ABSTRACT

Title of dissertation: Wave Chaos Studies
and The Realization of
Photonic Topological Insulators

Bo Xiao, Doctor of Philosophy, 2022

Dissertation directed by: Professor Steven Anlage
Department of Electrical and
Computer Engineering

Wave propagation in various complex media is an interesting and practical

field that has a huge impact in our daily life. Two common types of wave prop-

agation are examined in this thesis: electromagnetic wave propagation in complex

wave chaotic enclosures, where I studied its statistical properties and explored time-

domain pulse focusing, and unidirectional edge modes propagating in a reciprocal

photonic topological insulator waveguide.

Several theories, e.g. the Random Matrix Theory and the Random Coupling

Model, have been developed and validated in experiments to understand the statis-

tical properties of the electromagnetic waves inside wave chaotic enclosures. This

thesis extends the subject from a single cavity to a network of coupled cavities

by creating an innovative experimental setup that scales down complex structures,

which would otherwise be too large and cumbersome to study, to a miniature version

that retains its original electromagnetic properties. The process involves shrinking

down the metal cavity in size by a factor of 20, increasing the electromagnetic wave



frequency by the same factor and cooling down the cavity by a dilution refrigera-

tor to reduce its ohmic loss. This experimental setup is validated by comparison

with results from a full-scale setup with a single cavity and it is then extended for

multiple coupled cavities.

In the time domain, I utilized the time-reversal mirror technique to focus

electromagnetic waves at an arbitrary location inside a wave chaotic enclosure by

injecting a numerically calculated wave excitation signal. I used a semi-classical

ray algorithm to calculate the signal that would be received at a transceiver port

resulting from the injection of a short pulse at the desired target location. The

time-reversed version of this signal is then injected into the transceiver port and an

approximate reconstruction of the short pulse is observed at the target port.

Photonic topological insulators are an interesting class of materials whose pho-

tonic band structure can have a bandgap in the bulk while supporting topologically

protected unidirectional edge modes. This thesis presents a rotating magnetic dipole

antenna, composed of two perpendicularly oriented coils fed with variable phase dif-

ference, that can efficiently excite the unidirectional topologically protected surface

waves in the bianisotropic metawaveguide (BMW) structure recently realized by

Ma, et al., despite the fact that the BMW medium does not break time-reversal

invariance. In addition to achieving high directivity, the antenna can be tuned con-

tinuously to excite reflectionless edge modes to the two opposite directions with

various amplitude ratios.

Overall, this thesis establishes the foundation for further studies of the uni-

versal statistical properties of wave chaotic enclosures, and tested the limits of its



deterministic properties defined by the cavity geometry. It also demonstrated in

experiment the excitation of a unidirectional edge mode in a Bianisotropic Meta-

waveguide, allowing for novel applications in the field of communications, for exam-

ple phased array antennas.



Wave Chaos Studies
and The Realization of

Photonic Topological Insulators

by

Bo Xiao

Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment

of the requirements for the degree of
Doctor of Philosophy

2022

Advisory Committee:
Professor Steven Anlage, Chair/Advisor
Professor Thomas Antonsen
Professor Thomas Murphy
Professor Neil Goldsman
Professor Yanne Chembo
Professor John Cumings, Dean’s Representative



c© Copyright by
Bo Xiao

2022



Acknowledgments

For the work on testing the Random Coupling Model predictions in scaled-

down cavities (Chapter 2 - 5), we acknowledge the assistance of Bisrat Addissie.

This work was supported by ONR under Grant No. N000141512134, AFOSR COE

Grant FA9550-15-1-0171 and ONR DURIP grant N000141410772, and the Maryland

Center for Nanophysics and Advanced Materials.

For the work on focusing waves at arbitrary location in a ray-chaotic enclosure

using time-reversed synthetic sonas (Chapter 6), we thank the group of A. Richter

(Uni. Darmstadt, Germany) for graciously loaning the cut-circle billiard, H.J. Paik

and M. V. Moody for use of the pulsed tube refrigerator, Martin Sieber for comments

on improving the short orbit calculation algorithm, and the High-Performance Com-

puting Cluster at UMCP for use of the Deepthought cluster. This work is funded

by the ONR under Grants No. N00014130474 and N000141512134, and the Center

for Nanophysics and Advanced Materials (CNAM).

For the work on exciting reflectionless unidirectional edge modes in a photonic

topological insulator medium (Chapter 7), the work was supported by ONR under

Grant No. N000141512134, AFOSR COE Grant FA9550-15-1-0171 and the National

Science Foundation (NSF) under Grants No. DMR-1120923, No. PHY-1415547 and

No. ECCS-1158644.

ii



Table of Contents

List of Tables vi

List of Figures vii

List of Abbreviations xv

1 Introduction and Overview 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Ray-chaotic Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Random Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.1 Random Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.2 Normalized Spacings Between The Nearest Eigenfrequencies . 8

1.5 Random Coupling Model . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5.1 S-parameters & Z-parameters . . . . . . . . . . . . . . . . . . 11
1.5.2 Normalized Impedance . . . . . . . . . . . . . . . . . . . . . . 12
1.5.3 Loss Parameter α . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.4 Extensions of RCM . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Testing The Random Coupling Model Predictions in a Scaled-Down Cavity
With Remote Injection Setup 16
2.1 The Need For Scaled Cavities . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Scaling of The Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Frequency Scaling . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 Cavity Wall Conductivity Scaling . . . . . . . . . . . . . . . . 21
2.2.3 Scaling Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 VNA & Extenders . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2 Cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.3 Free-space Propagation Path . . . . . . . . . . . . . . . . . . . 33
2.3.4 Mode Stirrer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4 Testing Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

iii



2.4.1 Overall Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.2 Free-space Propagation Path Measurement . . . . . . . . . . . 42
2.4.3 Collecting an Ensemble of Cavity Data . . . . . . . . . . . . . 42
2.4.4 Direct Injection . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.5 Remote Injection . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.6 Extension to Multiple Connected Cavities . . . . . . . . . . . 48
2.4.7 Further Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Introducing Radiation Efficiency to the Random Coupling Model 54
3.1 Radiation Efficiency η and RCM With Lossy Ports . . . . . . . . . . 55

3.1.1 Zrad, Zant, Zcav and Zavg . . . . . . . . . . . . . . . . . . . . . 56
3.1.2 Radiation Efficiency η in One-port Systems . . . . . . . . . . 58
3.1.3 Radiation Efficiency η in Multi-port System . . . . . . . . . . 61
3.1.4 Scaling Relation Between α and η . . . . . . . . . . . . . . . . 61

3.2 Applying Radiation Efficiency η to Experimental Data . . . . . . . . 64
3.2.1 Calculating Loss Parameter Through Time-domain Energy

Decay Time τ . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.2 Fitting Radiation Efficiency η . . . . . . . . . . . . . . . . . . 67
3.2.3 η’s Independence From Cavity’s α . . . . . . . . . . . . . . . 70
3.2.4 Apply η to Experimental Data . . . . . . . . . . . . . . . . . 71

4 Single Scaled Cavity Experiment 74
4.1 Full-Scale Cavity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1.1 Frequency Dependent Loss Parameter . . . . . . . . . . . . . . 76
4.1.2 Fitting Error . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 Miniature Cavity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.1 Frequency Dependency of η and α . . . . . . . . . . . . . . . . 80

4.3 Comparison Between Full-Scale and Miniature Cavity . . . . . . . . . 82
4.3.1 Full-Scale Cavity with 1/4 absorber for N = 1 Frequency

Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3.2 Full-Scale Cavity with 1/4 absorber for N = 10 Frequency

Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 α Range Limitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4.1 Difference in Re[ξ11] and Re[ξ22] . . . . . . . . . . . . . . . . . 86
4.4.2 Difference in Re[ξ12], Re[ξ21] and Im[ξ12], Im[ξ21] . . . . . . . . 87

5 Multiple Scaled Cavity Experiment 91
5.1 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.1.1 Design considerations . . . . . . . . . . . . . . . . . . . . . . . 93
5.1.1.1 Cost efficient construction . . . . . . . . . . . . . . . 93
5.1.1.2 Efficient use of space and material . . . . . . . . . . 93
5.1.1.3 Flexible configuration . . . . . . . . . . . . . . . . . 100
5.1.1.4 Perturbers . . . . . . . . . . . . . . . . . . . . . . . . 103
5.1.1.5 Antenna and lens alignment . . . . . . . . . . . . . . 103

5.1.2 Assembled setup . . . . . . . . . . . . . . . . . . . . . . . . . 107

iv



5.2 Lessons Learned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2.1 Design assistance tools . . . . . . . . . . . . . . . . . . . . . . 110

5.2.1.1 CST modeling tool . . . . . . . . . . . . . . . . . . . 110
5.2.1.2 3D printed models . . . . . . . . . . . . . . . . . . . 112

5.2.2 Methods to Construct a Metal Box . . . . . . . . . . . . . . . 112
5.2.3 External vendors . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6 Focusing Waves at Arbitrary Locations in a Ray-Chaotic Enclosure Using
Time-Reversed Synthetic Sonas 119
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.2.1 Calculation of Synthetic Sona . . . . . . . . . . . . . . . . . . 124
6.2.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2.3 Frequency Domain Experiment . . . . . . . . . . . . . . . . . 127

6.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.3.1 Reconstruction Quality . . . . . . . . . . . . . . . . . . . . . . 129
6.3.2 Effects of Loss and Mismatched Port Coupling . . . . . . . . . 131

6.3.2.1 Effect of Loss on Reconstruction . . . . . . . . . . . 131
6.3.2.2 Mismatched Port Coupling . . . . . . . . . . . . . . 134

6.3.3 Synthetic Sona Duration Constraint . . . . . . . . . . . . . . . 136
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7 Exciting Reflectionless Unidirectional Edge Modes in a Reciprocal Photonic
Topological Insulator Medium 142
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.2 Design and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.5 Ferrite Induced Non-reciprocity . . . . . . . . . . . . . . . . . . . . . 158
7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

8 Conclusions 160
8.1 Prior Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Appendices 165

Appendix A MATLAB Script For Generating Normalized Impedance Ensem-
bles For 2-port Systems 166

Bibliography 171

v



List of Tables

7.1 Summary of extreme transmisison values as deduced from experimen-
tal data where TL and TR are transmission amplitude to left and right,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

vi



List of Figures

1.1 Diagram of an electric power system; transmission system is in blue. [1] 2
1.2 Random Matrix Theory predictions for the Probability Density Func-

tions of (a) Re[z] and (b) Im[z] as a function of increasing α, for a
one-port time-reversal symmetric wave-chaotic cavity. Figure 2.3 in
Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1 RCM-predicted probability density function of power ratios RN(α =

6) ≡ P
(N)
in /P

(1)
in for chains of up to seven cavities: the high-loss regime

is assumed for all the (statistically identical) cavities in the chain.
This is Figure 7 in Ref. [3]. . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Electrical resistivity of copper vs. temperature. The different curves
on the plot are the experimental results for different aluminum sam-
ples and were measured with different techniques at different times
from 1908 to 1981. (which is Figure 1 in [4]) . . . . . . . . . . . . . . 24

2.3 The schematics for the scaled cavity experiment setup. . . . . . . . . 28
2.4 Block diagram for Virginia Diode Inc frequency extender modules.

