenna arrays composed of Hertzian dipole radiators and the expected performance of the lenses were also calculated. For both prototypes, this
expected performance was found to be very close to the measurement results. This suggests that
the responses of the spatial phase shifters of the lenses do not change significantly, from their ideal
responses, when placed in this non-periodic environment. In general, the performance of the second prototype, which provides a smoother phase variation, is found to be better than that of the
first one. This suggests that increasing the number of discrete phase shifting zones is advantageous
in designing this type of lens.
137
•o
O-10-
r r
^ts.
3
.G
C-20-
vTvft\
«
Q
•O-30-
«
iZ
<
-30
n-50 - — , — i — . — | — . — f —
-75
-60
-45
-30
-15
0
15
Angle [deg]
—#— Measurement
—A— Simulation
-15
0
15
Angle [deg]
(b)
(a)
•15
0
15
Angle [deg]
(d)
(c)
DO
0
- Measurement
- Simulation
-15
o
15
Angle [deg]
(e)
Figure 7.18 Calculated and measured power patterns on the focal arc of the first fabricated
prototype (having five zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz, (c) 10.5
GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves excitation with 0°
angle of incidence (normal incidence). These power patterns are normalized to the maximum
measured value, which occurs at the focal point when it is excited with a normally incident plane
wave operating at 10 GHz. Therefore, 0 dB refers to the power level received on the focal point of
the lens when it is excited with a normally incident plane wave operating at 10 GHz.
138
Measurement
Simulation
-30
-15
0
15
Angle [deg]
Angle [deg]
(a)
(b)
- Measurement
- Simulation
•15
0
15
Angle [deg]
(d)
(c)
m o
Measurement
Simulation
-15
o
15
Angle [deg]
(e)
Figure 7.19 Calculated and measured power patterns on the focal arc of the first fabricated
prototype (having five zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz, (c) 10.5
GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves excitation with 15°
angle of incidence. These power patterns are normalized to the maximum measured value, which
occurs at the focal point when it is excited with a normally incident plane wave operating at 10
GHz. Therefore, 0 dB refers to the power level received on the focal point of the lens when it is
excited with a normally incident plane wave operating at 10 GHz.
139
-75
-60
-45
-30-15
0
15
Angle [deg]
30
45
SO
75
-75
-60
-45
-30
15
30
45
Angle [deg]
(a)
(b)
-15
0
15
Angle [deg]
(e)
Figure 7.20 Calculated and measured power patterns on the focal arc of the first fabricated
prototype (having five zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz, (c) 10.5
GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves excitation with 30°
angle of incidence. These power patterns are normalized to the maximum measured value, which
occurs at the focal point when it is excited with a normally incident plane wave operating at 10
GHz. Therefore, 0 dB refers to the power level received on the focal point of the lens when it is
excited with a normally incident plane wave operating at 10 GHz.
140
(a)
(b)
(c)
(d)
g-'OH—.—i—r—i—.—r->O
"-
-75
-GO
-45
-30
1 .
-15
1 .
0
1 r"—,—.—i—.—i—,—I
15
30
45
60
75
Angle [deg]
(e)
Figure 7.21 Calculated and measured power patterns on the focal arc of the first fabricated
prototype (having five zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz, (c) 10.5
GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves excitation with 45°
angle of incidence. These power patterns are normalized to the maximum measured value, which
occurs at the focal point when it is excited with a normally incident plane wave operating at 10
GHz. Therefore, 0 dB refers to the power level received on the focal point of the lens when it is
excited with a normally incident plane wave operating at 10 GHz.
141
(a)
(b)
(c)
(d)
(e)
Figure 7.22 Calculated and measured power patterns on the focal arc of the first fabricated
prototype (having five zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz, (c) 10.5
GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves excitation with 60°
angle of incidence. These power patterns are normalized to the maximum measured value, which
occurs at the focal point when it is excited with a normally incident plane wave operating at 10
GHz. Therefore, 0 dB refers to the power level received on the focal point of the lens when it is
excited with a normally incident plane wave operating at 10 GHz.
142
00 0
T—I^TT—i—•—i—•—r-
Angle [deg]
(b)
•o
Jt/J
§-103
.Q
£-20-
vftw
\j^o^
«
Q
•O-30-
«
—•— Measurement
—*— Simulation
<
«-50-75
-60
-45
Angle [deg]
-30
-15
0
15
Angle [deg]
30
45
(d)
(c)
ca o
i—•—I—'—r-
- Measurement
- Simulation
•15
0
15
Angle [deg]
(e)
Figure 7.23 Calculated and measured power patterns on the focal arc of the second fabricated
prototype (having ten zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz, (c) 10.5
GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves excitation with 0°
angle of incidence (normal incidence). These power patterns are normalized to the maximum
measured value, which occurs at the focal point when it is excited with a normally incident plane
wave operating at 10 GHz. Therefore, 0 dB refers to the power level received on the focal point of
the lens when it is excited with a normally incident plane wave operating at 10 GHz.
143
(a)
(b)
(c)
(d)
"•
Angle [deg]
(e)
Figure 7.24 Calculated and measured power patterns on the focal arc of the second fabricated
prototype (having ten zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz, (c) 10.5
GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves excitation with 15°
angle of incidence. These power patterns are normalized to the maximum measured value, which
occurs at the focal point when it is excited with a normally incident plane wave operating at 10
GHz. Therefore, 0 dB refers to the power level received on the focal point of the lens when it is
excited with a normally incident plane wave operating at 10 GHz.
144
(a)
(b)
(c)
(d)
GO
° "I
'
1—'
1
'
1
•
1
'
1
'
1
•
1
•
1
•
1
r
(e)
Figure 7.25 Calculated and measured power patterns on the focal arc of the second fabricated
prototype (having ten zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz, (c) 10.5
GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves excitation with 30°
angle of incidence. These power patterns are normalized to the maximum measured value, which
occurs at the focal point when it is excited with a normally incident plane wave operating at 10
GHz. Therefore, 0 dB refers to the power level received on the focal point of the lens when it is
excited with a normally incident plane wave operating at 10 GHz.
145
m u
-30
-15
0
15
Angle [deg]
60
75
(a)
•
|
i
,
i
|
i
|
i
|
i
|
i
|
i J^I
/
o-10-
,
\
3
c-20 -
</)
•-O-300)
o-40-
—•— Measurement
—*— Simulation
<
"5-50-30
-15
0
15
30
45
60
Angle [deg]
Angle [deg]
(d)
(C)
CQ
0
Figure 7.26 Calculated and measured power patterns on the focal arc of the second fabricated
prototype (having ten zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz, (c) 10.5
GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves excitation with 45°
angle of incidence. These power patterns are normalized to the maximum measured value, which
occurs at the focal point when it is excited with a normally incident plane wave operating at 10
GHz. Therefore, 0 dB refers to the power level received on the focal point of the lens when it is
excited with a normally incident plane wave operating at 10 GHz.
