Electromagnetics in Characterizations
by
Bae-Ian Wu
B. Eng. Electronic Engineering
The Chinese University of Hong Kong, Hong Kong, June 1997
M. S. Electrical Engineering and Computer Science
Massachusetts Institute of Technology, Cambridge, June 1999
Submitted to the Department of Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2003
@
Massachusetts Institute of Technology 2003. All rights reserved.
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
JU
JU L
0
7 2003
LIBRARIES
Author
Department of Electrical Engineering and Computer Science
May 20, 2003
Certified by
-
Jin A. Kong
-- Professor of Electrical Engineering
Thesi Supervisor
Accepted by
Arthur C. Smith
Professor of Electrical Engineering
Chairman, Department Committee on Graduate Students
BARKR
Electromagnetics in Characterizations
by
Bae-Ian Wu
Submitted to the Department of Electrical Engineering and Computer Science
on May 20, 2003 in partial fulfillment of the requirements for the
Degree of Doctor of Philosophy
Abstract
A unique negative lateral shift is demonstrated in the study of a Gaussian beam either reflected from a grounded slab or transmitted through a slab with both negative
permittivity and permeability, which is distinctly different from the shift caused by a
regular slab. The incident beam is modeled as a tapered wave with a Gaussian spectrum. The waves inside and outside the slab are solved analytically from Maxwell's
equations by matching the boundary conditions at the interfaces. It is shown that the
electric and magnetic fields in all regions can be unambiguously determined. Numerical simulations are presented and the amplitudes of the fields as well as the power
densities are computed for all regions. A dramatic negative lateral shift of the beam
at the exit interface is observed when both c and p are negative.
Guided waves in an isotropic dielectric slab are analyzed and it is found that
modes with real and imaginary transverse wavenumbers can both exist depending
on the constitutive parameters of the slab. The guided modes with both real and
imaginary transverse wavenumbers inside in a symmetric dielectric slab with negative permittivity and permeability are solved. It is found that for real transverse
wavenumbers, there exist cutoffs for all modes. In addition, a guidance condition of
the modes with imaginary transverse wavenumbers in the slab is shown to exist, and
a graphical method of determining such imaginary transverse wavenumbers of the
guided modes is introduced. Propagation of guided waves inside a less dense negative
medium is shown to be possible. Time-averaged Poynting vectors in all regions are
derived and it is shown that the direction of power flow inside the slab is opposite to
the flow outside the slab.
Characterization of the differential guided mode of a coupled-strip transmission
line allows us to understand its behaviors in high frequency circuit applications. Sparameters of the differential mode of a coupled-strip transmission line on a multilayer silicon substrate extracted from 4-port measurements and simulations are deembedded by the impedance/admittance subtraction method. By accurately determining the input inductance of the connecting pads, the parameters of the transmission line itself can be de-embedded. For the specific substrate profile considered, it is
found that there is a practical upper limit on the value of the differential impedance.
Baseline estimation for synthetic aperture radar interferometry is used to refine
2
the height estimation of the resulting digital elevation map. Furthermore, preprocessing is used to reduce the effects of local phase inconsistencies caused by noise.
By incorporating the information of the ground control points in the height inversion process, the initial estimation of the baseline parameters based on the satellite
state vectors and the commonly used high order polynomial fitting can be improved.
In this study, a simulated interferogram of a 2-D terrain is generated, and different
levels of phase noise as well as uncertainties in baseline parameters are introduced.
Five control points are use in a 60 x 60 km area. The platform height is 500 km and
the frequency used is in the L-band. The residues weighted least squares method
is used to reconstruct the unwrapped phase and ground control points are used to
yield an improved baseline. Both the absolute and relative control points phase relations are used. The registered inverted height profiles are then compared to the
original terrain, and the accuracy of the inversion process is studied with respect to
the amount of injected noise. With enhancements in interferogram quality, numerical
solver, and baseline estimation, height retrieval is more accurate and the usable area
of the interferogram is increased.
Thesis Supervisor: Jin Au Kong
Title: Professor of Electrical Engineering
3
Acknowledgments
I would like to express my sincere thanks to Professor Jin Au Kong, who has given me
the opportunity to do electromagnetics research in his group. I also enjoy and treasure
being a teaching assistant in several of his classes. His passion in electromagnetics and
his kindness will always inspire me and his invaluable advices will never be forgotten.
Special thanks to my thesis readers Professor David H. Staelin and Dr. Tomasz
M. Grzegorczyk for their helpful suggestions, Dr. Yoshihisa Hara for his support of
the InSAR program, and Dr. Check F. Lee for his supervising and support during
my summer internship at Analog Devices.
I would also like to acknowledge the helps and suggestions on the the works in
this thesis from Dr. Eric Yang, Dr. Kung-Hau Ding, Dr. Jerome J. Akerson, Dr.
Shih-En Shih, Dr. Yan Zhang, Dr. Chi 0. Ao, Dr. Henning Braunisch, Dr. Fernando
L. Teixeira, and Dr. Tomasz M. Grzegorczyk.
The efforts of the members of my area and oral exam committee: Professor Jin
Au Kong, Dr. Tomasz M. Grzegorczyk, Professor Henry Smith, Professor Richard
J. Blaikie, Professor Qing Hu, Professor Alan J. Grodzinsky, and Professor Cardinal
Warde, are much appreciated.
I am grateful to the friendships and discussions from the CETA group members:
Joe Pacheco, Christopher D. Moss, Benjamin B. Barrowes, Jie Lu, Xudong Chen, and
James Chen. In particular, I want to thank Yan Zhang who guided me during his
years in the group, Chi 0. Ao who introduced me to Texture, and Henning Braunisch
who has always helped me out with my mathematics.
I want to thank my parents and my brother for everything they have done for me.
My father, in particular, has planted the seed of my interests in electromagnetics.
To Marian, my wife, and our daughter, Anna, thank you for your patience and love.
Finally, I would like to dedicate this thesis to my family.
4
C'est le temps que tu as perdu pour ta rose qui fait ta rose si importante
5
6
Contents
1
Introduction
19
2
Gaussian Beam Shift in Left-Handed Metamaterials
23
2.1
Introduction to Left-Handed Materials . . . . . . . . . . . . . . . . .
2.1.1
Plane wave propagation in LHM and reflection at an air-LHM
interface ......
2.1.2
3
..............................
25
Dispersive nature of materials with negative c and p and modeling of dispersive materials in 2-D FDTD . . . . . . . . . . .
30
2.2
Analytical solution of waves in a slab upon a Gaussian beam incidence
35
2.3
Lateral shift of a Gaussian beam
. . . . . . . . . . . . . . . . . . . .
40
2.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
Guided Modes in a Slab Waveguide with Negative Permittivity and
49
Permeability
4
23
................................
49
3.1
Introduction .......
3.2
Guidance conditions for a symmetric dielectric slab ..........
52
3.3
Guided modes with real transverse wavenumber . . . . . . . . . . . .
54
3.4
Guided modes with imaginary transverse wavenumber . . . . . . . . .
56
3.5
Fields and Poynting vectors of guided waves . . . . . . . . . . . . . .
60
3.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
Guided Waves in Coupled-Strip Differential Transmission Line
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
63
63
4.2
4.3
4.4
Measurements and calibrations
. . . . . . . . . . . . . . . . . . . . .
64
4.2.1
Geom etries
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.2.2
Measurements of 4-port S-parameters . . . . . . . . . . . . . .
68
4.2.3
Equipments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.2.4
Standard definition table and calibration . . . . . . . . . . . .
70
De-embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
4.3.1
Impedance/admittance matrix subtraction method
. . . . . .
73
4.3.2
Transfer matrix method
. . . . . . . . . . . . . . . . . . . . .
73
4.3.3
Verification
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
Results and analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.4.1
S-parameters
. . . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.4.2
Impedance and loss . . . . . . . . . . . . . . . . . . . . . . . .
86
4.5
2 1/2-D EM solver for interconnects designs
. . . . . . . . . . . . . .
92
4.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
5 Height Retrieval in SAR Interferometry
5.1
5.2
Introduction to Interferometric SAR processing
97
. . . . . . . . . . . .
97
5.1.1
Phase unwrapping
. . . . . . . . . . . . . . . . . . . . . . . .
99
5.1.2
Height inversion and foreshortening . . . . . . . . . . . . . . .
101
Weighted least squares phase unwrapping method with Preconditioned
Conjugate Gradient (PCG) solver . . . . . . . . . . . . . . . . . . . .
5.2.1
5.3
Comparison of PCG method with Picard iteration for ERS-1
data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
Preprocessing of interferogram . . . . . . . . . . . . . . . . . . . . . .
114
5.3.1
Phase unwrapping of SAR data from JERS-1
117
5.3.2
Improvement of noise tolerance level by preprocessing in conjunction with PCG method
5.4
104
. . . . . . . . .
. . . . . . . . . . . . . . . . . . . 122
Baseline estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
130
5.4.1
Addition of noise . . . . . . . . . . . . . . . . . . . . . . . . .
131
5.4.2
Baseline inversion . . . . . . . . . . . . . . . . . . . . . . . . .
131
8
5.5
5.4.3
Incorporation of absolute and relative GCP information . . . . 132
5.4.4
Simulation results on baseline inversion without preprocessing
5.4.5
Baseline inversion with preprocessing . . . . . . . . . . . . . . 140
134
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Summary
145
Bibliography
149
Biographical note
160
6
9
10
List of Figures
1-1
Relationships between the different chapters in the thesis . . . . . . .
20
2-1
Backward wave in a left-handed medium. . . . . . . . . . . . . . . . .
27
2-2
TE wave incidence upon a left-handed material. . . . . . . . . . . . .
28
2-3
Dependence of the permittivity as a function of frequency.
. . . . . .
32
2-4
Computational cell (i,
in the 2-D FDTD computational domain. .
34
2-5
Electric field amplitude upon a line source incidence on a slab with
j)
negative c and p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2-6
Lateral shift of a ray from geometrical optics.
36
2-7
Configuration of a Gaussian beam incident upon a slab with thickness
. . . . . . . . . . . . .
d = d2 - di . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-8
Reference time-averaged power density on the xz-plane for a 300 incidence of a Gaussian beam with ci = co and p = po. . . . . . . . . . .
2-9
37
41
Time-averaged power density on the xz-plane for a 30' incidence of a
Gaussian beam upon a slab of thickness d = 4A with ci = -Eo
P 1 = -po.
and
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2-10 Time-averaged power density on the xz-plane for a 30' incidence of a
Gaussian beam upon a slab of thickness d = 4A with Ei = -6o and
p = 0.1/o.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2-11 Comparison of time-averaged power density along x at z = d for a 300
incidence of a tapered beam upon a slab of thickness d. . . . . . . . .
43
2-12 Reference time-averaged power density on the xz-plane for a 30' incidence of a Gaussian beam with ci =
11
6o
and
[1i
= po. . . . . . . . . . .
45
2-13 Time-averaged power density on the xz-plane for a 300 incidence of a
Gaussian beam upon a grounded slab of thickness d = 6A with c, = -Co
and p, = - po. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2-14 Comparison of time-averaged power density along x at z = 0 for a 30'
incidence of a tapered beam upon a slab of thickness d. . . . . . . . .
46
3-1
Configuration of a slab waveguide of thickness d. . . . . . . . . . . . .
53
3-2
Graphical determination of k 2xd, even modes. Circles indicate cutoffs.
P1//P2
3-3
=
-0.5.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Graphical determination of k2xd, odd modes. Circles indicate cutoffs.
p1/P2
=
- 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
P2 < 0). . . . .
58
3-4
Different zones of possible constitutive parameters
3-5
Graphical determination of a 2 d for different values of
mode.
3-6
(E2,
-P1/P
2
,
Cosh
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Graphical determination of aC2d for different values of
mode.
4-1
55
-p1/P2,
59
Sinh
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
Typical cross section of a coupled-strip transmission line on a multilayer substrate. . . . . . . . . . . . . . . . . . . . . . . .
65
4-2
Planar layouts of structure C1 and CC1. . . . . . . . . .
66
4-3
Planar layouts of structure C5 and CC5. . . . . . . . . .
67
4-4
Schematics of a 4-port device. . . . . . . . . . . . . . . .
68
4-5
Coupled-strip transmission lines and measurement probes as seen un-
4-6
der m icroscope. . . . . . . . . . . . . . . . . . . . . . . .
. . .
70
Measurement of the coupled-strip transmission line. . . .
. . .
71
. . . . . . . . . .
. . .
72
4-7 Schematics of the measurement setup.
4-8
Calibration setup comparison. . . . . . . . . . . . . . . .
4-9
Admittance/impedance matrix subtraction method . . .
72
. . .
73
4-10 Cl: Comparison of S-parameters from de-embedding method I (admittance/impedance matrix subtraction) and method II (transfer matrix).
12
76
4-11 CC1: Comparison of S-parameters from de-embedding method I (admittance/impedance matrix subtraction) and method II (transfer matrix ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
4-12 C5: Comparison of S-parameters from de-embedding method I (admittance/impedance matrix subtraction) and method II (transfer matrix).
78
4-13 CC5: Comparison of S-parameters from de-embedding method I (admittance/impedance matrix subtraction) and method II (transfer matrix ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
. . . . . . . .
82
4-15 Comparison between simulation and experiment. C1-S 21 . . . . . . . .
82
. . . . . . .
83
4-14 Comparison between simulation and experiment. Cl-S
1
4-16 Comparison between simulation and experiment. CC1-
1
4-17 Comparison between simulation and experiment. CC1-S
21.
. . . . .
83
4-18 Comparison between simulation and experiment. C5-S
1
. . . . . . . .
84
4-19 Comparison between simulation and experiment. C5-S
21 .
. . . . . . .
84
4-20 Comparison between simulation and experiment. CC5-S . . . . . . .
85
. . . . . .
85
4-22 Impedance of the differential mode. (Cl, CC1).
. . . . . . . . . . . .
87
4-23 Impedance of the differential mode. (C5, CC5).
. . . . . . . . . . . .
87
4-24 Losses of the differential mode. (Cl, CC1). . . . . . . . . . . . . . . .
88
4-25 Losses of the differential mode. (C5, CC5). . . . . . . . . . . . . . . .
88
4-21 Comparison between simulation and experiment. CC5-S
21.
.
4-26 Impedance of coupled-strip transmission line with different separation d. 89
4-27 Impedance of the differential mode of coupled-strip transmission line
with different separation d. . . . . . . . . . . . . . . . . . . . . . . . .
91
4-28 Loss of differential mode of coupled-strip transmission line with differ. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
4-29 Configuration for comparison of errors as a function of w/h. . . . . .
93
ent separation d.
4-30 Percentage error of the 1-layer and 2-layer simulations using a 4-layer
simulation (10 GHz) as reference as a function of w/h (coupled-strip
transmission line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
94
4-31 Percentage error of the i-layer and 2-layer simulations using a 4-layer
simulation (10 GHz) as reference as a function of w/h (microstrip transm ission line).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
5-1
Flowchart of simplified InSAR processing.
. . . . . . . . . . . . . . .
98
5-2
Geometry for InSAR height inversion. . . . . . . . . . . . . . . . . . .
101
5-3
Foreshortening diagram. . . . . . . . . . . . . . . . . . . . . . . . . .
103
5-4
Plot of absolute path difference against range of a simulated height
profile and its inverted height with and without foreshortening. . . . .
104
5-5
Comparison of inverted and original profile.
105
5-6
Original interferogram, DEM, and rewrapped images from Picard iter-
. . . . . . . . . . . . . .
ation and PCG method of Alaska-i dataset. . . . . . . . . . . . . . .
5-7
109
Original interferogram, DEM, and rewrapped images from Picard iteration and PCG method of Alaska-2 dataset. . . . . . . . . . . . . . .
5-8
Height error distribution for Alaska-i dataset.
5-9
Height error distribution for Alaska-2 dataset. . . . . . . . . . . . . .
110
. . . . . . . . . . . . . 111
111
5-10 Height error histogram of the inverted height by Picard iteration on
Alaska-i dataset,
Uh =
20.1m. . . . . . . . . . . . . . . . . . . . . . .
112
5-11 Height error histogram of the inverted height by PCG method on
Alaska-I dataset,
rh
= 21.2m. . . . . . . . . . . . . . . . . . . . . . .
112
5-12 Height error histogram of the inverted height by Picard iteration method on Alaska-2 dataset,
Olh =
39.9m. . . . . . . . . . . . . . . . . .
113
5-13 Height error histogram of the inverted height by PCG method on
Alaska-2 dataset,
Oh =
25m. . . . . . . . . . . . . . . . . . . . . . . .
113
5-14 Convergence comparison between Picard iteration and PCG method
for Alaska-2 dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
5-15 Simulated interferogram in the presence of injected Gaussian phase
noise with a standard deviation of 720 before preprocessing.
. . . . .
116
5-16 Simulated interferograms in the presence of injected Gaussian phase
noise with a standard deviation of 72' after preprocessing.
14
. . . . . .
116
5-17 Original interferogram of Mt. Fuji (512 x 512 pixels). . . . . . . . . .
119
5-18 Filtered interferogram (512 x 512 pixels). . . . . . . . . . . . . . . . .
119
5-19 2-D inverted terrain height of Mt. Fuji using JERS-1 InSAR data and
2-D image of the truncated DEM of Mt. Fuji used for comparison. . . 120
5-20 Comparison of the inverted height of Mt. Fuji with DEM (Cross section
along the azimuth). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121
5-21 Comparison of the inverted height of Mt. Fuji with DEM (Cross section
along the range).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5-22 Grayling DEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5-23 Original and rewrapped interferogram with and without preprocessing
(Injected RMS phase noise:
- = 00).
. . . . . . . . . . . . . . . . . .
125
5-24 Original and rewrapped interferogram with and without preprocessing
(Injected RMS phase noise: o = 350). . . . . . . . . . . . . . . . . . .
126
5-25 Original and rewrapped interferogram with and without preprocessing
(Injected RMS phase noise: a
=
65'). . . . . . . . . . . . . . . . . . .
127
5-26 2-D plots of RMS height inversion error [m] for simulated X-band
SAR interferograms with increasing RMS phase error [deg] in the local
backscattering coefficient (without preprocessing). . . . . . . . . . . .
128
5-27 2-D plots of RMS height inversion error [m] for simulated X-band
SAR interferograms with increasing RMS phase error [deg] in the local
backscattering coefficient (with preprocessing). . . . . . . . . . . . . . 129
5-28 Simulated terrain. (60 x 60km.) . . . . . . . . . . . . . . . . . . . . . 131
5-29 Simulated interferogram with GCP. . . . . . . . . . . . . . . . . . . .
132
5-30 Simulated interferogram with 5% RMS phase noise. . . . . . . . . . .
133
5-31 Height error histogram for different levels of injected phase noise estimated using fi as cost function. . . . . . . . . . . . . . . . . . . . . . 136
5-32 Mean standard deviation of height error for 100 realizations of noise
levels estimated using fi as cost function . . . . . . . . . . . . . . . .
137
5-33 Mean RMS baseline error for 100 realizations of noise levels estimated
using fi as cost function. . . . . . . . . . . . . . . . . . . . . . . . . . 137
15
5-34 Mean percentage RMS error of baseline orientation angle for 100 realizations of noise levels estimated using fi as cost function.
. . . . . . 138
5-35 Mean standard deviation of height error for 100 realizations of noise
levels estimated using
f2
as cost function . . . . . . . . . . . . . . . .
138
5-36 Mean RMS baseline error for 100 realizations of noise levels estimated
using
f2
as cost function. . . . . . . . . . . . . . . . . . . . . . . . . .
139
5-37 Mean percentage RMS error of baseline orientation angle for 100 realizations of noise levels estimated using
f2
as cost function.
. . . . . .
139
5-38 Height error histogram for different levels of injected phase noise estimated using
f2
as cost function with preprocessing. . . . . . . . . . .
141
5-39 Mean standard deviation of height error for 100 realizations of noise
levels estimated using
f2
as cost function with preprocessing. . . . . .
142
5-40 Mean RMS error of baseline for 100 realizations of noise levels estimated using
f2
as cost function with preprocessing. . . . . . . . . . .
142
5-41 Mean percentage RMS error of baseline orientation angle for 100 realizations of noise levels estimated using
f2
as cost function with prepro-
cessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
143
List of Tables
. . . . .
57
. . . . . . . . . . . . . . . . . . . . . . . .
71
3.1
Supported waveguide modes of LHM slab in different zones.
4.1
Standard definition table.
5.1
JERS-1 SAR parameters . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2
Mt. Fuji DEM parameters. . . . . . . . . . . . . . . . . . . . . . . . . 118
5.3
Comparison of residues in the interferogram before and after preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118
. . . . . . . . . . . . . . . . . . 122
5.4
X-band SAR simulation parameters.
5.5
Grayling DEM parameters . . . . . . . . . . . . . . . . . . . . . . . . 122
5.6
Comparison of standard deviation of inverted height error between
Picard iteration and PCG method for simulated X-band InSAR data
with different levels of injected Gaussian noise. . . . . . . . . . . . . .
5.7
123
RMS error [m] of height inversion for simulated SAR interferograms
with increasing RMS phase error [deg] in the local backscattering coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8
Simulated SAR interferogram configuration.
17
124
. . . . . . . . . . . . . . 130
18
Chapter 1
Introduction
Electromagnetic waves interacting with physical media produce responses that can
be used to characterize the media. This thesis investigates several examples where
such responses are used to quantify the physical media's parameters. In particular,
spatially confined waves allows us to relate theoretical results to experimental data,
which are typically spatially localized.
In Chapter 2 and Chapter 5, spatially confined free space waves are used to study
the characteristics of constitutive relations of novel materials and terrain height. In
Chapter 3 and Chapter 4, spatially confined guided waves are used to study the
waves propagation in new materials and transmission line designs. The relationships
between the different chapters can be summarized in Fig. 1-1. Gaussian beams and
guided waves are typical examples of confined waves. Based on these well-known genres of waves propagation, characteristics of unexplored terrain, novel materials, and
new transmission line structures can be analyzed. With such studies, it is hoped that
we can understand more about the similarities and differences between these different
problems and relate new physical phenomena to the basics of electromagnetics.