(∗) Variable attenuator. (†) Isolator. Picture taken from VDI product
manual [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 A typical setup with VNA and VDI extenders for measurement. Pic-
ture taken from VDI product manual [5]. . . . . . . . . . . . . . . . 30

2.6 The drawings (left), the outside vacuum can (middle) and the cooling
plates inside of the cryostat (right). The cooling plates are cooled in
a staged manner from 50K at the top to 10mK at the bottom. The
sample will be mounted onto the 10 mK plate, also called the sample
plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.7 The setup for high frequency waves to propagate in free-space from
the frequency extender to the cavity, and from the cavity to the re-
ceiving frequency extender. The horn antennas can efficiently launch
and receive electromagnetic waves and the Teflon lens collimates the
waves into a parallel beam. . . . . . . . . . . . . . . . . . . . . . . . . 34

vii



2.8 Magnetically coupled mode stirrer powered by a cryogenic stepper
motor. The magnetic strip outside the cavity (lower yellow bar in the
left side) is coupled by static magnetic field with the magnetic strip
inside the cavity (upper yellow bar in the left side), eliminating the
need for any hole on the wall. The upper right inset is an example of
the mode stirrer placed inside the cavity. . . . . . . . . . . . . . . . . 39

2.9 The transmission amplitude in dB for free-space propagation mea-
surement. The x-axis and y-axis are the distance between the lens
to the transmitter antenna (d1) and the receiver antenna (d2) respec-
tively. The yellow dotted circles enclose the region for best d1, d2

values for optimal transmission. . . . . . . . . . . . . . . . . . . . . . 43
2.10 Experimentally measured reflection (top two plots) and transmission

(bottom two plots) magnitude for the s = 20 cavity for 5 realizations
created by rotating the magnetically coupled mode stirrer. The left
column plots the data in the full frequency range from 75 GHz to
110 GHz while the right column zooms in a 100 MHz window. Each
curve is wildly different from others but yet follow the same slowly
varying trend which is the non-statistical system specific feature. . . . 44

2.11 Power ratio Λ as a function of frequency and its histogram for a good
ensemble (left) and a bad ensemble (right) data. Λ should have a
large mean and a large dynamic range if the cavity modes are well
perturbed and uncorrelated, which gives us a good ensemble of data. 45

2.12 Loss parameter α values for the full scale cavity measured by collabo-
rators at U.S. Naval Research Laboratory. Each α value is calculated
from experimental data in a 0.1 GHz window at the specified fre-
quency. The data points are taken from Table 1 in [6]. . . . . . . . . 46

2.13 Probability Density Function (PDF) of the normalized impedance for
a direct injection experiment on a scaled-down s = 20 copper cavity
(dotted line), in comparison with the RCM Monte Carlo simulation
with α = 4.5 (solid line). . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.14 3D-printed model of the multiple connected cavities and the configu-
ration for a 3-cavity cascade drawn in CST. . . . . . . . . . . . . . . 49

2.15 The air region of the 3-cavity cascade model with cavity walls and
backplane omitted. The wave goes into the first cavity through the
antenna mounting point, then couples into the second cavity mounted
on the other side of the backplane (not shown) through an aperture.
It is then coupled to the third cavity through the other aperture, and
gets out through the antenna mounting point on the third cavity. . . 50

viii



2.16 A simple cascade of three scaled cavities. Top: the graph of the
cascade, consisting of three nodes (scaled cavities) and two edges
(apertures connecting two neighbor cavities). Bottom: the configu-
ration of scaled cavities mounted onto the two opposing sides of a
central plate. Green rectangles represent scaled cavities on the front
side and blue rectangles represent scaled cavities on the back side.
Orange circles represent apertures connecting the two cavities on the
opposing sides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.17 A network of seven scaled cavities. Top: the graph of the network,
consisting of seven nodes (scaled cavities) and eight edges (aper-
tures connecting two neighbor cavities). Bottom: the configuration
of scaled cavities mounted onto the two opposing sides of a central
plate. Green rectangles represent scaled cavities on the front side and
blue rectangles represent scaled cavities on the back side. Orange cir-
cles represent apertures connecting the two cavities on the opposing
sides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1 The circuit model for a lossless (left) and a lossy (right) antenna.
Zant is the input impedance of the antenna which is approximated
by Zavg (which also includes short orbit effect in addition to Zant) in
RCM normalization process, Zrad is the radiation impedance which
describes the feature of the port, and R is the real part of Zrad that
represent the power loss due to radiation. Figure 4, 5 in [7]. . . . . . 58

3.2 The inverse Fourier transform of the measured S-parameters give the
value of τ from each fit in log-scale versus time. a) For the case of
transmission and b) for the case of reflection of the s = 20 scaled en-
closure measured through remote injection. Data from 9 realizations
at room temperature are plotted with different colors. The purple
line is the average, and the green line is the linear fit for the energy
decay portion of the average. The slope of the fitted line is −1/2τ ,
where τ is the energy decay time of the cavity. The time domain re-
sponse inside the blue dashed box in (b), labeled ’Prompt response’,
arises from the impedance mismatch between the external transmis-
sion channel and the cavity. This information is captured in Zavg. . . 67

3.3 Comparison between the normalized impedance PDF for a 2-port
system from a RCM Monte Carlo simulation with α = 5.6, in solid
lines, and that from a normalization process of experimental data
with η11 = 0.14 and η22 = 0.19, in dotted lines. . . . . . . . . . . . . . 69

3.4 The comparison between the normalized impedance PDF from a di-
rect injection experiment with an empty cavity (solid lines), and that
from a one-sided remote injection experiment with an empty cavity
(dotted lines). The remote injection data is processed using the radia-
tion efficiency η11,absorber obtained from the one-sided remote injection
with an absorber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

ix



3.5 A comparison between α values calculated from different methods as
a function of time during a cycling of the dilution refrigerator hosting
the scaled cavity. Blue solid line: calculated from time domain energy
decay time method; Dotted lines: calculated from RCM normaliza-
tion process with various values of η. . . . . . . . . . . . . . . . . . . 73

4.1 The full-scale cavity α measured by Zachary Drikas and Jesus Gil Gil
at NRL with various number of absorbers inside. The left side plot is
the average α for the whole frequency range, i.e. number of frequency
segments N = 1. The right side plot is the α for N = 10 frequency
segments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 The fitting error for calculating the αN=10(f) values for the full-scale
cavity is mostly around 0.05. The few high error data points are for
α > αmax over the limit of the simulation α range. The right plot
shows a typical case of the comparison between the PDF curves for
the 5th frequency segment where the fitting error is 0.036164. . . . . . 78

4.3 The tunable range of α values by using different wall material and
varying temperature. The least lossy wall material is polished copper,
which gives a range of 2.6 ≤ α ≤ 4.2, including the αfull for full-scale
cavity with 1/4 absorber in its range. The plotted α is calculated
from the time-domain energy decay time τ . . . . . . . . . . . . . . . . 80

4.4 The horn antenna has higher gain at higher frequency band which
results in higher radiation efficiency. Left plot taken from Virginia
Diode, Inc documentation Nominal Horn Specifications [8]. . . . . . . 81

4.5 The frequency dependent loss parameter for the miniature cavity cal-
culated from the time-domain energy decay time method, αQ, and
from fitting with RCM simulation results, αfit. . . . . . . . . . . . . . 82

4.6 The comparison of PDFs for the normalized impedance Im[ξ21] for
the full-scale cavity with 1/4 absorber, the miniature cavity at 103
Kelvin and the RCM simulation with α = 3.0. The experimental data
is selected from the whole frequency range (N = 1), i.e. 3.75 ∼ 5.5
GHz for the full-scale cavity and 75 ∼ 110 GHz for the miniature
cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.7 Comparison of the probability density function (PDF) for the imagi-
nary part of the normalized impedance ξ12 and ξ21 between the full-
scale cavity (solid line based on data) and the miniature cavity (dot-
ted line based on data) for three different frequency bands (of the
full-scale cavity) at different temperatures (of the scaled cavity). Top
row shows the PDFs in linear scale while the bottom row shows the
same PDFs in log scale. (a) α = 2.83 within [4.45, 4.625] GHz at 130
Kelvin, (b) α = 4.19 within [4.975, 5.15] GHz at 217 Kelvin and (c)
α = 5.82 within [5.325, 5.5] GHz at 297 Kelvin. Notice that in each
plot, all four curves collapse into one because they match each other
very well. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

x



4.8 The comparison of all 8 PDFs for the normalized impedances, the
real and imaginary part of ξ11, ξ12, ξ21 and ξ22, for the full-scale cavity
with 1/4 absorber and the miniature cavity. The experimental data
are selected from [5.325, 5.5] GHz, the 10th segment of the N = 10
frequency segments, for the full-scale cavity and is selected from 297
Kelvin and [106.5, 110] GHz for the miniature cavity. . . . . . . . . . 87

4.9 The comparison of all 8 PDFs for the normalized impedances, the
real and imaginary part of ξ11, ξ12, ξ21 and ξ22, for the full-scale cavity
with 1/4 absorber and the miniature cavity. The experimental data
are selected from [4.275, 4.45] GHz, the 4th segment of the N = 10
frequency segments, for the full-scale cavity and is selected from 90.1
Kelvin and [85.5, 89] GHz for the miniature cavity. . . . . . . . . . . 88

4.10 The distribution functions of the real (solid line) and the imaginary
(dotted line) part of the normalized impedance ξ12 calculated from
the RCM simulation for various α. The real and the imaginary part
distribution functions are identical when α = 8 but start to differ
when α = 5 and lower. . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.1 The complete computer 3D model for the multiple interconnected
scaled cavity experiment setup. The details of each component visible
here are discussed below. . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2 Scaled interconnected cavity design with aperture, antenna mounting
hole and mounting screws. . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3 Cross section of copper box wall showing the antenna mounting ar-
rangement. The antenna is mounted to the box with an adapter plate.
Concealed PEM studs are used to avoid having protruding screws or
nuts in the box interior. . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.4 One scaled cavity with two antenna installed is mounted on the cen-
tral plate. Notice that the aperture is covered up since this is a single
cavity configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.5 The 3D models of one unit box with different insert configurations.
Insert plates, shown in the upper right, can have two apertures (al-
lowing the box to be coupled with two other boxes), one aperture
(allowing the box to be coupled with another box as the end of a
cascading chain of boxes) or no aperture at all (for a single cavity
configuration). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.6 3D model for a 3-cavity cascade setup. 3 unit boxes are mounted
to opposite side of the center plate, connected through the apertures
to form a cascade. Electromagnetic waves come in from a receiving
antenna, passes through all 3 boxes in the cascade and go out through
the other antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.7 Center plate model with apertures and clear screw holes. The notch
at the top of the plate is to accommodate the antenna and lens. . . . 102