146
CQ
0
-60
-45
-30
-15
0
15
Angle [deg]
Angle [deg]
(a)
(b)
-15
0
15
Angle [deg]
(e)
Figure 7.27 Calculated and measured power patterns on the focal arc of the second fabricated
prototype (having ten zones) due to plane wave excitation at (a) 9.5 GHz, (b) 10 GHz, (c) 10.5
GHz, (d) 11 GHz, and (e) 11.5 GHz. The power patterns are for plane waves excitation with 60°
angle of incidence. These power patterns are normalized to the maximum measured value, which
occurs at the focal point when it is excited with a normally incident plane wave operating at 10
GHz. Therefore, 0 dB refers to the power level received on the focal point of the lens when it is
excited with a normally incident plane wave operating at 10 GHz.
147
Chapter 8
Design of Planar True-Time-Delay Lenses Using Sub-Wavelength
Spatial Time-Delay Units
8.1
Introduction
In the previous chapter, we presented a new technique for designing planar microwave lenses.
One characteristic of this type of lens is that the location of the focal point moves as frequency
is changed. At frequencies below the design frequency, the focal point is closer to the lens aperture and it moves away from the lens aperture as frequency is increased. In this chapter, we will
address this issue and propose a different approach to achieve a wideband lens with a frequencyindependent focal length. Dielectric lenses are typical examples of such lenses [157]-[159]. They
were among the first structures that were investigated. Their drawbacks, however, are their large
size, heavy weight, and high cost especially at low microwave frequencies. Moreover, dielectric
lenses suffer from internal reflections losses. An alternative approach for designing planar wideband lenses is to use arrays of transmit and receive antennas connected together using true-timedelay lines [156], [160]-[161]. Different true-time-delay techniques have been reported to achieve
wideband scanning. These techniques can be divided into two main categories, electrical and optical. With regards to the electrical techniques, the time delay is achieved by changing the wave
propagation characteristics of the transmission line connected to the antenna elements. In one approach, true-time-delay transmission lines are periodically loaded with varactor diodes to achieve
different propagation constants [174]-[175], thereby achieving different time delays. In another
approach, a true-time-delay transmission line is obtained by incorporating MicroElectroMechanical System (MEMS) switches distributed periodically in the transmission line [176]-[178]. On
148
the other hand, optical true-time-delay techniques are achieved by modulating the RF signals onto
light beams. These beam are then delayed by different times through varying the optical path by
either using wave guiing medium (fibers) [180]-[181] or through free space [182]-[183].
In this chapter, we propose a new approach for designing true-time-delay units. These delay
units are used to design wideband planar microwave lenses. The MEFSSs discussed in Chapter
6 are used to design these lenses. In this case, the unit cells of the MEFSS are treated as truetime-delay units. This is achieved by designing the transmission phase to be a linear function
of frequency within the operational bandwidth. These true-time-delay units are populated over
the lens aperture. Depending on its location within the lens aperture, each true-time-delay unit is
designed to provide different time delay. This is achieved by changing the slope of the transmission
phase over the operational bandwidth of the lens. The main characteristic of this type of lenses is
that the location of the focal point is frequency independent. The focusing response of the lens is
calculated and it is shown that the lens operates over a relatively wideband of 25%.
Double-Convex
Dielectric Lens
Proposed MEFSS-Based
True-Time-Delay Planar Lens
o
u+
-4—<
r/3
•i-H
Q
S3
03
Pi
^rik
=3^
a->
Time Delay
Sub-wavelenth Spatial
True-Time-Delay Unit
\ /
(a)
k
(b)
Figure 8.1 (a) Topology of a conventional double-convex dielectric lens, (b) Topology of the
proposed MEFSS-based planar true-time-delay lens.
149
-9-8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8 9
Distance from center of the aperture [cm]
Figure 8.2 Relative time delay that different rays of a spherical wave experience (with respect to
the ray that passes through the center of the aperture) at the input of the lens aperture and the time
delay profile, which must be provided by the lens to achieve a planar wavefront at the output
aperture of the lens. The time delay is referenced to the time it takes the wave to propagate from
focal point to the center of the aperture. The results are calculated for a circular aperture lens with
aperture diameter of 18.6 cm and focal distance of 30 cm.
8.2
Lens Design and Principles of Operation
Figure 8.1 shows a comparison between a double-convex dielectric lens and the proposed true-
time-delay (TTD) MEFSS-based planar lens. For a given lens aperture size and focal distance, the
time delay profile that must be provided by the lens can be calculated. Let's assume a point source
located at the focal point radiating spherical wave towards the lens. The radiated spherical wave
arrives at different points of the lens input surface at different times. The shortest time is for the
ray that propagates from the focal point to the center of the lens. Using this time as reference, the
relative time delay at the input surface is of the lens is calculated, as depicted in Fig. 8.2. This
relative time delay is for a lens with an aperture size of D = 18.6 cm having a focal length of
/ = 30 cm. The time-delay profile that must be provided by the lens is then calculated, as depicted
in Fig. 8.2. This ensures that all the rays of incident wave, which arrive at different times, leave
the lens surface at the same time. Hence, The spherical wavefront launched from the focal point is
150
Sub-wavelength
Capacitive Patches
Dielectric
Substrate
Sub-wavelength
Inductive Grid
Figure 8.3 (a) 3D topology of an Nth-ordev MEFSS discussed in chapter 6. (b) 2D top view of
the unit cell of a sub-wavelength capacitive patch layer (top) and an inductive wire grid layer
(bottom). The two layers are used a building block for the Nth-order MEFSS. (c) Equivalent
transmission line circuit model of the Nth-order MEFSS.
converted to a planar wavefront at the output surface of the lens. Next two sections discuss how to
design the MEFSS-based planar lens to provide the calculated time-delay profile.
8.2.1
Miniaturized-Element Frequency Selective Surfaces as Spatial Time Delay Units
Figure 8.3 shows the three-dimensional (3-D) topology and the equivalent circuit model of a
generalized MEFSS with an iVt/l-order bandpass response discussed in chapter 6. A comprehensive
analytical design and synthesis procedure for this structure is also reported in that chapter. Within
its pass band, a bandpass MEFSS allows the signal to pass with little attenuation. However, the
transmitted signal experiences a frequency-dependent phase shift, which can be determined by the
phase of the MEFSS's transmission coefficient. In order to obtain a wideband collimating lens,
the unit cells populating the lens aperture need to operate as spatial time-delay units rather than
spatial phase shifters. This requires that each unit cell element is designed to have a constant group
delay over the desired frequency band of operation. In filter theory, the group delay is defined
as a measure of the slope of the transmission phase response within the desired band. Therefore,
151
achieving constant group delay requires designing these elements to have a linear transmission
phase within the desired band of operation.
This way, each unit cell acts as a spatial time delay unit (TDU). These TDUs can be designed to
have different time delays and be populated over the lens' aperture to provide the time delay profile
shown in Fig. 8.2. To achieve different time delays over the desired frequency range, the MEFSS
can be designed to have linear transmission phase with different slopes (or equivalently different
group delays). The steeper the slope of the transmission phase of the MEFSS within its bandpass
is, the larger the time delay it provides will be. Two approaches can be used to provide different
group delays. In the first approach, the group delay is controlled through the order of the MEFSS
transfer function. The higher the MEFSS's order is, the larger its group delay will be in the desired
band. This is attributed to the fact that a higher-order MEFSS provides larger transmission phase
variations within its pass band, compared to an MEFSS with a lower order. For example, within
the highlighted region of the band pass, the second-order MEFSS whose response is shown in Fig.
