ABSTRACT

Title of Dissertation: TOPOLOGICAL PHOTONICS:
NESTED FREQUENCY COMBS AND EDGE MODE TAPERING

Christopher J. Flower
Doctor of Philosophy, 2024

Dissertation Directed by: Professor Mohammad Hafezi
Department of Physics

Topological photonics has emerged in recent years as a powerful paradigm for the design

of photonic devices with novel functionalities. These systems exhibit chiral or helical edge states

that are confined to the boundary and are remarkably robust against certain defects and imper-

fections. While several applications of topological photonics have been demonstrated, such as

robust optical delay lines, quantum optical interfaces, lasers, waveguides, and routers, these have

largely been proof-of-principle demonstrations.

In this dissertation, we present the design and generation of the first topological frequency

comb. While on-chip generation of optical frequency combs using nonlinear ring resonators has

led to numerous applications of combs in recent years, they have predominantly relied on the

use of single-ring resonators. Here, we combine the fields of linear topological photonics and

frequency microcombs and experimentally demonstrate the first frequency comb of a new class

in an array of hundreds of ring resonators. Through high-resolution spectrum analysis and out-

of-plane imaging we confirm the unique nested spectral structure of the comb, as well as the



confinement of the parametrically generated light.

Additionally, we present a theoretical study of a new kind of valley-Hall topological pho-

tonic crystal that utilizes a position dependent perturbation (or “mass-term”) to manipulate the

width of the topological edge modes. We show that this approach, due to the inherent topologi-

cal robustness of the system, can result in dramatic changes in mode width over short distances

with minimal losses. Additionally, by using a topological edge mode as a waveguide mode, we

decouple the number of supported modes from the waveguide width, circumventing challenges

faced by more conventional waveguide tapers.



TOPOLOGICAL PHOTONICS:
NESTED FREQUENCY COMBS AND EDGE MODE TAPERING

by

Christopher J. Flower

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

2024

Advisory Committee:
Professor Mohammad Hafezi, Chair/Advisor
Professor Kartik Srinivasan
Professor Yanne Chembo
Professor Carlos A. Rios Ocampo
Professor Miao Yu



© Copyright by
Christopher J. Flower

2024



To my family.

ii



Acknowledgments

This dissertation and the research within would not have been possible without the support

of my advisor, Mohammad Hafezi, whose patience with me was nothing short of remarkable.

The main experimental work on topological frequency combs that I undertook during the course

of this PhD began in early 2020, just as the COVID-19 pandemic began and almost four difficult

years into my PhD already. The lab in the ensuing years was something of a microcosm of the

country at the time in that it was marked by frustrations and failures. It was not until May of

2022 that the first working samples arrived, and not until late Friday evening, August 4th, 2023,

that I observed the first topological frequency comb. The work was finally published in June the

following year. Along the way I churned through samples, lasers, ideas, and cash with little to

show for it. But at no point did I have to worry about the plug being pulled, the support continued.

I was able to focus on the science, and was allowed to double down even when I was miserable

and the work looked hopeless. For that belief in me, I am profoundly grateful.

Over the years I have also received significant counseling from senior colleagues, including

Sunil Mittal, Elizabeth Goldschmidt, Gregory Moille, and Sabyasachi Barik which was excep-

tionally valuable and often encouraging. I am indebted to Lida Xu for not only his hard work but

also for the youthful enthusiasm that he brought to the lab in the final years of my PhD, which

helped balance my own curmudgeonly, senior graduate student cynicism. My thanks extend to

my other lab mates as well, including Deric Session, Supratik Sarkar, Mahmoud Jalali Mehrabad,

iii



Daniel Gustavo Suarez Forero, Ghadah Alshalan, Beini Gao, Pranshoo Upadhyay, Bin Cao, Jon

Vanucci, Julia Sell, Vikram Orre, and Erik Mechtel whose collaboration and companionship was

always welcome. I am also thankful for the friendship of Fangli Liu, my first office-mate, whose

conversations always brightened my day.

I have acquired so many scientific collaborators that I could not possibly list them all here,

but they each deserve and have my deep gratitude. A few of them include Shahriar Aghaeimei-

bodi, Subhojit Dutta, Aziz Karasahin, Edo Waks, Ogulcan Orsel, Gaurav Bahl, Nicholas Martin,

David Sharpe, Arka Majumdar, Wade DeGottardi, and Niloy Acharjee. I am also, of course,

deeply grateful to the members of my thesis committee, Kartik Srinivasan, Yanne Chembo, Car-

los A. Rios Ocampo, and Miao Yu.

In the humid summer of 2016, as I prepared to begin graduate school, I was lucky enough to

come by a room for rent in one of the “physics houses”, occupied continuously by UMD physics

graduate students for quite a few years. I lived in that room for the next seven years and, while the

house itself was slowly falling apart around me, managed to create some of the best memories of

my life. I am forever indebted to that moldy old house on Cool Spring Road for introducing me

to some of the best friends I could have hoped for. Jon Curtis, Jaron Shrock, Nour El-Husseini,

Kevin Palm, Amanda Bosworth, Antony Speranza, Zac Castillo, Zach Eldredge and I shared

more memories than I could possibly recount here. I’ll never forget weathering the quarantine

days of the pandemic in that house with some of my best friends, occupying ourselves late into

the night with drinks, cards, music, grilling, and above all, laughter.

I am exceptionally lucky when it comes to friends who have supported me along the way,

including some who I have known for almost two decades now. I don’t think many people can say

that they still have friends from grade school who they speak with regularly, but somehow I still

iv



have more than my fair share. Arjun Rao, David Wu, Danny Iachan, Dan Golden, Keval Patel,

Sam Fishman and Tom Ko have been a constant source of companionship, fun, and support over

the years, even when separated by a continent. Looking forward to seeing you all at the bonfire

this year. Surgery residency is no excuse, Tom. Rahul Grover has been a friend who has shown

an incredible ability to pick up right where we left off after busy, work filled years in between.

Brandon Yeung has been supporting me since we became friends in preschool, and, outside of

family, is probably the only person to have attended my birthday parties in four separate decades.

Ben Gelman continues to make me laugh with atrocious puns.

Those early years in Baltimore also were marked by numerous excellent teachers. Mrs.

DiLaura, my 5th grade teacher, made me realize how much I loved school. I’m sorry I cried

when I saw I had you for homeroom. Mrs. Henderson, my 8th grade science teacher, was the

first to make me really think that I should be a scientist. Mr. Audlin, our freshman AP Bio

leader, challenged me to expect more of myself. Mr. Wagner, my history teacher twice over and

academic team coach showed me how entertaining and enjoyable learning can be. There are too

many to list.

Undergrad at Duke was also, far better to me than I could have hoped. I stumbled into

quality friendships immediately, when the Belltower crew took the biggest change in my life and

made it into a welcome one. Cooper House and all the friends within became the second family I

didn’t know I could have. To this day I feel like I have friends in just about any city I visit, thanks

to Cooper. Will Victor has brought energy and excitement and laughter into my life since the day

I met him, and has been a true brother to me through so many ups and downs. Academically, I am

indebted to Haiyan Gao, who accepted me into her group despite me knowing next to nothing,

and George Laskaris, who spent his valuable time giving me the opportunity to do some science.

v



I am exceptionally grateful to Bastiaan Driehuys and all of the folks at the CIVM, with whom

I published my first, first-author manuscript and shared my first academically motivated beers.

To this day I consider those sweltering Durham summers a high-water mark of camaraderie and

science in my career.

Claire Komyati, my loving girlfriend, has supported me steadfastly over the last few years,

especially during the writing of this thesis. She in particular has received the brunt of my stress

more than anyone else, and has guided me through those lows with grace and care. Our cat,

Nemo, has also helped me push through the difficult times, although he remains unaware.

Finally, I cannot even begin to emphasize how impossible this work would have been with-

out my family. My parents, Ken and Karen Flower, have supported me since before I was born.

As I write this I am thirty years old, but already I have received lifetimes worth of love and guid-

ance from them. Doing my graduate work only an hour or two away from the home they have

always been ready to welcome me back to was probably the wisest decision I have ever made.

My brother Nick has been my role model for as long as I can remember, and even as we navigate

adulthood, he has been that more than ever. I am also particularly indebted to him for making my

family even bigger by giving me an incredible sister, Jess, and a perfect niece, Isabelle, who has

filled the last two years with more joy and surprises than I thought possible.

There are so many others who have helped me over the years in so many ways, I regret not

being able to list every one of you. I trust that you know who you are. Thank you.

vi



Table of Contents

Dedication ii

Acknowledgements iii

Table of Contents vii

List of Tables ix

List of Figures x

List of Abbreviations xii

Chapter 1: Introduction 1
1.1 Introduction to Topological Photonics . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Ring Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Ring Resonator Lattice Fundamentals . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Simulation of Lattice Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Nonlinear Photonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 Basics of Kerr Nonlinear Media . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 Frequency Microcombs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.8 Modeling the Nonlinear Ring Resonator - the Lugiato-Lefever Equation . . . . . 25

Chapter 2: Design of Topological Lattices for Nonlinear Photonics 32
2.1 Material Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Waveguide Dimensions and Mode Properties . . . . . . . . . . . . . . . . . . . 33
2.3 Waveguide Bend Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 Splitting Ratio and Directional Couplers . . . . . . . . . . . . . . . . . . . . . . 55
2.5 Free Spectral Range and Add-Drop Filters . . . . . . . . . . . . . . . . . . . . . 62

Chapter 3: Observation of Topological Frequency Combs 68
3.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5 Device Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.6 Linear Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.7 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

vii



3.8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.9 Supplementary Materials for Observation of Topological Frequency Combs . . . 85