5.8 Perturber assembly 3D model. . . . . . . . . . . . . . . . . . . . . . . 104
5.9 Perturber assembly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

xi



5.10 Another view of a 3-cavity cascade setup. The 1X3 opening, the
lens and antenna are aligned with the cryostat window to ensure
maximum transmission of mm-waves. The grid of small green slots
are for rotating gears (shown as red gears in upper right figure) to
rotate magnetically coupled perturbers inside the cavity. A special
plate can be installed within the 1X3 opening to mount gear if needed.106

5.11 Antenna adapter model. An adapter plate (rectangular blue plate) is
required to mount the antenna (blue cone) and the lens holder (green)
onto the cavity (red box). The antenna and lens can be installed onto
the adapter plate first, as shown in the upper left figure. Then the
adapter plate is mounted onto the cavity by PEM studs (small red
cylinder in upper left figure), which do not require screw holes or any
protrusion into the cavity. . . . . . . . . . . . . . . . . . . . . . . . . 108

5.12 Stepper motor mounting location. . . . . . . . . . . . . . . . . . . . . 109
5.13 Smooth mirror-like finish from sheet copper. The part comes with a

protective plastic film that can be peeled away to reveal the metal
surface that is free of milling marks. . . . . . . . . . . . . . . . . . . . 114

5.14 A side by side comparison for a 3-cavity cascade setup with the scaled-
down cavities on left and full scale cavities on the right. Figure 6 in [9]116

5.15 Comparison of the PDF of the induced voltage on a 50 Ohm load
attached to the last cavity between the scaled cavity cascade and its
full-scale counterpart. Figure 4 in [9]. . . . . . . . . . . . . . . . . . . 118

6.1 Calculation of a brief synthetic sona from four simple orbits in a
representitive 2D 1/4-bow-tie billiard. We first calculate a scaled
and time-delayed version of the input signal, which is a Gaussian
pulse modulation of a 7 GHz carrier signal in this example. These
waveforms are summed to obtain the synthetic sona shown at the
bottom. The two ports are 17.5 cm apart. . . . . . . . . . . . . . . . 125

6.2 (a) Physically measured sona, and (c) synthetic sona signal calculated
from orbits less than 4 m (= 10

√
A) in length, and their time reversal

reconstruction signals, (b) and (d), respectively. Only the upper half
of the signals are plotted since they are essentially symmetric about
the time-axis. The upper inset shows closeups of the initial Gaussian
pulse (left, blue), the measured sona reconstruction (middle, green)
and the synthetic sona reconstruction (right, red). The bottom inset
plots some of the orbits used to calculate the synthetic sona. The
horizontal and vertical straight walls of the billiard have lengths of
43.18 cm and 21.59 cm, and the two ports are 17.5 cm apart. . . . . 128

xii



6.3 Measured signals in cut-circle cavity in the measured sona time-
reversal process. The sona signal in the cavity normal (green, shorter
in time) and superconducting (blue, longer in time) states, and time-
reversal reconstruction in the cavity normal (cyan, lower amplitude)
and superconducting (red, higher amplitude) states. The sonas are
generated by injection of a Gaussian pulse with 6σt = 3ns, modulat-
ing a 7 GHz carrier signal. Data is taken at 6.4 K (superconducting
state) and room temparature (normal state). . . . . . . . . . . . . . . 133

6.4 Reconstruction quality (Vpp and focus ratio) and |S21|2 statistics (µ
and σn) measured in the 1/4-bowtie billiard as a function of carrier
frequency: (a) (b) when no noise is added; (c) (d) when 2mV Gaussian
random noise is added to the sonas and reconstructions. (a) (c) use
the physically measured sona method, while (b) (d) use the synthetic
sona method. All quantities are plotted normalized to their maximum
values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.5 Normalized reconstruction quality (normalized to its saturation value)
for the physically measured and synthetic sona methods when a win-
dowing function is applied to the sona before being time reversed, so
that only the beginning part of the sona is used for the time backward
step. The peak-to-peak voltage of the reconstruction is shown in (a)
while the focus ratio is shown in (b). “MEA” and “SYN” refer to the
physically measured and synthetic sona methods, respectively. “in-
sert1” and “insert2” are two variations of the bowtie cavity geometry
when inserts are added, as shown in the inset. . . . . . . . . . . . . . 140

7.1 Figure 1 in Ref. [10]. BMW as a photonic topological insulator. (a)
Schematic of the BMW. Part of the top metal plate is removed to
reveal the bed-of-nails structure below. The enlarged regions on the
right illustrate the origin of the bianisotropic response. Right inset:
an equivalent way to produce bianisotropy by adding an asymmet-
rically placed metallic volume (washer) around the rod. (b) PBS
of spin-degenerate metawaveguide with TE and TM modes forming
doubly degenerate Dirac cones at K points. (c) Field profiles of the
degenerate TE and TM mode at the K point. Colors: energy density.
Arrows: electric field for the TM mode and magnetic field for the TE
mode. Yellow dashed line: metallic border. (d) PBS with the band
gap induced by the bianisotropy of the metawaveguide. Dashed lines
in (b) and (d): TE and TM bands of interest. . . . . . . . . . . . . . 148

7.2 (a): Numerically calculated band structure of the waveguide, where
four edge modes have been labeled. (b) to (e): Field profile |Hx−iHy|
of the four edge modes at 6.13GHz (shown in (a) as a dashed line).
It is clear that only the two spin-up states (from the two valleys),
but none of the spin-down states, can be excited by a right circularly
polarized magnetic dipole. . . . . . . . . . . . . . . . . . . . . . . . . 149

xiii



7.3 (a) Schematic of the experimental setup. ∆em is the bianisotropy
coefficient [10] and the interface is defined by the boundary of the
two regions with ∆em of opposite sign, denoted by a red dashed line.
(b) Photograph of the top of the BMW structure showing the cor-
responding elements in (a). The inset shows the arrangement of the
two loop coils which are in the air gap of a cylinder at the interface. . 150

7.4 Transmission amplitude at the (a) left and (b) right side of the BMW
interface as a function of frequency while varying the phase difference
φ of the two driving loop antennas. The probe is positioned at the
center of the edge. The BMW bulk band gap extends from 5.80 GHz
to 6.47 GHz, as shown with the vertical dashed lines. . . . . . . . . . 152

7.5 Simulated BMW structure with TPSW-supporting interface in CST
with 16 period long and 8 period wide graphene-like lattice structure
of rods. The two loops making up the rotating magnetic dipole an-
tenna are perpendicularly placed with two independent driving coax-
ial cable inputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.6 CST simulation results for transmission amplitude to the left and
right side of the BMW structure while varying both the phase differ-
ence φ and driving amplitude (parameterized by angle θ ∈ [0, π/2])
of the two loop antennas. Results are shown for (a) left, (b) right
6.47 GHz and (c) left, (d) right 6.08 GHz. . . . . . . . . . . . . . . . 155

7.7 Transmission amplitude to the (a) left and (b) right side when varying
both the phase difference φ and the input power (parameterized by
θ) of the two loop antennas as deduced from the data at a frequency
of 6.38 GHz. To examine the directivity, we plot the ratio of (c) left
to right side and (d) right to left side transmission amplitude (in dB)
as a function of θ and φ. . . . . . . . . . . . . . . . . . . . . . . . . . 157

xiv



List of Abbreviations

α Alpha. The dimensionless loss parameter in Random Coupling Model
ξ The normalized impedance. Sometimes also denoted as z
z The normalized impedance. Sometimes also denoted as ξ
Zcav Measured cavity impedance
Zavg Ensemble average of Zcav

Zrad Radiation impedance of the antennas, or ports
Zant Input impedance of the antennas, or ports

Zant includes both Zrad and the antenna’s internal loss and delays

BMW Bi-anisotropic Meta-Waveguide
BTRS Broken Time Reversal Symmetry
CNAM Center for Nanophysics and Advanced Materials
CST Computer Simulation Technology, a numerical simulation software
FDTD Finite-Difference Time-Domain
IACS International Annealed Copper Standard
IREAP Institute for Research in Electronics and Applied Physics
LCP Left Circularly Polarized
NRL Naval Research Laboratory
OFHC Oxygen-Free High Conductivity
ONR Office of Naval Research
QHE Quantum Hall Effect
QSH Quantum Spin Hall
RCM Random Coupling Model
RCP Right Circularly Polarized
RMT Random Matrix Theory
TE Transverse Electric
TM Transverse Magnetic
TPSW Topologically Protected Surface Wave
TRS Time Reversal Symmetry
VDI Virginia Diodes Inc
VNA Vector Network Analyzer

xv



Chapter 1: Introduction and Overview

1



Figure 1.1: Diagram of an electric power system; transmission system is in blue. [1]

1.1 Background

Wave propagation in various complex media is an interesting and practical field

that has a huge impact in our daily life. One basic type of wave propagation is

guided waves through carefully engineered and manufactured media, such as waveg-

uides and transmission lines. For example, high frequency electromagnetic waves

carrying encoded information in optical fibers can travel vast distances in seconds,

providing the backbone of internet that connects the world. As another example,

electric power transmission relies on a mesh of transmission lines to transfer electric

power generated from power plants to users across the globe, providing the crucial

infrastructure of modern society.

Over the decades, research and engineering efforts has been made to optimize

wave propagation properties in such media to better suit our needs. The most com-

monly seen highly-engineered wave propagation medium in our daily life is probably

the optical fiber, a widely used waveguide to transmit light over long distances for

high bandwidth communication. In this case, we want to have lower insertion loss

2



such that the signal quality can be maintained with less re-amplification [11–13],

thus traversing longer distances without loss of signal fidelity. People also explored

single-mode fiber vs. multi-mode fiber for different use cases considering the travel

distance and bandwidth, invented new materials to achieve better properties com-

pared with the conventional silicon dioxide, or even to acquire entirely new novel

properties [14–17].

Another common type of wave propagation is in a confined medium, such as

inside a room, a building or in general any partially open enclosure. Its applications

includes telecommunication, sensors and wireless power transfer. One major differ-

ence with guided waves is that we typically do not try to control the details of the

enclosure, which makes sense since we would like our WIFI to work regardless of

which room I am standing in and what objects are placed around. Instead, we are

interested in the generic statistical properties of the wave transmission in enclosures

categorized by generic properties, such as the volume or the material making up the

walls, as opposed to the exact shape or other specific conditions.