8.4(a) provides a phase variation of 65°. This corresponds to a group delay of 240 psec (with
~ ±5% variation around that value), as shown in Fig. 8.4(b). This phase shift range (group delay)
10
11
12
13
'
o
»
™
Frequency [GHz]
Frequency [GHz]
(a)
(b)
Figure 8.4 Calculated frequency response of three MEFSSs having second-, third-, and
fourth-order bandpass responses, (a) Magnitude and phase of transmission coefficients, (b)
Magnitudes of transmission coefficient and the corresponding group delays. As the order of the
MEFSS response increases, the phase of the transmission coefficient changes over a wider range
within the highlighted region of the MEFSS bandpass. Hence, a larger group is achieved within
that frequency range of interest.
152
10
11
12
13
'
»
»
nu
Frequency [GHz]
Frequency [GHz]
(a)
(b)
Figure 8.5 Calculated frequency responses of three MEFSSs having fourth-order bandpass
responses with different fractional bandwidths. (a) Magnitude and phase of transmission
coefficients, (b) Magnitudes of transmission coefficient and the corresponding group delays. As
the bandpass fractional bandwidth of the MEFSS response increases, the slope of the transmission
phase coefficient decreases within the highlighted region of the MEFSS bandpass. Hence, a
smaller group is achieved within that frequency range of interest.
increases to 95° (318 psec) and 120° (410 psec) for the third-order and fourth-order MEFSSs,
respectively. In the second approach, the group delay is controlled by changing the fractional
bandwidth of the MEFSS with the same order. This is illustrated in Fig. 8.5(a), in which the
magnitude phase of the transmission coefficient of MEFSS with a fourth-order bandpass response
is depicted. The larger the fractional bandwidth of the MEFSS response is, the slower the slope
of the transmission phase will be. Thus, the group delay of a wideband MEFSS is smaller than
that of a narrowband structure. As shown in Fig. 8.5(b), MEFSS with 25% fractional bandwidth
has a group delay of 320 psec (with less than ±5% variation around that value), whereas MEFSS
with 35% fractional bandwidth has a group delay of 220 psec. Therefore, the desired time delay
profile of a planar lens determined from its aperture size, D, and focal distance, / , determines the
orders and fractional bandwidths of the MEFSSs that need to be used to synthesize the required
time delay units.
153
8.2.2
Lens Design Procedure
Let's assume that the planar microwave lens, shown in Fig.8.6, is located in the x — y plane
and has a circular aperture with diameter of D. A point source located at the lens' focal point at
(x = 0, y = 0, z = —/) radiates a spherical wavefront that impinges upon the surface of the lens
and is transformed to a planar wavefront at the output aperture. First, the travel time that the wave
takes to arrive at the lens' input surface is calculated:
T(x, y,z = 0) = y/x* + y* + P/c
(8.1)
Where 0 < y/x2 + y2 < D/2. The time delay profile that needs to be provided by the lens can be
calculated as:
TD(x, y,z = h) = (V(£>/2) 2 + / 2 - r)/c
(8.2)
Whereas the phase profile at the operating frequency can be calculated from:
$(x, y) = -ko[r -f]
+ k0(y/(D/2)2
+ f2 - / ) + $ 0
(8.3)
Where $ 0 is a positive constant that represents a constant phase delay added to the response of
every TDU on the aperture of the lens, k — ^ is the free space wave number, Ao is the free
space wavelength, and r = y/x2 + y2 + f2 is the distance between an arbitrary point on the lens'
aperture specified by its coordinates (x, y, z = 0) and the focal point of the lens (x = 0, y =
0, z =
-f).
To ensure that the output aperture of the lens represents an equiphase surface, the lens should
be designed such that two conditions must be satisfied across its aperture: 1) time delay profile
calculated from (8.2) must be the same over the desired band of operation, and 2) the phase shift
profile at the operating frequency must be equal to that calculated from (8.3). Satisfying these
two conditions ensures that the signal carried by the incident wave is not distorted. Moreover,
it ensures that the wave at the output of the lens is spatially coherent over the desired frequency
range. Let's assume that the lens' aperture is divided into M concentric zones populated with time
delay units of the same type. If the coordinates of a point located at the center of zone m are given
154
|l T
, Mu, tTDI
M
A.
•
/ >
i
i
i
i
i
i
i
i
T.-HTD,
*
^-
i
il
'
iX
/
ii
i
i
i
i
'
.1
i
i
T.-KTID,
11
i
i
i
D
i
i
/
z=0 z=h
Figure 8.6 Top view of the proposed MEFSS-based true-time-delay planar lens. A spherical wave
is launched from a point source located at the focal point of the lens, (x = 0, y = 0, z — —/). To
transform this input spherical wavefront to an output planar one, T(x, y) + TD(x, y) must be a
constant for every point on the aperture of the lens. TD(x, y) is the time delay, which is provided
by the lens. T(x, y) is the time it takes for the wave to travel from the focal point of the lens,
{x = 0, y = 0, z = —f), to a point at the lens aperture, (x, y, z = 0).
by (xm, ym, z = 0) (where m = 0,1,..., M — 1), then the desired time delay required from TDUs
that populate this zone can be calculated from:
TD(xm,ym)
= (y/(D/2)* +
P-rm)/c
(8.4)
Also, the desired transmission phase required from a STDU can be calculated from:
${xm, ym) = -k0[rm -f} + ko(y/(D/2)* + P - f) + $D
(8.5)
155
Having determined the time delay and phase of each for each zone, the lens design procedure
can be summarized as follows:
1. Select the desired center frequency of operation, fiens, and operational bandwidth, BWiens,
of the lens.
2. Select the desired size of the lens aperture, D, and the focal distance, / .
3. Divide the lens aperture into M concentric discrete regions or zones, where M is an arbitrary
positive integer.
4. Determine the time delay and the phase delay profiles of the lens by calculating the time
delay and phase delay required from the TDUs populating each zone using (8.4) and (8.5),
respectively.
5. Depending on the maximum variation of TD(x, y) and <5(x, y) over the lens' aperture, determine the order of the MEFSS needed to implement the time delay units.
6. Use the synthesis procedure described in chapter 6 to design the spatial time delay unit
that populates the central zone of the lens. This TDU should provide a linear transmission
phase with the steepest slope (or largest time delay) over the operational bandwidth, BWiens.
From the phase response of this TDU, determine the value of <50 at the center frequency of
operating, ftens.
7. Use the synthesis procedure described in chapter 6 to design the spatial time delay units that
populate zone m = 1,2, ...,M — 1. These TDUs should provide the required time delay
TD(xm, ym) calculated from (8.4) and phase shift &(xm, ym) calculated from (8.5).
We applied the aforementioned procedure to design a planar lens prototype shown in Fig.8.7.