3.9.1 Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.9.2 Estimation of Device Parameters . . . . . . . . . . . . . . . . . . . . . . 86
3.9.3 Measurement Setup and Methods . . . . . . . . . . . . . . . . . . . . . 88
3.9.4 Pump Laser Characterization . . . . . . . . . . . . . . . . . . . . . . . . 91
3.9.5 Comb Contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.9.6 Nesting Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.9.7 Generation and Spectra of Imaged Modes . . . . . . . . . . . . . . . . . 95
3.9.8 Simulated Nonlinear Mode Profiles . . . . . . . . . . . . . . . . . . . . 96

Chapter 4: Topological Edge Mode Tapering 98
4.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.4.1 Topological Bandgap Engineering for Tapering . . . . . . . . . . . . . . 104
4.4.2 Photonic Crystal Design . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.5.1 Effective Hamiltonian and Analysis . . . . . . . . . . . . . . . . . . . . 105
4.5.2 Simulation of Bulk and Edge Mode Properties . . . . . . . . . . . . . . . 106
4.5.3 Simulation of Tapered Edge Mode Properties . . . . . . . . . . . . . . . 109

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.7 Funding Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.8 Supplementary Information for Topological Edge Mode Tapering . . . . . . . . . 112

4.8.1 Supplementary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.8.2 Taper Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.8.3 Connection to a Nano-Beam Waveguide . . . . . . . . . . . . . . . . . . 114
4.8.4 Robust Broadband Transmission Through Sharp Bends . . . . . . . . . . 116

Chapter 5: Conclusions and Future Work 118
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Bibliography 122

viii



List of Tables

4.1 Valley-Hall Photonic Crystal Parameters. . . . . . . . . . . . . . . . . . . . . . . 112

ix



List of Figures

1.1 Single-Ring Resonator In Add-Drop Configuration. . . . . . . . . . . . . . . . . 3
1.2 Analytical Transmission Spectrum for Add-Drop Filter . . . . . . . . . . . . . . 5
1.3 Integer Quantum Hall Lattice Schematic. . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Gauge Field for Photons in the Integer Quantum Hall Lattice. . . . . . . . . . . . 11
1.5 Anomalous Quantum Hall Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Gauge Field for Photons in the Anomalous Quantum Hall Lattice. . . . . . . . . 14
1.7 Simulated Drop Spectrum for Anomalous Quantum Hall Lattice. . . . . . . . . . 16
1.8 Simulated Spatial Mode Profiles for Anomalous Quantum Hall Lattice. . . . . . . 17
1.9 Lugiato-Lefever Intracavity Power Versus Detuning . . . . . . . . . . . . . . . . 28
1.10 Lugiato-Lefever Field Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.11 Lugiato-Lefever Time Domain Field Profiles . . . . . . . . . . . . . . . . . . . . 30

2.1 Effective Refractive Indices Versus Geometry . . . . . . . . . . . . . . . . . . . 35
2.2 SiN Waveguide Dispersion Versus Wavelength. . . . . . . . . . . . . . . . . . . 38
2.3 800 nm Thick SiN Waveguide Dispersion Versus Width. . . . . . . . . . . . . . 40
2.4 Schematic of Circular Ring Versus Racetrack Resonators. . . . . . . . . . . . . . 43
2.5 Overlap Integral of Fundamental TE Modes at a Straight-Bent Interface Versus

Waveguide Width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6 TE Mode Profiles of Straight and Bent Waveguides. . . . . . . . . . . . . . . . . 46
2.7 Overlap Integral of Fundamental TE Modes at a Straight-Bent Junction Versus

Bending Radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.8 Total Transmission Through Straight-Bent-Straight Waveguide. . . . . . . . . . . 48
2.9 Fundamental Mode Transmission Through Straight-Bent-Straight Waveguide. . . 49
2.10 Normalized Electric Field Intensity in Straight-Bent-Straight Junctions. . . . . . 50
2.11 Curvature Profile and Schematic of Round and Euler Bends. . . . . . . . . . . . 52
2.12 Analysis of Euler Versus Round Bends. . . . . . . . . . . . . . . . . . . . . . . 54
2.13 Euler Bend Directional Coupler Schematic. . . . . . . . . . . . . . . . . . . . . 56
2.14 Directional Coupler Transmission Versus Coupling Gap. . . . . . . . . . . . . . 57
2.15 Directional Coupler Transmission Versus Coupling Length. . . . . . . . . . . . . 58
2.16 Directional Coupler Transmission Versus Wavelength and Coupling Gap. . . . . 60
2.17 Fundamental Mode Transmission of Directional Coupler. . . . . . . . . . . . . . 61
2.18 Add-Drop Filter and Simulated Transmission. . . . . . . . . . . . . . . . . . . . 64
2.19 Simulated and Measured Drop Transmission of an Add-Drop Filter. . . . . . . . 66

3.1 Generation of the Topological Frequency Comb. . . . . . . . . . . . . . . . . . . 71
3.2 Experimental Characterization of the Topological Lattice. . . . . . . . . . . . . . 77
3.3 Formation of the Topological Frequency Comb. . . . . . . . . . . . . . . . . . . 79

x



3.4 High-Resolution Spectra of Individual Comb Teeth. . . . . . . . . . . . . . . . . 80
3.5 Spatial Imaging of the Topological Frequency Comb. . . . . . . . . . . . . . . . 81
3.6 Band Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.7 Modal Cross-Section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.8 Through Spectrum of an ADF Around One Resonance. . . . . . . . . . . . . . . 89
3.9 Detailed Schematic of the Nonlinear Measurement Setup. . . . . . . . . . . . . . 90
3.10 Laser Characterization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.11 Laser Background Drop Spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.12 Comb Contrast. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.13 High-Resolution Spectra of Individual Comb Teeth. . . . . . . . . . . . . . . . . 95
3.14 Counterclockwise, Clockwise, and Bulk Comb Spectra. . . . . . . . . . . . . . . 96
3.15 Simulated Nonlinear Mode Profiles. . . . . . . . . . . . . . . . . . . . . . . . . 97

4.1 Topological Relationship Between Bandgap And Edge Mode Width. . . . . . . . 101
4.2 Valley-Hall Photonic Crystal Parameters and Properties. . . . . . . . . . . . . . . 102
4.3 E Field Amplitudes Versus In-Plane Displacement. . . . . . . . . . . . . . . . . 107
4.4 The Topologically Tapered Waveguide System. . . . . . . . . . . . . . . . . . . 108
4.5 Transmission of Tapered Valley-Hall Photonic Crystal Waveguide. . . . . . . . . 113
4.6 Transmission of Valley-Hall Photonic Crystal to Nanobeam Waveguide. . . . . . 114
4.7 Transmission of Bent Valley-Hall Photonic Crystal Waveguides. . . . . . . . . . 116

xi



List of Abbreviations

2D Two-Dimensional
ADF Add-Drop Filter
CW Continuous Wave
TMM Transfer Matrix Method
FSR Free Spectral Range
FWHM Full-Width at Half-Max
IQH Integer Quantum Hall
AQH Anomalous Quantum Hall
SPM Self-Phase Modulation
XPM Cross-Phase Modulation
FWM Four-Wave Mixing
LLE Lugiato-Lefever Equation
MI Modulation Instability
CMOS Complementary Metal-Oxide-Semiconductor
TPA Two-Photon Absorption
SiN Silicon Nitride
EME Eigenmode Expansion
SiO2 Silicon Dioxide
GVD Group Velocity Dispersion
FDTD Finite-Difference Time-Domain
GDD Group Delay Dispersion
TM Transverse Magnetic
TE Transverse Electric
3D Three-Dimensional
HWHM Half-Width at Half-Max
DUV Deep Ultraviolet
IR Infrared
1D One-Dimensional

xii



Chapter 1: Introduction

1.1 Introduction to Topological Photonics

The field of topological photonics was born out of the discovery by Haldane and Raghu that

topological physics could arise generally from waves in a periodic medium, enabling the realiza-

tion of topological states of photons [1,2]. Topological physics itself began with the discovery of

the integer quantum Hall effect in 1980 by Klaus Von Klitzing, which in turn earned him the 1985

Nobel Prize in Physics [3]. The topological nature of the effect was quickly uncovered thereafter

by Thouless [4] and Kohmoto [5]. For this and other contributions, David Thouless (jointly with

F. Duncan Haldane and J. Michael Kosterlitz) received the 2016 Nobel Prize in Physics. These

works kicked off several decades of rich discovery and exploration, as more and more topological

phases, characterized by new and different topological invariants, were uncovered. The impor-

tance of the topological invariant, as it related to topological phases of matter, was elucidated

by Jackiw and Rebbi and others with the notion of bulk-edge correspondence [6–9]. In short,

bulk-edge correspondence states that at the interface of two materials characterized by distinct

topological invariants there must exist states localized at the boundary. The existence of these

localized states, often known as edge states in the two-dimensional (2D) case, is one of if not

the most important unifying property of topological insulators and will play a central role in this

work.

The first experimental realization of these edge states in the world of photonics came in

a magneto-optical microwave photonic crystal by Wang et al. [10], but the first demonstrations

in the optical regime soon followed. In one of these works, Hafezi et al. realized topological

1



edge states of light in a two-dimensional system for the first time, enabled by the invention of the

topological ring resonator array with a synthetic gauge field for photons [11, 12]. A key insight

in this work, and other concurrent works [13], was that strict time reversal symmetry breaking

was not required for developing topological photonic systems. This can also be understood as

the development of photonic analogs of quantum spin Hall systems [14, 15], rather than tradi-

tional quantum Hall systems, and circumvented the practical challenges of weak magneto-optical

effects in the optical regime.

While early work largely focused on demonstrating the existence and robustness of topo-

logical edge states in photonic systems, recent work has been increasingly interested in finding

applications for their unique properties. In many cases, this has also led to the development of

topological photonic systems that move further away from strict analogy to their electronic coun-

terparts. Examples of recent applied results in topological photonics include robust optical delay

lines [16], chiral quantum optics interfaces [17–19], slow light engineering [20], waveguides [21],

and reconfigurable routers [22].