1.2 Thesis Overview

In this thesis, I study both of the above two types of wave propagation with exper-

imentation and compare them with simulation and theoretical results.

For wave propagation within enclosures, we measured the transmission prop-

erties of electromagnetic waves inside metal cavities, launching and receiving mi-

crowave signals through antennas mounted on the cavity walls. To imitate the

3



propagation of microwave signals used in telecommunication frequency ranges (typ-

ically 1 ∼ 6 GHz) inside a typical building, we scaled down an otherwise enormous

metal structure spanning several rooms into small metal boxes that could be man-

aged, configured and integrated into our experimental setup. In this thesis, I discuss

the practical challenges of scaling down the cavities in Chapter 2, explore the prop-

erties of the antenna on the transmission measurements in Chapter 3, verify my

experimental methodology with a single cavity setup in Chapter 4, then applied it

on a more complex multiple connected cavities setup in Chapter 5.

Although it is mentioned above that our focus is on the cavity’s statistical

properties, which are independent of the exact geometric details, it is nevertheless

interesting to explore what we could achieve if the geometric information of the

cavity boundaries is known. Specifically, we’d like to see if we could construct a

signal based on the geometry information alone and broadcast it from one antenna

to another such that it would focus in space and time at the second antenna. At

first glance, the geometric information seems simple to obtain. After all, we man-

ufactured all the cavities according to our own design specifications. However, the

actual geometry could vary with lots of uncontrollable factors, e.g. room tempera-

ture changing the volume of the cavities, warping of the cavity walls, differences in

surface condition due to dust, oil, or other residues, etc. So the effectiveness of our

constructed signal decreases as it bounces off the cavity walls more and more as it

travels. We discuss our findings in Chapter 6.

Finally, as a case study for guided wave propagation, we looked into the emerg-

ing new type of materials, photonic topological insulators, which are an interesting

4



class of materials whose photonic band structure can have a band-gap in the bulk

while supporting topologically protected unidirectional edge modes. In Chapter 7,

we performed experiments on a 2D waveguide composed of hexagonal graphene-like

lattice. The resulting electromagnetic waves traveled from the source antenna, along

the waveguide, to the two sides in a reflectionless manner, i.e. there’s no reflection

or back scattering of waves.

But before we delve into the topics in this thesis, there are some fundamental

theories and prior work that need to be discussed first.

1.3 Ray-chaotic Cavities

The property of waves coupling into and propagating inside an electrically large

complex enclosure is of interest to several fields. In electromagnetic compatibility

(EMC) the goal is to minimize the coupling by providing better shielding and thus

lower the chance of a larger induced voltage on sensitive devices located inside the

enclosure, thus minimizing potential damage. In telecommunications one would like

to maximize the coupling and the induced voltage to boost up the transmitted signal

inside an enclosure, which could be a room with windows and doors. These enclo-

sures are modeled as ray-chaotic cavities, where waves are very sensitive to geometry

details and small changes, which is the case for most real life situations. Thus a

statistical description is more approachable and applicable than trying to calculate

an exact solution of the problem. In light of this, the Random Coupling Model

(RCM) was introduced by Zhang et al., as a method to predict the statistical dis-

5



tribution of the induced voltages at locations inside a ray-chaotic enclosure [18,19].

It combines the universal predictions from Random Matrix Theory (RMT), briefly

introduced below in section 1.4, and the system-specific properties characterized by

the system’s port radiation impedances.

Complex enclosures, such as computer cases with circuitry inside, or offices

filled with desks, chairs, and electronics, are examples of ray-chaotic systems. To

define what we mean by ray-chaotic, consider the case where the wavelength is

short and a ray description is appropriate. Consider two rays starting from the

same location in such an enclosure but with slightly different directions. As the rays

propagate, reflecting from either curved surfaces or the interior features of the enclo-

sure, their separation will tend to increase exponentially in time, and we call such a

situation ray-chaotic. Ray chaos leads to an extreme sensitivity to initial conditions

for the rays [20]. For waves propagating in highly over-moded ray-chaotic struc-

tures, the exact solution for the fields depends strongly on the geometric details of

the structure and is very sensitive to small changes in frequency or geometry. Thus,

in the presence of even small uncertainties in structure or frequency, a statistical

approach may be more appropriate than trying to obtain an exact solution for field

quantities inside the structure [21]. The Random Coupling Model (RCM) is one

such method to predict the statistical properties of the waves inside a ray-chaotic

enclosure [18, 19]. The RCM has been widely discussed and tested over the years,

with good agreement between theory and experimental results on individual complex

structures [2, 3, 6, 7, 22–24]. It has also been used to study the effect of a non-linear

port on the measured statistical electromagnetic properties of a ray-chaotic complex

6



enclosure in the short wavelength limit [25].

1.4 Random Matrix Theory

The RCM is based on Random Matrix Theory, originally proposed to model the

energy level statistics of heavy nuclei [26]. The idea is that if the wave system

is sufficiently complex then its fluctuating properties have the same statistics as

those of a suitable ensemble of random matrices [27] (hence the name ”Random

Matrix Theory”). Certain statistical properties, such as the distribution of the

normalized spacings between nearest neighbor eigenfrequencies (ε), follow a universal

behavior regardless of the system details. The type of random matrix is chosen

according to the general symmetry of the system in order to correctly represent

its dynamics. For instance, the Gaussian Orthogonal Ensemble (GOE) describes

systems with Time-Reversal Symmetry and the Gaussian Unitary Ensemble (GUE)

describes systems with Broken Time-Reversal Symmetry. It is later conjectured

[26,28–32], and validated in some cases [27,33–35], that most wave-chaotic systems

follow random matrix statistics for a properly chosen ensemble, giving Random

Matrix Theory a broad range of applications.

1.4.1 Random Matrices

Random Matrices are matrices whose elements are random variables. If we take one

realization for each of the random variables and group them together as a matrix,

then we get one realization of the random matrix. Taking many such realizations is

7



called an ensemble. The two random matrices relevant to electromagnetic systems

are Gaussian Orthogonal Ensemble (GOE) and Gaussian Unitary Ensemble (GUE)

corresponding to systems with Time-Reversal Symmetry (TRS) and Broken Time-

Reversal Symmetry (BTRS) respectively.

• For a GOE matrix, each element, Xi,j, is a real independent-identically-distributed

(i.i.d) Gaussian random variable with zero mean. All the off-diagonal ele-

ments have a variance of 1/2 while the diagonal ones have a variance of 1, i.e.

Xi,j ∼ N(0, 1
2
) for i 6= j and Xi,i ∼ N(0, 1).

• For a GUE matrix; It is a Hermitian matrix with real diagonal elements and

complex off-diagonal elements. Each diagonal element is a real i.i.d Gaussian

random variable with zero mean and unit variance, i.e. Xi,i ∼ N(0, 1). The

off-diagonal elements are complex with real and imaginary parts both being

i.i.d Gaussian random variable with zero mean and 1/2 variance, i.e Xi,j ∼

N(0, 1
2
) + iN(0, 1

2
) for i < j.

Once an appropriate random matrix is constructed, its eigenvalues can be calculated

and then used to study the statistical properties of the corresponding wave-chaotic

system, such as the normalized spacings between the nearest eigenfrequencies.

1.4.2 Normalized Spacings Between The Nearest Eigenfrequencies

In the case of electromagnetics, the normalized spacings between the nearest eigen-

frequencies is defined as ε = (k2
n+1− k2

n)/(∆k2
n) where kn is the nth eigenfrequency’s

wavenumber, and ∆k2
n = 〈k2

n+1 − k2
n〉 is the mean spacing between eigenlevels. For

8



a given enclosure whose size is much larger than the wavelength, the mean spacing

can be calculated as

∆k2
n
∼= 4π/A (2D)

∼= 2π2/(kV ) (3D)

(1.1)

where A and V are the area and volume of the 2D or 3D cavity. Then according

to the RMT, the probability density function (PDF) of ε for different enclosures

follows one of the two universal curves, depending on whether the system has Time

Reversal Symmetry (TRS) or Broken Time Reversal Symmetry (BTRS):

P (ε) ∼=
π

2
εe−πε

2/4 (TRS)

∼=
32

π2
ε2e−4ε2/π (BTRS)

(1.2)

These are called “Wigner distributions” and they are proven to be applicable to

various wave-chaotic systems. Note that P (0) = 0 in both cases, meaning that all

degeneracies are broken in pure RMT systems.

1.5 Random Coupling Model

Random Matrix Theory (RMT) provides a universal statistical prediction for the

generic properties of wave-chaotic systems, however it is difficult to identify these

universal statistical properties in experimentally measured data because it inevitably

contains system-specific features like the coupling between the ports and the cavity

modes and short orbits [36–38]. The Random Coupling Model (RCM) introduces

a framework to incorporate the non-universal features with the universal statistical

9



properties of appropriate random matrices to reproduce in the statistical sense the

experimentally measured cavity impedance matrices. The effect of uniformly dis-

tributed loss in the system is to create a sub-unitary scattering system [39], and this

effect is captured to very good approximation by a single loss parameter α [18, 19].

The RCM is formulated in terms of the impedance matrix Z of an N -port system

in analogy to the reaction matrix in nuclear scattering theory [40–45]. The ports

represent sources or sinks of radiation that introduce or absorb energy in the enclo-

sure. The impedance relates the voltage induced on one port to the currents at all

of the N ports, and is simply related to the N × N scattering matrix S through a

bilinear transformation S = Z
1/2
0 (Z + Z0)−1(Z − Z0)Z

−1/2
0 , where Z0 is a diagonal

real matrix whose elements are the characteristic impedance of the transmission line

modes connected to each port.

Similar to RMT, the loss parameter α defines the universal statistical property

of the normalized impedance matrix, z (sometimes also denoted as ξ), which is a key

parameter to describe any electromagnetic enclosure. α is defined as α = k2/(∆k2
nQ)

where Q is the unloaded cavity quality factor. If we plug in Equation (1.1) for 2D

and 3D systems, α can be written as

α =
k2A

4πQ
(2D)

=
k3V

2π2Q
(3D)

(1.3)

10



1.5.1 S-parameters & Z-parameters

For an electromagnetic system, waves are coupled into or out of the system through

ports. An N port passive electromagnetic network is usually described by a N -by-N

scattering matrix S (S-parameters) or impedance matrix Z (Z-parameters), which

are both well-established quantities in microwave engineering. The S-parameter

element Si,j is the complex ratio of the amplitude and phase of the outgoing wave

at port i to the amplitude and phase of the incoming wave at port j when only the

jth port has a nonzero incoming wave while all other ports are terminated with a

matched reflectionless load, i.e. Si,j = bi
aj
|am=0,m 6=j, where ai, bi are complex numbers

representing the incoming and outgoing wave at port i respectively. Similarly, the

Z-parameter element Zi,j is the ratio of the induced voltage at port i to the source

current at port j when only the jth port has nonzero source current, i.e. Zi,j =

Vi
Ij
|Im=0,m 6=j, where Vi, Ii are the voltage and current at the ith port respectively. In

a typical microwave experiment, one uses a vector network analyzer to measure the

cavity S-parameters, which can be converted into the cavity Z-parameters by

Z = Z
1/2
0 (I + S)(I− S)−1Z

1/2
0 ,

where I is an identity matrix, Z0 is a diagonal matrix whose elements, Zi,i, are

the characteristic impedances of the transmission line connecting to the ith port

(typically 50 or 75 Ohms for a coaxial cable). To couple waves into or out of an

enclosure, an antenna is required which usually presents an impedance mismatch

11



with the connected transmission line. The mismatch causes the incoming waves to

partially reflect back and also prevents the antenna from picking up waves inside

the enclosure efficiently. This is an inevitable system-specific feature that exists in

almost all microwave experiments, resulting in a deviation from the expected RMT

universal behavior when analyzing the experimental data.