The works in this thesis are mainly based on published articles contributed by the
author in journals and conferences. In Chapter 2 and Chapter 3, new characteristics
of Left-Handed Media (LHM) are studied. An introduction to these new materials
19
Chapter 1. Introduction
20
Free Space (Gaussian Beam)
Beam Shifting
in
Height Retrieval
in
LHM
InSAR
Inverse
Forward
Guidance
Differential
in
transmission line
LHM
characterization
Confined Space (Guided Wave)
Figure 1-1: Relationships between the different chapters in the thesis.
as well as some of their basic properties that have been demonstrated by other researchers are given. Chapter 2 covers the Gaussian beam shifting phenomena based
on [1] and [2]. Such phenomena, though counter-intuitive, are based on sound physics
and the fields and power propagation characteristics as implied by the negative refraction index of such LHM have been confirmed [3]. In Chapter 3, the properties of
guided waves in a slab of LHM are investigated. Two conclusions emerged are that
cutoffs exist for the regular modes, and depending on the constitutive parameters,
modes with high field densities on the slab surfaces that have no cutoffs may exist
[4].
In Chapter 4, the characterizations of the impedances and losses of coupled-strip
transmission lines on a multilayer substrate are carried out. In order to measure the
characteristics of the differential-mode, 4-port measurements and simulations have
been conducted. The subsequent data were used to extract the differential-mode and
the common-mode parameters. These data provide information on the characteristic impedance and dispersion relation of the transmission line. In order to obtain
21
such results, the effects of the connecting pads must first be de-embedded. The impedance/admittance matrix subtraction method is compared to the transfer matrix
method.
The de-embedded results obtained from the measurements indicate that
there is an upper limit on the practical impedance value attainable with the specific substrate profile considered and also demonstrate the limitations of a 2 1/2-D
simulation software.
In Chapter 5, the spaceborne height retrieval process is considered. The Interferometric SAR (InSAR) processing is broken down into several modules, and enhancement of height retrieval is made possible by refining its modular components.
The distributed contribution nature of the response of InSAR means that both the
amplitude and phase of the return signal are corrupted by noise. To combat this
noise, the unwrapping method is refined, the interferogram is pre-processed, and the
baseline estimation is improved. The method of residues weighted least squares first
introduced in [5], is enhanced by using a preconditioned conjugate gradient solver.
Denoising technique based on the work in [6] is incorporated into the height retrieval
process. A ground control points based baseline refinement method is also developed.
By using pre-processing in each individual area, the noise tolerance of the whole process is improved. The height retrieval and the registration of the height pixels are
more accurate, and the usable area of an interferogram is enlarged.
22
Chapter 1. Introduction
Chapter 2
Gaussian Beam Shift in
Left-Handed Metamaterials
2.1
Introduction to Left-Handed Materials
In a paper published in 1968, Veselago introduced a medium with negative permittivity and permeability [7] and called it left-handed medium (LHM) because the
wavevector, the electric field vector, and the magnetic field vector form a left-handed
system.
As Veselago has pointed out, whereas media with negative c within a certain
frequency range exist in nature (for example, atmospheric plasma below plasma frequency), there is no naturally occurring medium with a negative permeability. Hence,
there was not a lot of theoretical studies on LHM until recently. There was, however,
a lot of researches in the area of free standing periodic structures and their applications. Notably, the development of Frequency Selective Surfaces (FSS) [8, 9]. Early
on, planar periodic metallic meshes and patches were compared to the circuit and
waveguide elements of high-pass filters and low-pass filters, with the difference being
that FSS acts on free space waves instead of guided waves. The ground on which
such analyses were carried out was because, first of all, the structures are planar and
23
Chapter 2. Gaussian Beam Shift in Left-Handed Metamaterials
24
metallic, hence the effects of each layer can be clearly defined. Secondly, only a few
layers were used in practical applications, analogous to stages of circuit filters. The
characteristics of such FSS in the context of metamaterial was not clear.
With the help of modern computing, a general method for understanding waves
propagation in an infinite structure of periodic cell comprising metal and dielectric
was borrowed from solid state physics and applied to photonic crystals [10]. Photonic
crystal or Photonic Band Gap (PBG) materials is a more general term than FSS, as
true 3-D structures were custom built to meet the desired properties [11]-[16]. One
characteristic of photonic crystals is the presence of band gaps. In typical applications,
the band gaps are designed to act as a reflector, similar to the stopband of a filter.
But as efforts were made to understand such band gaps in the context of an effective
medium of metamaterial, it was found that one can interpret such band gaps as
the result of an effective plasma [17]-[19].
In a similar way, structures that have
an effective negative p were developed [19]. The frequencies at which the electric
band gap and magnetic band gap are existing can be made to coincide. Analysis
of such superimposed structure was presented in [20] and it was shown that such
material exhibits negative refraction index. The possibility of fabricating these lefthanded materials has been demonstrated [21]. Transmission experiments were also
carried out numerically and experimentally [22]. Whereas each individual structure
(electric and magnetic PBGs) possesses a stopband, the presence of transmission
around the frequency where the electric and magnetic band gaps coexist suggests
that a metamaterial with negative E and p was made. These materials that have
both negative c and p exhibit phenomena that are experimentally verifiable. For
example, the reversed light bending effect by a prism at microwave frequencies [23].
Similar experiments were able to reproduce the same results subsequently [24, 25].
One of the more exciting applications is the focusing effect of a slab with E = -E0
and p = -po,
as suggested in [26], which has caused a controversy on the existence
of such material [27]-[29]. Whereas a lossless, non-dispersive LHM does not exist, a
2.1. Introduction to LHM
25
lossy, dispersive one has much of the features demonstrated from an analysis based
on an ideal isotropic medium. The negative refraction and partial focusing effects
have been confirmed both theoretically and experimentally [30]-[33].
Theoretical studies of waves in such media [34]-[42] address various aspects of
the waves propagation in LHM as well as scattering [43]-[45] by objects made from
LHM. There have also been numerical [46]-[50] and experimental [51, 52] studies
carried out to understand existing designs and LHM in general. New designs [53][56] are being proposed to make LHM more isotropic and operate at higher frequency.
These materials have the potential to be used in novel applications [57]-[61].
In this chapter, we construct a Gaussian beam incident upon a homogeneous,
lossless, isotropic dielectric slab and provide the analytic solutions to the electric
and magnetic fields both inside and outside the slab. By examining the distribution
of the time-averaged power density on the interfaces where the transmitted and reflected Gaussian beam emerges, we can describe quantitatively the amount of lateral
displacement of the exit beam position. Gaussian beam, being spatially confined, has
the advantages of numerical convergence, conceptual similarities with ray optics, and
strict compliance to Maxwell's equations. It is an effect which can be experimentally
verified, and can serve as a means to positively identify an isotropic material that
possesses simultaneously negative values of c and p.
2.1.1
Plane wave propagation in LHM and reflection at an
air-LHM interface
The basic phenomena of plane wave propagation in unbounded LHM and reflection
from halfspace is first dealt with in this subsection, and the basic concepts developed
from such geometries will help us to understand the the Gaussian beam shift and the
guidance studies in this chapter and the next.
Consider a plane wave propagation in a free space source-free region. The Maxwell
Chapter 2. Gaussian Beam Shift in Left-Handed Metamaterials
26
equations in differential form are:
V x E = iAH
(2.1)
V x H = -iwE
(2.2)
=0
(2.3)
V D= 0
(2.4)
V
where the fields satisfy the wave equation:
(V2+ k2)
in which k2
=
2pE
(2.5)
=0
{
is the dispersion relation. For time harmonic plane waves in a
source free isotropic medium, we have
I x E=
(2.6)
/wpH
(2.7)
k x H = -wEE
1-E = 0
(2.8)
k-H = 0
(2.9)
For materials with positive E and p, the vectors k, E, and H form a right-handed
system. For materials with negative e and
[,
handed system. The dispersion relation k2
=
the vectors k, E and H form a leftW2pE
holds for both right-handed and
left-handed materials.
For a plane wave in a medium with E = -EO
and p = -- po, with the E field in
the x direction and the H field in the y direction, the phase is propagating in the
negative z direction and the Poynting power vector S =
E
x H is in the positive z
2.1. Introduction to LHM
27
x
H
Figure 2-1: Backward wave in a left-handed medium.
direction:
kx
3=
= -W/PoH
Ex =
1
Ex
(k x E )
I1
WI-to r(
kET
-
)+E
Tk~~(m
)
(2.10)
W/-tO
Hence the plane wave in an isotropic left-handed medium is a backward wave (Fig. 2.1.1).
Backward wave is known to exist in LC circuits and waveguides [62, 63], but it is
the the support of this wave in a free standing structure that make the idea of LHM
novel. Applications of 2-D backward waves and their relations with LHM have been
studied in [64, 65]. In these cases, instead of plane waves, it was the voltage wave
that has negative phase and refraction angle.
Understanding that k and - are in opposite directions, we now study the reflection
problem. Consider a
polarized TE wave incident upon a second medium where the
boundary is at z = 0 (Fig. 2-2). Applying phase matching at the boundary [63], we
can find the fields on both sides of the interface.
28
Chapter 2. Gaussian Beam Shift in Left-Handed Metamaterials
z
Hir
Fir
Eli
S1i
Ork-
k 1
E1,
IA
62,
A2
k2
Sir
Ot
E2
H
2
S2
Figure 2-2: TE wave incidence upon a left-handed material.
In region 1:
Ey = (RLT Eoeikl'z
+
Eoe-iklz)eikx
(2.11)
Hx = - k 1 z (RT Eoeik1z - Eoe-iklzz)eikxx
(2.12)
Hz = kx (RLT Eoeiklz + Eoe-ik zzeikxx
(2.13)
W1i
In region 2:
E2y= T1
2E
WI12
H
2Z
=
WAp2
Eoe-ik2.z+ikxx
(2.14)
e-ik2zz+ikxx
(2.15)
TTE E0 e-ik2,z+ikxx
(2.16)
z=TE
0
29
2.1. Introduction to LHM
The Fresnel reflection and transmission coefficients are:
P12
R12
(2.17)
+P12
2
T 12
(2.18)
+P12
in which
TE =
P 12
__k2
TM
(2.19)
k1
-
k2z
=
P12
(2.20)
2z
(2.21)
The sign of k 2 2 is important and
(2.22)
k2
k 2 =--k-
to ensure that the tangential E and H fields are continuous through the boundary.
The Poynting vector S is continuous also and
3=
E xH
(2.23)
remains valid. For propagating waves, the time-averaged Poynting vector for a TE
wave can be expressed as:
< Si>
=1
~12
ki
WA+l
< r>
<
t> =1
+
Wp
/-
kx)
112
2
2
RTE 2 Eo1
+ )TTE
2
Wp2
(2.24)
E 0 12
2IEo2
2
(2.25)
(2.26)
(2.27)
Chapter 2. Gaussian Beam Shift in Left-Handed Metamaterials
30
So, power conservation yields
(2.28)
< Si > = < Sr > + < St >
-
2
(-2W/10 )
EOI 2
2
wpo
R122E)
R
1EO1 2 + 1
2
(-
)
Wp-tt
IT1 2 1EO1 2
(2.29)
It is seen that in order to satisfy the boundary conditions of continuous tangential
electric and magnetic fields, (and hence the Poynting vector), a negative k 22 must be
used. The same argument is used in the reflection coefficients such that P21 for TE
and TM waves are positive and R and T will not go to infinity. The bending of the
the 3t to the same side of Sj imply that the refractive index n is negative. The effect
of a negative n is demonstrated as the negative lateral shift in this chapter, and the
resulting negative Goos-Hinchen shift for nil >
Intl,
which will be further explored
in the next chapter, has fundamental implication on the guidance characteristics of a
LHM slab.
2.1.2
Dispersive nature of materials with negative c and y
and modeling of dispersive materials in 2-D FDTD
One of the characteristics of LHM is that they are dispersive in nature [66], as nondispersive materials with negative permittivity and permeability would lead to a
non-causal medium and negative energy. Loss always occur to some extend, but it
is valid to discuss the internal energy of a body when the absorption is small. In
this subsection, the Poynting theorem for dispersive medium is reproduced and 2-D
FDTD is used to demonstrate the imperfectness caused by the dispersive nature of
LHM. The Poynting theorem for dispersive medium [67] gives:
-V-S= V-(7ExH)
E
D
at +
B
t
2.1. Introduction to LHM
4
where the
E
31
-
aD*
at
at
at
+H
at )
(2.30)
field is represented as
(E +*)
E=
(2.31)
in which F = To(t)e-"t. We can define an operator
-- = fE
at
=
f
such that
f(w)E
(2.32)
where f(w) = -iwc for a constant 1E7. Expanding To(t) into its Fourier components:
f (w)T = f(w)Eo(t)e-w -+ f (W + 6)7Eoe-i(w+6)t
(2.33)
Further expanding f (w + 6) and sum up the components, we have:
f)
(df (w) aEo(t) e-w
dw
at
t
f-gU+T + df(W)g e-w+)
dw
(2.34)
Thus,
aD
at
J7 +
atC ew
(2.35)
I d(wE) 72
2 dw
(2.36)
Substituting (2.35) into (2.30) yields
at
d(w+) r12
2 dw
which reduces to the regular Poynting theorem for a non-dispersive medium. Note
that (2.36) is valid only for the low loss regions in the frequency spectrum, and it is
in these regions where practical applications are pursued. The definition of Poynting
vector S
medium.
=
F x H being the radiation power density is indeed still valid for a dispersive
32
Chapter 2. Gaussian Beam Shift in Left-Handed Metamaterials
To illustrate the effect of the dispersive nature, we consider the implementation
of a time domain dispersive material [18, 19], which mimics an isotropic left-handed
dielectric slab at certain frequencies. One possible implementation of such dispersion
is with the plasma-like dispersion relation:
2p
(2.37)
f(w) = CO1 -
)= go
1
-
(2.38)
W"P
It is seen that as the incident frequency drops below the plasma frequency, the values
of both the permittivity and permeability become negative. The collision-free plasmalike dispersion relation has the advantages of being causal and at the same time
lossless. The dependence of the permittivity as a function of frequency is shown in
Fig. 2-3 for the case where Wep
= Wmp =
266.5 rad s- 1 .
10
0
-10
--
wp =266.5 Rad/s
1.2
1.4
E -20
-30-40
-50
0
0.2
0.4
0.6
0.8
1
1.6
1.8
2
Relative Frequency (w/ wo)
Figure 2-3: Dependence of the permittivity as a function of frequency.
With such dispersion relations, we can proceed to implement a dielectric slab
numerically which has E and 1-t equal to -co and -po respectively at the excitation
2.1. Introduction to LHM
33
frequency. The propagation equations are constructed with the Maxwell equations
[68] as shown below. To model a linearly dispersive media, we add the electric and
magnetic currents terms to the Maxwell equations. This technique has been used in
[32, 41] and is concisely described in [69].
V xE
Vx H=
aat
OK
Wep
(2.39)
+
(2.40)
at
= -FJ + Eowe2pE
8K2
-
at
In the above equations,
a=t-K
-
FK + CowMpH
(2.41)
(2.42)
and wmp are the electric and magnetic plasma frequencies
respectively, and F is the collision frequency. In this example, we will set F to zero.
We can use these equations to form the FDTD equations [70, 68], and the updating
sequence at each time step is as follows:
Ja = 1 - 0.5FZt
1 + 0.5FAt
Ka =
Jb=
K =
Hf2'i)
=
H'
H
=
HE('J) -
t
1
-
0.5FZat
0.5Ft
1 + 0.5rAt
(2.44)
t(2.45)
Cw
1 + 0.5FAt
2omAt
1 + 0.5FAt
(Efi'J - Ef-') - 0.5(Kx'j) + Kx'i-)z)
Eil
(2.43)
- E( -
+ 0.5(K"'j) + Kz(i-1',))A)
(2.46)
(2.47)
(2.48)
Chapter 2. Gaussian Beam Shift in Left-Handed Metamaterials
34
z
Ey
K,
--
y
x
)I X
y-
H
Figure 2-4: Computational cell (i, j) in the 2-D FDTD computational domain.
K
K
EY'i
= Eiy)
+
= Ka Kx'j) + 0.5Kb(Hx('j) + H ,j+1))
-=
Ka K$') + 0.5Kb(H$ '") + H +1,j))
(2.49)
(2.50)
c ((H',j+1) - Hx',j)) - (H(i+lJ) - H(P'i)) - AJ'(ij)
(2.51)
J(i') = Ja jy(') + JbE('j)
(2.52)
The above equations are the propagation equations for a 2-D FDTD simulation with a
TE incident wave. The primed variables are the updated values of the unprimed ones.
With an absorbing boundary condition implemented along the boundary, we can place
a dispersive material in the computational domain (the fields and currents positions
with respect to the unit cell is shown in Fig. 2-4) and study its characteristics. In
Fig. 2-5, a slab with c = -co
and
[
= -po
at the excitation frequency is place
between x = 450 and x = 550. The units are in A and the incident wavelength is set
to 100 A to ensure small numerical dispersion. A line source is placed at (400, 500),
and (600, 500) represents the coordinate where the image would occur. However, due
2.2. Analytical solution of waves in a slab upon a Gaussian beam incidence
35
0.8
0.6
0.4
-0.2
C,,
0
N
-0.2
-0.4
-0.6
-0.8
900
0
100
200
300
400
500
600
700
800
x-axis
Figure 2-5: Electric field amplitude upon a line source incidence on a slab with
negative e and p.
to the fact that the material is dispersive, the image point drifts in the horizontal
direction periodically over time. Moreover, the concentration of field amplitude on
the interfaces corresponds to new mode of guided waves, which will be addressed in
the next chapter.
2.2
Analytical solution of waves in a slab upon a
Gaussian beam incidence
Analysis of a light ray transmitting through a slab of negative material has been
considered by Veselago [71]. As shown in Fig. 2-6, we can calculate the lateral shift,
Chapter 2. Gaussian Beam Shift in Left-Handed Metamaterials
36
x
E1 A
E0,1 0
E21
o
0j
Ot2
2
1 1
A2
Z
kkt
d
Figure 2-6: Lateral shift of a ray from geometrical optics.
A, of the refracted ray:
A = d tan 6t
(2.53)
sin Oi = n sin Ot
(2.54)
where from Snell's law,
The refractive index, n, when negative, will imply that when both c and p are neg-
atives, the values Of
Ot2
and the lateral shift A are negative also. A Gaussian beam
[72, 73] is used in this case, since, by decomposing the beam into plane waves with
a prescribed spectrum, the fields in different regions can be computed and will sat-
isfy Maxwell's equations. Time-averaged power densities in each region can thus be
studied.
Consider a slab with permittivity E, and permeability pi as shown in Fig. 2-7 with
the following incident wave in region 0:
E, = Fd
k_- i(k.,x+kozz0(
kx)
( 2.55 )
2.2. Analytical solution of waves in a slab upon a Gaussian beam incidence
37
x
Region 0
60, 10
Region 1
Region 2
E, IAl
E2, /12
- z
01
Oi
Figure 2-7: Configuration of a Gaussian beam incident upon a slab with thickness
d = d2 - d 1 .
where
)( kx) =
24
0 e-
(2.56)
272
is the Gaussian spectrum which carries the information on the shape of the footprint
centered at x = 0, z = 0, (at z = 0, the amplitude at jxj = g is down to 1/e times
the level at the center.) The incident beam is centered about ki = Jkjx +
iko sin Oi +
kz
=
ko cos 63, where Oi is the incident angle of the central plane wave. The
total electric and magnetic fields upon the incidence can be expressed as follows:
In region 0,
0 dkxO(kx)(eikoz + Re-ikoz)eikxx
Eoy =
Hox
=
Hoz =
dkxO(kx) -koz (e
f
-00
dkx)(kx)
oz
- Re-iko z)e ikxx
x (ekozz + Reikoz )eikxx
W/1 0
(2.57)
(2.58)
(2.59)
Chapter 2. Gaussian Beam Shift in Left-Handed Metamaterials
38
In region 1,
E1, =
J0
k1 z (Aeiklzz -
d kx O( k-)
()(6
Hlx =
(2.60)
dkxO/(kx)(Aeikszz + Be-ikz )eikxx
0Jd
Be-iklzz
)eikx x
(2.62)
dkxO(kx) k (Aeik1zz + Be-ikzz)e ikxx
H1, =
(2.61)
In region 2,
E 2y =
J
(2.63)
dkx p(kx)Teik2.z+ikxx
dkxO(kx)
H2X =
00
H2z =
k
dkx
x
(kx)
2z Teik2zz+ikxx
(2.64)
'zikxx
Tel 2 zx
(2.65)
The coefficients R, A, B, and T can be found by matching the boundary conditions
for the tangential electric and magnetic fields at z = dl and z = d2
[63].
Upon solving,
we have,
R
A
=
Ro1 + R12 ei2k1.(d2-d1) ei2kozd1
1 + RoiR12 ei2kiz(d2 -di)
=
2
e-i(kiz-ko)d(
(1 + poi) [1 +
B=
(2.66)
RoiRl 2 ei2kiz(d2-d1)]
R( ei(klzko)dei2kld268)
2
2
2
(1 + poi) [1 + RoR12 e
klz (d2 -di)]
4eikozd1 eik1z(d2-d1)e-ik2zd2
(1 + po1)(1 + P12) [1 + RoiR12 e2kiz(d2-di)
in which Rol and R
12
(
are the Fresnel reflection coefficients:
Rol =
1 + Poi
, R 12 = 1
1+
P12
P12
(2.70)
2.2. Analytical solution of waves in a slab upon a Gaussian beam incidence
39
where,
Poi =
pikoz
,Liko
P12 =
(2.71)
/-2k~z
The above expressions are for arbitrary c and p and the solution of the fields is
invariant with respect to the signs of kiz and k, for all k..