This dissertation will cover two such works in topological photonics. In particular, this

introductory chapter will cover the basics of single-ring resonators, linear ring resonator lattices,

Kerr nonlinear optics, and modeling of nonlinear ring resonators. Chapter 2 will describe the

design of topological ring resonator arrays for nonlinear photonics in detail. Chapter 3 will

present the published experimental work, Observation Of Topological Frequency Combs [23].

Chapter 4 will switch platforms from ring resonator arrays to photonic crystals and present the

published theoretical work, Topological Edge Mode Tapering [24]. Finally, Chapter 5 will cover

conclusions and future works.

For further reading on the field of topological photonics, see References [25–30].

2



Input

Add

Drop

Through

Figure 1.1: A single-ring resonator is depicted schematically with two bus waveguides coupled
in an add-drop filter configuration.

1.2 Ring Resonators

A ring resonator is an optical cavity formed by creating a simple loop out of a waveguide.

The condition for constructive interference is given by the following condition:

ω

c
neffL = 2πm, (1.1)

which is equivalent to when the path length of the ring is an integer multiple of the wavelength.

Here, ω is the frequency, c is the speed of light in vacuum, neff is the effective refractive index

of the waveguide mode, L is the circumference of the resonator, and m is an integer.

When coupled evanescently to one or two bus waveguides, the ring resonator is often called

an “all-pass” or “add-drop” filter, respectively.

As shown schematically in Figure 1.1, the add-drop filter (ADF) is characterized by two

3



coupling rates. First, the extrinsic coupling rate to the bus waveguides is denoted as κex. The cou-

pling for each waveguide is taken here to be equal for simplicity. Second, the intrinsic coupling

rate to all other modes including scattered modes is denoted κin.

The transmission spectra, assuming a continuous-wave (CW) input source at the Input

port, can be calculated in a variety of ways. One particular method, the Transfer Matrix Method

(TMM), yields the following relationship. Details of this calculation can be found in Reference

[31].

TT = |ET
E1

|2 = t2(1 + e−2αL − 2e−αL cos(βL))

1 + t4e−2αL − 2t2e−αL cos(βL)
, (1.2)

TD = |ED
E1

|2 = k4e−αL

1 + t4e−2αL − 2t2e−αL cos(βL)
. (1.3)

Here, t is the field transmission coefficient through the coupling region, k is the coupling

coefficient, α is the field decay per unit length, and β = neffω/c is the standard propagation

constant. The variables here can be related to those in Figure 1.1 in the following manner:

κex =
k2

2

vg
L
, (1.4)

κin = αvg, (1.5)

where vg is the group velocity of the mode.

4



0.8

0.6

0.4

0.2

190 192 194 196 198 200

Through
Drop

Frequency (THz)

Tr
an

sm
is

si
on

Normalized Through and Drop Transmission for Add-Drop Filter

Figure 1.2: Sample through and drop spectra plotted using analytical expressions derived from
the transfer matrix method. Transmission is normalized to the input transmission.

Figure 1.2 shows a sample drop and through spectrum calculated using the analytical ex-

pressions in Equation 1.3. These drop and through spectra are characterized by resonance peaks

or dips corresponding to different solutions of the phase matching condition for constructive in-

terference. The spacing, referred to as the free spectral range (FSR), is given by:

FSR = 2π
vg
L
. (1.6)

We also note here that a quantity of particular interest, the quality factor or Q, is given as

the ratio of the resonance frequency to the resonance full-width at half-max (FWHM). The Q is a

metric of how sharp the resonance is, and can also be interpreted as the ratio of the energy stored

in the ring to the rate of energy loss. It can be expressed as follows:

5



Q =
ω0

∆ω
= mF, (1.7)

F =
πvg

L(2κex + κin)
, (1.8)

where m is the mode number of the resonance. F gives a similar quantity, the finesse, which can

be thought of the ratio of the photon lifetime in the ring to the round-trip time. These quantities

will be of great importance later in this work due to their relationship with the amount of power

that builds up in a given ring resonator.

At this point, the basic properties of the ring resonator have been established. As one

approaches a resonance frequency, optical power will build up within the ring. With this in mind,

we can consider a single-ring in a simplified picture where it can be described by couplings and

some energy amplitude.

1.3 Ring Resonator Lattice Fundamentals

We can now consider more complicated models of coupled ring resonators by employing

the abstraction established in the previous section. In particular, treating a single-ring resonator

near a resonance as a site to which photons are bound with some resulting energy amplitude

suggests a natural connection to the tight-binding models of solid-state physics. Tight-binding

models assume that electrons in a solid tend to be trapped near the atomic cores, but can hop

from lattice site to lattice site via tunneling. As a result, the effective Hamiltonian for an atomic

6



lattice in the tight-binding approximation and second quantization can be written as a real-space

summation of creation and annihilation operators, hopping strengths, and phases.

This section will focus on two particular models from topological physics and their cor-

responding tight-binding Hamiltonians. The first is known as the integer quantum Hall (IQH)

effect, and is one of the seminal models of topological physics [3].

The tight-binding IQH Hamiltonian [7, 32] can be written:

ĤIQH =
∑
x,y

â†x,yâx,y − J(
∑
x,y

â†x+1,yâx,ye
−iyϕ + â†x,yâx+1,ye

iyϕ + â†x,y+1âx,y + â†x,yâx,y+1), (1.9)

where ax,y is the annihilation operator for a lattice fermion at site (x, y). The second and third

terms result from hopping along the x direction, while the fourth and fifth terms result from

hopping along the y direction. J gives the hopping, or tunneling rate, in units of the on-site

potential. Note that the term for hopping in the x direction includes a position and direction

dependent phase.

The eigenvalues of this Hamiltonian as a function of the magnetic field strength make up

the well-known Hofstadter butterfly spectrum [33]. For an infinite lattice, this spectrum exhibits

bandgaps in energy where no states exist. When reduced to a finite lattice however the bandgaps

become filled with localized states known as “edge states” which are confined to the boundaries

of the lattice. Furthermore, these states circulate around the boundary of the lattice with a well

defined group velocity. When the magnetic field is chosen such that there are two bandgaps, the

edge states in either bandgap circulate with opposite group velocities. Due to the separation in en-

7



ergy between the two cases, coupling or scattering between the clockwise and counterclockwise

circulating edge states is suppressed. This is often referred to as topological robustness.

The topological nature of these edge states has been the subject of intense study, and as a

result will not be explored in this work. For further consideration, consult References [31,32,34]

A second model of interest is known as the anomalous quantum Hall (AQH) model, which

differs significantly from the integer quantum Hall effect in that it is characterized by zero net

flux through the unit cell [14, 15, 35]. As outlined by Haldane, the inclusion of a periodic, local

magnetic flux density results in the acquisition of a phase only for nearest-neighbor hoppings,

with no phase for next-nearest neighbor hoppings. As a result, the effective Hamiltonian can be

written:

ĤAQH =
∑
m

â†mâm − J(
∑
⟨m,n⟩

â†mâne
−iϕm,n +

∑
⟨⟨m,n⟩⟩

â†mân), (1.10)

where (m,n) are real-space indices on a lattice, with single brackets ⟨m,n⟩ representing nearest-

neighbors and double brackets ⟨⟨m,n⟩⟩ representing next-nearest neighbors. In the original work,

the lattice considered was a honeycomb lattice made up of two interspersed A and B triangular

sublattices. The work also considered an inversion-symmetry breaking on-site energy term, M .

For simplicity, that term is taken to be zero and is excluded here.

The staggered magnetic-flux density here breaks time-reversal symmetry and opens a bandgap

that supports robust topological edge states, similar to that of the integer quantum Hall Hamilto-

nian above.

8



With these models in mind, we can begin to imagine how one might realize similar physics

in a ring resonator array. By coupling a set of identical rings, referred to as the “site-rings”, we

can create a system described by the amplitude of the field at each ring, or site, and the coupling

rate, or hopping strength, between sites. A second set of detuned rings, referred to as “link-rings”

can be employed to engineer more flexible geometries and introduce control over hopping phases.

These typically can be detuned by introducing a slight difference in the circumference compared

to the site-rings. In particular, in order to create maximal detuning, the resonance shift should be

approximately half of the FSR. This is equivalent to light circulating in the link-ring acquiring

an additional π phase shift compared to light in the site-rings. Mathematically, this is expressed

as: β(LSR − LLR) = ±π. As a result, the effective Hamiltonian can again be constructed as a

summation over lattice sites of creation and annihilation operators, hopping strengths, and phases,

exactly as in the case of the tight-binding model for electronic systems.

The square lattice of the integer quantum Hall model can be reconstructed with a relatively

simple scheme of site-rings (red) and link-rings (black), depicted in Figure 1.3.

As seen in Figure 1.3, each site-ring in the bulk has a set of four nearest-neighbors to which

they are coupled via link-rings. Intrinsic and extrinsic coupling rates are indicated, as well as the

hopping strength between site-rings, J . While the hopping scheme resembles that of the desired

tight binding model (Equation 1.9), at this stage there is no direction-dependent hopping phase

or, by extension, gauge field.

The link-rings offer a convenient mechanism to introduce a direction-dependent hopping

phase in this arrangement, depicted in Figure 1.4.

9



Figure 1.3: A schematically depicted lattice of ring resonators to emulate the integer quantum
Hall model for photons. Site-rings are shown in red, while detuned link-rings are shown in black.
Coupling variables, κex, κin and J are indicated

10



Long Path

Short Path

Figure 1.4: Two site-rings are depicted, coupled via a single link-ring. A vertical shift ξ, of the
link-ring introduced a path length differential for photons hopping from left to right versus right
to left.

As shown, the introduction of a vertical shift of the link ring, denoted ξ, creates a path

length difference when hopping from the left to the right site-rings as opposed to hopping right

to left. The result is a phase shift ±ϕ depending on the propagation direction of the light. When

combined into a lattice as in Figure 1.3, this results in the acquisition of a phase of ±ϕ when

traversing a plaquette in either the clockwise or counterclockwise direction. However, in order to

maintain this relationship, the vertical shift must increase in units of ξ with each row resulting in

a breaking of the translational symmetry of the lattice. While this is not infeasible with modern

fabrication techniques, it does pose additional practical considerations.