The lens is designed to operate over a frequency range from 8 —10.5 GHz and has a focal length of
/ = 30 cm and aperture diameter of D = 18.6 cm. With these specifications, the maximum time
delay variation required over the aperture of the lens is 46.95 psec. The desired time delay profile of
the lens, along the aperture radius is shown in Fig.8.2. This time delay range can be achieved from
156
a fourth-order MEFSS that is designed to have a linear phase response across the frequency range
of interest. The different time delays are obtained by changing the center frequency of operation
and the fractional bandwidth of the fourth-order MEFSS. In achieving that, all the capacitive and
inductive layers of the unit cell of a fourth-order MEFSS must be optimized.
In this lens prototype, the lens' aperture is divided into 12 discrete zones, as illustrated in Fig.
8.7. Each zone is populated by TUDs of the same type with unit cell physical dimensions of 6 mm
x 6 mm. As can be seen, this lens can provide a smooth time delay variation over its aperture. The
time delay units used in this lens are designed based on the assumption that they will operate in an
infinite two-dimensional periodic structure. However, the proposed lens is inherently non-periodic,
since it uses a number of different TDUs over its aperture. Therefore, the frequency responses of
the TDUs used in the design of the lens are expected to change when placed in this operational
environment.
Following the aforementioned design procedure, the required time delay and phase shift values
are first calculated for each zone. Then, the time delay unit (MEFSS unit cell) that populates
Zone 0 of the lens is designed. The structure is then simulated using full-wave EM simulations
in CST Microwave Studio. In doing this, the MEFSS unit cell is simulated as part of an infinite
periodic structure by placing it inside a waveguide with periodic boundary condition (PBC) walls.
The structure is then excited by a plane wave and the magnitude and phase of its transmission
coefficient are calculated. The physical and geometrical dimensions of the STDU that populate
Zone 0 are used as a reference (Z0-TDU) for designing the TDUs that populate other zones of
the lens. Since the TDUs of each zone must have a different time delay and phase shift, they
are designed by slightly de-tuning the operating frequency and fractional bandwidth of the Z0TDU to achieve different slopes of the transmission phase. Each slope corresponds to a different
time delay. Therefore, for a given frequency range, a relative time delay and phase shift can be
achieved. Having achieved the desired relative time delays and phase shifts, the time delay units
are then populated over their corresponding zones. The magnitude and phase responses of each
TDU are functions of angle and the polarization of incidence of the EM wave. Therefore, when
designing TDUs, these effects are taken into account. Since all these TDUs operate over relatively
157
18.6 cm
Figure 8.7 Topology of the MEFSS-based true-time-delay lens prototype with a circular aperture.
The lens aperture is divided into a number of concentric zones populated with identical spatial
time-delay units.
small incidence angles (less than 20°), they provide almost identical phase responses under oblique
incidence angles for the transverse electric (TE) and transverse magnetic (TM) polarizations. This
design process is followed to design true-time-delay units that operate over a frequency range
8 — 10.5 GHz. Over this frequency range, the transmission phase response of the designed TDUs
are approximately linear. The insertion loss of these TDUs varies within 0 — 2 dB. The 8 — 10.5
GHz frequency range spans a fractional bandwidth of 27% and is considered as the bandwidth over
which the lens will operate as true-time-delay lens.
158
Figure 8.8 Calculated received power in the vicinity of the expected focal point of the circular
true-time-delay lens prototype. The received power is calculated over a rectangular grid centered
at the expected focal point (x = 0 cm, y = 0 cm, z = —30 cm). In all of the figures, the
horizontal axis is the x axis with units of [cm] and the vertical axis is the z axis with units of
[cm]. The color bar values are in dB. The x marker in all of these figures represents the exact
coordinate where the received power maxima occur, (a) 8.0 GHz, (b) 8.5 GHz, (c) 9.0 GHz, (d)
9.5 GHz, (e) 10.0 GHz, (f) and 10.5 GHz.
8.3
Numerical Verification
The lens prototype discussed in the previous section is modeled using the same modeling
method discussed in Chapter 7. In this case, however, a circular two-dimensional antenna array is
assumed. Each element of this array is assumed to be an electrically-small Hertizan dipole with a
given amplitude and phase. The amplitude and phase of each radiating element of this array can be
determined from the simulated values of insertion loss and transmission phase of each TDU. The
dimensions of the spatial TDUs are 6 mm x 6 mm or equivalently 0.2A0 x 0.2A0. The overall array
gain of this model is evaluated in the vicinity of the expected focal point. The received power in
the vicinity of its expected focal point (x = 0, y = 0, z = —30 cm) as function of frequency are
159
0-10-
0-
-•-8=0°
—•—6=15°
/
- ^ — 8 =30°
A /x/A^^S^
-10-
-•-8=0°
- * - 8 =30°
-20- —T— 8 =45°
— * — 8 =60°
- 2 0 - — T - 8 =45°
— * — 8 =60°
-30-
-30-
-40-
-*o-
Offw>i\li V V
tf™ ii i
-50-60-
'
1
v•
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T A A/x*^ s 'v^
—•—8=15°
-50-
/Vinrwrl
MkK/u
^
1l A V2^
II J V y L*W \Y( W 1? V J I
-60-
^-
-15
0
15
Angle [deg]
30
45
60
75
-75
-60
-45
-30
-15
0
15
Angle [deg]
30
45
60
75
(b)
(a)
0-10-
-«-e=o°
— • — e =15°
/
—=v— e =30°
A A /Y^**-*
-20- — ^ — 8 = 4 5 °
— * — 8 =60°
-30-40-50-
^^fh^^^mfxmlM
V
"WtmnfWW ¥ W f
-60-75
Y
'
-60
-45
+
-30-15
0
15
Angle [deg]
—•—8=15°
/
8 =30°
-20- —T— 8 =45°
— * — 8 =60°
/ l
I
f\
I Ii4(l
-20- -»«— 8 =45°
—*— 8 =60°
JKlx
-40-
Sflfflli^^Mflf «
-50-
Tf*¥ v r w 1
-60-75
ji
-60
-45
Y
*V
a
-
60
75
V \r
-30-15
0
15
Angle [deg]
-30-
-50-
30
45
60
/
— * - 8 =30°
(\m
if A /vu^-'-s
y^J \f$T Wf\J*
*
f\L V\J )n AIVT^WJ V ^ \ •
-40-
i ii
(e)
— • - 8 =15°
AA A/^^S
-10-
AJL fiMftLlAfiOwytn\ *>/*\~
-30-
45
-»-8=0°
-•-8=0°
—±-
30
(d)
(c)
-10-
V \f~
75
-60-75
?/]fO/lll}V\[ I V
-30
-15
0
15
Angle [deg]
60
75
(f)
Figure 8.9 Calculated received power pattern on the focal arc of the circular true-time-delay lens
prototype when excited with plane waves arriving from various incidence angles, (a) 8.0 GHz, (b)
8.5 GHz, (c) 9.0 GHz, (d) 9.5 GHz, (e) 10.0 GHz, (f) and 10.5 GHz.
plotted over a square 8 cm x 8 cm grid centered at the expected focal point. Figure 8.8 shows the
calculated focusing response of this prototype over the frequency range of 8 — 10.5 GHz with 0.5
GHz increments. At each frequency, the location where the maximum value of the received power
is achieved is indicated. As is observed from Fig.8.8, the focal point of the lens remains stationary
as frequency is changed within the entire operational bandwidth of this lens.