Thus the field values
can be unambiguously determined. By the principle of duality, the TE case and the
TM case are dual of each other. We can obtain the expressions for the TM case by
the replacements E, -
Hn, 7n -
-En, and p, +-+ E, in (2.55)-(2.71), where the
subscript n = 0, 1, 2 denotes the region. The above expression for the coefficients
R, A, B, and T are for general dl and d2 . In the following discussion, dl is set to 0
such that the incident interface would be at z = 0.
Two special cases are of considerable interests. Firstly, let el = -Eo
Also, let E2
=
Eo and
12 =
and pi = -po.
po. The total electric and magnetic fields can be simplified
as follows:
In region 0,
Eoy = J
dkxV(kx)eikoz+ikxx
(2.72)
HoX
dkx O(kx )
1 eikoz
(2.73)
=
z
J
eikozzikxx
dkxV)(kx)
(2.74)
In region 1,
Ey =
H1x =
H1z =
J
J
dkxV)(kx)e-ikozz+ikxx
(2.75)
dkxO(kx) -koz e-k'zz+ik~x
(2.76)
dkx (kx)
00
k e-ikozz+ikxx
W/-10
(2.77)
Chapter 2. Gaussian Beam Shift in Left-Handed Metamaterials
40
In region 2,
E2y =
H
=
H2z =
J
]
0
dkxV/(kx )e-ikodeikozz+ikxx
(2.78)
p
dkx4'(kx)
(2.79)
f 00
-
z&Ize-
WI' 0o
(kx) kX
70 dkx
e koz
e-I2 kozdeikozz+ikxx
(2.80)
WI-Lo
Secondly, consider a similar case in which ei = -co and pi = -/o. Also, let region 2
be perfectly conducting. The total electric and magnetic fields are simplified by the
following coefficients:
R
=
2
R12 e-i kozd
(2.81)
(2.82)
A =1
B
=
2
R12 e-i kod
(2.83)
(2.84)
T =0
where R
12 =
-1
for the TE case and R 12
=
1 for the TM case. In the absence of the
slab,
RTE
RTM
=
-ei2kozd
(2.85)
ei2 kozad
(2.86)
and they correspond to the phase shift due to the position of the PEC ground plane.
2.3
Lateral shift of a Gaussian beam
The time-averaged power density can be expressed as
(9n)
=
[Re(Eny . H*z)] 2 + [Re(En, . H*)]
2
(2.87)
2.3. Lateral shift of a Gaussian beam
41
10
1.2
8
1.0
4
A
0.8 ClI
2
0V
-2
0.6
-4
-0.4
-4
-6
0.2
-8
-10
-8 -6 -4 -2
0
2
4
6
8
N
10 12
z/%
Figure 2-8: Reference time-averaged power density on the xz-plane for a 300 incidence
of a Gaussian beam with E, =
Co
and pi = po.
where the subscript n = 0, 1, 2 denotes the region. The time-averaged power density
in region 2,
(s2
,
has in general a Gaussian amplitude distribution along x on a
plane parallel to the slab. For the transmission case, the position of the Gaussianshaped power density,
(52
,
with respect to x, for a particular value of z > d, is
spatially shifted in the presence of a slab when compared to the one without the slab.
It is this shift that we are interested in. In particular, when the slab has both negative
permittivity and permeability, the peak of the distribution along x at z = d2 is seen
to be positioned before x = 0 (the reference position of the footprint). Defining the
x-coordinate of the peak of IKS2) at z = d as the lateral shift, we can see that for a
regular slab, the shift is positive, whereas for a slab with negative c and p, the shift
is negative.
To illustrate the effect of the lateral shift, we have constructed numerically an
incident Gaussian beam with tapering factor g = 2. The prescribed footprint is
centered at x = 0 on the plane z = 0, and the angle of the incidence is 30' with
respect to the z-axis. The magnitude of the electric field of the central wave is
Chapter 2. Gaussian Beam Shift in Left-Handed Metamaterials
42
K]
1.2
1.0
_
A
0.8 ClI
V
0.6
3
0.4
*
3
0.2
2
4
z/.
Figure 2-9: Time-averaged power density on the xz-plane for a 300 incidence of a
-po.
Gaussian beam upon a slab of thickness d = 4A with E = -Co and p,
10
8
6
2.0
4
A
1.5 CDI
V
2
0
1.0
-2
-4
0.5
-6
-8
-10
-8 -6 -4 -2
0
2
4
6
8
10 12
zIk
Figure 2-10: Time-averaged power density on the xz-plane for a 30' incidence of a
Gaussian beam upon a slab of thickness d = 4A with E, = -co and pi = O.lgo.
43
2.3. Lateral shift of a Gaussian beam
4
60
10
60
- - - -
~1.2E
~-- 1
.±0.6
1 0.4
V
- 0.2
0
-10 -8
-6
-4
x/X
6
4
2
0
-2
8
10
Figure 2-11: Comparison of time-averaged power density along x at z
incidence of a tapered beam upon a slab of thickness d.
=
d for a 300
assumed to be of 1 V/rn.
Fig. 2-8 to Fig. 2-10 show the time-averaged power density as a function of x and
z for the cases where ei and pu1 are (60, 110),
In Fig. 2-8, where ei
=
co, p1
=
(-60,
-1p0), and (-60, 0.1pio) respectively.
110, the beam propagates at an incidence angle of
300, reaches a maximum of power density at z
=
0, and for z > 0, the center of the
beam is above the reference position.
In Fig. 2-9, ei
= -co,
1 = -110, and the thickness of the slab is 4A, where A is
the wavelength of the incident wave in region 0. The beam emerges from the slab
at z
=
d below the reference position, prior to the incident footprint, exhibiting a
dramatic negative lateral shift. In this case, the reflection coefficient is 0, and the
transmission coefficient has a magnitude ITI
=
Ie- 2 koz
9, the time-averaged power density distribution from z
distribution of the incident beam from z
=
-4A to z
=
=
=
1. Furthermore, in Fig. 20 to z
=
4A mirrors the
0.
Indeed, the negative lateral shift is unique to materials having both negative c
Chapter 2. Gaussian Beam Shift in Left-Handed Metamaterials
44
and p. For example, consider a case where c is negative and p is positive as shown in
Fig. 2-10, where ci = -co
and All = 0.1po, simulating a slab of electric plasma. The
transmission coefficient is very small, as most of the energy is reflected. The positive
lateral shift is miniature.
For a more quantitative illustration, the time-averaged power density along x at
z = d for two cases are plotted in Fig. 2-11. When E, = Co, pi1 = po, there is a
positive shift. While a negative shift is observed for E, = -Co,
Ati = -po.
It is remarkable that when Ei = -Co and pi = -po, the electric field and magnetic
field at z = 2d are identical to the incident fields at z = 0. At z = 2d, for example,
the electric field
E2yz=2d = J
=
J
dke-2kodei(kxx+ko2d),O4(kx)
(2.88)
dkxeikxxo (kx)
(2.89)
(2.90)
=EjyI2=o
Hence the prescribed footprint, both magnitude and phase, is restored at z = 2d.
For the reflection case, the time-averaged power density in region 0 along the
x-axis,
K3o)
,0'is in general a superposition of the power density of the incident
and reflected beam, each exhibiting a Gaussian-shaped profile. The position of the
reflected Gaussian-shaped power density, (30)
,
with respect to x, for a particular
value of z < 0, is shifted in the presence of a slab when compared to the one without
the slab. Similar to the transmission case, the peak of the distribution along x at
z = 0 is seen to be positioned before x = 0 (the reference position of the footprint).
Defining the x-coordinate of the peak of (So)
of the reflected beam at z = 0 as
the lateral shift, we can see that for a regular slab, the shift is positive, whereas for
a slab with negative E and p, the shift is negative. To illustrate the effect of the
lateral shift, we have constructed numerically an incident Gaussian beam with the
same specifications as in the transmission case.
45
2.3. Lateral shift of a Gaussian beam
Fig. 2-12 and Fig. 2-13 show the time-averaged power density, 1(o) , as a function
of x and z for the cases where ci and fit are (Eo, po) and (-Eo, -to)
respectively. In
Fig. 2-12, where ci = Eo, pi = i-to, the beam propagates at an incidence angle of 300,
reaches a maximum power density at z = 0, and the center of the reflected beam is
above the reference position.
In Fig. 2-13, Ei = -co,
si = -po,
and the thickness of the slab is 6A, where A is
the wavelength of the incident wave in region 0. The reflected beam emerges from the
slab at z = 0 below the reference position, prior to the incident footprint, exhibiting
a dramatic negative lateral shift.
For a more quantitative illustration, the time-averaged power density along x at
z = 0 for two cases are plotted in Fig. 2-14. When El = Eo, iti = i-to, there is a
positive shift. A negative shift is observed for Ei = -Eo,
.2
i-ti
= -- to.
10
8
E
1.5
6
E
4
A
C)n
V
2
0.5
0
-2 1
-2
0
2
4
6
z/k
Figure 2-12: Reference time-averaged power density on the xz-plane for a 300 incidence of a Gaussian beam with ei = co and IL, = i-o.
46
Chapter 2. Gaussian Beam Shift in Left-Handed Metamaterials
2
~
H
2
E
0
1.5
-2
E
A
IC/)
V
-6
0.5
-8
-10
-2
6
4
2
z/k
0
Figure 2-13: Time-averaged power density on the xz-plane for a 300 incidence of
a Gaussian beam upon a grounded slab of thickness d = 6A with Ei = -Eo and
Al = -Po-
1.4
-
E
incident
amplitude
1.2
N
P oEo
-
-
-
reflected
amplitude
reflected
amplitude
1
E
-/
0.81- -/to,-fo)
(Io,Eo)
0.6
A
0.4
C
V 0.2
0
0
-10 -8
/
-6 -4
-2
0
2
4
6
8
10
Figure 2-14: Comparison of time-averaged power density along x at z = 0 for a 300
incidence of a tapered beam upon a slab of thickness d.
2.4. Conclusion
2.4
47
Conclusion
In conclusion, LHM is a novel material and its characteristics are based on the fact
that its permittivity and permeability are negative. The resulting negative refractive
index exhibits itself as a backward wave in an LHM. For a halfspace problem, the
transmitted Poynting vector is seen to be on the same side of the normal as the
incident vector, hence negative refraction. Due to the dispersive nature of LHM, field
amplitudes vary slowly over time, and instead of a stationary image beyond an LHM
slab upon a line source incidence, a partial focusing is observed.
To illustrate the concept of negative lateral shift as predicted by simple ray tracing,
the problem of a tapered beam, constructed by a superposition of plane waves with
a Gaussian amplitude spectrum, incident upon a dielectric slab with arbitrary f and
p is solved. The electric and magnetic fields inside and outside the slab are provided.
It is seen that, whereas the beam emerged from a regular slab exhibits a positive
lateral shift, the beam emerged from a slab with negative c and t experiences a
unique negative lateral shift. Furthermore, for the transmission case, the incident
fields along the x-axis are reproduced at z = 2d by an isotropic slab with -Eo
-Ao.
and
48
Chapter 2. Gaussian Beam Shift in Left-Handed Metamaterials
Chapter 3
Guided Modes in a Slab
Waveguide with Negative
Permittivity and Permeability
3.1
Introduction
One of the applications of LHM not yet fully explored is their ability to guide waves
in an unusual fashion. Waves in a slab of uniaxial negative medium were discussed in
[74], in which it was demonstrated that the directions of the Poynting vectors inside
and outside the slab are opposite. Yet, guided waves with imaginary transverse
wavenumbers were not addressed in details. Guidance with a slab of plasma was
discussed in [75], and similar configurations were also discussed in [76].
In both
instances, only the permittivity (c) of the media could assume negative values. In this
chapter, we investigate the guidance conditions of a slab waveguide with both negative
permittivity (c) and permeability (p) and we demonstrate that first, guided modes
with real transverse wavenumber always have cutoffs and second, that guided modes
with imaginary transverse wavenumbers exist. We present a graphical solution of the
transcendental equations involved to determine the values of the imaginary transverse
49
Chapter 3. Guided Modes in a Slab Waveguide
50
wavenumbers, which helps locating the poles of the Green's function, representing the
modes. Upon using this graphical technique, we can also determine the number of
possible modes and whether a particular mode has cutoff or not. These modes can be
experimentally measured, and may serve as a means to identify an isotropic material
that possesses simultaneously negative values of c and 1t over a certain frequency
range.
Guided waves can be formed when there is an interface between two dielectrics,
and the possibly of having one between a right-handed medium and a left-handed
medium has recently been suggested [77]. We can consider a
y polarized
TE surface
wave propagating in the x direction along the interface between two dielectric media:
Ei =
yEoekxxe-
z,
(z > 0)
;ikxx +a2z
, (z < 0)
Eoe
e
E2 =
(3.1)
and by Faraday's law,
H, =
e
H2 = .
+ ikx) e-O"z, (z > 0)
(kx(ai
(3.2)
+ ikx) e 2 z, (z < 0)
(3.3)
(--2
1 =k
a=
1 =k
k - k2=k
w2 61,t
w2 2 pP 2
(3.4)
Equation (3.4) is derived from the dispersion relations of the upper and lower media.
The boundary conditions give:
Ev(z = 0+) = Ey(z = 0-)
(3.5)
Hx(z = 0+) = H_(z = 0-)
(3.6)
3.1. Introduction
51
al
ao
Ai
P2
(3.7)
The last equality is possible due to the magnetic plasma effect. (Note: both al > 0
and a 2 > 0 must be satisfied to have a surface wave.) Therefore, there is a TE surface
wave. TM surface waves at the interface of two dielectric media can be analyzed in
the same manner:
Hi = yHoeik xe-'1Z, (z > 0)
H2
= Hoeike
+a2z,
(z < 0)
(3.8)
(3.9)
By Ampere's law,
H o=
eikxxI (Jiai + ik) e -'iz, (z > 0)
1W
(3.10)
E
E2= Hoe
- (-ia 2 + ikx) eQ Z, (z < 0)
2
(3.11)
Boundary conditions give
H_(z = 0+) = Hy(z = 0-)
(3.12)
Ex(z = 0+) = Ex(z = 0-)
(3.13)
a
El
=
_
a-2
(3.14)
E2
Since a 1 and a 2 are positive, if Ei > 0, 62 has to be negative. It is known that
electric plasma yield negative permittivity when driven at a frequency lower than the
plasma frequency. But by having both negative E and p, we have the flexibility of
having a generally polarized surface wave propagating along the interface between a
left-handed medium and a right-handed medium.
Chapter 3. Guided Modes in a Slab Waveguide
52
3.2
Guidance conditions for a symmetric dielectric
slab
Whereas the guidance condition for the wavevector along the propagating direction
for an one-interface problem is rather straightforward, guided waves satisfying the
guidance condition in a slab of LHM are more complicated. We consider a homogeneous infinite slab of thickness d, permittivity E2, and permeability P2 in free space,
as shown in Fig. 3-1. For guided TE waves (results for TM waves can be obtained
from duality) with a wavevector k
kx + kz, the electric and magnetic fields in the
=
three regions are written as:
Ei =
9Eie" 1e
(3.15)
E2=
(Aeik2xX + Be-ik2xx )ikz
(3.16)
E3=
E 3e
(3.17)
R, =
1(-iikz+ .ai) Eie"xeikzz
iWPi
1
72 =
(-:ikz
1(-iikz
iWPI
where kz is the wavevector in the
(3.18)
+ .ik 2x) Aeik2xxeikzz +
Il(-ikz
11
773 =
'eikzz
-
2ik 2x)
-
.ai) E 3 e 1xeikz z
Be-ik2xeikzz
(3.19)
(3.20)
direction, k 2x is the transverse wavenumber in
region 2, which can either be real or imaginary, and a 1 is a positive real number
corresponding to the evanescent waves outside the slab.
To find the guidance condition, we match the boundary conditions at x = 0 and
53
3.2. Symmetric dielectric slab
z
Region 1
Regio 2
Region 3
AlI, El
A216E2
A16,1
d
0
Figure 3-1: Configuration of a slab waveguide of thickness d.
x = d to obtain:
(3.21)
Ei = A+B
Aeik2xd
+ Be-ik2.d = E 3 e-1d
(3.22)
-p 2 1E1 = A - B
(3.23)
Aeik2.d - Be-ik2xd = P 21E
3 e-aid
(3.24)
where we define
(3.25)
P21-A2
Upon solving, we have
1
-P21
ik2.d
eim"r
(3.26)
1 + P21)
which imposes restrictions on the values that the transverse wavevector can assume.
Chapter 3. Guided Modes in a Slab Waveguide
54
3.3
Guided modes with real transverse wavenumber
For guided modes with real transverse wavenumber k2,, the guidance condition can
be rewritten as
(klxd) tan
aijd =
P2
k2,d
2
mr(3.27)
2
For even m, we have
ald = Pi (k2xd) tan
k2xd
2
P2
(3.28)
and for odd m, we have
(k 2xd) cot
ald =
92
k2xd
(2
(3.29)
The dispersion relations in the two different regions are:
(kid) 2
(3.30)
(kzd) 2 + (k 2 xd) 2 = (k 2 d) 2
(3.31)
(kzd) 2
-
(aid)2
=
which yield a set of circles defined by:
(k 2 xd) 2 + (aid)2 = (k2 - k )d2
(3.32)
The solution of the transcendental equations can be obtained graphically [63].
For even modes (Fig. 3-2), it is seen that the lowest mode is m = 2 and for odd
modes (Fig. 3-3), the lowest mode is m = 1. Note that whereas in a positive index
medium, the lowest mode has no cutoff, there is a cutoff frequency for the lowest mode
in a LHM slab. The cutoffs are determined by the minimal distances between the
origin and the tangent-like curves, as indicated by the circles in Fig. 3-2 and Fig. 3-3.
Thus, the cutoff condition is different from a regular slab, which is determined by
3.3. Guided modes with real transverse wavenumber
55
aid = 0 (for propagating waves).
aid
k
m=6
m=2 m=4
P- k2 xd
57r
37r
7r
m=0
Figure 3-2: Graphical determination of k2.d, even modes. Circles indicate cutoffs.
[1/[P2 = -0.5.
The cutoff of the lowest mode with real transverse wavenumber can also be understood by considering the Goos-Hinchen phase shift at the interfaces [78, 79]. We
can write the reflection coefficient as
1 - P21
1 + P21'
P21
/2iai
-
(3.33)
or
(3.34)
R21= ea
in which
021 =
- tan-1 p2 ai
(3.35)
is the Goos-H~inchen phase shift. From (3.26), we can rewrite the guidance condition
as
20
21
+ k2 xd
= m7r
(3.36)
Chapter 3. Guided Modes in a Slab Waveguide
56
m
m
=3
=5
m=7
I
I
I
I
II
I
I
I
I
i 27r
47r
*-
67r
k2ld
1!
Figure 3-3: Graphical determination of k2.d, odd modes. Circles indicate cutoffs.
p1/A2 = -0.5.
For a positive p2,
A 2,
however,
3.4
#21
0 21
is negative and hence the lowest mode is m
is positive and the lowest mode is m
=
=
0. For a negative
1.
Guided modes with imaginary transverse wavenumber
Although regular modes always have cutoffs for a LHM slab, energy may still be
propagating at all frequencies because of the existence of other types of modes, which
may not have cutoffs. If the transverse wavenumber is imaginary, (k2,
=
ia
2
, where
a 2 is positive real), the guidance condition (3.26) becomes:
-=
e
+1
(3.37)
where
p 2ai
P21 =
1 1a
1 - e2d
=
1 t ea2d
<0
(3.38)
3.4. Guided modes with imaginary transverse wavenumber
57
The guidance condition can be rewritten as
aid =
/11 (a 2d) tanh (a 2 d
\ 2(3
A2
(3.39)
aid =
I'(a 2d) coth
a2d
(2
(3.40)
d2
The dispersion relations in the two regions give
(kzd) 2 - (aid)2
=
(kid) 2
(3.41)
(kzd) 2 - (a 2 d) 2
=
(k2 d) 2
(3.42)
which give a set of hyperbolas:
(a 2 d) 2
=
(k2 - k2)d
2
(3.43)
(a 2 d) 2 - (aid)2
=
(k2 - k2)d
2
(3.44)
(aid)2
-
We can determine a 2 d graphically. First we identify the possible values of the constitutive parameters and determine the type of modes supported. Since the even and
odd functions of the regular modes of the electric fields, cos and sin [63], become cosh
and sinh for imaginary k2 -, we will call the modes with such imaginary transverse
wavenumbers "Cosh modes" and "Sinh modes". Fig. 3-4 shows the different zones
of possible constitutive parameters and Table 3.1 shows the supported modes in the
different zones.
Zone
1
2
3
4
Supported Mode
Sinh
Cosh
Cosh, Sinh
Cosh, Sinh
Table 3.1: Supported waveguide modes of LHM slab in different zones.
Chapter 3. Guided Modes in a Slab Waveguide
58
P2
/#1
P1[ E2
-1
4
12
0
1
Figure 3-4: Different zones of possible constitutive parameters
(E2,
A2 <
0).
We can plug (3.39) or (3.40) into (3.43) and plot (k' - k')d2 against a 2 d for
the Cosh modes and Sinh modes, as shown in Fig. 3-5 and Fig. 3-6 respectively.
The vertical axes in Fig. 3-5 and Fig. 3-6 correspond to the values of (k2 - k2)d2
or w 2 A 2 E2 (1 -
[1E1/
2 62 )d
2
. The horizontal axes correspond to the values of a 2 d.
The shaded regions correspond to the cases where k2 > k2 and the unshaded regions
correspond to the cases where k2 > k2.
Note that in either case, k' or k 2can
be
negative, indicating a plasma-like medium.