An additional scheme has been developed that avoids this challenge through simulating

the tight-binding Hamiltonian of the anomalous quantum Hall effect (Equation 1.10) [36–38].

A modified lattice that emulates the hopping scheme of the AQH tight-binding Hamiltonian is

depicted in Figure 1.5.

11



Figure 1.5: A topological ring resonator array that emulated the anomalous quantum Hall effect
is depicted schematically. Each site-ring has a set of four nearest-neighbors and two next-nearest
neighbors with identical hopping strengths, J .

12



As can be seen, the new scheme is characterized by each bulk site-ring having a set of four

nearest-neighbors and two next-nearest neighbors. As indicated, the hopping strength, J , remains

the same for both nearest and next-nearest neighbors. A direction dependent hopping phase is

achieved in this scheme without any modification of the periodic structure.

Figure 1.6 shows an inset of the lattice of Figure 1.5 with the hopping phases indicated.

In particular, if we consider the clockwise circulating mode in the site-rings, a photon in ring

A can hop to two nearest-neighbors, labeled B and D, or one next-nearest neighbor, labeled C.

To hop from A to B, the photon will traverse one quarter turn of the link-ring and accumulate

a phase of ϕ/4. To hop from A to C, the photon will traverse one half turn of the link-ring and

accumulate a phase of ϕ/2. And finally, to hop A to D, the photon will traverse three quarters of

the link-ring accumulating a phase of 3ϕ/4. Up to a global phase of ϕ/2, the photon then sees

a direction dependent hopping phase of ±ϕ/4 when hopping to nearest-neighbors, and no phase

when hopping to next-nearest neighbors, as seen in the AQH Hamiltonian (Equation 1.10).

By choosing the detuning of the link-rings to be one half the FSR of the site-rings, the

round trip phase accumulation results in ϕ = π [37].

1.4 Simulation of Lattice Behavior

The characteristics of these lattices can be studied via several standard techniques. The

first, which relies on the single mode approximation, is known as coupled-mode theory. With

this approach, coupling to input and output fields can be treated with standard input-output for-

13



AA

B

C

D

Figure 1.6: Four site-rings are depicted, coupled via a single link-ring. Each site ring is labeled
A through D. A photon circulating in the clockwise mode in site-ring A will acquire propagation
phases of ϕ/4 when hopping to site-ring B, ϕ/2 when hopping to site-ring C, and 3ϕ/4 when
hopping to site-ring D via the link-ring. Up to a global shift of ϕ/2, this results in a ±ϕ/4 phase
shift when hopping to nearest-neighbor site-rings B and D, and no phase shift when hopping to
the next-nearest neighbor site-ring C.

14



malism which allows one to study the transmission through the lattice [39]. In particular, the time

evolution of the energy amplitudes in the rings can be expressed as a set of coupled differential

equations. By assuming a simple plane wave form for the input field, the coupled differential

equations can be simplified, linearized, and solved with standard matrix methods. For a more

detailed description of this approach, see Reference [31].

A standard “drop” spectrum for an anomalous quantum Hall lattice calculated using this

method is shown in Figure 1.7. The parameters used here are based on actual parameters used in

experiment, described in Chapter 3. In particular, they are J = 2π × 25 GHz, κex = 2π × 12.5

GHz, and κin = 2π×0.35 GHz. The lattice size is 10 by 10, however, due to the coupling scheme

as shown in Figure 1.5, this consists of a total of 180 site ring resonators in two interspersed 10

by 9 grids.

Figure 1.7 shows three distinct regions. In particular, there are two bulk bands showing

highly variable and disordered transmission surrounding a central edge band with a bandwidth of

approximately 2J . Although only partially resolved, the edge band consists of a large number of

individual edge modes. The number of modes within the edge band corresponds to the size of the

lattice. Notably in this calculation, the bulk bands have higher transmission than the edge bands.

Generally speaking, this is not the case in experiment as the bulk bands are highly susceptible to

imperfections and disorder.

While the transmission properties of the lattice do suggest the existence of edge and bulk

bands, examining the field or mode profiles of the lattice helps clarify the directionality and

confinement properties implied by the names “bulk” and “edge”. In particular, by choosing the

15



Frequency (J)
-4 -2 0 2 4

Simulated Transmission Spectrum of AQH Lattice

Po
w

er
 (m

W
)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Figure 1.7: A sample drop transmission spectrum for an anomalous quantum Hall lattice made
up of two bulk bands and a single, central edge band. Input power is 1 mW.

16



Edge and Bulk Mode Profiles

Edge Mode - Detuning: 0.10 J Bulk Mode - Detuning: 2.41 J
Y 

C
oo

rd
in

at
e

X Coordinate
0 5 10 15 20

0

5

10

15

20

0 5 10 15 20 0.0

0.2

0.4

0.6

0.8

1.0
A B

Figure 1.8: A) Simulated mode profile for an anomalous quantum Hall lattice with a pump detun-
ing of 0.10 J from the center of the edge band depicting a highly confined and circulating edge
mode. B) Simulated mode profile for an anomalous quantum Hall lattice with a pump detuning
of 2.41 J from the center of the edge band depicting a disordered bulk mode.

frequency of the input field and plotting the resulting field in the lattice, one can extract the spatial

dependence of various modes.

Two examples are shown in Figure 1.8. Figure 1.8A shows an edge mode excited with an

input field detuning (from the center of the full transmission band) of 0.10 J in order to remain

squarely in the edge band. As is clear, the excited mode is tightly confined to the lattice edge.

Furthermore, the linear loss profile due to the nonzero κin indicates that the mode is circulating

in the counterclockwise direction. Figure 1.8B shows a mode excited via an input field with a

significantly larger detuning, 2.41 J , such that the excited mode will be squarely in one of the bulk

bands. In this case, a highly disordered mode profile is apparent with no degree of confinement

and no discernable direction of propagation from linear losses.

17



While these calculations serve as an introduction to the linear behavior of these topological

lattices of ring resonators, we have only scratched the surface here. For more detailed discussions

of these lattices, see References [11, 12, 16, 23, 36–38, 40–44]. For more detailed discussion of

methods used to calculate the properties of thee lattices, see Reference [31].

1.5 Nonlinear Photonics

With the advent of the laser in the 1960s, performing physics experiments at significantly

higher optical intensities than ever before became possible. The experiments that followed led to

a number of realizations about the nature of light in media. In particular, it was observed that the

refractive index of a material typically depends on the intensity of the light, and that the frequency

composition of light in a material can also undergo changes as a function of the intensity. The

first observation of the latter of these two effects in 1961 by Franken et al. is often considered as

the starting point of nonlinear optics [45].

In linear optics, a dielectric medium is described by an electric susceptibility tensor, χ,

which relates the electric field to the polarization density:

P = ϵ0χ
(1)Ẽ(t), (1.11)

where ϵ0 is the permittivity of free space. For isotropic materials, this susceptibility is just a

18



constant of proportionality. In the case of a nonlinear material, this expression becomes:

P = ϵ0(χ
(1)Ẽ(t) + χ(2)Ẽ(t)2 + χ(3)Ẽ(t)3 + ...). (1.12)

Equation 1.12 contains the fundamental mathematical relationship that underlies nonlinear opti-

cal phenomena. The reason that the polarization, which is the dipole moment per unit volume of

the material, gives insight into nonlinear optical effects is because a time-dependent polarization

can take the role of a source term in the wave equation formulation of Maxwell’s equations.

∇2Ẽ − n2

c2
∂2Ẽ

∂t2
=

1

ϵ0c2
∂2P̃NL

∂t2
. (1.13)

As a result, it is directly responsible for the generation of new frequency components.

In this section, we will begin with a basic introduction to some of the optical effects that

arise from Kerr nonlinear media. Following this, we will discuss how these effects are modeled

in the case of a nonlinear ring resonator. Generally, ϵ0 will be dropped for simplicity beyond this

point.

1.6 Basics of Kerr Nonlinear Media

Kerr nonlinear media, which are materials that have a nonzero third-order susceptibility,

host a variety of nonlinear optical effects. In order to gain some familiarity with these effects,

one can consider the response of the polarization to various input fields, following the treatment

19



in [46].

The simplest input field to consider is a monochromatic input field: Ẽ(t) = Ee−iωt + c.c.

We note here that polarization is taken to be linear, and the tensor nature of the susceptibility is

suppressed for simplicity.

The nonlinear contribution PNL can then be written:

PNL = χ(3)Ẽ(t)3 = χ(3)(Ee−iωt + c.c.)3 (1.14)

= χ(3)((E3e−3iωt + c.c.) + (3E2E∗e−iωt + c.c.)) (1.15)

= χ(3)((E3e−3iωt + c.c.) + 3E∗E(Ee−iωt + c.c.)). (1.16)

The first of the two terms here possesses the same form as the monochromatic input field,

except at frequency 3ω instead of ω. This term corresponds to the phenomenon of third-harmonic

generation, where three photons of frequency ω are destroyed and one photon of frequency 3ω is

created.