160
The power patterns of the lens prototypes are also calculated at the focal arc and under oblique
angles of incidence. These power patterns are calculated under plane wave illumination with incidence angles of 0°, 15°, 30°, 45°, and 60°. These calculations are then conducted over the
frequency range 9.5 GHz-11.5 GHz with 0.5 GHz increments. Figure 8.9 show the calculated
power patterns of the circular lens prototype on the focal arc over the operational bandwidth and
for angles of incidence 0°- 60°. All the values of the calculated power patterns are normalized to
the peak value of the focusing at 8 GHz and under normal plane wave illumination. As shown
in Fig. 8.9, the power pattern on the focal arc is steered by an angle equal to that of the incident
angle. The peak values of these patterns decrease as the angle of incidence increases. As can be
seen, the focusing performance of the true-time-delay lens does not deteriorate over the operational
bandwidth of the lens. It is also observed that the beam-width of the power pattern on the focal
arc becomes narrower as the frequency of the plane wave excitation is increased, as shown in Fig.
8.9(a)-(f).
8.4
Conclusions
A new technique for designing true-time-delay planar microwave lenses is introduced in this
chapter. This technique relies upon using true-time-delay units rather than spatial phase shifters
that provide a constant phase shift. The proposed lens uses the unit cells of miniaturized-element
frequency selective surfaces. These unit cells are designed to operate as spatial true-time-delay
units. This is achieved by designing the unit cell such that the phase of its transmission coefficient
is linear over the frequency range of interest. This will result in a constant group delay as a
function of frequency. Different group delay values are achieved by changing the slope of the
transmission phase of the MEFSS unit cells. A prototype that approximates the time-delay profile
of a conventional convex dielectric lens was designed following the proposed procedure. In this
lens, the prototype is divided into twelve discrete zones. The prototype operates over a relatively
wide band of operation with a fractional bandwidth of 27%. A simplified model to predict the lens
performance is used. In this model, the lens was treated as two-dimensional antenna array with a
circular shaped aperture composed of Hertzian dipole radiators. The simplified model predicts that
161
a stationary focal distance is maintained over the entire operational bandwidth of the true-timedelay lens.
162
Chapter 9
Study of Power Handling Capability of Miniaturized-Element Frequency Selective Surfaces
9.1
Introduction
Despite the significant body of research that has been conducted in the area of periodic struc-
tures and frequency selective surfaces, these investigations have primarily been focused on developing new design techniques and improving the FSS performances. Studying the power handling
capability of frequency selective surfaces has not received much attention despite its importance
for high power microwave applications [57] and [184]. Munk has addressed this aspect in his book
(see chapter 10 in [57]). He points out that FSSs have limited power-handling capability due to
two main reasons: dielectric breakdown and heat dissipation. This is in part due to the use of
resonating elements as their building blocks. Inside a typical FSS, the local electric field intensity can be higher than that of the incident wave by several orders of magnitudes. He concluded
that to improve the power handling capabilities of FSSs with resonant elements, the inter-element
spacing should be as small as possible. However, in the case of FSSs populated with dipole-type
elements, the inter-element spacing should not be reduced to a point where the tips of the elements
are very close to each other. This close spacing will significantly enhance the electric field inside
the structure, thereby reducing its power handling capability.
In this chapter, we study the power handling capabilities of the frequency selective surfaces
presented in Chapters 5 and 6. The power handling capability is limited by the failure of the
material used in these structures. This failure is caused by two factors: 1) heat dissipation and
2) dielectric breakdown. With regards to the heat dissipation, failure occurs when the structure
163
operates under sustained high power levels. In this case, lossy materials will heat up and the
surface will eventually melt or burn. This type of failure happens in communication systems
when the average power of the transmitted signal becomes too high. On the other hand, dielectric
breakdown occurs when the structure is under very high peak power levels. In this case, the electric
field strength in the dielectric material exceeds break down level of the dielectric material. This
failure mechanism is expected to occur pulses with high peak power levels are transmitted. An
example of this is high-power microwave directed energy weapons, where pulses with high peak
powers but short durations are used. Therefore, the main goal of this study is to give an insight into
the effect of various design parameters on the peak and average power handling capability of the
FSS discussed in Chapters 5 and 6. To achieve this goal, combined electromagnetic and thermal
simulations in CST Microwave Studio and Multiphysics are used. In the next two sections, the
dielectric breakdown and heat dissipations issues are discussed in detail.
9.2
Dielectric breakdown caused by the electric field
When the electric field intensity inside the FSSs discussed in Chapters 6-8 exceeds a certain
value, breakdown will eventually occur. This is due to the fact that the electric field strength within
the FSS unit cell can be much larger than that of the incident field.
First, we will establish a relationship between the electric field intensity of the incident wave
and that at/inside the FSS. To achieve that, we define two field enhancement factors. The first
factor, maximum field enhancement factor (MFEF), is defined as the ratio of the maximum electric
field intensity at/inside the FSS to that of the incident wave. The second factor, surface field
enhancement factor (SFEF), is defined as the ratio of the electric field intensity at the surface of the
FSS to that of the incident field. Therefore, if values of MFEF and SFEF are equal, this means that
maximum electric field enhancement occurs at the surface of the structure. These two factors can
be used to approximate the power levels at which breakdown is expected to occur. Additionally,
they can be used to compare the power handling capability of different FSS designs to one another.
The lower the values of MFEF and SFEF are, the higher the power handling capability of the FSS
will be. Next, we study the effect of different parameters on the values of MFEF and SFEF. The
164
BW%
2nd-order
3rd-order
44fc-order
10
38.9/38.9
34.9/27.0
25.7/19.6
20
13.3/13.3
17.6/9.7
16.5/8.1
30
10.6/10.6
11.6/10.3
13.1/6.1
Table 9.1 Field enhancement factors of MEFSSs with different Butterworth responses. The field
enhancement factor is calculated for different response orders and fractional bandwidths.
BW%
2nd-order
3rd-order
4th-order
10
46.5/46.5
39.6/39.6
42.9/42.2
20
15.3/15.3
23.6/23.6
20.4/19.5
30
7.6/7.6
12.9/11.8
12.7/9.2
Table 9.2 Field enhancement factors of FSSs with different Chebyshev responses. The field
enhancement factor is calculated for different response orders and fractional bandwidths.
parameters studied in this section are the types of transfer functions, their orders, and fractional
bandwidths. The design procedure presented in Chapter 6 is used to design the different FSSs.
RO4003 dielectric substrates are used to separate the different inductive and capacitive layers.
The unit cell sizes in all these structures are kept the same with dimension of 6.5 mm x 6.5
mm. The details of the physical parameters of these FSSs are listed in Appendix A. Values of
MFEF and SFEF for different fractional bandwidths and orders are listed in Tables 9.1 and 9.2 for
miniaturized elements frequency selective surfaces (MEFSSs) with Butterworth and Chebyshev
frequency responses, respectively. In these tables, the first (second) number refers to the value
of MFEF (SFEF). For both types of responses, as the fractional bandwidth increases, both MFEF
and SFEF decrease. This is due to the fact that small fractional bandwidths require high quality
factor resonators that are coupled to each other. For a parallel LC resonator, higher quality factor
is achieved by increasing the capacitor value. Hence, a smaller gap is needed to achieve higher
capacitance values. Based on this study, we conclude that the wider the bandwidth of operation of
these structures, the higher the transient power that the proposed FSS can handle.