1.5.2 Normalized Impedance

To remove the system-specific features, the Random Coupling Model uses a non-

statistical impedance matrix, Zavg, which is obtained by averaging the measured

cavity Z-parameters Zcav over many realizations, to characterize the system-specific

properties and to normalize the experimental data. It is expected that the universal

fluctuations cancel out in the ensemble average because 〈Re[Z]〉realization = 1 and

〈Im[Z]〉realization = 0. Realizations of the same cavity are created by perturbing the

cavity modes while retaining the same cavity volume, usually by rotating a large

irregularly-shaped metal panel. Zavg mainly includes the radiation impedances of

the antennas which characterize the mismatch mentioned above, and it also includes

the influence of non-statistical short orbits between the ports, as discussed in [38].

Once we obtain Zavg = 〈Zcav〉realizations, the RCM suggests normalizing the measured

Zcav by

ξ = (Re[Zavg])−1/2(Zcav − jIm[Zavg])(Re[Zavg])−1/2 (1.4)

This quantity ξ, sometimes also simply called z in some papers, is expected to have

the statistical properties predicted by RMT for some value of loss parameter α.

12



1.5.3 Loss Parameter α

Similar to Random Matrix Theory, the statistics of the normalized impedance ξ

follows a universal curve defined by the loss parameter α, which can be calculated

according to Equation (1.3). Figure 1.2 (which is Figure 2.3 in [2]) illustrates the

statistics of ξ for a one-port system with different α values. The statistics are not

calculated by traditional electromagnetic simulations that solve Maxwell’s equations

inside a particular enclosure. Instead, the statistics are obtained by Monte Carlo

simulations by constructing many random matrices following RMT, and examining

their statistics. In practice, it is usually difficult to obtain an accurate estimate for

the unloaded quality factor of the cavity, thus alternative methods are employed to

determine α.

For example, in the high loss regime (α > 5), the variance of the normalized

impedance can be expressed as

σ2
Re[ξ] ≈ σ2

Im[ξ]
∼=

1

2πα
(BTRS)

∼=
1

πα
(TRS)

(1.5)

Hence, one can conduct experiments to measure Zcav for many realizations, nor-

malize them to get ξ then use Equation (1.5) to calculate α. More generally, α

can always be determined by finding the universal curve defined by α that best fit

the statistics of ξ, which is obtained through experiment. The universal impedance

statistics curves only depend on α values and can be calculated by methods pre-

sented in [2] Chapter 2.5.

13



Figure 1.2: Random Matrix Theory predictions for the Probability Density Func-
tions of (a) Re[z] and (b) Im[z] as a function of increasing α, for a one-port time-
reversal symmetric wave-chaotic cavity. Figure 2.3 in Ref. [2].

A complete list of methods to estimate cavity α is presented in Ref. [2] Ap-

pendix B.

1.5.4 Extensions of RCM

The RCM provides a framework to analyze the statistical properties of wave-chaotic

systems, taking into account the universal properties, the loss parameter α, and the

system-specific properties. However, it only covers single-cavity linear systems. In

real life scenarios, this means that RCM can be used to study the wave properties

inside, say, a room, but it requires additional steps to study interconnected cavities,

such as the floor of a building consisting of multiple rooms connected with doors

and hall ways, with each room potentially having its own unique geometry and

properties. In Chapter 5, we look into this case by extending RCM to model multiple

connected cavities.

There are also situations where nonlinear behavior is added to the mix, for

14



example when an amplifier or p/n junction-based electronics is present in the room.

Nonlinearity adds complexity to the system, making the modeling of the system

more difficult, but is nevertheless an interesting property that often opens avenues

for novel applications not only in electromagnetic waves [46], but also in other areas

such as elastics waves [47] and gravity waves [48]. Extending RCM into nonlinear

systems, Min Zhou added an active nonlinear circuit which generates strong second

harmonic waves into a quasi-2D wave-chaotic billiard in [49]. The study showed that

the statistics of the harmonics can be predicted by modeling the nonlinear cavity as

two cascaded linear cavities connected by a nonlinear circuit.

15



Chapter 2: Testing The Random Coupling Model Predictions in a

Scaled-Down Cavity With Remote Injection Setup

16



There is interest in using the RCM to understand the wave properties of more

complex structures, such as a cascade or a network of coupled cavities [50, 51]. It

becomes increasingly difficult to experimentally test these structures due to their

large size and the difficulty in managing and reconfiguring them in a typical labo-

ratory environment. To solve this problem, we propose miniaturizing the complex

structure while maintaining the statistical properties of the waves by carefully scal-

ing the frequency and the quality factor of the system. Electromagnetic geometric

scale modeling has been used extensively in simulations and modeling of large struc-

tures for decades [52–54]. It is also routinely done in other fields such as ship model

testing [55] and wind tunnel testing [56] where a miniature model is constructed

and studied for design verification . The idea of scaling down the geometric size is

not new in modeling, but the challenge is to make other electromagnetic properties

scale appropriately as well.

In Chapter 2, we discuss the practical challenges of testing the theory in ex-

periment, explain how the scaled cavity setup works and introduce the experimental

setup. In order to analyze the experimental data, we need to modify the RCM to

accommodate the case with lossy ports, which is discussed in Chapter 3. Then in

Chapter 4 we validate this setup by comparing the single miniature cavity result

with the single full-scale cavity result obtained by our collaborators at the Naval

Research Laboratory (NRL). Finally we expand the setup to have multiple cavities

coupled through apertures, and present the experimental result in Chapter 5

17



2.1 The Need For Scaled Cavities

Many RCM predictions have been successfully tested in experiments and it is still

expanding to include more complex scenarios. In this project, we are interested in

the RCM prediction for connected cavities, as discussed in [3,24]. A chain of coupled

ray-chaotic cavities represents many common situations, such as the compartments

in a ship and offices and hallways in a building. RCM predicts the statistics of the

induced voltages for ports located in each cavity along the chain as a function of

the loss and the coupling. Figure 2.1 shows the predicted ratio of the power coupled

into the N th cavity over the power coupled into the first cavity, as a function of

N for serially connected α = 6 cavities. However this theoretical result is hard to

verify in experiment since it requires a huge space to host the connected cavities, and

many decades of dynamic range of microwave power measurement. The previous

experimental setup for a single over-moded cavity with volume V ≈ (20λ)3 is about

1m3 and it would be difficult to fit in even a few cavities in the limited university lab

space. Hence we introduced a scaled cavity experiment setup that can host a chain

of cavities with total volume V ≈ (450λ)3. The basic idea is to scale down the cavity

size and, in order to keep Maxwell’s equations unchanged, scale up the frequency

of the electromagnetic waves and conductivity of the cavity walls. We’ll discuss the

details of the scaling in Section 2.2, its implementation in Section 2.3 and present a

series of progressive experiments to help understand and test the operation of this

setup.

18



Figure 2.1: RCM-predicted probability density function of power ratios RN(α =

6) ≡ P
(N)
in /P

(1)
in for chains of up to seven cavities: the high-loss regime is assumed

for all the (statistically identical) cavities in the chain. This is Figure 7 in Ref. [3].

19



2.2 Scaling of The Cavity

The main purpose of the scaling is to shrink the cavities down in linear size by a

factor of s, i.e. V ′ = V/s3, and create a scaled-down-in-size version with the same

statistical electromagnetic properties. Note that each individual mode in the full

scale enclosure will not be reproduced precisely in the miniature version. Instead,

the statistical properties should be identical before and after the scaling. We are

targeting scaling factors of s = 20 and s = 40. We would also like to make this a

manageable, reconfigurable and extendable experimental setup. In this section, we

will discuss the necessary scaling of the frequency and cavity wall conductivity in

order to maintain the same α value, and thus the same statistical properties.

2.2.1 Frequency Scaling

The need for frequency scaling is obvious: when the cavity is scaled down in size

by a factor of s the wavelength must also scale down by s in order for the cavity

to remain the same electrical size. If volume is scaled down to V ′ = V/s3 and

V = (Nλ)3, we need V ′ = (Nλ′)3, where λ and λ′ are the wavelength before and

after the scaling respectively. Thus wavelength is scaled down by s, λ′ = λ/s, and

the frequency f = c/λ is scaled up by s.

Frequency scaling can be achieved by using frequency extenders which are

frequency multipliers that convert signals from 0 ∼ 10 GHz (microwave) to the

several hundred GHz range (mm-wave). After scattering, the signals are received

and then mixed down to 0 ∼ 10 GHz so that they can be measured by a microwave

20



Vector Network Analyzer (VNA). The proposed setup uses extenders that are in the

range of 75 ∼ 110 GHz (WR10 band) and 220 ∼ 330 GHz (WR3.4 band), which

correspond to the scaling of s = 20 from 3.75 ∼ 5.5 GHz and s = 40 from 5.5 ∼ 8.25

GHz.

2.2.2 Cavity Wall Conductivity Scaling

As stated in Equation (1.3), since α ∝ k3V/Q must remain unchanged, and that k3V

remains the same for the given cavity, the quality factor Q must be the same as the

full-scale cavity. The quality factor is defined in general as the ratio of the energy

stored to the energy dissipated per cycle on resonance. When the wavelength is

much smaller than the system size, the quality factor of an empty metallic enclosure

with loss dominated by ohmic loss in the walls can be estimated as

Q ≈ 3

2

V

Sδ
,

where S is the wall surface area, δ =
√

2/(ωµσ) is the skin depth in the local limit,

and σ is the electrical conductivity.