For a specified set of (61, pi) and (E2, A2), and a operating angular frequency w,
we can draw a horizontal line in Fig. 3-5 or Fig. 3-6 where the intersections with the
curves of the ratio
(--P1/P2)
give the values of a 2 d, characterizing the propagating
mode with imaginary transverse wavenumber a 2 . As the vertical position of such
horizontal line varies with w, departures of the horizontal line from 0 in either vertical
directions signify a increase in frequency. Cutoff frequencies can thus be identified
for media with different constitutive parameters, and it is seen that Cosh modes,
when supported, have no cutoffs, whereas Sinh modes, depending on the constitutive
3.4. Guided modes with imaginary transverse wavenumber
59
I
parameters, may possess cutoffs.
5
4
1.6
3
2
1*4 1.2
.... . .. . ..
.. .
1
0
-1
-2
-3
1.0
1/1 .0363
-4
0.4
0
1
2
0,8
.
4
3
5d 6
7
8
9
10
Figure 3-5: Graphical determination of a 2d for different values of -
/p92 , Cosh mode.
1
The unshaded regions of the graphs ((k2 - k )d2 < 0) correspond to an optically
less dense slab (k 2 being real) or a plasma-like medium (k2 being imaginary). In
both cases, the expression (k2 - k)d
2
is always negative. The support of the "Cosh
mode" and "Sinh mode" for such plasma media is well known [75], and the special
case of a plasma slab gives the same Cosh and Sinh modes as a slab with negative
real k 2 where Ik21 < Ikil. Therefore the lower regions of the graphs ((k2 - k )d 2 < 0)
represent the dual of the case in [75] (TE in this case vs. TM in [75]). Furthermore,
when the Cosh or Sinh modes are supported, we can infer from the graphs that
multiple intersections are possible for certain constitutive parameters in a specific
frequency range. The dashed line in Fig. 3-5, corresponding to
-p
1
/P 2
=
1/1.0363,
shows the point where the transition from single intersection to multiple intersection
takes place, and is in agreement with [75].
For a specific angular frequency w larger than the cutoff of the lowest regular
Chapter 3. Guided Modes in a Slab Waveguide
60
15
Cs'
1'2
1.4
1.6i
10
5
0
09
-5
0.4
0
1
2
0.6
0.8
3
4
5
6
7
8
9
10
Figure 3-6: Graphical determination of a 2 d for different values of -t1/P2, Sinh mode.
mode as shown in Fig. 3-2-Fig. 3-3, regular modes may coexist with the Cosh or Sinh
modes, and we must take them into account when considering a guidance problem.
3.5
Fields and Poynting vectors of guided waves
It is conducive to study the fields amplitude and the direction of the Poynting power
in such waveguides. For real k2 ,, the E and H fields are:
E, = 9Eie"1xeik.,z
E2= pE 1 (1 -pl)
(3.45)
2
cos [k 2 x (x -
) +
2
eikz
(3.46)
z
E3= i9Eiejd e"xeik
(3.47)
3.5. Fields and Poynting vectors of guided waves
ai) Eiee ek
(-h-fk,+
I
H1
61
(3.48)
1
R
2
P E1(I - pl)2
=
d
n,]
-2 + 2
d
e ik;
211
-k 2xsin k 2 x(X
-2 )
1
(-.Hk, - .aj) Eiealde-1"xeikz,
I
{--ikz Cos k2x (X
H3
The time-average power
=
(s)
(3.49)
(3.50)
in the three regions are:
2 2ax
E e
2 w1Ei
K51)
=
(32)
=
2/1
-LE1
-2li
2 W1p2
d
-2)
(3.52)
21
Ee-2a(x-d)
(3.53)
Cos 2 k 2x(X
)= 2
wpi,
(3.51)
For k 2 x = ia 2 , E 2 , H 2 , and (K2) becomes:
E2
=
(1
E1
-
psi)2 cosh
(pji - 1)i sinh
}a
2 (X -
d ) eikzz
2
(3.54)
H2
-
eik z
(1
-
pi)I cosh
{iHkz
(psi
-
1)i sinh
}
Ia2 (X -
d
-)2
Chapter 3. Guided Modes in a Slab Waveguide
62
+ a2
(1 - pii)2 sinh
2
(P21
KS2)
=
-
sn
1)1 cosh
a2(X -
d1
-)
2
(3.55)
-P i
,E2- k- E~
Ep
2 111 _ 2
2
sinh 2
d
1
(3.56)
a2(X - 2
It can be seen that in either case, the power flows of the waves inside and outside the
slab are in opposite direction and are determined by the signs of
3.6
/12.
Conclusion
In conclusion, guided waves with high surface field intensity are supported at the
interface between a right-handed medium and a left-handed medium. For a single
interface, the fields decay away from the surface and there is only one mode. For
a slab, however, the transverse wavevector must obey the guidance condition and
hence discrete modes are formed. The guided waves in a symmetric slab waveguide
of negative isotropic medium are analyzed and the guidance conditions are solved
for both real transverse wavenumbers and imaginary transverse wavenumbers. The
electric and magnetic fields inside and outside the slab are also derived.
It is shown that the cutoff condition is different from a regular slab and for real
transverse wavenumbers, there exist cutoffs for all modes, whereas for imaginary
transverse wavenumbers, depending on the constitutive parameters, modes with no
cutoffs can exist. The field amplitudes assume the shape of Cosh or Sinh functions for
such imaginary transverse wavenumbers. Finally, the power flows of the waves inside
and outside the slab are in opposite direction for both real and imaginary transverse
wavenumbers.
Chapter 4
Guided Waves in Coupled-Strip
Differential Transmission Line
4.1
Introduction
Transmission lines are used in high-speed analog integrated circuit designs [62] to
connect and isolate high gain stages as well as to improve signal integrity. Although
in many cases 50 Q lines are used in the input and output stages to facilitate off-chip
communications, the optimum transmission line impedance used in the intermediate
stage is not necessarily 50 Q. In some cases, one may prefer higher impedance to
reduce the current driving requirement.
Therefore, it is desirable to characterize
what are the available transmission line parameters for a given substrate, so that
design engineers can perform proper optimizations and tradeoffs.
One way to determine the transmission line parameters is to perform simulations.
Characterization by simulations, however, owe to their inherent assumptions and limitations, need to be validated for a particular configuration. For on-chip applications,
the concerns are thick lossy metal, lossy substrate, and space charge layers in semiconductors. Compared to an idealized case, these imperfectness introduce non-linearities
as well as increase the number of unknowns in typical numerical schemes. Rather than
63
Chapter 4. Guided Waves in Differential Coupled-StripLine
64
solving the exact problem approximately, in many cases, we seek an exact solution to
an approximative formulation.
Another way to solve the on-chip transmission line problem is by measurements.
Measurements, however, can only cover a few discrete points in the parameter space.
The overlapping of parameter space between experiment and simulations allows us to
use these few data points to validate or calibrate the EM solvers for a broader use.
Two different kinds of solvers are commonly used to characterize circuits. The first
kind can handle metal with finite thickness and the second kind uses an infinitely thin
metal layer in simulations, which is also known as 21/2D approximation. Whereas
in small structures, having a full 3-D simulation yield more accurate results, it is not
as efficient in characterizing large circuits.
This chapter summarizes the measurements and analyses of differential coupledstrip transmission lines on a multilayer substrate. The objectives of the study are two
folds. The first is to determine the range of the available transmission line parameters.
The second is to calibrate an EM solver for on-chip interconnects designs.
4.2
4.2.1
Measurements and calibrations
Geometries
The common parameter space between experiments and calibration in this characterization study are four coupled-strip transmission lines with corresponding calibration
structures. Fig. 4-1 shows the cross-section of the substrate. Fig. 4-2 and Fig. 4-3
show the basic planar geometries with the corresponding test structures to be used
for de-embedding purpose. Geometries Cl and CCl consist of a pair of coupled lines.
The separation d for geometries C1 and CC are 7 pm and 10 pm respectively. Geometries C5 and CC5 consist of a pair of coupled lines with a parasitic strip between
them. The separation d for geometries CI and CC1 are 7 pm and 10 pm respectively.
For on-chip differential transmission lines, the limitation is often on the upper end.
4.2. Measurements and calibrations
65
The finite dimensions of the metal traces pose an upper limit on the impedance that
the transmission lines can achieve. The differences among the structures C1, CC1,
C5, and CC5 are designed to cover a wide enough parameter space to explore the
on-chip coupled-strip transmission line characteristics.
r = 1, r = 0
Er = 3.9,
3im
a=0
Al
-
10000
Er = 3.9, o = 0
Er = 11.9, a
Al
a =3.3x
107
5/m
Er = 3.9, a = 0
Er = 11.9, a
~3pm
- 20
2.3pm
1pm
700pm
Figure 4-1: Typical cross section of a coupled-strip transmission line on a multilayer
substrate.
Chapter 4. Guided Waves in Differential Coupled-Strip Line
66
1200pm
100pm
100pm
::
10pm
100pm
SHORT
100lpm
1000p-m
NORMAL
L/
3pm
d
OPEN
H
Il
Figure 4-2: Planar layouts of structure CI and CC.
67
4.2. Measurements and calibrations
1200m
100tm
1OOtm
-10011m
SHORT
10011m
OOm
1000M
NORMAL
di
T
30pm3m
t j
3OAt
OPEN
Figure 4-3: Planar layouts of structure C5 and CC5.
Chapter 4. Guided Waves in Differential Coupled-Strip Line
68
4.2.2
Measurements of 4-port S-parameters
The characterization of the coupled-strip transmission lines was carried out by using a
2-port network analyzer. Full 4-port measurements were taken and subsequently de-
2
O-M
4
Figure 4-4: Schematics of a 4-port device.
embedded and converted to differential mode for analysis. The 4-port S-parameters
corresponding to a 4-port network (Fig. 4-4) are defined by the following expression:
V1-
S 11 S12 S 1 3 S 14
V1+
V2
S 2 1 S22 S 23 S 2 4
V 2+
V_
S31 S32 33 S 34
V3+
V4-
S 4 1 S 42 S 43 S 44
V 4+
or
=
-+
(4.2)
In the above expression, every matrix element represents a ratio of the complex amplitude of a scattered wave to the complex amplitude of an incident wave with incident
waves at other ports being absent. This means that we can use two wideband loads
to terminate two of the four ports while measuring the other two ports. For example,
when measuring S11, S 21 , S 12 , and S22, we terminate port 3 and 4 by using wideband
50 Q loads and connect port 1 and port 2 to the network analyzer. Mathematically,
this is identical to a 4-port measurement; in practice, ideal 50 Q loads are difficult to
obtain and that degrades the measurement quality.
As the network analyzer has only two-port capability, six two-port measurements
are needed to yield one set of four-port result for one of the twelve structures. Once
69
4.2. Measurements and calibrations
the 4-port S-parameters are obtained, we can decomposed it into the common and
differential modes using a standard procedure. We define
1 -1
M =
0
0
0
0
1 -1
v21
1
0
0
0
0
1
1
1
(4.3)
and we can transform (4.2) to
M V+ = M-S-M
where M - S - M
(4.4)
*M.V
corresponds to the mixed-mode S-parameter matrix [80]:
Vid-1
Sddll Sdd12 Sdcll Sdc12
Vdl
Vid72
Sdd21 Sdd22 Sdc2l Sdc22
Vd1d2
Vc-1
Scan1 Scd12 Secni Sc12
V1
V-2
Scd21 Scd22 Scc2l Scc22
Vc2
(4.5)
The upper left block of the mixed-mode S-parameter matrix contains the differential
mode S-parameters. To fill in the value of the matrix requires precise measurements
of the responses of the geometries. In general, in order to extract the parameters of
the Device Under Test (DUT), effects from the connection patches, used to facilitate
on-wafer measurements, need to be de-embedded.
4.2.3
Equipments
The experiments were performed using a set of HP 8720C vector network analyzer
in conjunction with a Summit probe station. The probes used are of the variant
G-S-S-G from Picoprobe. The four landing tips facilitates landing on the grounding
patches of the wafers (Fig. 4-5 and Fig. 4-6) . Fig. 4-7 represents the schematics of the
Chapter 4. Guided Waves in Differential Coupled-Strip Line
70
Figure 4-5: Coupled-strip transmission lines and measurement probes as seen under
microscope.
measurement setup, where the labeled circles correspond to the 4 labelled connections
to the coaxial cables. In the following descriptions, we will use these labels to outline
the procedure.
4.2.4
Standard definition table and calibration
To begin the calibration procedure, a standard definition table has to be formed
(Table 4.1). It models the discrepancies of the calibration chip as standard offset
segments of transmission lines with offset impedance, delay, and capacitance. The
fringing fields of the open circuit structure, for example, can thus be compensated.
Notice that the value for the offset delay of the 'thru' structure has two values.
The first one, 2.68 ps, corresponds to the calibration setup of the 'thru' as shown
in Fig. 4-8a. The resulting measurement shows a non-physical peaking of S21 above
0 dB in subsequent measurements. Arrangement as shown in Fig. 4-8b was adopted
71
4.2. Measurements and calibrations
Figure 4-6: Measurement of the coupled-strip transmission line.
Standard
1 Short
2 Open
3 Load
4 Thru
Fixed/Sliding
Co
-_
4.3 fF
Fixed
--
Offset Zo
Offset Delay
500 Q
0.052 ps
50 Q
0
500 Q
0.0337ps
50 Q
2.68ps/1.55ps
Table 4.1: Standard definition table.
to yield a better calibration.
Chapter 4. Guided Waves in Differential Coupled-Strip Line
72
2
--
G
S
4
G
S
S
G
S
G
1
3
qs
GGB SN9721
GGB SN9734
Figure 4-7: Schematics of the measurement setup.
G
S
S
G
G
S
S
G
02
(a)
40
0
G
G
S
G
S
G
(b)
Figure 4-8: Calibration setup comparison.
0
73
4.3. De-embedding
4.3
4.3.1
De-embedding
Impedance/admittance matrix subtraction method
The first way to de-embed the connecting pads is to model them as series impedance
and shunt admittance. Fig. 4-9 [81] gives a concise description of the method.
r
,i
PORT.2
POET-1
POW-2
PORT.I
2
I
V2
VI
I3
LI jJ
* De-embedding Procedure
-
Substract capacitive component
-
Substroct reactive component[z'MEAS
and
[ZDUT] = [EMEAS-
[zSHORT]
[YMEAS]
-
[YOPEN]
- [ZSHORT];
where
[zMEAS
have been corrected for capacitive component.
[zSHORT] =*([YMEASY-OPEN'
[YSH-
OtTYOPElf'}'
[YDUT]
Figure 4-9: Admittance/impedance matrix subtraction method.
The modelling of connecting pads as series inductance and shunt capacitance is
accurate for low frequencies, when the structure is small compared to the wavelength.
The method is robust and is less susceptible to noise.
4.3.2
Transfer matrix method
We can also model the connecting patches as a section of transmission line with a
characteristic impedance Zo and phase <5 = k. Modelling the calibration patches as
equivalent open and short circuits enables us to calculate these two values from the
74
Chapter 4. Guided Waves in Differential Coupled-Strip Line
S-parameters, and the impedance Zo is given by:
1(4.6)
Zo =
whereas the complex wavenumber k can be found from:
-1
tan2 (k) =
(4.7)
in which Z11 and Y11 are the first entries of the matrices Z and Y defined by:
S=(W-7).(Z+I)-1
,=
(4.8)
Thus, the response of the DUT can be de-embedded by the use of transfer matrix:
=
TDUT=
=patch 'Tmeas
1
Tpatch
(4.9)
The conversions between S and T are given by:
I__L
2 1
Vi+
_][S
2
=
1
y(
(4.10)
and
V
V~.
[- 2a
-Ti
_ 2
-1T1
-(T
1 2 T 2 1 -T 1 1 T 22
Th
- TT11
v1+
)
.
(4.11)
V2+
where the superscript + indicates an incident amplitude and the superscript
-
indi-
cates a scattered amplitude. The transfer matrix method performs better in a purely
simulated environment but in experiment, discontinuities of geometries are encountered and the impedance/admittance matrix subtraction method proved to be more
robust, especially when frequency range is not very high (< 20 GHz).
4.3. De-embedding
4.3.3
75
Verification
Numerical simulations have been carried out using the SonnetEM simulation program.
For each design, we carried out four simulations. The first two are the simulations
of the calibration structure (open/short), which are used to determine the input
inductance and capacitance of the connecting pads. The third simulation is the
structure itself, without the connecting pads. This result will be used as the reference
with which the de-embedded results are compared. The fourth simulation is on the
coupled-strip with connecting pads. This provide the responses corresponding to
simulated measurements. In all cases, the metal is assumed to have a conductivity of
a-= 3 x 107 Sm-1 and an effective thickness of 3 pm. This is implemented by having
two layers of metal, each with an effective thickness of 1.5 Mm. They correspond to
the top surface and the bottom surface of the rectangular metal trace.
For all cases, the simulation domain comprises the side walls and ground planes,
which are perfectly conducting, and the top of the simulation domain is terminated
with a matched waveguide load. The reference planes are at the edge of the corresponding structures.
The simulations are set up to yield 4-port S-parameters responses from 1 GHz to
50 GHz in 1 GHz step. The simulated results are first converted to 2-port parameters
and are subsequently de-embedded by both methods. The magnitude and phase of
S11 and S2 1 are plotted for the different designs and are shown in Fig. 4-10-Fig. 4-13.
Within each design, the simulated measurements as well as the de-embedded results
are compared to the reference value.
As seen from the simulation results, both de-embedding methods yields accurate
magnitude and phase responses, comparable to the reference solution, up to around
20 GHz. For higher frequencies, it is seen that as wavelength becomes smaller, the
transfer matrix method becomes more accurate. In practice, the reference position of
the ports are not at the edges of the structures, but are at the center of the patches.
The impedance/admittance matrix subtraction method is hence more robust for low
Chapter 4. Guided Waves in Differential Coupled-Strip Line
76
The transfer matrix method will be useful as measurement frequency
frequencies.
goes higher.
Reference
Simulated Meas.
- DUT (Method 1)
--- DUT (Method 2)
-
200
10
0
-
100
- -10
-o
c -20
0
-.
00
0
10-30
-A
-40
0
/ ............
1
.100
. . . . . . .. . . .
2
3
Frequency (Hz)
4
-
. .-.
-200
5
0
1
x 1010
2
3
Frequency (Hz)
4
5
x 10
0
1
.-..
....
-..
.
0
....
-..
-20
.-.
-40
_0
o-1
- . . . . . . . .. . . . . . . .
CD
--.
2
.......
.4:
-60
C\D
-4
0
1
3
2
Frequency (Hz)
4
5
x 1010
-100
-.-.-.-.
......-
-80
-3
0
1
2
3
Frequency (Hz)
4
5
x 1010
Figure 4-10: Cl: Comparison of S-parameters from de-embedding method I (admittance/impedance matrix subtraction) and method II (transfer matrix).
77
4.3. De-embedding
Reference
-
Simulated Meas.
DUT (Method 1)
- -
---200
10
................
0 ..........
S 10
1001
(D
4)
..
.
..
.
. . .
.
. .
.
... ..............
.. .. .. ..... ... ... .. ... ... .. .. ... ...
R-30
-40
W
................
-50
0
S ..
-100 [
...........................
-60
0
1
3
2
Frequency (Hz)
-200 '
0
5
4
1
x 1010
2
3
Frequency (Hz)
4
5
x 1010
0
1
-........ ......
-20
(D
0)
v-1
CM
,
. ... ... ..
... .. .. ... . . ... ..... .. ... .
'0 -20
DUT (Method 2)
-..
....
- - ...
....
-.
....-.-. -.
-2
-.
-.
. -.-.-.
-3 -........
- -... .. . .
.. .
(D
0)
-4
0
1
2
3
Frequency (Hz)
4
5
x 1010
-40
-
. ......
-60
-.
. . ..
..
-.. . . ..
-80
-100
1
3
2
Frequency (Hz)
5
4
x 10
Figure 4-11: CC1: Comparison of S-parameters from de-embedding method I (admittance/impedance matrix subtraction) and method II (transfer matrix).
Chapter 4. Guided Waves in Differential Coupled-Strip Line
78
Reference
-
Simulated Meas.
DUT (Method 1)
--- DUT (Method 2)
- -
10
200
0
-. -10
a
-.. .. . ~.
.-. .. .. .
-20
-- - ...........
-
0
1
0)
V
0)
0)
C
0
.
-
.
CD~
-3a
-40
......
........... --.
.....-..
100
-
2
3
Frequency (Hz)
4
5
-100
F
-200
0
.-..
. . .. . .-.
...
2
1
... . .
3
5
4
Frequency (Hz)
x 1010
X 10
0
1
...
-.
........
-.
-.
- .... ........
......
--1
C
0)
.-.-.-.-.-. -20 -.
-.
-......
- L.
........... - -..
-2
-3L
A?
0)
-40
04
U)
-60
...
- -... .........
-..
-.
...-.
-A
-80
/)-3
A
4
0
1
3
2
Frequency (Hz)
4
5
x 1010
-100
0
1
2
3
Frequency (Hz)
4
5
x 1010
Figure 4-12: C5: Comparison of S-parameters from de-embedding method I (admittance/impedance matrix subtraction) and method II (transfer matrix).
79
4.3. De-embedding
Reference
Simulated Meas.
- DUT (Method 1)
---- DUT (Method 2)
-
Ir I
0
[
100
-10
S-20
CD
0
/
O -30
-40
[
CD, 100
-50
-60
3
1
2
3
Frequency (Hz)
4
-200
5
0
1
x 1010
2
3
Frequency (Hz)
4
5
x 1010
0
1
-20
~0
. . . .. . . .. . . .
. ..
-..
(D
'0
-40
v-1
0)
.. .
-80
)-3
-4
-.. . . .. . .. . ..