165
Unit Cell Size
Butterworth
Chebyshev
5 mm
33.5/33.5
42.2/42.2
5.5 mm
26.1/26.1
29.0/29.0
6 mm
18.6/18.6
22.4/22.4
6.5 mm
13.3/13.3
15.3/15.3
Table 9.3 Field enhancement factor of MEFSSs as a function of unit cell size. All the MEFSSs
are designed to have a second-order bandpass response operating at 10 GHz with 20% bandwidth.
We also studied the effect of unit cell size on the MEFSS power handling capabilities. For
this study, we used a second-order MEFSS operating at 10 GHz with a 20% fractional bandwidth
and either Butterworth or Chebyshev response. Different unit cell sizes are used to obtain the
same frequency response. The field enhancement factors are calculated for both responses and
the results are listed in Table 9.3. Despite the fact that the frequency response has not changed,
both enhancement factors decrease as the unit cell size increases. This is attributed to the fact
that larger gaps between the capacitive patches are required, for larger unit cell size, to achieve
the same capacitance value. This concludes that gap spacing between the capacitive patches is the
determining factor of dielectric breakdown in these structures.
To understand how to use MFEF and SFEF in estimating the power handling capabilities of the
MEFSS, let's use the Butterworth fourth-order MEFSS with a fractional bandwidth of 20%. For
this MEFSS, the values of MFEF and SFEF are 16.5 and 8.1, respectively. Since these two values
are different, this means that maximum field enhancement occurs inside the MEFSS. Hence, the
power handling capability must be studied inside and at the surface of the MEFSS. For this MEFSS,
the maximum power it can handle is limited by either the breakdown field intensity of RO4003
substrate (31.2 MV/m) or the breakdown field intensity of air (3 MV/m). Since the exact value of
the electric field intensity that causes the dielectric breakdown is not accurately known, a safety
margin of 20 is chosen. This means that the maximum electric field intensity enhanced at/inside
the MEFSS is not allowed to exceed more than 5% of the breakdown field intensity. Using the
breakdown field intensity of RO4003 and the value of MFEF, the maximum electric field intensity
166
inside the MEFSS is 95 KV/m. This corresponds to an incident electromagnetic wave with a
maximum transient power density of 11.8 MW/m2. Whereas, using the breakdown field intensity
of air and the value of SFEF, the maximum electric field intensity at the surface of the MEFSS
is 18.5 KV/m. This corresponds to an incident electromagnetic wave with a maximum transient
power density of 0.45 MW/m2. Therefore, the maximum transient power that this MEFSS can
handle should not exceed the minimum value of the two power densities, which is 0.45 MW/m2.
In another example, let's use the Chebyshev second-order MEFSS with a fractional bandwidth
of 10%. This structure has the highest field enhancement factors of 47.3, and thus, it has the lowest
power handling capability. Since both MFEF and SFEF are equal, the maximum electric field
enhancement occurs right at the surface of the MEFSS, and thus, it is limited by the breakdown
field intensity of the air (3 MV/m). With a safety margin of 20, the maximum electric field intensity
at the surface of the structure is 150 KV/m. This corresponds to an incident electromagnetic wave
with a maximum transient power density of 13 KW/m2. To increase the maximum power density
rating, a very thin dielectric superstrate (A0/200) is added on both sides of the structure, where
A0 is the free-space wavelength. By doing so, the field enhancement factor at the surface of the
MEFSS is reduced to 20, thereby increasing the maximum transient power density of 75 KW/m2.
Therefore, the MEFSSs of the type discussed in Chapter 6 are expected to be capable of handling
very high transient power levels.
9.3
FSS failure due to heat dissipation
Under sustained high power excitation, the lossy dielectric substrates as well as the conduc-
tance of metallic structures of the MEFSSs generate heat. If this heat is not dissipated, the temperature of the MEFSS rise and the material will eventually burn. The greatest amount of heat is
usually developed in the region where the electric current density is maximized.
Similar to studies conducted in the previous section, we studied the effect of the same parameters on the temperature rise inside the MEFSSs. At room temperature, 300° K, an average incident
power of 2.2 KW/m2 is incident on the different MEFSSs. The temperature rise inside the MEFSSs
are then calculated using CST Multiphysics thermal simulation. The results are listed in Tables 9.4
167
and 9.5 for MEFSSs with Butterworth and Chebyshev frequency responses, respectively. As is observed, when the fractional bandwidth increases, the temperature rise decreases. This is attributed
to the fact that, for the same incident power level, the electric field intensity inside the narrow-band
MEFSS is enhanced more than that of a wider band (see Table 9.1 and 9.2). Hence, more power is
dissipated in the narrow band MEFSSs.
The effect of unit cell size on the temperature rise of the MEFSSs is also studied. In this study,
a second-order MEFSS operating at 10 GHz with 20% fractional bandwidth is used. Operating
under sustained power density of 2.2 KW/m2, the temperature rise of the MEFSS is calculated for
both Butterworth and Chebyshev responses, and the results are listed in Table 9.6. The temperature
rise is relatively the same (about 32° K), despite the change in the unit cell dimension. As unit cell
size increases, the average power pumped into each unit cell increases. However, as shown in
Table 9.3, the electric field enhancement decrease as the unit cell size increases. Hence, for a given
incident power pumped into each unit cell, the dissipated power is decreased. So as the unit cell
size increases, the effect of these two factors tend to cancel each other, and thus, the level of the
dissipated power remains relatively the same.
To estimate the maximum continuous power that these structures can handle before melting,
let's use the Chebyshev fourth-order FSS with 10% fractional bandwidth. Under the same incident
power density, this structure has the largest temperature rise. For an electromagnetic wave with
average power density of 2.2 KW/m2 is incident on this FSS, the maximum temperature rises inside
BW%
2nd-order
3rti-order 4th -order
10
74.5
61.25
87.7
20
31.75
30.0
44.8
30
20.17
21.9
33.67
Table 9.4 Temperature rise of MEFSSs with different Butterworth responses. The temperature
rise is caused by an average power of 2.2 KW/m2 that is incident on the different MEFSSs. The
temperature rise is calculated for different response orders and fractional bandwidths.
168
BW%
2nd-order
3rd-order
4t/l-order
10
79.9
71.15
109
20
32.5
33.17
49.7
30
20.7
25.1
34.1
Table 9.5 Temperature rise of MEFSSs with different Chebyshev responses. The temperature rise
is caused by an average power of 2.2 KW/m2 that is incident on the different MEFSSs. The
temperature rise is calculated for different response orders and fractional bandwidths.
the structure to 409° K (138° C). However, this temperature is still below the melting point of the
RO4003 material which is about 698° K (425° C).