δ is the depth that the electromagnetic waves can penetrate into the metal

before the induced current screens out the incident wave. Ohmic loss within the

skin depth is the main cause of energy loss in our cavities. In the local limit, the

skin depth is related to the conductivity σ by δ =
√

2/(ωµσ), where ω = 2πf is the

angular frequency, µ is the magnetic permeability of the conductor. Putting this

into the expression for Q and then putting Q into the expression for the RCM loss

21



parameter α ∝ k3V/Q, we have

α ∝ k3S
√
ωµσ

. (2.1)

If we only look at the scaling with respect to s, we can further apply k ∝ s, ω ∝ s,

S ∝ 1/s2, which leads to

α ∝ s3(1/s2)√
sσ

=
√
s/σ. (2.2)

Therefore, the conductivity of the wall must also be scaled by a factor of s in order

to keep the RCM loss parameter α the same after the scaling. This is required to

have the same fluctuating statistical properties in the scaled and un-scaled cavities.

In theory, the electrical resistance of a metal changes with temperature in two

regimes [57]:

• At room temperature the change in resistance is proportional to the change

in temperature: ∆R = R0αR∆T , where ∆R is change in resistance, ∆T is

change in temperature, R0 is resistance at the reference temperature and αR

is the temperature coefficient of resistance (not to be confused with the loss

parameter α in the Random Coupling Model). Typically a metal’s resistance

goes up as temperature goes up, i.e. αR is positive, since collisions between

vibrating atoms and conduction electrons happen more frequently.

• At low temperature the resistance is dominated by impurities or defects in the

material and becomes almost constant with temperature.

In experiment, conductivity scaling can be achieved by a combination of meth-

22



ods:

• Cool down the cavity to low temperatures using a cryostat,

• Change the cavity wall material to a better conductor, such as from aluminum

to Oxygen Free High Conductivity (OFHC) copper,

• Reduce surface roughness by physical polishing of the cavity interior surfaces.

First we put the metal cavity in a cryostat that can bring down the temperature

to about 10 Kelvin. The electrical resistivity of aluminum ρAl vs. temperature

is shown in Figure 2.2 (Figure 1 in [4]) with different samples and measurement

techniques. It is clear that ρAl is decreasing as temperature goes down but the

curve eventually saturates at some value ρsat
Al . The saturation value are different for

different curves, which are the result of measuring different samples. In general,

the higher the purity, the lower the ρsat
Al value. So ideally by choosing a suitable

aluminum material, we can achieve more than s = 40 scaling by cooling down

to a certain low temperature, before reaching the superconducting transition of

aluminum (which happens around 1.2 Kelvin).

During the cavity design, we discovered that commercially available aluminum

may not have a high enough conductivity scaling range to match the s = 40 scaling,

i.e. their ρsat
Al value is not low enough. This is mainly due to the added impurities in

the aluminum alloy that lead to higher ρsat
Al but are required to maintain certain de-

sirable mechanical properties. So instead of using aluminum as the cavity material,

we chose OFHC copper which has a lower ρsat
Cu value.

23



Figure 2.2: Electrical resistivity of copper vs. temperature. The different curves
on the plot are the experimental results for different aluminum samples and were
measured with different techniques at different times from 1908 to 1981. (which is
Figure 1 in [4])

24



After changing to OFHC copper, it was found in experiment that the α value

at the lowest temperature is still higher than what we expected, prompting more

research into the factors that affect surface conductivity of metals. One simple

method to increase the surface conductivity is to reduced surface roughness [58,59]

by physically polishing the inside walls of the cavity. The 5-sided rectangular cavity,

or the cup, is milled out from a piece of bulk metal, leaving a milling pattern on the

inside walls. Although the surface feels smooth the milling pattern indicates that

there are micro-structures that increases surface roughness. If the surface roughness

is at the scale of the skin depth, the microwave surface losses will be enhanced [58,59].

After polishing it to a mirror finish, we observe a further decrease in the α value.

An important constraint is that the polishing work is not at its best performance

because it is difficult to put proper tools inside the solid 5-sided box. Thus in

the future design of scaled cavities, it is best to start with a well-polished sheet

metal and then fold it into a 5-sided cavity, which is the basis for my design for the

multiple coupled cavities. The copper used in the 5-sided cavity is C10100 Oxygen-

Free Electronic (OFE) copper, 99.99% pure copper with 0.0005% oxygen content

and a minimum 101% IACS (International Annealed Copper Standard) conductivity

rating. The copper used in the multiple-cavity design is C11000 Electrolytic-Tough-

Pitch (ETP), the most common copper with 0.02% to 0.04% oxygen content and

usually meets or exceeds the 101% IACS specification, and is mechanically polished

to mirror-like finish.

With these three methods combined, we are able to reduced the α value from

about 6.5, at room temperature for the miniature aluminum cavity, to about 2.8, at

25



the lowest temperature for the polished copper cavity. The comparison is shown in

Figure 4.3

2.2.3 Scaling Limits

How far can we scale the cavities down in size? The current limiting factor for s

to go beyond 40 is mostly the difficulty in further frequency scaling. As we will

discuss in the following sections about the experimental setup, the corresponding

frequency band for s = 40 is 220 ∼ 330 GHz, which is a very high frequency range

for millimeter wave products such as frequency extenders, waveguides and horn

antennas. Perhaps as millimeter-wave technology advances and matures, we could

reach a scaling factor of 60, 80 or beyond.

Another issue with scaling with larger values of s is the need to increase the

wall conductivity by even greater amounts in the scaled structure. This is certainly

possible if we start with full-scale systems having large values of RCM loss parameter

α, for example 10 and larger.

2.3 Experimental Setup

In our setup, we scale down a 66 cm by 122.5 cm by 127.5 cm aluminum “full-scale”

cavity designed for the 3.7 ∼ 5.5 GHz range (WR187 band) by a factor of 20 in each

dimension, i.e. s = 20. The scaled cavity is a rectangular box (6.375 cm X 6.125

cm X 3.300 cm with rounded corners) containing a perturber of irregular shape

that can be rotated by motor control. The new (scaled) frequency range becomes

26



75 ∼ 110 GHz (WR10 band), which can be measured by using a Keysight network

analyzer (KT-N5242A 10 MHz to 26.5 GHz PNA-X ) working together with two VDI

frequency extenders (Tx/Rx WR10 module). To achieve higher Q, the miniature

cavity is made of oxygen-free high-conductivity (OFHC) copper, with mechanically

polished inner wall surface to reduce the surface resistance [58, 59]. We then use a

custom-built BlueFors BF-XLD400 cryogen-free dilution refrigerator system, which

can reach a base temperature of 10 mK under minimum heat-load conditions, to cool

the cavity and further increase Q. The available volume for samples is a cylinder of

50 cm in diameter and 50 cm in height, that has a total volume of V ≈ (150λ)3 at

100 GHz, providing abundant space for larger structures.

The following sections describe the detail of the above experimental setup in

four parts:

• Measurement system including a VNA and frequency extenders (2.3.1) to ex-

tend the frequency range of VNA from 26.5 GHz to 330 GHz.

• Cryostat to keep the sample in low temperature in order to achieve conduc-

tivity scaling (2.3.2).

• Free-space propagation path to inject the source signal (75 ∼ 110 GHz or

220 ∼ 330 GHz frequency range) into the sample cavity without direct waveg-

uide connection (2.3.3), also called remote injection.

• Magnetically coupled mode stirrer powered by a cryogenic stepper motor to

generate multiple realizations of the cavity by perturbing the cavity modes

(2.3.4).

27



Figure 2.3: The schematics for the scaled cavity experiment setup.

The schematics of the setup is shown in Figure 2.3.

2.3.1 VNA & Extenders

As mentioned in Section 2.2.1, we need frequency extenders to up convert the VNA

source signal from around 10 GHz to hundreds of GHz, and then down convert the

transmitted signal for the VNA to measure. The VNA that we are using is a 4-port

Keysight N5242A PNA-X, a high performance model that works up to 26.5 GHz.

The frequency extenders are made by Virginia Diode Inc, models WR10TxRx and

WR3.4TxRx, whose block diagram is shown in Figure 2.4 (Figure 1 in [5]). The

28



Figure 2.4: Block diagram for Virginia Diode Inc frequency extender modules. (∗)
Variable attenuator. (†) Isolator. Picture taken from VDI product manual [5].

“RF (L)” and “RF (H)” are input microwave signals generated by the VNA with

relatively lower (L) or higher (H) frequency. The RF signal is first multiplied in

frequency by a factor of N2 (for “RF (H)” input) or N1N2 (for “RF (L)” input)

and then attenuated and sent through an isolatior to become the mm-wave input

signal. A portion of the mm-wave signal is sampled out before sending to the test

port, in order to be mixed down by the local oscillator (LO) signal, which is also

multiplied in frequency by a factor of M . The resulting intermediate frequency

(IF) fIF,Ref. = N1N2fRF(L) ±MfLO is in the range of 10 GHz, measurable by the

VNA, and is measured at the “Ref.” port. The reflected mm-wave signal from the

test port is sampled and mixed down in a similar fashion and is measured at the

“Meas.” port. As discussed in Section 1.5 the S-parameters are complex ratios of

amplitudes of the outgoing waves, measured at the “Meas.” port, to the incident

waves, measured at the “Ref.” port.

29



Figure 2.5: A typical setup with VNA and VDI extenders for measurement. Picture
taken from VDI product manual [5].

A typical setup with VNA and VDI extenders for measurement is shown in

Figure 2.5 (Figure 10 in [5]). These models are chosen because they provide the best

dynamic range in the global market, and since the loss during the transmission from

one extender to the other is expected to be fairly high, especially when multiple

cavities are connected, we must ensure that the dynamic range is large enough for

the transmitted signal to stay above the noise level.

2.3.2 Cryostat

A cryostat is an apparatus for maintaining a very low temperature. In our setup,

we are using a custom made BlueFors BF-XLD400 cryogen-free dilution refrigerator

system that can reach a base temperature of 10 mK under minimum heat-load

30



conditions. The space available for mounting samples is a cylinder of 50 cm in

diameter and 50 cm in height, giving us a volume of V ≈ (450λ)3 at 300 GHz. For

s = 40, we are scaling the frequency from 5.5 ∼ 8.25 GHz to 220 ∼ 330 GHz, and

shrinking a geometry as large as V ≈ 253m3 into our cryostat. This is extremely

difficult to achieve in full scale and also hard to operate and manage. Our scaled

setup is an elegant solution to electromagnetic problems in such large scales.

Figure 2.6 shows the drawings, the outside vacuum can and the cooling plates

inside of the cryostat. The horizontal circular plates will be cooled down to 50K,

4K, 1K and 10mK from the top plate to the bottom plate, respectively. Our sample

cavity will be mounted onto the bottom plate (the sample plate) which should reach

the lowest temperature. There are several thermal shields designed to thermally

isolate the cooling plate and a vacuum can at the outermost layer to provide a

single vacuum space. Due to the need for a special free-space propagation path from

extender to the sample cavity, there are windows on the side of the cryostat shields

allowing electromagnetic waves to propagate in and out of the cryostat interior.