-60
-2
0
1
2
3
Frequency (Hz)
4
5
x 1010
-100
0
1
2
3
Frequency (Hz)
5
4
x 10
Figure 4-13: CC5: Comparison of S-parameters from de-embedding method I (admittance/impedance matrix subtraction) and method II (transfer matrix).
Chapter 4. Guided Waves in Differential Coupled-Strip Line
80
4.4
4.4.1
Results and analysis
S-parameters
Fig. 4-14 to Fig. 4-21 show the differential mode S-parameters of the coupled-strip
transmission lines. The solid, dotted, and dashed lines represent the measurements
before de-embedding, the de-embedded measurement results, and results from simulations respectively. The measured results were processed from raw measured data
using 20-points moving window complex averaging to smooth out fluctuations. The
de-embedding was done using the standard impedance/admittance matrix subtraction method. The simulations were performed using SonnetEM.
From the experimental results, it can be seen that the de-embedding methods
perform well and physical parameters are successfully extracted to yield reasonable
comparison between simulations and experiments. Agreements of the phase values of
S21
between experiments and simulations indicate that the calibration has been per-
formed accurately. The linear phase responses of S21 further show that the corrected
calibration procedure outlined in the previous section has eliminated the problem
caused by the coupling of the fixtures.
Comparisons of the magnitudes of S21, however, show that the simulations predict
larger losses than experiments. But the loss from the de-embedding of the experiments
is affected by the termination mismatches. This is due to two factors. Firstly, the
matched loads connected to the unused ports are not perfect, and the VSWR for
such terminations are typically at around 1.1. Secondly, the measurement ports can
only be calibrated for either port (1, 3) or port (2, 4). When the coaxial cables
are not connected in the calibrated configurations, mismatches at the measurement
ports occur. The final extracted magnitudes of S21 from experiment, however, do give
physical results as they do not exceed 0 dB in general.
For S11 , again we see more agreements between the phase than the magnitudes.
This can be explain by similar reasonings as with S21. Note that the experimental
4.4. Results and Analysis
81
values of S11 of the coupled-strip transmission lines are lower than the simulated
ones. It indicates that the characteristic impedance of the transmission lines measured
from the experiments are lower than that from the simulations, and hence are better
matched to the 100 Q system differential impedance.
Chapter 4. Guided Waves in Differential Coupled-StripLine
82
0
-5
-..
-10
-
200
Meas.(before de-em)
Meas.(after de-em)
Simulation.
Meas.(before de-em)
. . Meas. (after de-em)
--- Simulation
-
150
100
m -15
.--
-..-.
-8 -20
50
0)
a)
c1 -25
-
-30
0
-50
U) -35
-40
-100
-
-150
-45-
-50
0
0.2 0.4 0.6
1.2 1.4
0.8 1
Frequency (Hz)
1.6
-200
1.8 2
x 10 10
0
0.4 0.6 0.8
0.2
1
1.2
1.4
1.6
Frequency (Hz)
1.8
2
x 1010
Figure 4-14: Comparison between simulation and experiment. C1-S 11 .
1
10
Meas.(before de-em)
Meas.(after de-em)
---Simulation
-
0.5
0
-- --
6 -0.5
_0
.-
Meas.(before de-em)
Meas.(after de-em)
0
-10
-20
-
-
Simulation
.
-30
1
(D
E -1.5
-40
< -50
-2
c -60
-70
o -2.5
-3
-80
-3.5
-90
-4
0
1.2
0.2 0.4 0.6 0.8 1
1.4
Frequency (Hz)
-100 1.6
1.8
X 10'"
2
0
0.2
0.4 0.6 0.8
1
1.2
Frequency (Hz)
1.4
1.6
1.8
X 10,"
Figure 4-15: Comparison between simulation and experiment. Cl-S 21 -
2
83
4.4. Results and Analysis
0
;.uu
Meas.(before de-em)
-5
Meas.(after de-em)
- Simulation
-10-
150100-
. - .....
-15
50
-20
_0
-25
Cn
-30
U, -50
b -35-
0~'
-100
-40 -
-50
0
Meas.(before de-em)
Meas.(afterde-em)
-150
-45 -
0.2 0.4
0.6 0.8 1 1.2 1.4
Frequency (Hz)
1.6
Simulation
0
1.8 2
x 1010
0.2
0.4 0.6 0.8
1.2
1
1.4
1.6
Figure 4-16: Comparison between simulation and experiment. CC1-S
Meas.(before de-em)
de-em)
0.5
0
-
-.--.-
m -0.5
01
---Simulation
d-eMea
(afer
....
-50
-2
c -60
-70
-2.5
-3
-
--
-20
-30
A? -40
'E -1.5
.
Meas.(before de-em)
im
de-em)
0
.Meas.(after
-10
Simulation
1
to
10
1
1.8 10 2
x 10
Frequency (Hz)
-80
-90
-3.5
-100
-4
0
0.2
0.4 0.6 0.8
1
1.2 1.4
Frequency (Hz)
1.6
1.8
x 1010
2
0
0.2
0.4
0.6
1.2 1.4
0.8 1
Frequency (Hz)
1.6
2
1.8
x 1010
Figure 4-17: Comparison between simulation and experiment. CCi-S 21 .
Chapter 4. Guided Waves in Differential Coupled-Strip Line
84
0
200
-
-5
---
-10
Meas.(before de-em)
Meas.(after de-em)
Simulation.
--.
-
Simulation
100
S-15
-20
A?
-25
C
2 -30
U)
-35
- - -.
50
...
0
-50
-100
-40
-150
-45
-50
Meas.(before de-em)
Meas.(after de-em)
150
0
0.2 0.4
0.6 0.8
1
1.2
-200
1.4
1.6
Frequency (Hz)
1.8
0
2
0.2 0.4
0.6
0.8
1
1.2
1.4
1.6
Frequency (Hz)
x 1010
1.8
2
x 1010
Figure 4-18: Comparison between simulation and experiment. C5-S 11 .
10
1
0.5
Meas.(after de-e)
0
@-0.5
a
'0
as
iuato
S-30
-1
-40
*E-1.5
2
Simulation
-10
-20
-- Simulation.
..
- .. --...
Meas.(before de-em)
Meas.(after de-em)
0
Meas.(before de-em)
-
c -50
-2
o -2.5
ci -60
-70
-3
-80
-3.5
-90
-4
.....
le-Mes.
ater
100[
0
0.2
0.4 0.6 0.8
1
1.2
Frequency (Hz)
1.4
1.6
1.8
x 1010
2
0
0.2
0.4 0.6 0.8
1
1.2
Frequency (Hz)
1.4
1.6
1.8
x 1O
Figure 4-19: Comparison between simulation and experiment. C5-S 2 1 .
0
2
85
4.4. Results and Analysis
0
v ,im)
Meas.(before de-em)
Meas (after de-em)
-- Simulation
-5
-10
,
,
,
Meas.(before de-em)
Meas.(after de-em)
150
---
Simulation
1
1.2
100
6-15
-20
50
-25
0
2 -30
-50
M)
u) -35
-100F
-40
-150
-45
-50
0
0.2
0.4 0.6 0.8 1
1.2 1.4 1.6
Frequency (Hz)
1.8
-200 '
0
2
0.2 0.4 0.6 0.8
1.4
1.6
Frequency (Hz)
x 1010
1.8
2
x 1010
Figure 4-20: Comparison between simulation and experiment. CC5-Sjj.
1
10
Meas.(before de-em)
de-em)
Simulation
S--
0.5
0
0
.Meas.(after
-10
-20
-0.5
-
Meas.(before de-em)
-..... Meas.(after de-erm)
--Simulation
aa) -30
-
F -1.5
S-2
-40
-50
4C -60
-2.5
-70
-3
-3.5
-4
0
0.2 0.4
0.6 0.8
1
1.2
Frequency (Hz)
1.4
1.6
1.8
x 1010
2
-80
-90
-100L
0
1.2 1.4
0.2 0.4 0.6 0.8 1
Frequency (Hz)
1.6
1.8
x 1010
Figure 4-21: Comparison between simulation and experiment. CC5-S,.
2
Chapter 4. Guided Waves in Differential Coupled-Strip Line
86
4.4.2
Impedance and loss
The characteristic impedance of the transmission lines can be obtained from the
Z matrix by modelling the de-embedded differential transmission line as a 2-port
network:
V1
cos kz
I1
[jYosinkz
z
cosikz
1
V2
12
The results are shown in Fig. 4-22-Fig. 4-23, in which the experimental results are
compared to simulation results.
Fig. 4-24 to Fig. 4-25 shows the measured and simulated losses of the structures for
length f = 1000 pm. Whereas S2 1 give us the transmission amplitude, its magnitude
is affected by the mismatch between the system impedance and the characteristic
impedance. By plotting the imaginary part of the wavenumber times length kif, we
can assess the inherent losses of the structures due to finite conductivity and dielectric
loss:
Loss per mm = 20 loglo(e-ki)
(4.13)
The total loss of power is bounded at around 0.5 dB for a length of 1000 ,am,
where higher loss is loosely associated with higher impedance.
The discrepancies
between the experiments and simulations is quite high percentage-wise, because the
inherent loss of the transmission line is small with respect to the experimental setup.
Furthermore, the de-embedded results suffer from noise due to various mismatches.
87
4.4. Results and Analysis
250
250
Real(Expt)
-----Real(Simulatfion)
200
200
E150
150
Real(Expt)
- - -Real(Simulat
E'
on)
a 100
C
.M
E
0
50
---- -----
---------------
----
CL0
-50
-100
-100
cu -150
lmag(Expt)
Imag(Simulation)
-200
-
-250
0
0.2
0.4
0.6
0.8 1
1.2 1.4
Frequency (Hz)
1.6
m-150 lmag(Expt)
Imag(Simulation)
-200
-
1.8 2
x 1010
0
0.2
0.4
0.6
1.2
0.8 1
Frequency (Hz)
1.4
1.6
1.8 2
10
x 10
Figure 4-22: Impedance of the differential mode. (C1, CC1).
250
250
Real(Expt)
Real Simulaton)
200
E 150
0
200
-
-
Real(Expt)
Real(Simulation)
E
.C 150
0 100
oi
c-
50
ca
-10
.-
50
. --------- -- --
-;100
.E0
5Q0
-50
CD -100
t -100
Z6 -150
-200
.t
Imag(Expt)
rnag(Simulaton)
0
0.2
0.4
0.6
0.8
1
1.2
Frequency (Hz)
1.4
1.6
1.8 2
x 1010
-150
Imag(Expt)
-200
-250
-- --
0
0.2
0.4
0.6
1.2
0.8 1
Frequency (Hz)
lmag(Simulation)
1.4
Figure 4-23: Impedance of the differential mode. (C5, CC5).
1.6
1.8 2
x 1010
Chapter 4. Guided Waves in Differential Coupled-Strip Line
88
1
1
0.5
0.51
0
0
E
E
-0.5 -
-0.5 4
GO
V)
to
-j
£0
£0
o -1
o0-
-J
-1.5 k
-1.5 I
Expt
Simulation
___-
---
-2
0
0.2 0.4 0.6 0.8
1
1.2
1.4
1.6
Frequency (Hz)
1.8
-
----2
0
0.2
0.4 0.6
x 100
1
1.2 1.4
0.8
Frequency (Hz)
Expt
Simulation
1.6
2
1.8
x 1010
Figure 4-24: Losses of the differential mode. (CI, CC).
1
1
0.5 F
0.5 I-
0
0
E
E
E
E -0.5 I-E
a
£0
£0
_3
0
_j3-1
-
-1.5 [
-
-2
-0.5
0
0.2 0.4 0.6 0.8
1
1.2
Frequency (Hz)
-1.5
Expt
Simulation
1.4 1.6
1.8
x 10
10
F
- xpt
Simulation
2
-2
0
0.2 0.4
0.6 0.8 1
1.2 1.4
Frequency (Hz)
Figure 4-25: Losses of the differential mode. (C5, CC5).
1.6
1.8
x 1010
2
4.4. Results and Analysis
89
As we have shown that the simulation results are accurate compared to the experiments, we can use the simulation tool to investigate the impedance and loss for a
differential line. Analytic trend of the impedance using a parallel wire approximation
is:
Zo =
(4.14)
cosh- (d/2a)
where d is the center to center separation of the wires and a is the effective radius of
the line. The real part of the impedance is shown in Fig. 4-26.
200
180
160
0
w140
120
100
80
60
W 40
20
0
C
5
10
25
20
15
Separation (Micron)
30
35
40
Figure 4-26: Impedance of coupled-strip transmission line with different separation
d.
We see that as the separation becomes larger, the rate of increase of the impedance decreases. With the ground plane in place, the capacitance in practice will
be dominated by the wire-to-ground capacitance as the separation increase. As frequency goes higher, the rate of increase of impedance also changes. The small time
constant
(T = L
~- ) of the bulk of the epi-layer implies that the substrate is resistive
within the frequency range. Hence we expected the impedance to vary as
for
f
<
f,
log(d/2a)
because for low frequency, the resistive effect is dominant, and varies as
Chapter 4. Guided Waves in Differential Coupled-Strip Line
90
log(d/2a) for
f
>
f, because
the capacitive effect becomes dominant and the electric
field will penetrate the epi-layer.
The loss, however, is proportional to the dimension where the induced current is
flowing, and hence grows faster than impedance at high frequencies for an increased
separation. For low frequency, closer spacing between the conductors imply a higher
field intensity and current distribution near and inside the wires and, consequently,
results in higher loss. The cross-over point is at around 15 GHz.
Based on C1 and CC1, we increase the separation to 15pm, and 30pm to investigate the practical limit in impedance and determine the bound on the loss as well.
Fig. 4-27 and Fig. 4-28 show the impedance and loss with respect to different frequencies for different separations. It can be seen that when the separation of the traces
is 30pm, the real part of the impedance is around 170 Q and the loss for the 1 mm
section is 0.35 dB, which suggest a practical limit of 180 Q for a bounded 0.5 dB loss
per mm for such differential coupled-strip transmission lines on the given substrate.
4.4. Results and Analysis
91
250
.4
E
0
200
7--------------
150
-
-
-
-
C
.L
-
100
-------
.g 50
5
d 30pm (Real)
d
15pm (Real)
d = 10pm (Real)
-
7pm (Real) .d -
0
d = 30pm (Imag.)
5prn (Imag.)-d
d = 10pm (Imag.)
d 7pm (Imag.)
Q -50
-100
0
0.2
0.4
0.6
0.8 1
1.2 1.4
Frequency (Hz)
1.6
-
1.8 2
x 1010
Figure 4-27: Impedance of the differential mode of coupled-strip transmission line
with different separation d.
0
d: = 30pm
------- d =15p~m
-0.05
-0.1
........
d =7pmn
-0.15
EF
EI
-0.2
E
E -0.25
COCn
9
..........
..... I.....
..............
-------
-0.3
-0.35
-0.4
F
-0.45
-0.5
0
0.2
0.4
0.6
1.4
1.2
0.8 1
Frequency (Hz)
1.6
1.8
2
x 10
Figure 4-28: Loss of differential mode of coupled-strip transmission line with different
separation d.
Chapter 4. Guided Waves in Differential Coupled-Strip Line
92
4.5
2 1/2-D EM solver for interconnects designs
From the previous section, it is seen that results predicted by SonnetEM correlate well
with measurements. As seen from the impedance calculations, SonnetEM predicts
accurate imaginary part but consistently overestimate the real part. This suggests
that it can predict the phase or dispersion accurately and over-predict the magnitude.
In other words, it will overestimate the inductance and underestimate the capacitance.
This is also true for microstrip structures but with different amount of error. This
phenomenon is common for 2 1/2D EM solvers. One way to solve this problem is to
use multiple layers to model thick metals as in the previous section, but this requires
substantially more computer resources and may not be even feasible for complex
interconnects. An alternative approach is to find out what are the errors and the
corresponding correction factors empirically. Obviously, the error would depend on
the metal width to metal thickness ratio.
As simulations, in which thick metals are modelled as multiple layers, agree with
the experiments closely, it is important to further understand the behavior of the
numerical solutions. We compare the error percentage of the inductance L, the capacitance C, the loss ki, and the impedance Zo of the one-layer solution and the
two-layer solution with the four-layer solution in Fig. 4-30 (coupled-strip) and Fig. 431 (microstrip). To characterize the effect due to the modelling of the layers, we first
assume the 4-layer results as the numerically converged solution to the fields in the
geometries.
We use two types of geometries: the first case is a pair of coupled strips and the
second case is a microstrip, both of which are commonly used. For each geometry,
we further change the aspect ratios of the metal traces (Fig. 4-29).
It is seen that the two layer solution in general yields a more accurate result at
the expense of time. The inductance from using less layer is overestimated, whereas
the capacitance is underestimated, resulting in a higher characteristic impedance.
Furthermore, for both cases, the percentage errors drop almost linearly as a function
4.5. 21/2-D EM solver for interconnects designs
c, = 3.9, o=
:
W
Al
o=3.3 x 107
7
93
0
pm
3 pm
c, = 3.9,
Al
=3.3x
3 pm
coupled-strip
0
Al
= 3.3 x 107
3
3 pm (h)
=
W
:
W
a
pm
(h)
microstrip
Figure 4-29: Configuration for comparison of errors as a function of w/h.
of log(w/h). This a a very useful feature, because it allows us to interpolate the
results for other w/h ratio and facilitates accurate evaluations of new simulations.
The calculation of the impedance and loss are based on the calculation of inductance and capacitance of the transmission lines:
Zo =
R,
VG + jC
k =w /(R + jwL)(G + jwC), ki ~ - R-+
21 L
GC
(4.15)
in which the value of ki is based on the assumption of a low loss line. The overestimation of L and underestimation of C will still produce a relatively stable k, hence
the phase velocity variation is small from the different models used. The addition of
more layers results in a more accurate estimation of capacitance because the stray
capacitance between the sides of the metal traces and the ground can be captured
more accurately; and as the ratio w/h increases, the structure tends toward a pair
of parallel plates, thus the stray capacitance has little effects. Similarly, the multilayer approximation gives a more distributed current densities on the effective cross
sections of the metals, resulting in a more accurate inductance. The error for the
microstrip case is considerably smaller and can be explained by the lack of coupling
between the metal traces which occurs for the case of coupled strips.
Chapter 4. Guided Waves in Differential Coupled-Strip Line
94
Z Error
L Error
70
60
0
40
030 -I
30
20
w
CL
a)
...
.. ...
40
50
....-....-....-.
-.
-.
.. ..
-~.-.
20
0
0
0
3
2
1
0
Loss Error
C Error
80
-
..-.
.
....
60
-.
. . . . -.-.. . . . .
0
0
-.......
....
.....
-..
-..
CM
-.
........... ........
) -30
-40 -50
3
Log2 W/H
0
-20
2
1
Log2 W/H
-10
. - ...
-.
-.
.......... ..
-.-.-.-.-.-.-.
10
10
. ....... ... ..-
0
.
--
2
1
Log2 W/H
3
a)
0Y)
cc
t
a)
0
(D
a-
40
20
f.
0.
-20
0
.......-
..-...-.
.-..
.
..
.
2
1
3
Log2 W/H
1 layers
------
2 layers
Figure 4-30: Percentage error of the 1-layer and 2-layer simulations using a 4-layer
simulation (10 GHz) as reference as a function of w/h (coupled-strip transmission
line).
4.5. 21/2-D EM solver for interconnects designs
95
Z Error
L Error
30
50
....
--.. ..
40
-..
--.
.-...
-.
25
.............
....-. .. . -.....
..
...
15
..-..-..
.-. ..
-.
-.
--.
10 ......-.
20
.....
-..
..
..
....
. .. ....
- ....
0
10
0.
U.
0
2
1
-5
3
0
CD
.-.
-.
. .. . . -.
-30
..
-.
-40
-50
80
2
..... ..........
. .. ............
-20
0
..
-.-.-.-.-
-.
- -- - -
1
2
Log2 W/H
Loss Error
1 nn.
--- ---. ..
--.
. ....
-10
3
Log2 W/H
C Error
n ....
2
1
Log 2 W/H
(
. ....
20 -....... ......
2
w
30
. ....
.
w 60
0
0)
0
0
U
0
0~
40
-77 .
20
..--
0
3
-20
2
1
3
Log2 W/H
1 layers
- -----
2 layers
Figure 4-31: Percentage error of the 1-layer and 2-layer simulations using a 4-layer
simulation (10 GHz) as reference as a function of w/h (microstrip transmission line).
96
4.6
Chapter 4. Guided Waves in Differential Coupled-Strip Line
Conclusion
Experiments of six 2-port measurements performed on each of the 12 structures representing the four different designs of the differential transmission lines have been
successful. Upon de-embedding, the S-parameters of the four designs are compared
with the simulation values. From the phase comparisons of S21 , the validity of the improved calibration is confirmed. Comparisons of the magnitudes of the S-parameters
yield reasonable results with the uncertainly in the experiment due to mismatch
bounded at ~< 0.5 dB. The close agreements between the experimental values of
Zo and the simulated values prompted us to examine the relationship between the
the layers of metal used in the simulations and the aspect ratio of the metal. Both
differential and microstrip structures yield a percentage error response that is approximately linear with respect to log(w/h). This allow us to accurately estimate the
errors while simulating a new structure in an time-efficient manner.
Chapter 5
Height Retrieval in SAR
Interferometry
5.1
Introduction to Interferometric SAR processing
Interferometric Synthetic Aperture Radar (InSAR) [83]-[87] allows the digital elevation model (DEM) to be generated based on the technique of phase unwrapping
[88]-[91]. The process of InSAR processing can be briefly described by the flowchart
in Fig. 5-1. The phase unwrapping process converts the wrapped phase to the abso-
lute phase difference, which is a result due to the position difference of the observation
platforms, separated by a distance known as the baseline. The baseline is normally
calculated from the orbits of the platforms and with this, the height inversion process
can be performed.