The second term is more relevant to this work, and can be understood as an a Kerr induced

intensity dependent refractive index due to E∗E = |E|2 being proportional to the intensity of the

field. Thus, the total polarization at frequency ω can be written:

PTotal(ω) = χ(1)Ẽ(t) + 3χ(3)|E|2Ẽ(t) = χeff Ẽ(t), (1.17)

20



where

χeff = χ(1) + 3χ(3)|E|2. (1.18)

Using the relationship χeff = n2 − 1 one can relate this change in susceptibility to a change

in refractive index. First, allowing for some intensity depenent refractive index term as follows:

n = n0 + n
′
2|E|2 one can write:

(n0 + n
′

2|E|2)2 = 1 + χ(1) + 3χ(3)|E|2, (1.19)

which, by matching terms of the same order in |E|2 yields:

n2
0 = 1 + χ(1), (1.20)

n
′

2 =
3χ(3)

2n0

. (1.21)

(1.22)

Typically, the nonlinear refractive index is expressed in terms of the intensity, I = 2n0ϵ0c|E|2,

such that n = n0 + n2I . We can relate the quantity n2 to the previously written n′
2 as:

n2 =
n′
2

2n0ϵ0c
=

3χ(3)

4n2
0ϵ0c

. (1.23)

This intensity dependent refractive index ultimately leads to a large host of effects, although

21



the one of most relevance to this work is self-phase modulation (SPM). Self-phase modulation

refers to the fact that the direct result of a nonzero n2 is the generation of a nonlinear phase shift

ϕNL as the field propagates. This phase shift is given as ϕNL = −n2Iω0L/c where L is the

propagation length.

More complex phenomena resulting from the Kerr nonlinearity can be seen when consid-

ering a more complex input field with three consituent frequencies:

Ẽ(t) = E1e
−iω1t + E2e

−iω2t + E3e
−iω3t + c.c. (1.24)

Expanding Ẽ(t)3 results in 44 distinct frequency components. Following the treatment in

Boyd [46], and defining P̃ (3)(t) =
∑

n P (ωn)e
−iωnt, they are:

P (ω1) = ϵ0χ
(3)(3E1E

∗
1 + 6E2E

∗
2 + 6E3E

∗
3)E1, (1.25)

P (ω2) = ϵ0χ
(3)(6E1E

∗
1 + 3E2E

∗
2 + 6E3E

∗
3)E2, (1.26)

P (ω3) = ϵ0χ
(3)(6E1E

∗
1 + 6E2E

∗
2 + 3E3E

∗
3)E3, (1.27)

P (3ω1) = ϵ0χ
(3)E3

1 , (1.28)

P (3ω2) = ϵ0χ
(3)E3

2 , (1.29)

P (3ω3) = ϵ0χ
(3)E3

3 , (1.30)

P (ω1 + ω2 + ω3) = 6ϵ0χ
(3)E1E2E3, (1.31)

22



P (ω1 + ω2 − ω3) = 6ϵ0χ
(3)E1E2E

∗
3 , (1.32)

P (ω1 + ω3 − ω2) = 6ϵ0χ
(3)E1E3E

∗
2 , (1.33)

P (ω2 + ω3 − ω1) = 6ϵ0χ
(3)E2E3E

∗
1 , (1.34)

P (2ω1 + ω2) = 3ϵ0χ
(3)E2

1E2, (1.35)

P (2ω1 + ω3) = 3ϵ0χ
(3)E2

1E3, (1.36)

P (2ω2 + ω1) = 3ϵ0χ
(3)E2

2E1, (1.37)

P (2ω2 + ω3) = 3ϵ0χ
(3)E2

2E3, (1.38)

P (2ω3 + ω1) = 3ϵ0χ
(3)E2

3E1, (1.39)

P (2ω3 + ω2) = 3ϵ0χ
(3)E2

3E2, (1.40)

P (2ω1 − ω2) = 3ϵ0χ
(3)E2

1E
∗
2 , (1.41)

P (2ω1 − ω3) = 3ϵ0χ
(3)E2

1E
∗
3 , (1.42)

P (2ω2 − ω1) = 3ϵ0χ
(3)E2

2E
∗
1 , (1.43)

P (2ω2 − ω3) = 3ϵ0χ
(3)E2

2E
∗
3 , (1.44)

P (2ω3 − ω1) = 3ϵ0χ
(3)E2

3E
∗
1 , (1.45)

P (2ω2 − ω0) = 3ϵ0χ
(3)E2

2E
∗
2 , (1.46)

with the remaining 22 terms being negative frequency equivalents. While overwhelming, several

of these terms should be easily recognizable. First, terms of the form P (ωn) can again be in-

terpreted as resulting in an intensity dependent refractive index as in the previous example. The

23



only difference here is that there is a more complex, multiple intensity dependence reflecting the

self-phase modulation but also cross-phase modulation (XPM) from the other frequencies ωm

where n ̸= m. Terms of the form P (3ωn) are also recognizable as the previously discussed third-

harmonic generation. Structurally similar terms at frequency ωi + ωj + ωk also appear, which

describe a phenomenon similar to third-harmonic generation known as triple-sum generation.

The remaining terms posses some form ∝ EiEjE
∗
ke

−i(ωi+ωj−ωk)te−i(βi+βj−βk)z where the

temporal and spatial phase has been added back in to emphasize that these terms, known collec-

tively as four-wave mixing (FWM), are “phase sensitive” terms. In particular, the FWM process

requires phase matching such that β ≈ βi + βj − βk. Generally, this term can be interpreted as

a process by which two photons are annihilated while two others are created, according to the

conservation of energy: ωi + ωj = ωk + ω.

1.7 Frequency Microcombs

In this section we will introduce the idea of the optical frequency comb and a few key

results from the field. Broadly speaking, an optical frequency comb is a source that consists

of equidistant frequency components [47, 48]. Applications of these sources range from optical

clocks [49,50] and precision spectroscopy [51,52] to distance measurement [53] and microwave

signal synthesis [54, 55], among many others. Pioneering work developing the first optical fre-

quency combs led to Theodor Hänsch and John “Jan” Hall sharing half of the 2005 Nobel Prize

in Physics [56].

24



Much of this work was originally done with femtosecond laser systems requiring a large

footprint with relatively low repetition rates. However, in work done by Del’Haye et al., it was

shown that combs could also be produced from CW sources through the use of nonlinear opti-

cal cavities [57]. These early experiments, typically done in microscale silica resonators with

extremely high Q factors, constituted enormous advances in the field but still suffered from dis-

persion limited bandwidth and limited coherence. These difficulties were largely overcome with

the discovery of the dissipative Kerr soliton regime by Herr et al. [58] which allowed for gener-

ation of broadband and highly coherent on-chip frequency combs. Since this result, the field of

optical frequency microcombs has taken off with applications in spectroscopy [59,60], precision

timekeeping, on-chip signal synthesis, ranging and detection [61, 62], optical neural networks,

and more.

In the following section, we will introduce how nonlinear effects are modeled in a ring

resonator and, by extension, how optical frequency microcombs are formed. For further reading

on the field of optical frequency microcombs, see References [63, 64].

1.8 Modeling the Nonlinear Ring Resonator - the Lugiato-Lefever Equation

Modeling of driven Kerr nonlinear ring resonators with dispersion and dissipation was

historically carried out in the frequency domain due the natural fit of coupled mode equations

to optical parametric oscillators where only a few lines are generated [65]. The essence of this

approach is to develop a set of equations, each corresponding to one mode, with a series of

25



coupling terms derived from nonlinear four-wave mixing. This formalism was then extended to

Kerr frequency combs with many coupled modes. A derivation of this model/approach can be

found in Reference [66].

While the frequency domain picture is a useful one for modeling of Kerr nonlinear ring

resonators, modeling in the temporal domain has become more widespread in recent years. In

1987, Lugiato and Lefever developed a model for a continuous-wave driven mirror cavity with

Kerr nonlinearity and diffraction that exhibited spatially-dissipative states [67]. This model was

extended in 1992 into the temporal domain by including dispersion as opposed to diffraction,

which in turn exhibited temporally localized states [68].

The resulting Lugiato-Lefever Equation (LLE) takes the following form:

τr
∂E(t, τ)

∂t
= [−α− iδ0 + i

β2

2

∂2

∂τ 2
+ iγL|E(t, τ)|2]E(t, τ) + rEin. (1.47)

Here, E(t, τ) is the cavity field envelope at spatial coordinate z = 0, and is a function of two

time variables. The first, τ , is standard time. The second, known as “slow” or “retarded” time

is defined as t = τ − z/vg where vg is the group velocity of the light. τR is the round trip time

of the cavity. α is the linear cavity loss, δ0 is the phase cavity detuning of the center frequency

of the E field spectrum with no intensity in the cavity, β2 is the group velocity dispersion, γ is

the effective nonlinearity or nonlinear coefficient, L is the cavity length, and r is the reflection

coefficient of the composite bus waveguide and ring system, equivalent to t in Equation 1.3. Ein

is the input field. We note here that dispersion has been truncated to second-order.

26



While there have been works studying analytical descriptions of solutions of the LLE [58],

the most straightforward way to get familiar with the dynamics of the LLE is to solve it numer-

ically. Here we will present some representative solutions of the LLE, generated using the free

and open source Python library “pyLLE” [69]. Here, the simulated ring has a radius of 23 µm,

intrinsic and extrinsic quality factors of 106, and an effective nonlinearity of γ = 3.2 1/Wm. The

dispersion profile used is an anomalous dispersion profile in the region of the pump but notably

also includes higher order contributions. This profile is available from the pyLLE documentation.

In order to achieve a more physical result, the dispersion is not truncated to second order.

Using a pump power of 130 mW and sweeping blue to red-detuned through a resonance,

the following plot of intracavity power (with the pump removed) is obtained:

As can be seen in Figure 1.9, there are several distinct regions. Starting blue detuned,

intracavity power undergoes a rapid increase as the primary comb is formed, labeled as region

A. Quickly, the intracavity power becomes strongly dependent on detuning and exhibits extreme

variation as the modulation instability (MI) or chaotic regime is accessed, labeled as region B.

Finally, following an abrupt drop in intracavity power, a smooth plateau is observed. This plateau

is known as the soliton-step and is labeled region C.

The corresponding spectra to each of the distinct regions are displayed in Figure 1.10. In

panel A, the spectrum of the primary comb is characterized by a small number of distinct peaks at

a spacing corresponding to some integer multiple of the resonator FSR. Within the MI region, the

spectrum exhibits a disordered structure and lack of smooth envelope. Following the soliton-step,

the spectrum changes dramatically, exhibiting a smooth envelope with a hyperbolic secant form.