9.4
Conclusions
A numerical study investigating the power handling capability of the MEFSSs discussed in
Chapters 5 and 6 is presented. The power handling capability of MEFSSs are limited by the failure
of the material used in these structures. Material failure occurs due to either dielectric breakdown
or heat dissipation. The study on the dielectric breakdown reveals that the gap between the subwavelength rectangular patches of the capacitive layers is the the main factor in determining the
maximum power handling of the structure. The smaller the gap between these patches are, the
lower the power these structures can handle will be. Nevertheless, the MEFSSs are expected to
Unit Cell Size
Butterworth
Chebyshev
5 mm
32.6
31.9
5.5 mm
32.03
33.35
6 mm
31.68
32.11
6.5 mm
31.76
32.5
Table 9.6 Temperature rise of MEFSSs with two different response types as a function of unit cell
size for an incident power density of 2.2 KW/m2. All the MEFSSs are designed to have a
second-order bandpass response operating at 10 GHz with 20% bandwidth.
169
be able to handle high-peak pulsed power up to 75 KW/m2. Since the exact field levels that cause
breakdown are not known in calculating this number, a safety factor of 20 is used. The study on
the heat dissipation breakdown, however, reveals that high-Q FSSs produce more heat than others.
Based on the thermal simulation conducted using CST Multiphysics Studio, a sustained average
RF power of 2.2 KW/m2 could rise the temperature of the MEFSS to 138° C. This temperature is
well below the melting temperature of the dielectric material used, but high enough to require the
use of a suitable thermal management system.
170
APPENDIX
Physical dimensions of MEFSSs reported in Chapter 8
171
Parameter
BW%
Value
™1,2
si = s 2
10%
3.3 mm
0.15 mm
0.315 mm
Value
20%
2.7 mm
0.5 mm
0.592 mm
Value
30%
2.26 mm
0.91mm
0.836 mm
ha{l,2}
~ hj3{l,2}
Table A. 1 Physical parameters of a second-order MEFSS with a Butterworth response operating
at 10 GHz with different fractional bandwidths. The unit cell dimensions of all designed
prototypes are 6.5 mm x 6.5 mm. The substrates have a dielectric constant of er= 3.38. w1>2, si,
$2, ha{it2}, andfyg{i,2}refer to the geometrical parameters identified in Fig. 6.1 of Chapter 6.
Parameter
BW%
Value
_
Wl,2
Sl = s 2
10%
3.325 mm
0.115 mm
0.272 mm
Value
20%
2.67 mm
0.385 mm
0.51 mm
Value
30%
2.22 mm
0.81mm
0.721 mm
^a{l,2}
^8(1,2}
Table A.2 Physical parameters of a second-order MEFSS with a Chevyshev response operating at
10 GHz with different fractional bandwidths. The unit cell dimensions of all designed prototypes
are 6.5 mm x 6.5 mm. The substrates have a dielectric constant of er= 3.38. u>i)2, si, s 2 , ^a{i,2},
and /i/3{i,2} refer to the geometrical parameters identified in Fig. 6.1 of Chapter 6.
Parameter
D
™1,2
si = s 2
Value
5 mm
2.4 mm
0.135 mm
Value
5.5 mm
2.55 mm
0.2 mm
Value
6 mm
2.62 mm
0.32 mm
Value
6.6 mm
2.7 mm
0.5 mm
Table A.3 Physical parameters of a second-order MEFSS with Butterworth response operating at
10 GHz with a fractional bandwidth of 20%. The physical parameters are for different unit cell
dimensions. The substrates have a dielectric constant of er= 3.38 and a thickness of
^a{i,2}==fys{i,2}=0.592 mm. D, tuli2, si, s 2 , hQ{ii2}, andfyg{i,2}referto the geometrical parameters
identified in Fig. 6.1 of Chapter 6.
172
Parameter
D
W\,2
s\ = s2
Value
5 mm
2.4 mm
0.1 mm
Value
5.5 mm
2.55 mm
0.165 mm
Value
6 mm
2.625 mm
0.245 mm
Value
6.6 mm
2.67 mm
0.385 mm
Table A.4 Physical parameters of a second-order MEFSS with a Chebyshev response operating at
10 GHz with a fractional bandwidth of 20%. The physical parameters are for different unit cell
dimensions. The substrates have a dielectric constant of er= 3.38 and a thickness of
^oi{i,2}=^/3{i,2}=0.51 mm. D, wij2, s\, s2, foa{i,2}, and /i/3{i,2} refer to the geometrical parameters
identified in Fig. 6.1 of Chapter 6.
Parameter
D
Value
6.5 mm
0.421 mm
0.722 mm
0.722 mm
Parameter
fys{2,3}
er
si
S2
Value
0.421 mm
3.38
0.24 mm
0.34 mm
Parameter
S3
Wi}2
W2,3
Value
0.24 mm
3.12 mm
3.12 mm
ha{i,2}
V{1,2}
h
a{2,3}
Table A.5 Physical parameters of a third-order MEFSS with a Butterworth response operating at
10 GHz with a fractional bandwidth of 10%. D, wlj2, w2t3, s1, s2, s 3 , /ja{i,2},fys{i,2}>^a{2,3}, and
hp{2,3} refer to the geometrical parameters identified in Fig. 6.1 of Chapter 6.
Parameter
D
^a{l,2}
hj3{i,2}
^a{2,3}
Value
6.5 mm
0.815 mm
0.628 mm
0.628 mm
Parameter
fya{2,3}
er
si
S2
Value
0.815 mm
3.38
0.83 mm
0.38 mm
Parameter
S3
Wl,2
^2,3
Value
0.83 mm
2.34 mm
2.34 mm
Table A.6 Physical parameters of a third-order MEFSS with a Butterworth response operating at
10 GHz with a fractional bandwidth of 20%. D, w1:2, w2f3, s\, s2, s 3 , /ia{i,2},fys{i,2},^a{2,3}, and
hi3{2,3} refer to the geometrical parameters identified in Fig. 6.1 of Chapter 6.
173
Parameter
D
Value
6.5 mm
Parameter
ha{i,2}
fys{l,2}
^a{2,3}
1.23 mm
0.505 mm
0.505 mm
fy(3{2,3}
er
Sl
S2
Value
1.23 mm
3.38
1.26 mm
0.48 mm
Parameter
•S3
W\,2
^2,3
Value
1.26 mm
1.86 mm
1.86 mm
Table A.7 Physical parameters of a third-order MEFSS with a Butterworth response operating at
10 GHz with a fractional bandwidth of 30%. D, u>i}2, u>2,3, Si, s 2 , s 3 , /^{i^},fys{i,2}>^{2,3}, and
fys{2,3} refer to the geometrical parameters identified in Fig. 6.1 of Chapter 6.
Parameter
D
ha{i,2}
fya{i,2}
^a{2,3}
Value
6.5 mm
0.288 mm
0.734 mm
0.734 mm
Parameter
^3(2,3}
er
Sl
S2
Value
0.288 mm
3.38
0.145 mm
0.32 mm
Parameter
•S3
Wl,2
^2,3
Value
0.145 mm
3.14 mm
3.14mm
Table A.8 Physical parameters of a third-order MEFSS with a Chebyshev response operating at
10 GHz with a fractional bandwidth of 10%. D, iu1)2, w2,3, s l5 s 2 , s 3 , /ia{i,2}, ^{i,2}j ^Q{2,3}, and
fys{2,3} refer to the geometrical parameters identified in Fig. 6.1 of Chapter 6.