However, the processes of phase unwrapping and baseline estimation are sensitive
to noise and the resulting effects are inaccurate height retrieval/registration and unusable portions of the interferograms. The origin of the phase noise is a combination
of many factors, for example, the sharp variation of terrain height, shadowing effect of
97
98
Chapter 5. Height Retrieval in SAR Interferometry
Interferogram generation
- PCG
Phase Unwrapping ---
-
Baseline estimation
---
-
-Preprocessing
GCP
Height inversion
Figure 5-1: Flowchart of simplified InSAR processing.
the terrain, and the decorrelation between the scattering amplitudes from the terrain
due to the time difference between observations. The phase noise exhibits itself in
the interferogram as local inconsistencies of the data, or residues, which slow down
the convergence of typical unwrapping algorithms. The error of baseline estimation,
on the other hand, is often due to the deviation of the platform's flight path from an
ideal orbit. As height inversion is very sensitive to baseline length, accurate estimate
of baseline is very important. To reduce the effects of noise, various method have
been developed, for example, the addition weights to the noisy interferogram pixels
[5, 89] and the use of ground control points [92].
As the flowchart in Fig. 5-1 suggests, each step of the InSAR processing can be
treated in a modular manner. Thus preprocessing of interferograms will enhance
both the phase unwrapping method and the baseline estimation. In this chapter, we
apply Preconditioned Conjugate Gradient (PCG) method to the residues weighted
least squares phase unwrapping algorithm. We also use ground control points to
obtain an accurate baseline estimation.
In both cases, a developed technique of
denoising is used for the preprocessing of the interferogram.
The objective is to
improve the noise tolerance of the InSAR processing, and to obtain more accurate
5.1. Introduction to InSAR processing
99
terrain height information. Height retrieval examples using data from ERS-1, JERS1, and simulated X-band SAR are presented to illustrate the effectiveness of this suite
of methods.
5.1.1
Phase unwrapping
The phase unwrapping problem can be stated as follows:
=
(5.1)
Arg {eiij
where i and j denote the position of the pixel in the interferogram. 4 is the wrapped
phase, $ is the absolute phase, and the operation 'Arg' yield the phase angle of its
and 7r. The objective of phase unwrapping is to
argument which lies between -7r
solve for q based on 0. The phase unwrapping algorithm used here is based on the
least squares method [93]. Defining
(5.2)
W{#i} = 0i'j
where Oi,j = qij +2k7r (k being an integer here) is the wrapped phase (-7r <4,,5
and
4 is the absolute phase.
< 7r)
Since it is not possible to compare the two directly, we try
to optimize the gradient of the phase. To find the expression for such optimization,
we let
j = W{ ,, -
i-ij} and AY = W{oij - O',j-1}
(5.3)
and the boundary conditions at the edges of the interferogram is of the Neumann
type:
AXj =±1
=
1
±1
= 0
(5.4)
where m and n are the numbers of pixels in the x and y directions. Least squares
Chapter 5. Height Retrieval in SAR Interferometry
100
solution to
will minimize the cost function:
/ij
n
m
n
m
YO ij - oi_1, - A )2 +
i=2j=1
Eo (#
,_
-
-
(5.5)
- AY 2
i=1j=2
where the dimensions of the matrices A-" and AY are (m - 1, n) and (m, n - 1)
respectively. Rearranging the terms, it is seen that the solution of the least squares
problem satisfies the equation:
oi+1,, + 0i1,j
+ 5,jj+i + oij-1 - 4,-
=
Pi,j
(5.6)
where pi,j can be interpreted as the discrete Laplacian operator. By using integral
transformation, (5.6) can be solved efficiently. We can define the forward 2-D discrete
cosine transform (DCT) for the transform pair formed between the elements of the
M x N sized matrices:
fi, j +-+Fa,,
(5.7)
as:
M
Fa,
=
N
E 4fij cos
7
a(2i + 1) COS [
(2j + 1)]
(5.8)
L
(5.9)
and its inverse 2-D DCT as:
1
fij
=
MN
M
N
S Fowi(a)w2 (3) cos [
1
a(2i + 1) cos
1M2
MNa=1 0=1
where
wi(a)
1
if a
2
= -
w 2 (j) =
=
0, else wi(a) = 1
1
if 3 = 0, else w2 (/)
2
-
=
1
(2j + 1)
5.1. Introduction to InSAR processing
101
By transforming both side of (5.6), the unwrapped phase can be found by
1
i
=
M
NE
MNa=1
[
N
0=1
' wi (a)w2()
and we can equate y and q
(3
-
Cos
1MI
p and q-
2(cos !
a(2) cos
2
/3(2j + 1)] (510)
0-are the DCT pairs of p and 0):
+ cos
-2)
The boundary conditions are taken care of by the cosine transform, which forces the
derivatives at the boundaries to be zero and the least squares solution can thus be
obtained from the inverse transform of q5'0.
5.1.2
Height inversion and foreshortening
The last steps in the height retrieval process are the height inversion and foreshortening. The cross section of the geometry on which InSAR height inversion is based is
illustrated below (Fig. 5-2) where p1 and P2 are the path lengths between the sensors
z
B
P1
P2
H
h
I
Figure 5-2: Geometry for InSAR height inversion.
Chapter 5. Height Retrieval in SAR Interferometry
102
and the terrain, B is the baseline, a and 9 are the baseline orientation angle and the
looking angle respectively, h is the terrain height, and H is the platform height. The
inversion of the height information [87] is based on the equation:
sin(O - a) -
6
B
B
2p1
+
62
-,
(5.12)
=P2-1P
2Bp1
h = H -P cos 9
(5.13)
with the parameters shown in Fig. 5-2.
For spaceborne SAR, the platform height is of the order hundreds of kilometers,
and the baseline is of the order hundreds of meters. The second and third terms on
the right hand side of (5.12) are often dropped to yield:
6
B
sin(-a)=-,
6
(5.14)
=P2-P1
The height can thus be computed based on the path difference calculated from the
unwrapped phase In terms of 6, we have:
H - p1 cos sin-1
1
B
(B
2p1
-
62
+
J +a
2Bp1)
(5.15)
or
h= H- p1 cos sin-i
.B
(-) + a
(5.16)
From(5.16), it is seen that in order to obtain an accurate height, accurate unwrapped
phase, as well as baseline and baseline orientation angle are needed. These parameters also affect the dynamic range of the inverted height and will in turn affect the
registration of the height pixels.
In a SAR interferogram, the orientation of the pixels are arranged in a (range,
cross range) manner rather than in an (x, y) manner. For spaceborne SAR platform,
the cross range dimension can be mapped to the y direction linearly, but we cannot
5.1. Introduction to InSAR processing
103
C
2
-
---------9
3
+
B
Figure 5-3: Foreshortening diagram.
perform the same mapping for the range direction because the range is a function of
height. This necessitates the foreshortening process and can be explained by considering Fig. 5-3. The horizontal axis is in the x direction and the regions 1, 2, 3, and
4 are in the range direction. Whereas by mapping the height information directly
to range we would arrange the pixels in the order C, A, and B; the correct order is
A, B, and C, with B and C co-located. It is noticed that points A and B are not
affected as they are of zero height. The error of point C is due to its height and the
looking angle 6. The correction factor g can be computed by using the height h of
point C:
7r
g = htan
-
-
)
(5.17)
Thus a positive value of g would shift the pixel in the positive x direction and a
negative value of g would shift the pixel in the negative x direction.
Fig. 5-4 shows the absolute path difference against the range direction, the inverted
height and the foreshortened height profile. The effect of the shifting of registered
height pixels along the x axis is illustrated. In this noiseless setting, the foreshortened
height profile matches the original terrain profile (Fig. 5-5). The limit within which
(5.17) is useful is governed by the amount of noise present and the shadowing effect
of the terrain. If the noise level is very high, not only the height would be erroneous,
Chapter 5. Height Retrieval in SAR Interferometry
104
-114.5
-115.5
-
-116
-
..
...
-
-117.5
-----
-118
- --
-- . -
Inverted profile(foreshortered)
- - - -.
- -.-.-.-.
- - -.-.-.-
- ......
......................
....-.
-117
-119
-
-.
-.
. ..-.-.-.-.
. .... .. . .
-116.5
-118.5
1Iverted profile
.-.
. . . . . -....- . .-.-... .. .. .
..
...
..
..
..
..
.....
.-...
.. ..
-115
......-
-- --..
--. ...
- - ...
~
---
- --- - -- - -
-
..
-.-.-.-. ..-.-.. . -.- -.-.
-119.5
1.0 77 1.0775 1.078 1.0785 1.079 1.0795
P1(m)
......
E
1.08
.1
,.....
...
.....
-.
.'
..-
- - - - -......
.
- .. ...
I........
.
.
.
.-.1.0805 1.081
0
X 104
10
20
30
40
50
60
70
80
90
100
Registered position (m)
Figure 5-4: Plot of absolute path difference against range of a simulated height profile
and its inverted height with and without foreshortening.
the position of the registered height pixel of the DEM would also be deviated.
In an ideal case, an interferogram would be free of residues and noise. Unwrapping
algorithms would perform flawlessly and the retrieved height profile would correspond
to the DEM exactly. The following suite of different techniques will thus address the
issue of height retrieval in the presence of noise.
An improvement to the phase
unwrapping process is introduced first, followed by the preprocessing technique and
its applications to both phase unwrapping and baseline estimation.
5.2
Weighted least squares phase unwrapping method with Preconditioned Conjugate Gradient
(PCG) solver
In the residues weighted least squares method, the weight of either 0 or 1 is assigned
to each pixel. The decision of placing the weights is determined by the existence of
105
5.2. WLS phase unwrappingmethod with PCG solver
4
3
Original profile
2
0000000
-
Ec
,nVOMOprofile
-
r
000o
1
01
-2 -3
-A
0
10
20
30
40
60
50
70
80
90
100
Normalized Range (m)
Figure 5-5: Comparison of inverted and original profile.
residues formed by the associated pixels:
resij =
where 6 = V4i,j- Oi-1,j.
6
"+1,j + 6 i+i,j+
If (round{resi, /27r}
-
6
+1,j+1 - 6Yj+1
(5.18)
0), there is a residue and the associated
pixels will be weighted accordingly. We can represent (5.6) as
(5.19)
and the weighted normal equation of the least squares method as
(5.20)
where Q = 7 + D1 represent the weighted operator and c is the weighted wrapped
phase difference. Hence we can solve (5.20) iteratively by Picard iteration with the
Chapter 5. Height Retrieval in SAR Interferometry
106
initial weighted wrapped phase difference T.
- Ok+1 = T - D ' Ok
(5.21)
The difference matrix D represents the difference between the weighted difference and
the un-weighted difference of the wrapped phase:
D
O
=
'
(Q
(5.22)
The disadvantage of this Picard iteration is that in order to monitor the convergence,
we can rely only on setting a predetermined number of iterations or by computing
the norm of D -k
at the end of each iteration, which is a measure of how good the
weighted discrete Laplacian of the solution compares with the unweighted one. It is
arbitrary and does not guarantee convergence.
We can also solve the weighted least squares problem by using the Preconditioned
Conjugate Gradient (PCG) solver. The description of the algorithm can also be found
in [93]. Starting with
#o =
0 and To = -, we solve for fo in
P -zO = k, where k = 0, 1, 2, ...
(5.23)
which forms the conditioning of the matrix to be solved by the conjugate gradient
method. At each iteration, we compute the vector Pk and the coefficient ak and
3
k,
and minimize the residue To. For k > 1,
Pk = Zk_1 + Okpk-1
(5.24)
where
=T
Tk-
2
(5.25)
- -k-2
5.2. WLS phase unwrapping method with PCG solver
107
The iterative solution is given by
Ok =
k-1
+ akpk
(5.26)
where
ak
Tkl1T-T=
Pk -Q'Pk
The iterative process is repeated by updating
rk =Tk-1 -
(.7
(.7
rk,
kQ 'Pk
(5.28)
The implementation of PCG algorithm utilizes the residue of the normal equation directly in the formulation, and is thus superior to Picard iteration. In our simulations,
a residue of 10-4 is used as the criterion to terminate the unwrapping program.
5.2.1
Comparison of PCG method with Picard iteration for
ERS-1 data
To illustrate the improvement of PCG method over Picard iteration, the results from
the height inversion of ERS-1 L-band datasets near Mount Phillips of Alaska are
presented here. The study comprises two datasets. The size of each interferogram is
256 x 256 pixels. Alaska-1 dataset has an average fringe density and Alaska-2 dataset
has a region of high fringe density, which poses a challenge to the phase unwrapping
method. Whereas for branch-cut phase unwrapping method [94]-[97], the rewrapped
phase is always the same as the original wrapped phase, and hence visual judgement
of the rewrapped phase is not feasible; the rewrapped phase is not the same as the
original wrapped phase for the least squares method. In the presence of noise, the
rewrapped phase of the more accurate solution will yield a closer visual resemblance to
the original wrapped phase. Hence the abilities of the two methods to preserve fringe
density can be compared.
Fig. 5-6 and Fig. 5-7 show the original interferograms,
Chapter 5. Height Retrieval in SAR Interferometry
108
DEMs, and the rewrapped images from Picard iteration and PCG method of the two
datasets. It is seen that when the fringe density is not very high, Picard iteration
and PCG method perform on a similar level. But as fringe density is increased, PCG
method is able to maintain a dense rewrapped phase (lower right corner of Fig. 5-7).
The inverted and registered height pixels are also compared to the DEM, and
the errors of the pixels are shown as 2-D plots in Fig. 5-8 and Fig. 5-9. Again, the
2-D height error distributions of both methods are similar for Alaska-1 dataset. The
estimation of height around the area of high fringe density by PCG method is more
accurate.
Fig. 5-10 and Fig. 5-11 show the height error histograms of Alaska-1 dataset by
the two methods and Fig. 5-12 and Fig. 5-13 show the height error histograms of
Alaska-2 dataset by the two methods. The inverted height are adjusted so that they
have a 0 mean in these cases. The histograms show a similar distributions for Alaska1 dataset, but the width of the distribution of the error by the PCG method on
Alaska-2 dataset is considerably narrower.
To show the convergence of both methods quantitatively, the RMS height error
of both methods are plot against iterations in Fig. 5-14. Both methods are set to
compute for 50 iterations, and it is seen that the PCG method converges faster and
to a more accurate solution.
5.2. WLS phase unwrappingmethod with PCG solver
Original interferogram
Digital elevation map
Rewrapped image
(Picard iteration)
Rewrapped image
(PCG)
109
Figure 5-6: Original interferogram, DEM, and rewrapped images from Picard iteration and PCG method of Alaska-1 dataset.
Chapter 5. Height Retrieval in SAR Interferometry
110
Original interferogram
Digital elevation map
Rewrapped image
(Picard iteration)
Rewrapped image
(PCG)
Figure 5-7: Original interferogram, DEM, and rewrapped images from Picard iteration and PCG method of Alaska-2 dataset
ill
5.2. WLS phase unwrapping method with PCG solver
250m
(Picard iteration)
-250m
(PCG)
Figure 5-8: Height error distribution for Alaska-1 dataset.
200m
(Picard iteration)
-200m
(PCG)
Figure 5-9: Height error distribution for Alaska-2 dataset.
Chapter 5. Height Retrieval in SAR Interferometry
112
800
,
700
600
500
w 400
300
200
100
0
-200
-150
-100
-50
0
Error(m)
50
100
150
200
Figure 5-10: Height error histogram of the inverted height by Picard iteration on
Alaska-i dataset, O'h = 20.1 m.
?5UU
700 .
600.
500
400
z)
300
200
100
0|L-200
-150
-100
-50
0
50
Error(m)
100
150
200
Figure 5-11: Height error histogram of the inverted height by PCG method on Alaska1 dataset, Uh = 21.2 m.
5.2. WLS phase unwrapping method with PCG solver
113
800700
600
500 .8
400 -
zC
300 200
100
n
-200
-150
-100
-50
0
Error(m)
50
100
150
200
Figure 5-12: Height error histogram of the inverted height by Picard iteration method
on Alaska-2 dataset, c-h = 39.9 m.
.
800
.
.
.
.
700
600
x
500
400
z
.
300 r
200
100
0h
-200
..
I,
-150
-100
.
-50
0
Error(in)
50
100
150
200
Figure 5-13: Height error histogram of the inverted height by PCG method on Alaska2 dataset, -h= 25m.
Chapter 5. Height Retrieval in SAR Interferometry
114
50
45
40
C
+'--
Picard iteration
5
30
-
PCG
0
10-
Number of iterations
Figure 5-14: Convergence comparison between Picard iteration and PCG method for
Alaska-2 dataset.
5.3
Preprocessing of interferogram
The two processes, phase unwrapping and height inversion, are sensitive to noise.
Hence removing the effect of noise while preserving the fringe density will result in
better retrieval of height.
A simple method to perform the filtering would be to
average the phase of the pixels in the interferogram within a pre-defined window size.
The disadvantage of such method is that it would correspond to a low pass filter.
As the noise are filtered out, so is the phase information. The dynamic range of the
inverted height and the slope of the terrain will be underestimated.
The method
presented here is based on [61 and the complex averaging method is chosen for its
simplicity and speed.
We can formulate the filtering process as a transformation operator T:
= T [kij]
(5.29)
5.3. Preprocessingof interferogram
115
which in the context of the phase unwrapping equation (5.1) yields
= T {Arg [ei[oiji+noise
}
(5.30)
The operation is on real-valued data or the phase of the pixels, which is limited to
(-7r,
7r). Such filtering of
4
ij
in the presence of discontinuities results in blurring.
We can, however, use an operator T that works on the complex interferogram exp(iO)
and obtain the filtered real phase 0' as
= Arg tT
[e"'J] 4
(5.31)
Comparison of (5.31) with (5.30) shows that this method process the complex phase
value instead of the real phase. As the values are continuous on the complex plane,
that is no discontinuity. The windowed-average filter is one example of such method.
A 3 x 3 window applied to the interferogram yields a filtered version which contains
the phase of the complex average.
The effect of preprocessing on the interferogram in the presence of injected Gaussian phase noise with a RMS value of 720 (Fig. 5-15) can be seen in Fig. 5-16. The
preprocessed image yields improved visual quality while preserving the fringe density.
Using the preprocessed pixels for phase unwrapping and retrieving phase information
of ground control points (GCP) is expected to result in improved baseline estimation
and height inversion.
Chapter 5. Height Retrieval in SAR Interferometry
116
x10
x14
6
3
5
2
4
1
3
0
E
0
0
-1
2
-2
0
-3
5.85
5.9
5.95
6.05
6
6.1
6.15
x 10
Range (m)
Figure 5-15: Simulated interferogram in the presence of injected Gaussian phase noise
with a standard deviation of 720 before preprocessing.
x W
6 IM
--
3
2
1
4
E
0
0
U
-1
-2
0
-3
5.85
5.9
5.95
6
Range (m)
6.05
6.1
6.15
x 10
Figure 5-16: Simulated interferograms in the presence of injected Gaussian phase
noise with a standard deviation of 720 after preprocessing.
5.3. Preprocessingof interferogram
5.3.1
117
Phase unwrapping of SAR data from JERS-1
As an example of InSAR preprocessing, the SAR interferogram from the Japan Earth
Resources Satellite (JERS-1) based on the data pair dated September 7, 1995 and
October 21, 1995 is used. The satellite is an L-band system with wavelength A =
0.23530 m. The baseline is 567.9 m and the baseline orientation angle a is 580. The
mean range is 729, 542 m. A summary of the JERS-1 SAR parameters are included
in Table 5.1.
The DEM used for registration and comparison with the inverted height is based
on the survey data dated 1987 (first surveyed in 1971 and corrected in 1977). The
DEM has vertical and horizontal accuracies of 0.1 m and 50 m respectively. A summary of the Mt. Fuji DEM parameters are included in Table 5.2.
To begin the registration process, the height is inverted from the InSAR data
and then foreshortened. Foreshortening readjusts the position of each pixel along
the range direction to compensate for the fact that range pixels are not necessarily
in order as far as the ground dimension is concerned. Fig. 5-19 shows the rectified
image of the DEM of Mt. Fuji. The size of the data is interpolated and scaled to
270 x 270 pixels. Fig. 5-19 also shows the rectified inverted and foreshortened terrain
map computed from the InSAR data. The RMS error (standard deviation) for the
registered dataset (270 x 270 pixels) is 118 m. Fig. 5-20 shows the cross section cut
along the azimuthal direction. The profile is reproduced accurately except at the
peak, where the crater may have caused some shadowing. Fig. 5-21 depicts the cross
section cut along the range direction.
Fig. 5-17 and Fig. 5-18 are the images of the original interferogram and the interferogram after the application of complex averaging. It is noticed that the fringe
density remains the same while perceptively, the preprocessed image is clearer. The
preprocessed interferogram contains less residues as tabulated in Table 5.3. The preprocessing of the interferogram also improved the convergence of the PCG method
which converges in 28 steps instead of 48 steps.
118
Chapter 5. Height Retrieval in SAR Interferometry
Data Size:
Range Resolution:
Azimuthal Resolution:
Baseline:
Range:
Wavelength:
Incident angle (0) :
Baseline orientation angle (a):
512x 512
25 m
25 m
567.9 m
729542 m
0.23530 m
390
580
Table 5.1: JERS-1 SAR parameters.
Data Size:
N-S Resolution 2:
E-W Resolution i:
Lower Left Corner (SE):
Lower Right Corner (NE):
Upper Left Corner (SW):
Upper Right Corner (NW):
Minimum height:
Maximum height:
Horizontal accuracy:
Vertical accuracy:
400 x 400
46.3 m
56.9 m
35.2500'N
35.2500'N
35.4167*N
35.4167 0 N
240 m
3771 m
50 m
0.1 m
138.6250 0 W
138.8750 0 W
138.6250'W
138.8750'W
Table 5.2: Mt. Fuji DEM parameters.