27



30

25

20

15

10

5

0

In
tra

-c
av

ity
 P

ow
er

 (m
W

)

Pump Detuning (GHz)
1.0 -6.5-0.5 -2.0 -3.5 -5.0

A

B

C

Nonlinear Ring Resonator Intra-Cavity Power vs Detuning

Figure 1.9: Intracavity power versus detuning, calculated from the Lugiato-Lefever Equation.
The region A corresponds to the formation of a primary comb, B corresponds to modulation
instability, and C corresponds to the soliton-step. The pump contribution to intracavity power is
removed.

28



A B C

Po
w

er
 (d

bm
)

20

0

-20

-40

-60

-80

-100
200 250 300 350 400 200 250 300 350 400 200 250 300 350 400

Frequency (THz)

Nonlinear Ring Resonator Field Spectra vs Pump Detuning

Figure 1.10: Field spectra from the Lugiato-Lefever Equation with distinct values of pump detun-
ing. A) Primary comb spectrum. B) Modulation instability spectrum. C) Kerr soliton spectrum.

In this case, dispersive waves are also generated due to higher-order dispersion, leading to the

octave-spanning bandwidth of the comb [70].

While not obvious from the spectrum alone, the Kerr soliton state in C is a coherent state

characterized by phase synchronization of the comb lines, low phase noise, and a stable, local-

ized structure. In particular, optical solitons exhibit a solitary pulse that maintains its shape and

energy in spite of the presence of losses and dispersion. This arises from a balancing of the Kerr

parametric gain (from FWM) with the linear losses of the system, as well as a balancing of the

intensity dependent phase shift (from SPM and XPM) and dispersion [66].

The corresponding temporal field profiles are shown in Figure 1.11.

Panel A of Figure 1.11 shows the waveform of the primary comb. In particular, the primary

comb resembles a modulated CW solution of the LLE. As subcomb formation generates more

teeth in the spectrum, this waveform can approach what are known as “Turing rolls”, which can

29



Po
w

er
 (d

bm
)

1.0

0.8

0.6

0.4

0.2

0.0
-0.4 -0.2 0.0 0.2 0.4

Time (ps)
-0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4

A B C
Nonlinear Ring Resonator Field Time Domain vs Pump Detuning

Figure 1.11: Time domain field profiles from the Lugiato-Lefever Equation with distinct values
of pump detuning. A) Primary comb profile. B) Modulation instability profile. C) Kerr soliton
profile.

correspond to a low phase noise state. Panel B shows the noisy, unstructured waveform of an

example MI state. Panel C shows the expected solitary pulse corresponding to the dissipative

soliton state.

The LLE and coupled mode formalism was extended to the AQH lattice geometry by Mit-

tal et al. in work titled Topological Frequency Combs and Nested Temporal Solitons [38]. To

summarize, it was shown that topological lattices, in the presence of nonlinearity and dispersion,

could host coherent optical frequency combs including nested temporal solitons and Turing rolls.

In essence, the super ring resonator formed by the edge states could host nonlinear states qual-

itatively similar to the single-ring case discussed above despite the significantly more complex

system. One key difference between the lattice case and the single-ring case, however, was that

the lattice supports two distinct timescales. First, there is the single-ring round-trip time which

sets the single-ring FSR. However, due to the circulating nature of the edge states, there is also

30



the full lattice round-trip time which depends on J . This corresponds to a second, much smaller

frequency scale. As a result, the topological frequency combs predicted numerically in [38] ex-

hibited a two-fold spectral structure with both a large and small FSR.

A second key difference was particular to the nested temporal soliton state. It was shown

that the mode conversion efficiency from a CW source could be as high as 50%, far outperforming

the theoretically limited mode conversion efficiency for a single-ring, single soliton state [71,72].

While practical difficulties abound for experimental realization of a topological nested soliton

state, this result is particularly encouraging for applications that may require efficient operation,

as the limited mode conversion efficiency in the single-ring case, approximately 5%, is funda-

mental as opposed to practical.

Additionally, it was shown that the topological soliton states also inherit the topological

protection of their linear counterparts. In other words, the states are robust to certain defects in

the lattice.

31



Chapter 2: Design of Topological Lattices for Nonlinear Photonics

In this section, we will outline a series of considerations that must be accounted for when

designing topological ring resonator lattices intended for use in the regime of nonlinear optics.

2.1 Material Selection

Much of the work to date in topological lattices of ring resonators has been carried out in sil-

icon photonic systems for a plethora of reasons. The main reason is that technologies supporting

silicon photonics are mature and readily available. Silicon-on-insulator platforms using CMOS-

compatible fabrication processes are currently easily accessible through advanced foundries. Sil-

icon also has a particularly high refractive index, leading to favorable waveguiding properties.

Absorption at telecommunications relevant wavelengths (around 1550 nm) is generally quite low

because the photon energies are less than the silicon bandgap. However, when designing lattices

for nonlinear photonics, the situation changes dramatically.

In particular, while silicon has a reasonable Kerr nonlinearity due to the non-zero third

order susceptibility, it is also affected by the nonlinear process known as two-photon absorption

(TPA). TPA, first discussed in the graduate dissertation of Maria Göeppert-Mayer [73], comes

32



from the imaginary part of the third order susceptibility (in contrast to those third order nonlinear

effects discussed in Chapter 1). The presence of this effect leads to generation of free carriers

in the material as well as heating, both which result in modifications of the material refractive

index. These effects, generally speaking, are considered detrimental for efficient nonlinear optics.

However, there are several applications for TPA including spectroscopy, optical data storage, and

microfabrication [74–76].

In order to minimize nonlinear absorption but retain as many favorable properties as pos-

sible, using silicon nitride (SiN) is a common choice in nonlinear photonics due to its signifi-

cantly larger bandgap. In particular, SiN has only a slightly lower refractive index (n = 2.00)

and nonlinearity (n2 = 2.4 × 10−19W−1m2) [77] and still benefits from commercially available

CMOS-compatible fabrication and low absorption at telecommunication wavelengths.

2.2 Waveguide Dimensions and Mode Properties

When designing topological photonic lattices for operation in the linear regime, choosing

a waveguide cross-section is a fairly straightforward process. The main consideration is ensuring

that the waveguide will operate in the single mode regime. The primary reason for this is that

designing the hopping phases discussed in Chapter 1 relies directly on the effective refractive

index of the mode, so the lattice can only be designed with one mode in mind for a particular

Hamiltonian. In particular, the propagation phase can be written ϕ = βz where β = neffω/c and

z is a spatial coordinate or propagation distance. In the second expression, neff is the effective

33



refractive index of the waveguide mode in question (which generally depends on frequency, ω)

and c is the speed of light in a vacuum. As a result, this effective refractive index is necessary to

calculate the path length differential for detuning the link-rings as well as the vertical link-ring

shift for a lattice of integer quantum Hall design.

The effective refractive index of a mode depends on both the material and geometry of

the waveguide, including both the core and cladding, and the field distribution of that particular

mode. Additionally, any curvature of the waveguide can result in a shift in effective refractive

index, although that will not be considered here. As a result, the most straightforward way

to access these values is through numerical simulation. EigenMode Expansion (EME) methods,

such as the one available through the commercial software Lumerical, are particularly well suited

for this [78]. Figure 2.1 shows a series of effective refractive indices for the fundamental TE mode

of a SiN waveguide in a silicon dioxide (SiO2) cladding at a wavelength of 1550 nm, generated

with Lumerical EME.

Generally, the effective refractive indices shown in Figure 2.1 do vary with the width of the

waveguide as well as the height, but fall largely in the range 1.6-1.8 for the geometries tested.

The inset shows a sample normalized mode profile for a 1200 by 800 nm waveguide. Calculating

the required path length difference for the link-rings with effective refractive indices in hand is

straightforward using the previously stated expression for ϕ and setting it to π. The extra length

of the link-rings is given as:

η =
c

ωneff

π, (2.1)

34



2.0

1.9

1.8

1.7

1.6
1.81.0 1.2 1.4 1.6

n ef
f

400 nm
600 nm
800 nm

Core Height

neff vs Waveguide Cross-Section

Core Width (     )

m2

m

Figure 2.1: Effective refractive indices at a wavelength of 1550 nm for various SiN waveguide
cross sections are shown.

35



which, when using 1.78 for neff and 2π × 194.3 THz for ω, gives η ≈ 440 nm for the 1200

by 800 nm waveguide cross-section. Similarly, the shift ξ, used in the integer quantum Hall

lattices as depicted in Figure 1.4 is found to be 110 nm for a ϕ of π/2. Notably, this is because

light traversing either the long or short path will see a path length difference of twice the shift.

As an aside, propagation losses can also be influenced by the waveguide dimensions, although

this is generally a secondary concern when interested in linear physics since it can be easily

counteracted with proportionally increased laser power.

In the nonlinear case, a second critical consideration comes into play, which is the dis-

persion of the waveguide. Dispersion in a resonator comes from two sources. The first is the

material itself, and the second is the waveguide geometry, or cross-section. For our purposes,

we are only interested in the dispersion up to second order, also known as the group-velocity

dispersion (GVD). The propagation phase constant introduced above can be expanded around ω0

as follows:

β =
n(ω)ω

c
= β0 + β1(ω − ω0) + β2

(ω − ω0)
2

2
+ ... (2.2)

where βn = ∂nβ/∂ωn|ω0 . The first two terms in this expansion can be directly related to the phase

(β0 = 1/vϕ) and group (β1 = 1/vg) velocities of the mode. The third term, the GVD, can be

written as β2 =
∂
∂ω

1
vg

. This term is responsible for the common effect of pulse broadening, where

different frequency components of a pulse spread out as a result of differing velocities. When β2

is positive, the system is described as having “normal” dispersion, where higher frequencies travel

36



slower. The case we are most interested in however, is the case of “anomalous” dispersion, where

higher frequencies travel faster. Anomalous dispersion is required for Kerr comb generation

specifically because it serves to detune the modes that will host the comb sidebands with the

opposite sign of the detuning from pump induced cross-phase modulation. In other words, the

intensity dependent nonlinear resonance shift in Kerr media works to compensate for anomalous

dispersion.