These windows, allowing outside radiation to get inside, puts an extra heat load on

the cryostat, which changes the lowest temperature that it can reach from 10 mK

to about 10 K. However, this higher base temperature is still lower than we need

for the scaled structure measurement.

31



Figure 2.6: The drawings (left), the outside vacuum can (middle) and the cooling
plates inside of the cryostat (right). The cooling plates are cooled in a staged manner
from 50K at the top to 10mK at the bottom. The sample will be mounted onto the
10 mK plate, also called the sample plate.

32



2.3.3 Free-space Propagation Path

Since the miniature cavity is sitting inside the evacuated cryostat at low temper-

ature, it is not possible to employ an input connection from the signal source to

the cavity via a coaxial cable or waveguide. Accordingly, we use a quasi-optical

free-space propagation path similar to that of a collimated beam in an optical ex-

periment.

The high frequency electromagnetic wave coming out of the frequency extender

is launched into air by a horn antenna, then collimated by a Teflon lens. The

output is a collimated beam propagating in free-space like a plane wave which has

significantly lower loss compared to transmission in any waveguide or transmission

line. The receiving end has a focusing lens, identical to the one on the source side,

and a receiving horn antenna which is mounted on the wall of the cavity to transmit

the received wave into the cavity. Two such free-space propagation paths are used

for the two cavity ports, one path for each port. Figure 2.7 shows the experimental

setup highlighting the free-space propagation path, the frequency extenders, the

horn antennas, and the lenses.

This free-space propagation path has its advantages and challenges compared

to other methods.

First of all, there is no direct thermal contact between the source and the

cavity. This is the most crucial reason why we choose this setup. The cavity needs

to be cooled to a low temperature inside the cryostat in order to achieve conductivity

scaling. A direct connection, such as waveguide, will certainly prevent the cavity

33



Figure 2.7: The setup for high frequency waves to propagate in free-space from
the frequency extender to the cavity, and from the cavity to the receiving frequency
extender. The horn antennas can efficiently launch and receive electromagnetic
waves and the Teflon lens collimates the waves into a parallel beam.

34



from cooling down by thermal conduction. Whereas this remote injection method

only introduces some thermal radiation which can be lowered by proper filtering.

Secondly, it is very low loss. Transmission lines and waveguides suffer from

high loss at high frequency due to the scaling of ohmic loss and dielectric loss

with frequency. The ohmic loss in metal is proportional to the surface resistance

Rs = 1/(σδ) where σ is the conductivity and δ =
√

2/(ωµσ) ∝ 1/
√
ω is the skin

depth. So Rs ∝
√
ω increases as frequency goes up, leading to higher ohmic loss.

In addition to ohmic loss, transmission lines also have dielectric loss which also

scales with frequency. In transmission lines, waves propagate as E(z) = E0e
−(α+iβ)z,

where α is the attenuation constant (not the loss parameter in RCM) given by

α ≈ (ωε′′/2)
√
µ/ε′ in which ε = ε′−jε′′ is the complex permittivity of the dielectric.

So α ∝ ω and the wave decays much faster when frequency goes up.

However, it is a challenge to properly collimate the free-space beam. For free-

space propagation, the dielectric loss in air is negligible, and the reflection at the

horn antenna due to the impedance mismatch is also very small, about -20 dB (1%

of power is reflected). The only significant factor for losing signal strength is the im-

perfect collimation. Since the frequency is not in the visible light spectrum range, it

is not easy to properly position the antenna and lens to guarantee best collimation.

But through careful experimentation, we can achieve an average transmission am-

plitude of -5 dB across a 1 meter transmission distance for WR3.4 band (220 ∼ 330

GHz). For comparison, the loss of a 1 meter long gold-plated waveguide (about

$1000 per 5 cm piece)would be 14 ∼ 20 dB, according to the Virginia Diode Inc

waveguides data sheet [60].

35



2.3.4 Mode Stirrer

Since the RCM is a statistical theory, an ensemble of scattering systems is required

to determine the system-specific features and the statistical properties of the en-

closures. Consequently, we need to perturb the cavity modes while maintaining

the volume of the cavity such that each measurement is a unique realization of the

cavity with the same loss parameter. A typical method to create many realizations

is to rotate a large metal panel inside the cavity (a “mode stirrer”), as used in

Refs. [6, 23, 61, 62]. But for our setup, which has very high frequency and is sitting

inside the cryostat in low temperature, we need to consider the following issues:

• The motor that powers the panel must be able to work in high vacuum and

low temperatures. The heat generated by the motor coil can quickly build up

in vacuum with normal motors, and the motor material and design must be

carefully chosen such that thermal contraction won’t cause mechanical prob-

lems.

• The cavity must be tightly enclosed by metal walls since any small gap or hole

can leak out the electromagnetic waves reverberating inside. Normally The

hole on the wall to host the panel axis is not a problem since the microwave

wavelength (a few centimeter) is much larger than the hole diameter (a few

millimeter). But with the wavelength scaled into the millimeter range, the

leakage through the hole is substantial.

To meet these requirements, we designed a magnetically coupled mode stirrer

36



powered by a cryogenic stepper motor, as shown in Figure 2.8. The cryogenic stepper

motor (Phytron VSS 52.200.2.5UHVC) is suitable for ultra high vacuum (up to 10−11

mbar) and cryogenic temperature environment, typically used in space applications.

The motor rotates a magnetic strip outside the cavity which is magnetically coupled

to another magnetic strip inside the cavity, thus eliminating the need for an opening

on the wall or direct mechanical contact. The metal mode-stirring panel is attached

to the inside magnetic strip and rotates when the stepper motor rotates.

It is worth noting that we do not need to control the exact location or the

shape of the perturber for any given measurement, which would be an extremely

difficult task in such a high frequency (and short wavelength) regime. The goal is to

create an ensemble of scattering systems which differ in detail, but all have the same

underlying statistical properties. Since the sole purpose of introducing the perturber

is to scatter the waves in a different way without changing the statistical properties

of the cavity (i.e. the loss parameter α) the precise location and orientation of

the perturber is not important. However, it is important that we rotate and move

the perturber to uniquely different orientations and positions, so that a diverse set

of cavities with different scattering properties is created, enabling a high quality

statistical ensemble.

In experiments, the motor rotates a small step then waits for the Vector Net-

work Analyser (VNA) to measure the S-parameters of the cavity in the current

realization. When the VNA measurement is complete, the motor rotates again, and

this process is repeated. Representative S-parameter measurements for two nearby

realizations of the cavity and perturber are shown in Fig. 2.8 (b). In this way, data

37



for 200 highly uncorrelated realizations of the full-scale cavity is collected and used

to obtain statistics of the electromagnetic properties, and to calculate the ensem-

ble average required by the RCM to characterize system-specific properties. After

collecting the ensemble S-parameter data, we check to see if each realization is sta-

tistically independent to a significant degree from all the others by looking at their

correlation coefficient. The Matlab function “corrcoef” is used for this purpose. We

then construct the impedance matrix for the cavity data Zcav as described in Sec-

tion 1.4. The ensemble averaged impedance is obtained as Zavg = 〈Zcav〉realizations for

each measured frequency point. A histogram of the impedance values is constructed

by taking the real or imaginary part of a matrix element of the Z-matrix over the

entire ensemble (9 realizations for the miniature cavity or 200 realizations for the

full-scale cavity) and over the whole frequency range (75 - 110 GHz for the miniature

cavity or 3.7 - 5.5 GHz for the full-scale cavity). For the miniature cavity, since the

cavity is gradually changing temperature, also changing the cavity loss, we can only

measure 9 realizations in a half hour window before the loss between the first and

last measurement differ significantly. It turns out that the large number of data

points for many modes within the broad frequency bandwidth compensates the lack

of realizations and we still get good statistics.

38



Figure 2.8: Magnetically coupled mode stirrer powered by a cryogenic stepper
motor. The magnetic strip outside the cavity (lower yellow bar in the left side) is
coupled by static magnetic field with the magnetic strip inside the cavity (upper
yellow bar in the left side), eliminating the need for any hole on the wall. The upper
right inset is an example of the mode stirrer placed inside the cavity.

39



2.4 Testing Experiments

In a larger picture, our final goal is to test the RCM prediction of the statistics of the

energy coupled into the N th cavity in a cascade of connected cavities using the setup

introduced in Section 2.3. This work is in collaboration with U.S. Naval Research

Laboratory (NRL). As a first step, our colleagues at NRL measured a full-scale 66

cm X 122.5 cm X 127.5 cm aluminum cavity at 3.7 ∼ 5.5 GHz (WR187 band) and

5.5 ∼ 8.25 GHz (WR137 band). We measured 2 scaled cavities with s = 20, scaling

from 3.7 ∼ 5.5 GHz to 75 ∼ 110 GHz, and s = 40, scaling from 5.5 ∼ 8.25 GHz to

220 ∼ 330 GHz, respectively. After confirming that the scaled-down setup produces

the same statistical impedance for the same α as the full-scale setup, we can proceed

to investigate the statistical property of the connected cavities.

In this section, we perform a series of experiments that can verify the operation

of our experimental setup. Since the combination of the frequency extenders, remote

injection and cryogenic environment was never explored before, we take progressive

steps to study each component individually. Then later in section 4 we present the

single cavity experiment result to confirm that the setup produce the same statistics,

and in section 5 we discuss the multiple cavity experiment result.

2.4.1 Overall Plan

Since this experimental setup is rather complex and has not been studied by any

research group known to us, we take the following incremental steps towards the

final goal making sure that we understand the results at each stage.

40



1. Obtain good transmission at mm-wave frequencies for the free-space propaga-

tion path between two horn antennas.

2. Directly measure the scaled cavity, without the free-space propagation path,

perturb the cavity modes using the magnetically coupled mode stirrer and

repeat the measurements. Confirm that the collected realizations represent a

good ensemble of the cavity. In all following experiments, a “measurement”

means collecting an ensemble of data rather than measure the cavity at one

single state.

3. Directly measure the scaled cavity, without the free-space propagation path,

and calculate its loss parameter α at room temperature. This is called a “direct

injection” measurement.

4. Measure the scaled single cavity at room temperature with the free-space

propagation path installed and compare the calculated α with direct injection.

This is called a “remote injection” measurement.

5. Put the remote injection setup in the cryostat to perform the same measure-

ment at low temperatures. Note that this involves propagating the mm-wave

signals through windows on each free-space path. We expect to see a de-

crease of α with decreasing temperature. We compare the obtained α values

to the result of the full-scale setup measurement done by our collaborators at

NRL making sure that the scaled setup works well for a single cavity. This is

discussed in section 4.

41



6. Extend the scaled single cavity setup to multiple connected cavities. Investi-

gate the statistics of the power coupled into the N th cavity and test the RCM

predictions in [3, 24]. This is discussed in section 5

The rest of the section 2.4 discuss items 1 to 4 on this list.