Positive residues
Negative residues
PCG-WLS iterations
Averaged
Original
916
917
28
2301
2299
48
Table 5.3: Comparison of residues in the interferogram before and after preprocessing.
5.3. Preprocessingof interferogram
Figure 5-17: Original interferogram of Mt. Fuji (512 x 512 pixels).
Figure 5-18: Filtered interferogram (512 x 512 pixels).
119
120
Chapter 5. Height Retrieval in SAR Interferometry
3500
3000
2500
2000
1500
Inverted height (m)
3500
3000
2500
2000
1500
DEM (m)
Figure 5-19: 2-D inverted terrain height of Mt. Fuji using JERS-1 InSAR data and
2-D image of the truncated DEM of Mt. Fuji used for comparison.
121
5.3. Preprocessingof interferogram
3800
3600
I
~
3400
3200
Ec
L)
3000
2800
2600
2400
2200
0
2 00
100
150
Cross range (pixels)
50
250
300
Figure 5-20: Comparison of the inverted height of Mt. Fuji with DEM (Cross section
along the azimuth).
4000
3800
-SAR
-M
3600
3400
-c
0)
0
I
3200
3000
2800
2600
2400
2200
2000
0
50
200
150
100
Range (pixels)
250
300
Figure 5-21: Comparison of the inverted height of Mt. Fuji with DEM (Cross section
along the range).
Chapter 5. Height Retrieval in SAR Interferometry
122
5.3.2
Improvement of noise tolerance level by preprocessing
in conjunction with PCG method
This example applies complex interferogram filtering with subsequent phase unwrapping and height inversion to simulated data with different levels of injected noise. An
actual DEM is used as input to a SAR simulation system (EMSARS) which allows
simulation of terrain features such as vegetations. The DEM is shown in Fig. 5-22
and the interferogram resulted are further processed to estimate the original terrain
height and to compute the RMS height error which requires a registration procedure
to match the two DEMs. The X-band SAR simulation parameters are tabulated in
Table 5.4 and the Grayling DEM parameters are shown in Table 5.5
Data Size:
Range Resolution:
Azimuthal Resolution:
Baseline:
Range:
Wavelength:
Incident angle (0) :
Baseline orientation angle (a):
240 x 300
10 m
10 m
141.4214 m
694962.69 m
0.0320 m
34.9680
450
Table 5.4: X-band SAR simulation parameters.
Data Size:
N-S Resolution 2:
E-W Resolution D:
Vertical accuracy:
610x 710
6m
6m
2m
Table 5.5: Grayling DEM parameters.
Before we apply preprocessing to the interferograms, we first compare the noise
tolerance performance between Picard iteration and PCG method. Since the PCG
method take into account the convergence of the residue of the normal equation, it
123
5.3. Preprocessingof interferogram
100
200
300
300
400
500
150
600
100
50
700 .m
0
100
200
300
400
500
600
Figure 5-22: Grayling DEM.
is expected to perform better in the presence of high noise level. Table 5.6 shows
the RMS height error from the retrieval process of the X-band simulated InSAR
images with different levels of RMS phase noise. It is seen that whereas both method
perform well for low noise level, PCG has a higher noise tolerance. Note that without
preprocessing, high level of injected phase noise (350 upward) does not allow proper
registration and comparison.
RMS phase noise
00
100
200
Picard iteration
PCG
21.02 m
22.33 m
25.22 m
m
26.56
33.32 m
25.99 m
Table 5.6: Comparison of standard deviation of inverted height error between Picard
iteration and PCG method for simulated X-band InSAR data with different levels of
injected Gaussian noise.
With preprocessing, the quality of the noisy interferograms is improved as seen in
Fig. 5-23-Fig. 5-25. The RMS height errors of the DEM are tabulated in Table 5.7.
124
Chapter 5. Height Retrieval in SAR Interferometry
The first row in Table 5.7 gives the results when there is no additive noise. The other
rows are for more noisy interferograms with the noise injected as phase error of the
local backscattering coefficient before calculation of the SAR image. It is seen that,
for noisy data, the complex interferogram filtering methods yield improved height
inversion accuracy and that the preprocessed results deteriorate more slowly at the
extreme noise level end. Fig. 5-26-Fig. 5-27 show the errors in 2-D for various noise
levels introduced into the interferogram.
Noise
0
10
20
35
50
65
Noisy
20.9
24.4
33.2
38.8
63.1
98.1
Averaged
23.9
24.3
24.6
27.7
34.2
54.9
Table 5.7: RMS error [m] of height inversion for simulated SAR interferograms with
increasing RMS phase error [deg] in the local backscattering coefficient.
5.3. Preprocessingof interferogram
125
Original interferogram
Rewrapped phase without preprocessing
Rewrapped interferogram with preprocessing
Figure 5-23: Original and rewrapped interferogram with and without preprocessing
(Injected RMS phase noise: o = 0*).
126
Chapter 5. Height Retrieval in SAR Interferometry
Original interferogram
Rewrapped phase without preprocessing
Rewrapped interferogram with preprocessing
Figure 5-24: Original and rewrapped interferogram with and without preprocessing
(Injected RMS phase noise: o = 350).
5.3. Preprocessingof interferogram
127
Original interferogram
Rewrapped phase without preprocessing
Rewrapped interferogram with preprocessing
Figure 5-25: Original and rewrapped interferogram with and without preprocessing
(Injected RMS phase noise: o = 65*).
Chapter 5. Height Retrieval in SAR Interferometry
128
0
100
350
50
0
650
Figure 5-26: 2-D plots of RMS height inversion error [m] for simulated X-band SAR
interferograms with increasing RMS phase error [deg] in the local backscattering
coefficient (without preprocessing).
129
5.3. Preprocessingof interferogram
00
100
350
65
Figure 5-27: 2-D plots of RMS height inversion error [m] for simulated X-band SAR
interferograms with increasing RMS phase error [deg] in the local backscattering
coefficient (with preprocessing).
Chapter 5. Height Retrieval in SAR Interferometry
130
5.4
Baseline estimation
A simulated SAR interferogram is generated using the SAR configuration given in
Table 5.8. The interferogram is constructed by first generating a simulated mountain
terrain as shown in Fig. 5-28 where the dimensions are 60 x 60 km with x in the range
direction and y in the cross range direction. The dynamic range of the terrain is
about 500 m. By using such SAR configuration, the phase of the signal at each sensor
is calculated for each pixel using
0o ij = 2kop i a, o = 1, 2
(5.32)
where ko is the free space wavenumber, a indicates the sensor used, and pij, is the
distance from the sensor to the simulated terrain height for the pixel at (i, j). The
interferogram phase value of the pixel at (i, j) is obtained by wrapping the phase
difference
02 ij
-
#1 ij -
Baseline
0
a
500 m
33.40260
36.86980
Pi
5.9913 x 10 5 m
H
A
500km
0.2353 m
Table 5.8: Simulated SAR interferogram configuration.
Fig. 5-29 shows the generated interferogram with the flat-phase removed. In this
case there is no injected noise and it can be seen that the interferogram is very clear
and free of inconsistencies. Five ground control points (GCPs) are inserted into the
interferogram and are indicated as dots on the fringe map.
131
5.4. Baseline Estimation
500
400
600
400
-
300
200
0
x10
4
6
100
TA
tp 3
6
5
2
1
x
04
0
3
0
Figure 5-28: Simulated terrain. (60 x 60 km.)
5.4.1
Addition of noise
The noise is added by
#(i,7j) = #0(iij)+
where
ON(i,
N(i,j)
(5.33)
j) is a random variable that is Gaussian distributed and its standard
deviation reflects the percentage (of 27r) phase noise injected. Fig. 5-30 shows the
same interferogram with 5% phase noise injected and it is seen that the fringes are
visually distorted. Such injected noise will affect both the unwrapped phase and
height calculation.
5.4.2
Baseline inversion
For known baseline orientation angle, the baseline can readily be computed by solving
the quadratic equation in B:
Bi
=
-p 1lAi ± I(p Ai)2 + If + 2 plz 6 i
(5.34)
Chapter 5. Height Retrieval in SAR Interferometry
132
x10
6
3
5
2
4
0
3
0
2
-2
0
-3
5.85
5.9
5.95
6.05
6
Range (m)
6.15
6.1
x 10,
Figure 5-29: Simulated interferogram with GCP.
where
Ai =sin cos-
1 H - hi
- a
(5.35)
in which the subscript i denotes the ith co-registered pixel of the GCP and the
subscript of p indicates that the range as measured from the first sensor position is
used. Since 6i and pli are taken from the SAR measurement, hi is the known height
of GCP, and the baseline orientation angle a is assumed to be known in this case,
the baseline can be calculated exactly.
5.4.3
Incorporation of absolute and relative GCP information
Sensor platform, at each pass over the target terrain, in general contains ambiguities
in x, y, and z direction, which translates into uncertainties in baseline length and
orientation angle. To obtain both B and a based on GCP, a cost function is needed
133
5.4. Baseline Estimation
x 104
6
3
4
5
H2
4
1
3
0
2
-1
1
-2
a)
0)
C
Cu
Ci,
C,,
0
0
-3
0
5.85
5.9
5.95
6
6.1
6.05
Range (m)
6.15
5
x 10
Figure 5-30: Simulated interferogram with 5% RMS phase noise.
to be formulated, which upon minimization will yield a least squares error for the
GCP parameters.
We can use either the absolute height of the GCP as the cost function, yielding
fi as shown below, or we can incorporate the relative positions of the GCP into the
cost function also, resulting in
5
f2
as shown below.
(
hi - H - pi cos [sin-
f, (a, B)
(
hi-H- P1i cos Isin-1
2(a, B) =
- 2piB
(
6
o5
2
2Bpji)
(5.36)
+
-
B +
2Bp 1 ;j
2p+
(xi
-
B
_Z
5
+ E [(pli sin Oi - pi-i1 sin Oi-1)
-
Xi_1)12
(5.37)
i=2
The task is to form an initial guess of B and a, with which the optimization routine
Chapter 5. Height Retrieval in SAR Interferometry
134
can search for the best solution. In this case the optimization is done in MATLAB, and
it is found that the initial guess required is robust, and accurate baseline parameters
can be obtained for low injected noise levels.
5.4.4
Simulation results on baseline inversion without preprocessing
The baseline estimation method is carried out based on the optimization of the two
cost function fi and f2. For the same terrain map (Fig. 5-28), 100 realizations of
noisy interferograms with different levels of additive Gaussian noise are used in each
case. The GCPs are identified in the interferograms and the baseline as well as the
baseline orientation angle are estimated.
In order to verify the effects of the baseline estimation algorithms, the inverted
height from the interferogram is compared to the ground truth. Since the SAR and
the DEM images are offset in the range direction, the SAR and the DEM images must
be registered before any comparisons can be made. The registration process includes
all steps necessary to synchronize the two images so that a point by point comparison
can be made.
Fig. 5-31 shows the height error histogram for different levels of injected phase
noise with B and a estimated by using fi as the cost function. It can be seen that
as the amount of injected noise increase, the height error spreads out. Averaging
over 100 realizations, the mean standard deviation of height error is plotted in Fig. 532. The mean standard deviation of height error is seen to increase linearly as noise
increases and change from 0 m for the noiseless case to around 55 m for 20% RMS
phase noise, which correspond to approximately 10% of the dynamic range.
Fig. 5-33 and Fig. 5-34 shows the mean RMS baseline error and mean percentage
RMS error of baseline orientation angle for 100 realizations of noise levels with B and
a estimated by using fi as the cost function respectively and both are also seen to
vary linearly as a function of injected noise.
5.4. Baseline Estimation
135
To try a different method for the estimation of the baseline, we introduce
f2
as
the cost function which take into account the relative positions of the GCPs as well as
the height information. Fig. 5-35 to 5-37 shows the corresponding results on height
error, baseline length and orientation angle. Overall, the performance of
f2
is the
same as fi.
Using just one pixel to represent the GCP results in poor performance in the
presence of high noise level and preprocessing can be used in this case to raise the
noise tolerance.
Chapter 5. Height Retrieval in SAR Interferometry
136
2.5
percentage phase noise
percentage phase noise
5
x 104
2.5
10
X 104
2-
2
Cn
(D
x
1.5%6.
0 a1.5
. --
.0
1
E
E
C
0.5
0.5-
~FlK
-20(
2.5
-100
100
0
error (m)
01
-20()
200
2.5
200
20
x 104
2
2
Cn
a)
X
100
0
error (m)
percentage phase noise
percentage phase noise
15
x 104
I
-100
CO
x
* 1.5
1.5
a)
.0
E
1
E
n1
C
0.5
0
-20(
0.5
'-"'
-100
I
'
'
'
0
error (m)
'
6
100
200
01
-20 )
I I I
_ ___j
-100
, I
0
error (m)
--
I-
100
200
Figure 5-31: Height error histogram for different levels of injected phase noise estimated using fi as cost function.
137
5.4. Baseline Estimation
60
50
40
C
30
CO
> 20
-
10
C
Sa
ch
0 01
2
4
6
8
10
12
14
16
18
20
percentage phase noise
Figure 5-32: Mean standard deviation of height error for 100 realizations of noise
levels estimated using fi as cost function.
2
1.8-
E)
0,
E)
1.6 1.41.21 0.8
0.6
E 0.4
0.2
00
2
4
6
8
10
12
14
16
18
20
percentage phase noise
Figure 5-33: Mean RMS baseline error for 100 realizations of noise levels estimated
using fi as cost function.
138
Chapter 5. Height Retrieval in SAR Interferometry
0.04
-
0.035
a)
0.03
CO
.0
0.025
0
0.02
0.015
0.01
C
CD
0.005
0
0
.
I .
2
4
6
j
8
I
10
12
14
16
18
20
percentage phase noise
Figure 5-34: Mean percentage RMS error of baseline orientation angle for 100 realizations of noise levels estimated using fi as cost function.
50
.~40
30
D 20
-0
as
-v
-
10
00
2
4
6
8
10 12 14
16
percentage phase noise
18
20
Figure 5-35: Mean standard deviation of height error for 100 realizations of noise
levels estimated using f2 as cost function.
139
5.4. Baseline Estimation
2
.
.
2
4
.
.
.
.
.
.
.
16
18
1.8
0 1.6
C',
1.4
1.2 -
0.6
0.4
$
0.2
00
6
8
10 12 14
percentage phase noise
20
Figure 5-36: Mean RMS baseline error for 100 realizations of noise levels estimated
using f2 as cost function.
0.04
C
0)
.C
0.035
0.03
.0
0.025
0
0.02
E
a)
CM
cc
0.015
0.01
0.005
0
0
2
4
6
8
10
12
14
16
18
20
percentage phase noise
Figure 5-37: Mean percentage RMS error of baseline orientation angle for 100 realizations of noise levels estimated using f2 as cost function.
Chapter 5. Height Retrieval in SAR Interferometry
140
5.4.5
Baseline inversion with preprocessing
With preprocessing, a 3 x 3 complex averaging window is centered at each GCP to
filter out the noise. Fig. 5-38 shows the height error histogram for different levels
of injected phase noise with B and a estimated by using
f2
as the cost function.
It can be seen that whereas the amount of injected noise increases, the height error
does not spread as much as the case without preprocessing.
Averaging over 100
realizations, the mean standard deviation of height error is plotted in Fig. 5-39. The
mean standard deviation of height error is seen to increase linearly as noise increases
and change from 0 m for the noiseless case to around 20 m for 20% of phase noise,
which correspond to approximately 4% of the dynamic range and improve the error
by a large margin.
Fig. 5-40 and Fig. 5-41 shows the mean RMS baseline error and mean percentage
RMS error of the baseline orientation angle for 100 realizations of noise levels with
B and a estimated by using
f2
as the cost function respectively and both are also
seen to vary linearly as a function of injected noise. The error introduced, however
is only about one third of what was observed in the case for the unprocessed image,
suggesting preprocessing can be used both to increase the accuracies of unwrapping
and inversion of baseline. Furthermore, it increases the noise tolerance of the DEM
retrieving process and can yield useful information even when the coherence of the
pixels is small.
5.4. Baseline Estimation
141
percentage phase noise
percentage phase noise
5
5
C/)
x 104
-
-
5
-il
3
2
1
0'
-20 0
5
x 104
-100
0
error (m)
100
4Co
0.
E
X
10
-
-
.
2
1
F " --
01
-20 )
200
-100
0
error (m)
'
100
200
percentage phase noise
percentage phase noise
15
.
3
C
.
5
20-
x 104
4
4
Co
104
X
4
4
z
.0
C
E
-
-
X
3
3
U)
E.
E
.0
2
1
1
0'
-20 )
2
E
'
-100
' ' ' ' '
0
error (m)
'
100
200
O'
-20
0
-100
re
0
100
200
error (m)
Figure 5-38: Height error histogram for different levels of injected phase noise estimated using f2 as cost function with preprocessing.
Chapter 5. Height Retrieval in SAR Interferometry
142
20
E 18
0
16
14
C
12
10
.9 8
(D 6
cd
4
2
0c
2
4
6
8
10
12
14
16
18
20
percentage phase noise
Figure 5-39: Mean standard deviation of height error for 100 realizations of noise
levels estimated using f2 as cost function with preprocessing.
2
1.8
1.6
1.4
0
1.2
fi
0.8
as
.o
E
0.6
G
0.4
0.2
0
0
2
4
6
8
10
12
14
16
18
20
percentage phase noise
Figure 5-40: Mean RMS error of baseline for 100 realizations of noise levels estimated
using f2 as cost function with preprocessing.
143
5.4. Baseline Estimation
0.04
C
(D
0.035
Cn
0.03
(%
0.025
0.02
a)
0.015
Cn
a)
0.01
0.005
0c
2
4
6
8
10
12
14
16
18
20
percentage phase noise
Figure 5-41: Mean percentage RMS error of baseline orientation angle for 100 realizations of noise levels estimated using f2 as cost function with preprocessing.
Chapter 5. Height Retrieval in SAR Interferometry
144
5.5
Conclusion
Height retrieval based on InSAR processing is studied and each sub-process is treated
as a modular unit. Improvements are obtained for phase unwrapping and baseline
estimation. Furthermore, preprocessing is applied to the interferogram to yield a
cleaner image.
The basics of InSAR processing is reviewed and the first improvement to the process is the use of preconditioned conjugate gradient solver in the residues weighted
least squares phase unwrapping method. It improves the dynamic range of the inverted height and performs better than the previous method (Picard iteration). Unwrapping examples using ERS-1 data are used to show that PCG method converges
faster and to a more accurate solution. It is also able to preserve the fringe density in
highly dynamic regions. Unwrappings performed on X-band simulated interferograms
with various levels of injected phase noise show that PCG method has a higher noise
tolerance.
Preprocessing by using a 3 x 3 complex averaging window cleans noisy interferogram without suffering the loss of fringe density. Examples using the JERS-1 data
shows that it reduce the number of residues and increase the rate of convergence
of the PCG method. Preprocessings performed on X-band simulated interferograms
with various levels of injected phase noise results in more accurate height retrieval
and increase the usable area of the interferogram, noise tolerance is also increased.
Baseline estimation using the absolute height and relative positions of the ground
control points as cost function yield an improved baseline. Noise effects again can be
suppressed by using preprocessing. A three-fold improvement is typically obtained for
the height retrieval, baseline estimation, and baseline orientation estimation. The objectives of improving the accuracy of height retrieval and increase the noise tolerance
of InSAR process are achieved.
Chapter 6
Summary
In Chapter 2, left-handed media with negative permittivity and permeability are studied. Such materials are often periodic structures with periodicity much smaller than
the operating wavelength. Whereas a microscopic analysis will yield more insights
into the mechanism of such left-handedness, a macroscopic approach is taken. Such
theoretical analysis allows us to relate the new phenomena to known effects of regular
media. A typical example is the bending of a ray and it can be seen that much of
the LHM's phenomena can be conceptually understood with the aid of this concept.
However, to study the field characteristics and in particular the Poynting vector in
such medium, a formulation that would satisfy Maxwell's equations is essential. Gaussian beam has such advantage while being able to be related to a ray conceptually.
Indeed, rigorous analysis confirm the prediction of the negative lateral shift of a ray.
The extra information gained from this analysis is that firstly, for a matched negative
slab, the incident footprint is reproduced in the transmitted region. Secondly, the
Poynting power component parallel to the slab is opposite in direction compared to
the one outside the slab. The analysis of such slab by a Gaussian beam can also
provide theoretical comparison with experiments, and facilitates the characterization
of LHM.
In Chapter 3, guided waves in a slab of left-handed metamaterial are analyzed. A
145
Chapter 6. Summary
146
macroscopic approach is taken and we analyze the case for TE waves. By matching
the boundary conditions of the fields at the interfaces, the guidance condition is
found. Normally, for a regular dielectric slab waveguide, where the permittivity and
permeability are positive, it is seen that the transverse wavenumber can only assume
a real value. The fields of such modes has an amplitude distribution of a cos or
sin function inside the slab, and decay exponentially outside the slab. The decaying
fields result in a mode that has no lower cutoff. The real transverse wavenumbers also
imply that the wavenumber along the propagation of the guided wave is smaller than
the wavenumber of the dielectric. This is important in radiation problem because
the study of guided waves aids the location of poles in the layered Green's function.
When a slab with negative permittivity and permeability is considered, it can be seen
that in order to satisfy the guidance condition, the transverse wavenumber can be
real as well as imaginary. The imaginary transverse wavenumber corresponds to the
cosh and sinh like field amplitudes distribution and is well known for its existence
in plasma. However, regular modes do not propagate in plasma. In a slab of LHM,
depending on the constitutive relations, both regular modes and these plasmon modes
can coexist. It also imply that the wavenumber along the propagation of the guided
wave can have a value larger than the wavenumber of the LHM. Furthermore, the
Poynting vectors inside and outside the slab are in opposite direction for both modes.