In order to determine a suitable waveguide geometry, accurate values for the refractive

index of the material (in this case, SiN) are crucial. When using experimentally measured values,

improper fitting can result in spurious higher-order derivative terms which can cause large errors

in the GVD (which is dependent on the second frequency derivative of the refractive index) as

well as higher-order dispersion terms. As a result, it is advisable to use the Sellmeier equation

for the material whenever available. For SiN, the Sellmeier equation is written:

n2
SiN = 1 +

3.0249λ2

λ2 − 135.34062
+

40314λ2

λ2 − 12398422
, (2.3)

as derived in Reference [79]. This can be used to create a custom, analytical material profile

in commercial finite-difference time-domain (FDTD) solvers such as Lumerical, which can in

turn be used to calculate the dispersion as a function of wavelength or frequency. Numerically

calculated dispersion values in SiN for a series of waveguide cross-sections are plotted in Figure

2.2.

We note that here, the definition of dispersion is using convention from fiber optics, and

37



Waveguide Dispersion vs Wavelength

Waveguide
Cross-Section (nm)

1000x400
1200x400
1400x400
1000x600
1200x600
1400x600
1000x800
1200x800
1400x800

1400 1500 1600 1700165015501450
Wavelength (nm)

-1000

0

-800

-600

-400

-200

D
is

pe
rs

io
n 

(p
s/

nm
/k

m
)

Anomalous
Regime}

Figure 2.2: Various numerically calculated dispersion profiles for SiN waveguides with width
1000 - 1400 nm and thickness 400 - 800 nm.

38



is slightly different from the β2 given above. In particular, dispersion in Figure 2.2 is defined as

the group delay dispersion (GDD) written Dλ = ∂τg
∂λ

where τg = L/vg is the group delay for a

given length, L. To convert between the two, one can write Dλ

L
= −2πc

λ2 β2. Most importantly,

the sign is different, which results in the anomalous dispersion regime in Figure 2.2 requiring

dispersion to be positive as opposed to negative in the case of β2. As is apparent, the condition for

entering anomalous dispersion in a rectangular waveguide of SiN is fairly strict. Of the above nine

profiles, only the widest (1200 - 1400 nm) and thickest (800 nm) waveguides support anomalous

dispersion, with all other tested parameters falling well below in the normal dispersion regime.

In order to determine an optimal waveguide cross-section, we can further examine the dispersion

as a function of waveguide width for the commercially available SiN thickness of 800 nm. Figure

2.3 shows numerical values for this at a wavelength of 1550 nm. Dispersion increases generally

as a function of waveguide width, and crosses into the anomalous dispersion regime around a

width of 1100 nm.

Notably, due to the 800 nm thickness of the waveguide, all of these cross-sections support

transverse magnetic (TM) as well as transverse electric (TE) modes. All reported dispersion

parameters and effective refractive indices here refer to the fundamental TE mode. Second order

TE modes are supported above widths of approxiamtely 1050 nm. While the 1000 nm wide

waveguide does fall into the single-mode regime (for each polarization), it also falls back into

the regime of normal dispersion. According to simulation, zero dispersion at 1550 nm and 800

nm SiN thickness is achieved around a waveguide width of 1065 nm, where second order modes

are supported. As a result, in order to design topological photonic lattices in SiN for Kerr comb

39



60

-20

0

20

40

D
is

pe
rs

io
n 

(p
s/

nm
/k

m
)

Dispersion at 1550 nm for 800 nm Thick SiN Waveguide Vs Width

1000 1400130012001100
Waveguide Width (nm)

Anomalous

Normal

Single
TE

Mode

Figure 2.3: Waveguide dispersion as a function of waveguide width for SiN thickness 800 nm
and wavelength 1550 nm. Red dotted line indicates the crossover from normal (negative) to
anomalous (positive) dispersion.

40



generation, it is required to tolerate the presence of two polarizations and at least one higher-order

mode.

From a practical perspective, a second polarization is quite easily accommodated experi-

mentally. While the lattice can only be designed with one mode in mind as previously stated, the

coupling to one polarization can be relatively easily achieved with standard polarization optics.

Additionally, a wider waveguide will preferentially guide the TE polarization, or equivalently, the

TE polarization of a wider waveguide will have a larger effective refractive index that should al-

low an experimenter to distinguish the two. However, as the waveguide cross-section approaches

symmetry (800 by 800 nm), the difference in the effective refractive indices between the TE and

TM modes will approach zero, which may make distinguishing the two experimentally more chal-

lenging. Notably, controlling the input polarization is only sufficient when there is no mechanism

to couple the two polarizations within the device. Higher-order modes, in contrast, are experi-

mentally far more challenging to accommodate. As will be discussed in detail in the following

section, simply ensuring that the input laser beam is in the appropriate mode is insufficient, as it

is relatively easy to couple the fundamental and higher-order modes within these devices.

2.3 Waveguide Bend Analysis

Now that it is established that some multi-mode behavior is difficult to avoid when op-

erating in the anomalous dispersion regime in SiN waveguides, it is important to explore the

implications of this for both straight and bent waveguides. Although the term “ring” was used

41



to describe the building blocks of the topological lattices introduced in Chapter 1, in reality the

resonators have a racetrack geometry. That is, rather than a circular geometry, they are composed

of distinct sections of straight and bent waveguides, illustrated schematically in Figure 2.4A-B.

The reason behind this requirement can be observed in panels C and D, where coupling regions

are shown between two ring resonators and two racetrack resonators, respectively. In the case

of the circular ring resonator, the coupling gap changes along the propagation direction of the

waveguide due to the curvature of the ring. As a result, the coupling strength depends on both

the gap between the resonators as well as the size of the ring. Practically, this makes achieving

strongly coupled resonators challenging. In the case of the racetrack resonator however, a con-

stant coupling gap can be maintained within a region of variable length. Altering the length of

the straight regions, known as the coupling length, can modify the coupling strength significantly

and only slightly alter the size of the resonator. Furthermore, this change in size could be com-

pensated by a change in the radius of the bent regions. The extra control over coupling strength

offered by the racetrack geometry is directly related to the ability to design a topological lattice

of resonators with a particular hopping strength, J .

However, the racetrack geometry does introduce additional complexity, which becomes

particularly important in the case of large waveguide cross-sections that support multiple modes.

When a waveguide is curved, the electric field profiles of the modes change depending on the

radius of curvature. When interfacing a straight region (no curvature) and a bent region (constant

curvature) there is a discontinuity in the curvature profile along the direction of propagation. For

certain waveguide cross-sections and radii of curvature, this can result in a spatial mismatch in

42



A B

C D

Figure 2.4: A) Depiction of a circular ring resonator. B) Depiction of a racetrack resonator com-
prised of straight and bent waveguide segments. C) Depiction of the coupling region between two
circular ring resonators. D) Depiction of the coupling region between two racetrack resonators.

43



Overlap Integral of Straight and Bent Fundamental Modes vs Width

1.00

0.90

0.98

0.96

0.94

0.92

1000 1200 1400 1600 1800 2000

R = 20

A

C

Waveguide Width (nm)

B

Width
800 nm SiN

SiO2

m

O
ve

rla
p 

In
te

gr
al

Figure 2.5: A) Depiction of the waveguide cross-section, embedded in a silicon dioxide cladding.
B) Depiction of a straight-bent waveguide interface with a 20 µm radius of curvature. C) The
overlap integral between the fundamental modes of a straight and a bent SiN waveguide with a
20 µm radius of curvature, calculated at a wavelength of 1550 nm.

the fundamental modes of each segment. This phenomenon is illustrated in Figure 2.5. Panel A

schematically depicts the waveguide cross-section, where thickness is fixed at 800 nm. Panel B

schematically depicts the straight and bent waveguide junction, where the bent region has a radius

of curvature of 20 µm. Panel C shows the overlap integral between the fundamental modes of

the bent and straight regions as as a function of waveguide width.

The overlap integral, which can be used as a proxy for the coupling efficiency of the

44



straight-bent interface, falls off steeply as the waveguide width increases, indicating significant

coupling to higher-order modes. While a reduction of a few percent may not seem like a large

effect, it is important to note that this is the effect of a single straight-bent interface. Light circu-

lating in a racetrack resonator will pass through such an interface eight times in a single roundtrip,

and may complete many such roundtrips. As a result, coupling to higher-order modes can occur

readily in racetrack resonators and, by extension, topological photonic lattices.

This effect can be further understood by examining the mode profiles themselves. In Figure

2.6, the profiles of the first and second order TE modes of 1800 by 800 nm straight and bent

SiN waveguides are plotted along with values of their respective overlap integrals. Bending the

waveguide with a 20 µm radius of curvature results in a shifting of the mode, reducing the overlap

between straight and bent fundamental modes and increasing the overlap between fundamental

and higher-order modes.

Unsurprisingly, this mode mixing also depends strongly on the radius of curvature of the

bent region. Figure 2.7 shows the relationship of the overlap integral between fundamental modes

of straight and bent waveguides of increasing radii of curvature. Again, the cross-section of the

waveguide is 1800 nm wide by 800 nm thick.

The complexity that results from this phenomenon can be explored with more involved

structures by utilizing three-dimensional (3D) FDTD simulation, which solves Maxwell’s equa-

tions numerically [80]. Notably, some common approximation methods used to reduce the com-

putational requirements of solving Maxwell’s equations in complex structures fail to capture this

effect. One such example is Lumerical “2.5D varFDTD”, which collapses a 3D geometry into a

45



Fundamental TE Second Order TE

Straight

Bent

1000 nm

93.2% 89.4%

6.7%

6.7%

0.5

0.0

1.0

Figure 2.6: Mode profiles for straight and bent 1800 x 800 nm SiN waveguides embedded in
SiO2. For the bent waveguide, the radius of curvature is 20 µm. All profiles are calculated for a
wavelength of 1550 nm.