2.4.2 Free-space Propagation Path Measurement

The purpose of this measurement is to determine the best distance between the

antenna and the Teflon lens in order to get the optimal transmission through free-

space. The two antennas are placed at a fixed distance, about 110 cm apart, and

the lenses are placed at a distance of d1 from the transmitter horn antenna and

d2 from the receiver horn antenna. All elements are aligned to be on a single line

and only d1, d2 are varied. The average transmission is plotted in Figure 2.9 with

the best values of d1, d2 enclosed in the yellow dotted circle. For the WR10 band,

d1 = d2 = 10.45 cm and for the WR3.4 band, d1 = d2 = 15.45 cm.

2.4.3 Collecting an Ensemble of Cavity Data

This experiment tests if the magnetically coupled mode stirrer can effectively per-

turb the cavity modes in the cryogenic environment. An example of the measured

reflection (S11) and transmission (S21) magnitude data for the s = 20 cavity is plot-

ted in Figure 2.10 for 5 realizations. Notice that the full frequency range plot for

|S11| on the top left shows that all curves follow the slowly varying trend which is

the non-statistical system specific feature, while the zoom-in plot on the top right

42



Figure 2.9: The transmission amplitude in dB for free-space propagation measure-
ment. The x-axis and y-axis are the distance between the lens to the transmitter
antenna (d1) and the receiver antenna (d2) respectively. The yellow dotted circles
enclose the region for best d1, d2 values for optimal transmission.

shows that each curve differs dramatically from each other, which is the result of

the mode stirrer perturbing the cavity modes.

It is clear from Figure 2.10 that the cavity modes are indeed perturbed but

we need to further ensure that realizations are uncorrelated with each other and

represent a statistically sound ensemble. To do this, we adopt the use of the power

ratio Λ defined in Ref. [2] as the ratio of the maximum transmitted power to the

minimum transmitted power at each frequency for all mode stirrer positions, i.e. Λ =

max(|S21|2)/min(|S21|2). Λ is a function of frequency and should have a large mean

and a large dynamic range if the cavity modes are well perturbed and uncorrelated,

which gives us a good ensemble of data. To see how Λ distinguishes a good ensemble

from a bad one, we plot two Λ spectra calculated from 2 ensembles, each consisting

of 9 realizations, and their histograms in Figure 2.11. The Λ in the left figure, which

represent a good ensemble, has a mean of about 15 dB and a dynamic range of 40 dB,

43



Figure 2.10: Experimentally measured reflection (top two plots) and transmission
(bottom two plots) magnitude for the s = 20 cavity for 5 realizations created by
rotating the magnetically coupled mode stirrer. The left column plots the data in
the full frequency range from 75 GHz to 110 GHz while the right column zooms in
a 100 MHz window. Each curve is wildly different from others but yet follow the
same slowly varying trend which is the non-statistical system specific feature.

44



Figure 2.11: Power ratio Λ as a function of frequency and its histogram for a good
ensemble (left) and a bad ensemble (right) data. Λ should have a large mean and a
large dynamic range if the cavity modes are well perturbed and uncorrelated, which
gives us a good ensemble of data.

indicating large field fluctuations as the mode stirrer rotates. On the contrary, the

Λ on the right figure has an almost zero mean and 15 dB dynamic range, suggesting

high correlation between realizations. In fact, the data in the right figure were taken

when the mode stirrer was stuck and not moving.

2.4.4 Direct Injection

The purpose of the direct injection experiment is to get an accurate value for the cav-

ity loss parameter α before applying conductivity scaling and without the influence

of the free-space propagation path.

As shown in Equation (2.2) if we only scale the cavity size and measurement

frequency, then the loss parameter α ∝
√
s/σ. Based on the previous RCM ex-

periment on the full-scale cavity [6], α(s = 1) ≈ 2.05 for 3.7 ∼ 5.5 GHz and

α(s = 1) ≈ 5.95 for 5.5 ∼ 8.25 GHz as shown in Figure 2.12. Thus we are expecting

45



Figure 2.12: Loss parameter α values for the full scale cavity measured by col-
laborators at U.S. Naval Research Laboratory. Each α value is calculated from
experimental data in a 0.1 GHz window at the specified frequency. The data points
are taken from Table 1 in [6].

α(s = 20) ≈
√

20α(s = 1) = 9.17 for the WR10 band and α(s = 40) ≈
√

40α(s =

1) = 37.6 for the WR3.4 band. As mentioned in Section 2.2.2, the scaled cavity ma-

terial is copper (and the full-scale cavity material is aluminum) so there is a scaling

up of conductivity
√
σCu/σAl =

√
5.96/3.50 = 1.30. Thus the α we should see for

the s = 20 (WR10 band) cavity is 9.17/1.30 = 7.05, and for the s = 40 (WR3.4

band) cavity is 37.6/1.30 = 28.9. Notice that these are just estimates, based on

the metal bulk conductivities for pure metal. In experiments, the actual ohmic loss

depends on the surface resistance and the penetration depth, which is influenced by

the impurity, surface roughness [58, 59], temperature and is frequency-dependent.

The PDF of the normalized impedance for a direct injection experiment on a

scaled-down s = 20 copper cavity is shown in figure 2.13. By fitting with the RCM

predictions for the PDF for various α, which is obtained through a series of Monte

Carlo simulations with different α values, we determined that α = 4.5 for the direct

46



Figure 2.13: Probability Density Function (PDF) of the normalized impedance for
a direct injection experiment on a scaled-down s = 20 copper cavity (dotted line),
in comparison with the RCM Monte Carlo simulation with α = 4.5 (solid line).

injection setup. This value is close to what we expect, which is about 7.

It was later discovered from the time-domain method, discussed in section

3.2.2, that this α = 4.5 is actually an overall α that includes both the cavity’s

loss and the loss in the short waveguide section in the cavity wall. The copper

cavity has a quarter inch thick wall and the mm-wave enters and exits the cavity

through two rectangular openings on the wall, which are essentially two quarter

inch long waveguides. The waveguides in the wall add up the total loss and need

to be excluded from the total loss, characterized by α = 4.5 in the direct injection

experiment, to reveal the true cavity loss. This is done by introducing the radiation

efficiency η into the RCM, which is discussed in chapter 3.

47



2.4.5 Remote Injection

The remote injection measurement is similar to the direct injection except that

the two free-space propagation paths, one at the transmitter end and one at the

receiver end, must be installed, as shown in Figure 2.7. The α value calculated from

Equation 1.4 is around 30, much higher than the direct injection α value because of

the influence from the lossy free-space propagation path. The RCM normalization

procedure in Section 1.5 can only normalize out the system-specific features that are

lossless. Without any modification to Equation 1.4, the obtained α represent the

overall lossyness, including the cavity and the lossy free-space path. If we consider

the lossy free-space path together with the horn antennas as a complex “port” or a

composite “antenna”, then this can be generalized into the problem of RCM with

lossy ports. The solution is to use the radiation efficiency to quantify the lossyness

of the ports and then properly exclude it from the overall α. This is discussed in

detail in Chapter 3.

2.4.6 Extension to Multiple Connected Cavities

The experimental setup and the results for multiple scaled cavities are discussed

mainly in Chapter 5. As a brief preview, the life-size 3D-printed model of the

system and the configuration for a 3-cavity cascade is shown in Figure 2.14. This

design is for s = 20 and has the advantage of being reconfigurable, extendable and

is compatible the remote injection setup in the cryostat.

The backplane is a thin metallic plate, which is vertically attached to the cold

48



Figure 2.14: 3D-printed model of the multiple connected cavities and the configu-
ration for a 3-cavity cascade drawn in CST.

plate in the cryostat, and has screw holes for mounting the rectangular cavities. It

also has apertures to couple the cavities mounted on one side to the ones mounted

on the other side. If we view the backplane as a grid, where each grid cell has a

circular aperture at its center, then the cavity is a 2-cell block having 5 walls. And

once mounted to the backplane, it will have 2 open apertures. There are two special

cavities that have a rectangular opening on the side, as highlighted in Figure 2.14

as ”ports”. The rectangular opening is covered by a special adapter that has the

Teflon lens attached to it, positioned at a certain distance to the antenna and is well

aligned. The three blocks attached to the backplane in Figure 2.14 is an example of

a simple 3-cavity cascade. In the left figure, the 2 gray cavities at the front are the

first and third cavity which will have the horn antennas installed on the rectangular

apertures. They both couple to the green cavity on the other side through large

circular apertures on the backplane. The 3-cavity cascade model is also shown in

Figure 2.15 with cavity walls and backplane omitted.

With this grid-cell model we can easily compose any N -cavity cascade and

49



Figure 2.15: The air region of the 3-cavity cascade model with cavity walls and
backplane omitted. The wave goes into the first cavity through the antenna mount-
ing point, then couples into the second cavity mounted on the other side of the
backplane (not shown) through an aperture. It is then coupled to the third cavity
through the other aperture, and gets out through the antenna mounting point on
the third cavity.

place it in the remote injection setup to measure its characteristics. For this s = 20

scaled cavity size, we can easily put 20 cavities on each side of the backplane, more

than enough for the connected cavity analysis. Furthermore, we can put additional

backplanes parallel to the existing one to host additional cavities. This design allows

us to investigate many interesting coupled cavity configurations beyond the theory

prediction.

2.4.7 Further Studies

The remote injection scaled measurement setup described in Section 2.3 combined

with the grid-cell cavity design mentioned above opens up the opportunity to study

RCM predictions in a network of connected cavities.

In the grid-cell cavity design, cavities are basically 2-cell blocks and thus are

50



2-port cavities with the apertures acting as the port connecting to other cavities.

We can extend the cavities to be 3-cell or 4-cell blocks that covers 1X3 cells or

2X2 cells making them 3-port or 4-port cavities. Their volume can be maintained

by changing the height of the cavities. If we consider the cavities as nodes, we can

arrange them on the backplane such that they form a network, such as a tetrahedron

(3-branch nodes) or a pentagon (4-branch nodes). The linear cascade that we are

going to study initially is the case of the simplest network, as illustrated in Figure

2.16. By changing the cavities from 1X2 blocks to 2X2 blocks, we can use the similar

setup to study complicated graphs as shown in Figure 2.17. The example network

has a total of seven connected scaled cavities that are of difference sizes and have

anywhere from one to four apertures. Each cavity is represented as a node and each

aperture is represented as an edge in the graph. Limited only by the size of the

cryostat interior, the number of different networks that we could study with this

experimental setup is orders of magnitude larger than that with a conventional full

scale cavity setup.

Furthermore, the volume inside the cryostat is so large compared to the scaled-

down cavities that we can potentially shrink even larger structures. For example,

the cryostat’s cylindrical space of diameter d = 50 cm and height h = 50 cm is

equivalent to a full scale cylinder of