In Chapter 4, the study of differential transmission lines implemented by coupled
strips on a multilayer silicon substrate is presented. Experiments were conducted
on four designs, and measurements from the 12 structures are de-embedded and
converted to the differential parameters. This study bridge the parameter space between simulations and experiments and provide a common space where the limitations
of each aspect can be assessed.
The main hurdle in this project is the measure-
ment/calibration step. The lack of 4-port network analyzer prevented direct 4-port
measurement, instead, six 2-port measurements were performed on each structures.
In addition, non-availability of a 4-port calibration substrate with a compatible pitch
147
size required a correction on the calibration. For the simulations, assessing the accuracy of EM solvers capable of simulating large planar circuit, it was found that
such ability to compute a large number of unknowns is at an expense of accuracy.
However, by comparing with measurements, we can find out the trend of such error
and compensate them accordingly.
In Chapter 5, improvements to the processing of interferometric synthetic aperture radar data are made in a modular manner. Preconditioned conjugate gradient
method enhances the phase unwrapping and baseline estimation based on the formulation of a cost function that take into account the positions and heights of the ground
control points yield more accurate baseline. Preprocessing by complex averaging is
used in both cases to provide a cleaner interferogram. The objective of this suite of
methods is primarily to overcome the effects of noise. Finite aperture size, shadowing
effect of terrain, change of terrain over time, and many other effects contribute to the
noise in the interferogram. The noise exhibits itself as mathematical inconsistencies,
or residues. By complex averaging, we essentially change the problem from matching
the rewrapped phase to the original interferogram to matching the rewrapped phase to
the modified phase. Although the problem has changed, the results as demonstrated
in the different examples in the chapter, show that a better performance is obtained
in all cases. The process of matching the the rewrapped phase to the original wrapped
phase, is further enhanced by residues weighting and preconditioned conjugate gradient method, which incorporate such matching into its formulation (Picard iteration
does not). This suite of methods allow a more accurate height to be retrieved, yield
a larger usable area of the interferogram, and improved the noise tolerance of the
InSAR height retrieval process as a whole.
In conclusion, in this thesis, examples of characterization of physical parameters
via the application of spatially confined electromagnetic waves are discussed. Forward
theoretical examples include the Gaussian beam shift in LHM and the development of
new guided modes in such materials. Inverse/experimental examples include Interfer-
148
Chapter 6. Summary
ometric SAR processing and on-chip coupled-strip transmission line characterization.
By developing both forward and inverse electromagnetic studies in these topics, it is
hoped that the results may find applications in new area and improve the characterization of physical media.
Bibliography
[1] J. A. Kong, B.-I. Wu, and Y. Zhang, "A unique lateral displacement of a Gaussian beam transmitted through a slab with negative permittivity and permeability," Microwave Opt. Technol. Lett., vol. 33, no. 2, pp. 136-139, Apr. 2002.
[2] J. A. Kong, B.-I. Wu, and Y. Zhang, "Lateral displacement of a Gaussian beam
reflected from a grounded slab with negative permittivity and permeability,"
Appl. Phys. Lett., vol. 80, no. 12, pp. 2084-2086, Mar. 2002.
[3] J. Pacheco, T. M. Grzegorczyk, B.-I. Wu, Y. Zhang, and J. A. Kong, "Power
propagation in homogeneous isotropic frequency-dispersive left-handed media,"
Phys. Rev. Lett., vol. 89, no. 25, pp. 257401, Dec. 2002.
[4] B.I.-Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, "Guided modes with
imaginary transverse wavenumber in a slab waveguide with negative permittivity
and permeability," J. Appl. Phys., June 2003.
[5] J. J. Akerson, Y. E. Yang, Y. Hara, B.-I. Wu, and J. A. Kong,
"Automatic
phase unwrapping algorithm in synthetic aperture radar (SAR) interferometry,"
IEICE Trans. Electron., vol. E83-C, no. 12, pp. 1896-1904, Dec. 2000.
[6] H. Braunisch, B.-I. Wu, and J. A. Kong, "Phase unwrapping of SAR interferograms after wavelet denoising,"
in Proc. IEEE Int. Geosci. Remote Sensing
Symp. (IGARSS), Honolulu, July 24-28, 2000, vol. 2, pp. 752-754.
149
[7] V. G. Veselago, "The electrodynamics of substances with simutaneously negative
values of e and p," Soviet Physics USPEKHI, vol. 10, no. 4, pp. 509-514, Jan.Feb. 1968.
[8] T. K. Wu, Frequency Selective Surface and Grid Array, Wiley & Sons, New
York, 1995.
[9] J. C. Vardaxoglou, Frequency Selective Surfaces, Research Studies Press Ltd.,
1997.
[10] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals, Princeton
University Press, 1995.
[11] E. Yablonovitch,
"Inhibited spontaneous emission in solid-state physics and
electronics," Phys. Rev. Lett., vol. 58, pp. 2059, 1987.
[12] E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and
J.
.D.
Joannopoulos, "Donor and acceptor modes in photonic band structure,"
Phys. Rev. Lett., vol. 67, pp. 3380, 1991.
[13] D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, "3D wire mesh photonic crystals," Phys. Rev. Lett., vol. 76, no. 14, pp. 2480-2483, Apr. 1996.
[14] M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,"
Phys.
Rev. B, vol. 62, pp. 10696, 2000.
[15] B.-I. Wu, "Analysis of photonic crystal filters by the finite-difference time-domain
technique," M.S. thesis, Massachusetts Institute of Technology, 1999.
[16] E. Yablonovitch, T. J. Gmitter, and K. M. Leung, "Photonic band structure:
The face-centered-cubic case employing nonspherical atoms," Phys. Rev. Lett.,
vol. 67, pp. 2295, 1991.
150
[17] J. B. Pendry and F. J. Garcia Vidal, "Computational studies of photonic band
gap in metals," in IEE colloquium on semiconductor optical microcavity devices
and photonic bandgaps, London, UK., 1996, pp. 5/1-5/6.
[18] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Extremely low
frequency plasmons in metallic mesostructures,"
Phys. Rev. Lett., vol. 76, no.
25, pp. 4773-4776, June 1996.
[19] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency
plasmons in thin-wire structures,"
J. Physics-Condensed Matter, vol. 10, pp.
4785-4809, 1998.
[20] D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, "Direct calculation of permeability and permittivity for a left-handed metamaterial," Appl. Phys. Lett., vol.
77, no. 14, pp. 2246-2248, Oct. 2000.
[21] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz,
"Composite medium with simutaneously negative permeability and permittivity," Phys. Rev. Lett., vol. 84, no. 18, pp. 4184-4187, May 2000.
[22] R. A. Shelby, Smith D. R., S. C. Nemat-Nasser, and S. Schultz,
"Microwave
transmission through a two-dimensional, isotropic, left-handed metamaterial,"
Appl. Phys. Lett., vol. 78, no. 4, pp. 489-491, Jan. 2001.
[23] R. A. Shelby, Smith D. R., and S. Schultz,
"Experimental verification of a
negative index of refraction," Science, vol. 292, pp. 77-79, Apr. 2001.
[24] A. A. Houck, J. B. Brock, and I. L. Chuang,
"Experimental observations of
a left-handed material that obeys snell's law,"
Phys. Rev. Lett., vol. 90, pp.
137401, 2003.
151
[25] C. G Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian,
"Experimental verification and simulation of negative index of refraction using
snell's law," Phys. Rev. Lett., vol. 90, pp. 107401, 2003.
[26] J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett., vol.
85, no. 18, pp. 3966-3969, Oct. 2000.
[27] P. M. Valanju, R. M. Walser, and A. P. Valanju, "Wave refraction in negativeindex media: Always positive and very inhomogeneous," Phys. Rev. Lett., vol.
88, pp. 187401, 2002.
[28] N. Garcia and M. Nieto-Vesperinas, "Is there an experimental verification of a
negative index of refraction yet?," Optics Letters, vol. 27, pp. 885, 2002.
[29] N. Garcia and M. Nieto-Vesperinas,
"Left-handed materials do not make a
perfect lens," Phys. Rev. Lett., vol. 88, pp. 207403, 2002.
[30] S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, "Refraction in media
with a negative refractive index," Phys. Rev. Lett., vol. 90, pp. 107402, 2003.
[31] J. Pacheco, C. D. Moss, B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong,
"Theoretical and numerical simulations of propagation through left-handed materials," in PIERS, 2002, p. 359.
[32] R. W. Ziolkowski, "Superluminal transmission of information through an electromagnetic metamaterial," Phys. Rev. E, vol. 63, pp. 046604, 2001.
[33] C. D. Moss, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, "Numerical studies
of left handed metamaterials," PIER, vol. 35, pp. 315-334, 2002.
[34]
G. Gomez-Santos, "Universal features of the time evolution of evanescent modes
in a left-handed perfect lens," Phys. Rev. Lett., vol. 90, pp. 077401, 2003.
152
[35] D. R. Fredkin and A. Ron, "Effectively left-handed (negative index) composite
material," Appl. Phys. Lett., vol. 81, no. 10, pp. 1753, 2002.
[36] A. K. Iyer and G. V. Eleftheriades,
"Negative refractive index metamaterials
supporting 2-D waves," in IEEE MTT-S Int. Microwave Symp. Digest, June
2002, vol. 2, pp. 1067-1070.
[37] I. S. Nefedov and S. A. Tretyakov, "Theoretical study of waveguiding structures
containing backward-wave materials," in URSI XXVIIth General Assembly, August 2002, p. 1074.
[38] J. B. Pendry and S. O'Brien, "Very-low-frequency magnetic plasma," Journal
of Physics: Condensed Matter, vol. 14, pp. 7409-7416, 2002.
[39] A. L. Pokrovsky and A. L. Efros, "Electrodynamics of metallic photonic crystals
and the problem of left-handed materials," Phys. Rev. Lett., vol. 89, pp. 093901,
2002.
[40] S. Anantha Ramakrishna, J. B. Pendry, D. Schurig, D. R. Smith, and S. Schultz,
"The asymmetric lossy near-perfect lens," Journal of Modern Optics, vol. 49,
no. 10, pp. 1747-1762, 2002.
[41] R. Ziolkowski and E. Heyman,
"Wave propagation in media having negative
permittivity and permeability," Phys. Rev. E, vol. 64, pp. 056625, 2001.
[42] R. W. Ziolkowski,
"Double negative metamaterial design, experiments, and
applications," in IEEE Antennas Propagat. Int. Symp., June 2002, vol. 1, pp.
396-9.
[43] V. Kuzmiak and A. A. Maradudin, "Scattering properties of a cylinder fabricated
from a left-handed material," Phys. Rev. B, vol. 66, pp. 045116, 2002.
153
[44] A. N. Lagarkov and V. N. Kisel, "Electrodynamic properties of simple bodies
made of materials with negative permeability and negative permittivity," Doklady Physics, vol. 377, pp. 40-43, 2001.
[45] J. Pacheco, B. E. Barrowes, and J. A. Kong,
"Electromagnetic scattering by
canonical objects composed of left-handed materials," in PIERS, 2002, p. 453.
[46] P. Markos and C. M. Soukoulis, "Transmission studies of left-handed materials,"
Phys. Rev. B, vol. 65, pp. 033401, 2002.
[47] P. Markos and C. M. Soukoulis, "Numerical studies of left-handed materials and
arrays of split ring resonators," Phys. Rev. E, vol. 65, pp. 036622, 2002.
[48] F. J. Rachford, D. L. Smith, P. F. Loschialpo, and D. W. Forester, "Calculations
and measurements of wire and/or split-ring negative index media," Phys. Rev.
E, vol. 66, pp. 036613, 2002.
[49] D. R. Smith, D. Schurig, and J. B. Pendry, "Negative refraction of modulated
electromagnetic waves," Appl. Phys. Lett., vol. 81, no. 15, pp. 2713-2715, 2002.
[50] T. Weiland, R. Schuhmann, R. B. Greegor, C. G. Parazzoli, A. M. Vetter, D. R.
Smith, D. C. Vier, and S. Schultz, "Ab initio numerical simulation of left-handed
metamaterials: Comparison of calculations and experiments,"
J. Appl. Phys.,
vol. 90, no. 10, pp. 5419-5424, 2001.
[51] R. B. Greegor, C. G Parazzoli, K. Li, and M. Tanielian, "Origin of dissipative
losses in negative index of refraction materials," Appl. Phys. Lett., vol. 82, no.
14, pp. 2356-2358, 2003.
[52] K. Li, J. McLean, R. B. Greegor, C. G Parazzoli, and M. Tanielian, "Free-space
focused-beam characterization of left-handed materials," Appl. Phys. Lett., vol.
82, no. 15, pp. 2535-2537, 2003.
154
[53] P. Gay-Balmaz and 0. J. F. Martin, "Efficient isotropic magnetic resonators,"
Appl. Phys. Lett., vol. 81, no. 5, pp. 939-941, 2002.
[54] P. Gay-Balmaz and 0. J. F. Martin, "Electromagnetic resonances in individual
and coupled split-ring resonators," J. Appl. Phys., vol. 92, no. 5, pp. 2929-2936,
2002.
[55] S. O'Brien and J. B. Pendry, "Magnetic activity at infrared frequencies in structured metallic photonic crystals," Journal of Physics: Condensed Matter, vol.
14, pp. 6383-6394, 2002.
[56] G. Shvets, "Photonic approach to making a material with a negative index of
refraction," Phys. Rev. B, vol. 67, pp. 035109, 2003.
[57] N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas and Wireless
PropagationLetters, vol. 1, no. 1, pp. 10-13, 2002.
[58] S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A metamaterial
for directive emission," Phys. Rev. Lett., vol. 89, pp. 213902, 2002.
[59] J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic band gap from a stack
of positive and negative index materials," Phys. Rev. Lett., vol. 90, pp. 083901,
2003.
[60] J. B. Pendry and S. Anantha Ramakrishna, "Near-field lenses in two dimensions," Journal of Physics: Condensed Matter, vol. 14, pp. 8463-8479, 2002.
[61] B. Gralak, S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic
crystals," J. Opt. Soc. Am A, vol. 17, no. 6, pp. 1012-1020, 2000.
[62] R. E. Collin, Foundationsfor Microwave Engineering, McGraw-Hill, Inc., New
York, second edition, 1992.
155
[63] J. A. Kong, Electromagnetic Wave Theory, EMW, Cambridge, MA, 2000.
[64] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive
index media using periodically L-C loaded transmission lines,"
IEEE Trans.
Microwave Theory Tech., vol. 50, no. 12, pp. 2702-2712, 2002.
[65] G. V. Eleftheriades, A. K. Iyer, A. Grbic, and 0. Siddiqui, "Negative refractive
index metamaterials based on L-C loaded transmission lines," in PIERS, 2002,
p. 354.
[66] D. R. Smith and N. Kroll, "Negative refractive index in left-handed materials,"
Phys. Rev. Lett., vol. 85, no. 14, pp. 2933-2936, Oct. 2000.
[67] L. D. Landau, E. M. Lifshitz, and Pitaveskil, Electrodynamics of Continuous
Media (Vol. 8 of Course of Theoretical Physics), Pergamon, Oxford, UK, 2nd
edition, 1984.
[68] A. Taflove,
Computational Electrodynamics: The Finite-Difference Time-
Domain Method, Artech House, Boston, 1995.
[69] J. L. Young and R. 0. Nelson, "A summary and systematic analysis of FDTD
algorithms for linearly dispersive media," IEEE Antennas Propagat. Mag., vol.
43, no. 1, pp. 61-77, Feb. 2001.
[70] K. S. Yee,
"Numerical solution of initial boundary value problems involving
Maxwell's equations in isotropic media," IEEE Trans. Antennas Propagat., vol.
AP-14, no. 3, pp. 302-307, May 1966.
[71] V. G. Veselago, "Properties of materials having simultaneously negative values
of the dielectric (e) and the magnetic (p) susceptibilities)," Soviet Physics-Solid
State, vol. 8, no. 12, pp. 2854-2856, June 1967.
[72] H. Braunisch, Y. Zhang, C. 0. Ao, S.-E. Shih, Y. E. Yang, K.-H. Ding, J. A.
Kong, and L. Tsang, "Tapered wave with dominant polarization state for all
156
angles of incidence," IEEE Trans. Antennas Propagat., vol. 48, pp. 1086-1096,
July 2000.
[73] L. Tsang, J. A. Kong, K.-H. Ding, and A. 0. Chi, Scattering of Electromagnetic
Waves, Vol. 2: Numerical Simulations, Wiley, New York, 2001.
[74] I. V. Lindell and S. Ilvonen, "Waves in a slab of uniaxial BW medium,"
J.
Electromagn. Waves Applicat., vol. 16, no. 3, pp. 303-318, 2002.
[75] A. A. Oliner and T. Tamir, "Backward waves on isotropic plasma slabs," J.
Appl. Phys., vol. 33, pp. 231-233, 1962.
[76] B. Prade, J. Y. Vinet, and A. Mysyrowicz,
"Guided optical waves in planar
heterostructures with negative dielectric constant," Phys. Rev. B, vol. 44, no.
24, pp. 13556-13572, 1991.
[77] R. Ruppin, "Surface polaritons of a left-handed medium," Physics Letters A,
vol. 277, pp. 61-64, 2000.
[78] Von F. Goos and H. Hinchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. der Phys., vol. 6, pp. 333-346, 1947.
[79] P. R. Berman, "Goos-Hiinchen shift in negatively refractive media," Phys. Rev.
E, vol. 60, pp. 067603, 2002.
[80] D. E. Bockelman and W. R. Eisenstadt, "Combined differential and commonmode scattering parameters: theory and simulation," IEEE Trans. Microwave
Theory Tech., vol. 43, no. 7, pp. 1530-1539, July 1995.
[81] J. J. Hajjar, "S-parameter measurements: improved de-embedding techniques,"
Analog Devices Internal Document, 1999.
[82] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, Wiley & Sons, New York, 1995.
157
[83] F. K. Li and R. M. Goldstein, "Studies of multibaseline spaceborne interferometric synthetic aperture radars," Proc. IEEE Int. Geosci. Remote Sensing Symp.
(IGARSS), vol. 28, no. 1, pp. 88-96, 1990.
[84] V. Mrstik, G. VanBlaricum, G. Cardillo Jr., and M. Fennell,
"Terrain height
measurement accuracy of interferometric synthetic aperture radars," Proc. IEEE
Int. Geosci. Remote Sensing Symp. (IGARSS), vol. 24, no. 1, pp. 219-228, 1996.
[85] N. Marechal, "Tomographic formulation of interferometric SAR for terrain elevation mapping," Proc. IEEE Int. Geosci. Remote Sensing Symp. (IGARSS),
vol. 33, no. 3, pp. 726-739, 1995.
[86] E. Rodriguez and J. M. Martin, "Theory and design of interferometric synthetic
aperture radars," IEE Proceedings-F,vol. 139, no. 2, pp. 147-159, 1992.
[87] H. A. Zebker and R. M. Goldstein, "Topographic mapping from interferometric
synthetic aperature radar observations," Journal of Geophysical Research, vol.
91, no. B5, pp. 4993-4999, 1996.
[88] D. Just, N. Adam, M. Schwaebisch, and R. Bamler,
unwrapping algorithms for SAR interferograms,"
"Comparison of phase
in International Geoscience
and Remote Sensing Symposium, Firenze, Italy, July 1995, vol. 3, pp. 767-769.
[89] C. R. Guarino, "Weighted two-dimensional phase unwrapping," in International
Geoscience and Remote Sensing Symposium, Firenze, Italy, July 1995, vol. 1,
pp. 193-195.
[90] H. Lim, W. Xu, and X. Huang, "Two new practical methods for phase unwrapping,"
in International Geoscience and Remote Sensing Symposium, Firenze,
Italy, July 1995, vol. 1, pp. 196-198.
[91] G. Fornaro, G. Franceschetti, R. Lanari, and E. Sansosti, "Theoretical analysis
on the robust phase unwrapping algorithms for SAR interferometry," in Inter158
national Geoscience and Remote Sensing Symposium, Lincoln, Nebraska, July
1996, vol. 4, pp. 2047-2049.
[92] H. Kimura, "Method to estimate baseline and platform altitude for SAR interferometry," in International Geoscience and Remote Sensing Symposium, Firenze,
Italy, July 1995, vol. 1, pp. 199-201.
[93] D. G. Ghiglia and L. A. Romero,
"Robust two-dimensional weighted and un-
weighted phase unwrapping that uses fast transforms and iterative methods," J.
Opt. Soc. Am. A, vol. 11, no. 1, pp. 107-117, 1994.
[94] J. R. Buckland, J. M. Huntley, and S. R. E. Turner, "Unwrapping noisy phase
maps by use of a minimum-cost-matching algorithm," Applied Optics, vol. 34,
no. 23, pp. 5100-5109, 1995.
[95] J. M. Huntley, "Noise-immune phase unwrapping algorithm," Applied Optics,
vol. 28, no. 15, pp. 3268-3270, 1989.
[96] R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry:
Two- dimensional phase unwrapping," Radio Science, vol. 23, no. 4, pp. 713-720,
1988.
[97] D. Gusfield, C. Martel, and D. Fernandez-Baca,
"Fast algorithms for bipartite
network flow," SIAM Journal of Computing, vol. 16, no. 2, pp. 237-251, 1987.
159
Biographical note
Bae-Ian Wu was born in Hong Kong on November
degree in electronic engineering from the Chinese
Kong, in 1997, and the M. S. degree in electrical
Institute of Technology, Cambridge, Massachusetts,
3, 1975. He received the B. Eng.
University of Hong Kong, Hong
engineering from Massachusetts
in 1999.
Since 1997 he has been a research assistant in the Department of Electrical Engineering and Computer Science and the Research Laboratory of Electronics at Massachusetts Institute of Technology, Cambridge, Massachusetts, where he has concentrated on electromagnetics wave theory and applications.
He was a TA in the EECS department of MIT for several undergraduate and
graduate EM courses (1998-2002).
160