Parameter
D
Value
6.5 mm
Parameter
ha{i,2}
fy?{l,2}
^a{2,3}
0.551 mm
0.664 mm
0.664 mm
fys{2,3}
er
Sl
S2
Value
0.551 mm
3.38
0.5 mm
0.4 mm
Parameter
•S3
Wl,2
^2,3
Value
0.5 mm
2.41 mm
2.41 mm
Table A.9 Physical parameters of a third-order MEFSS with a Chebyshev response operating at
10 GHz with a fractional bandwidth of 20%. D, tuli2, u>2j3, si> s 2, S3, ^a{i,2},fy?{i,2}»^«{2,3}, and
^{2,3} refer to the geometrical parameters identified in Fig. 6.1 of Chapter 6.
174
Parameter
D
Value
6.5 mm
Parameter
fy(3{l,2}
^a{2,3}
0.817 mm
0.573 mm
0.573 mm
fy?{2,3}
er
S\
S2
Value
0.817 mm
3.38
0.72 mm
0.37 mm
Parameter
«3
Wl>2
™2,3
Value
0.72 mm
1.84 mm
1.84 mm
ha{\,2}
Table A. 10 Physical parameters of a third-order MEFSS with a Chebyshev response operating at
10 GHz with a fractional bandwidth of 30%. D, tu1)2, IL>2,3, Si, s 2 , S3, ^{1,2},fys{i,2},^a{2,3}5 and
h/3{2,3} refer to the geometrical parameters identified in Fig. 6.1 of Chapter 6.
Parameter
ha{\,2}
h(3{l,2}
^{2,3}
fy8{2,3}
Value
0.548 mm
0.722 mm
0.737 mm
0.737 mm
Parameter
^Q{3,4}
fys{3,4}
er
Sl,S4
Value
0.722 mm
0.548 mm
3.38
0.325 mm
^12,^34
^23
D
3.05 mm
3.15 mm
6.5 mm
Parameter
Value
S2,S3
0.268 mm
Table A. 11 Physical parameters of a fourth-order MEFSS with a Butterworth response operating
at 10 GHz with a fractional bandwidth of 10%. D, w\t2, w2,3, ^3,4> si, s2, s3, s 4 , /iQ{i,2}, fyg{i,2}>
ha{2,3}> ^{2,3}, ^a{3,4}» and ^{3,4} refer to the geometrical parameters identified in Fig. 6.1 of
Chapter 6.
175
Parameter
^a{l,2}
fys{l,2}
^a{2,3}
fy3{2,3}
Value
1.07 mm
0.61 mm
0.656 mm
0.656 mm
Parameter
^a{3,4}
^/3{3,4}
er
S\, S4
Value
0.61 mm
1.07 mm
3.38
0.37 mm
Parameter
•S2,S3
^12,^34
W2Z
D
Value
1.1 mm
2.25 mm
2.3 mm
6.5 mm
Table A. 12 Physical parameters of a fourth-order MEFSS with a Butterworth response operating
at 10 GHz with a fractional bandwidth of 20%. D, iyli2,102,3, "^3,4> -Si> s2, s 3 , s 4 , /i^i^}, fy?{i,2}>
^•a{2,3}5fys{2,3}>^a{3,4}5 and ^{3,4} refer to the geometrical parameters identified in Fig. 6.1 of
Chapter 6.
Parameter
^a{l,2}
Value
1.63 mm
Parameter
^«{3,4}
Value
fys{l,2}
0.445 mm
^a{2,3}
fy(3{2,3}
0.689 mm
0.689 mm
j3{3,4}
€r
0.445 mm
1.63 mm
3.38
1.7 mm
Parameter
•S2,S3
^12,^34
W23
D
Value
0.47 mm
1.52 mm
1.8 mm
6.5 mm
h
Sl,
s4
Table A. 13 Physical parameters of a fourth-order MEFSS with a Butterworth response operating
at 10 GHz with a fractional bandwidth of 30%. D, iy1)2,102,3, ^3,4> Si5 s2> s 3 , s 4 , /i^i^}, fys{i,2}5
^a{2,3}> ^/3{2,3}5 ^a{3,4}5 and ^{3,4} refer to the geometrical parameters identified in Fig. 6.1 of
Chapter 6.
Parameter
ha{i,2}
fys{l,2}
^a{2,3}
fys{2,3}
Value
0.307 mm
0.747 mm
0.74 mm
0.74 mm
Parameter
ha{3,4}
^3(3,4}
er
Sl, s 4
Value
0.747 mm
0.307 mm
3.38
0.15 mm
Parameter
•S2,S3
Wi2, ™34
^23
D
Value
0.3 mm
3.2 mm
3.1 mm
6.5 mm
Table A. 14 Physical parameters of a fourth-order MEFSS with a Chebyshev response operating
at 10 GHz with a fractional bandwidth of 10%. D, wit2, w2,3, u>3)4, s l5 S2, S3, s 4 , ha{it2}, fys{i,2},
^Q{2,3},fyg{2,3},^a{3,4}, and ^{3,4} refer to the geometrical parameters identified in Fig. 6.1 of
Chapter 6.
Parameter
^a{l,2}
fy3{l,2}
^a{2,3}
V{2,3}
Value
0.585 mm
0.689 mm
0.66 mm
0.66 mm
Parameter
^a{3,4}
^/3{3,4}
er
Sl, s 4
Value
0.689 mm
0.585 mm
3.38
0.556 mm
Parameter
•S2,S3
^12,^34
™23
D
Value
0.374 mm
2.3 mm
2.36 mm
6.5 mm
Table A. 15 Physical parameters of a fourth-order MEFSS with a Chebyshev response operating
at 10 GHz with a fractional bandwidth of 20%. D, u>ij2, ui2,3, ^3,4, s\, s2, s 3 , s 4 , ha{1}2}, ^/?{i,2},
^a{2,3}, hp{2,3}, ^a{3,4}> and /i^{3,4} refer to the geometrical parameters identified in Fig. 6.1 of
Chapter 6.
Parameter
K{i,2}
Value
0.861 mm
Parameter
ha{3,4}
Value
0.611 mm
Parameter
Value
S2,S3
0.434 mm
fy3{l,2}
^Q{2,3}
h/3{2,3}
0.611mm
0.592 mm
0.592 mm
P{3A}
er
Sl, s 4
0.861mm
3.38
0.89 mm
^12,^34
W23
D
h
1.7 mm
1.84 mm
6.5 mm
Table A. 16 Physical parameters of a fourth-order MEFSS with a Chebyshev response operating
at 10 GHz with a fractional bandwidth of 30%. D, w1>2, w2,3, u>3,4, si, s2, s 3 , s 4 , ha{lt2}, fy3{i,2}>
ha{2,3}, hp{2,3}, ^a{3,4}5 and ^{3,4} refer to the geometrical parameters identified in Fig. 6.1 of
Chapter 6.
178
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