46



10 20 30 40 50
Bending Radius (     )m

1.00

0.75

0.95

0.90

0.85

0.80

O
ve

rla
p 

In
te

gr
al

Overlap Integral of Straight and Bent Fundamental Modes vs Bending Radius

1800 nm
800 nm SiN

SiO2

Figure 2.7: Overlap integral of the fundamental TE modes of a straight-bent interface as a func-
tion of bending radius. An inset shows a schematic of the straight waveguide cross-section.

2D set of effective refractive indices to allow for faster simulation [81]. However, this technique

assumes very little coupling between modes, and as a result does not accurately reproduce full 3D

results in this context. In order to understand the impact of the nonzero overlap integral between

straight waveguide fundamental TE modes and bent waveguide higher-order modes, we can sim-

ulate a slightly larger segment of a racetrack resonator. In particular, the inclusion of a second

straight region, as depicted in Figure 2.8A, is sufficient. As before, the waveguide is made from

800 nm SiN embedded in silicon dioxide.

Figure 2.8B shows the results from exciting the fundamental mode of the first straight

region and plotting total transmission through the second straight region as a function of bending

radius. This simulation captures bending losses in addition to other losses, such as propagation

47



1.00

0.90

0.98

0.96

0.94

0.92

To
ta

l T
ra

ns
m

is
si

on

Total Transmission vs Bending Radius

1.2        Width
1.8        Width

m
m

R

{Width
Input

Measurement

A B

R (     )m
10 20 30 40

L

L

Figure 2.8: A) Schematic depictiion of a straight-bent-straight waveguide with straight length
L and bending radius R. B) Simulated total transmission as a function of bending radius for
waveguides 1.2 and 1.8 µm wide. Transmission is averaged over 1400 - 1700 nm wavelengths
and normalized to the input transmission.

loss, that are expected to be less significant. Transmission here is average between wavelengths of

1400 - 1700 nm and is normalized to the input power. For the two waveguide widths shown, 1.8

and 1.2 µm, transmission increases dramatically as bending radius increases before effectively

reaching a plateau. In other words, bending loss is high for small radii of curvature, but is quickly

reduced. Notably, the wider of the two waveguides is less susceptible to bending losses, and may

seem favorable as a result. However, the picture immediately becomes more complicated when

we decompose the total transmission into constituent modes. Figure 2.9 displays data from the

same simulation, but plots the transmission of only the fundamental mode (now normalized to

the total transmission through the second straight region) versus the radius of curvature R.

Figure 2.9B shows the same general trend as the bending loss, transmission in the funda-

mental mode tends to increase as the bending radius increases. However, the transmission for

48



1.00

0.40

0.90

0.80

0.70

0.60

Fu
nd

am
en

ta
l M

od
e 

Tr
an

sm
is

si
on

Fundamental Mode Transmission vs Bending Radius

1.2        Width
1.8        Width

m
m

R

{Width
Input

Measurement

A B

R (     )m
10 20 30 40

L

L

0.50

Figure 2.9: A) Schematic depictiion of a straight-bent-straight waveguide with straight length L
and bending radius R. B) Simulated fundamental mode transmission as a function of bending
radius for waveguides 1.2 and 1.8 µm wide. Transmission is averaged over 1400 - 1700 nm
wavelengths and normalized to the input transmission. Strong oscillations in transmission can be
observed.

waveguides of both widths are strongly modulated, exhibiting oscillatory behavior.

The reason for this oscillation is that at the first straight-bent interface, the fundamental

mode of the straight waveguide excites both the fundamental and higher-order modes of the bent

waveguide, which then propagate through the bend with different phase velocities. This results

in a beat note phenomenon, where the combination of bent waveguide modes can arrive at the

second interface in or out of phase depending on the propagation length of the bend (which is

determined by the bending radius). As a result, for certain bending radii, the coupling into the

straight waveguide at the second interface can be almost entirely in the fundamental mode, even

if significant mode mixing occurred at the first interface. This phenomenon is explored in detail

in Reference [82].

49



0

-25

-5

-10

-15

-20

-30
0-25 -5-10-15-20-30 5

R1 R2

X (     )m

Y 
(  

   
)

m

Input

1800 nm
800 nm SiN

SiO2

0.5

0.0

1.0

Figure 2.10: Electric field intensity profiles are plotted for straight-bent-straight waveguides with
bending radii of 14 µm and 20 µm with a waveguide cross-section of 800 by 1800 nm. Despite
having a larger radius of curvature, the profile of the 20 µm bend waveguide exhibits significantly
more distortion in the second straight region.

Notably, despite having higher total transmission, the wider waveguide is significantly more

susceptible to these mode mixing effects, in agreement with the trend observed in Figure 2.5.

Despite the improvement for the thinner, 1.2 µm wide waveguide, the mode mixing in only three

waveguide segments is still significant.

Without analysis such as modal decomposition, this effect can still be observed in the elec-

tric field. Figure 2.10 shows the normalized electric field intensity through two different straight-

50



bent-straight SiN waveguides with cross-section 1800 by 800 nm at a wavelength near 1550 nm.

Each waveguide has a different radius of curvature, specifically, R1 = 14µm and R2 = 20µm.

Despite the second waveguide having a larger bending radius, which suggests that mode mixing

will be lower according to Figure 2.7, the apparent distortion of the field in the output waveguide

is significantly greater. This is a direct result of the beating phenomenon illustrated in Figure

2.9, where a 14 µm radius corresponds to a well matched path length where the fundamental and

higher-order modes return in-phase to the straight segment, resulting in very high transmission

in the fundamental mode. While the initial excitation of higher-order modes in the case of the 20

µm radius is lower, the result is increased mode mixing due to the different propagation length

of the bent region.

In principle, one could take advantage of the fact that the bending radius can be matched to

high transmission in the fundamental mode even in the presence of mode mixing, as in the case

of R = 14µm here. However, this would require a high degree of accuracy in the simulation

of the relative phase velocities of the fundamental and higher-order modes. Even slight error in

this could result in higher-order mode excitation in the straight waveguide region, which would

cascade as the light passes through hundreds or even thousands of straight-bent interfaces in

a topological lattice. As a result, it is clear that an informed choice of waveguide width and

bending radius may not be sufficient to avoid mode mixing in multi-mode topological lattices,

and a further modification of the geometry is required.

One such modification is to do away with the constant radius of curvature in bent regions

and introduce a curvature profile that changes as a function of position along the waveguide.

51



Reff

Rmin

Position Along Bend

1/Rmin

C
ur

va
tu

re

1/Reff

A B

Round
Euler

Figure 2.11: A) Curvature along the path of a round and Euler bend, showing a constant curvature
for the round bend and linearly changing curvature for the Euler bend. B) Schematic depiction
of a round and Euler bend with identical effective radii, Reff .

One common implementation, known as the Euler or clothoid bend, is characterized by a linear

increase in curvature from 0 to 1/Rmin, where Rmin is some minimum radius of curvature that

will be smaller than the effective radius of the bend, Reff [83]. Figure 2.11A shows the curvature

profiles of Euler and round bends, while B schematically shows the respective geometries of each

type of bend.

The potential advantage of the Euler bend is immediately apparent, as the curvature profile

starts from zero, thus avoiding any discontinuity at the interface of a straight bend, which by

definition also has zero curvature.

However, the Euler bend pays a price compared to an equivalent round bend (where Reff =

RRound. In particular, the maximum curvature is larger than that of the round bend. Generally

52



speaking, bending losses increase with curvature, so that must also be taken into account.

To evaluate the performance of the Euler bends, we perform the same analysis as in Figures

2.8 and 2.9 and display the results in Figure 2.12. In particular, Figure 2.12A and B show total

and fundamental mode transmissions for straight-bent-straight waveguides using Euler bends,

normalized to the input field and total transmission respectively. C and D show the same, with

results using round bends shown on the same plot for comparison.

Panels A and B show qualitatively similar results as seen previously, bending losses de-

crease with larger radii, and wider waveguides still suffer more from the mode mixing phe-

nomenon. However, panels C and D show that while bending losses did not change dramatically

for most radii between round and Euler bends, the magnitude of the mode mixing has been dra-

matically reduced. In particular, although the 1.8 µm wide Euler bend waveguide still exhibits

some mode mixing, it does appear to slightly outperform the 1.2 µm round bend. Most impor-

tantly, the 1.2 µm wide waveguide exhibits negligible mode mixing beyond effective radii of

approximately 15 µm.

At this point, a more complete picture is emerging. As explored previously, SiN thickness

of about 800 nm with waveguide widths of at least 1200 nm are required to enter into the anoma-

lous dispersion regime, with dispersion increasing as width is increased further. However, such

large waveguide cross-sections, when paired with sharp bends, are highly susceptible to detri-

mental mode mixing. While larger bending radii and more narrow waveguides can mitigate these

effects, modifying the curvature profile of the bent regions is necessary to eliminate mode mixing

and maintain anomalous dispersion and a reasonable system size.

53



1.00

0.95

0.99

0.98

0.97

0.96

Tr
an

sm
is

si
on

1.00

0.90

0.98

0.96

0.94

0.92

10 20 30 40 10 20 30 40

1.2        Width, Euler
1.8        Width, Euler

m
m

1.2        Width, Euler
1.8        Width, Euler

m
m

Transmission vs Effective Radius for Straight-Bent-Straight Waveguides

Total Transmission Fundamental Mode Transmission
A B

1.00

0.95

0.99

0.98

0.97

0.96

10 20 30 40

1.00

0.40

0.90

0.80

0.70

0.50

Reff (     )m
10 20 30 40

0.60

C D

1.2        Width, Euler
1.8        Width, Euler

m
m

1.2        Width, Round
1.8        Width, Round

m
m

1.2        Width, Euler
1.8        Width, Euler

m
m

1.2        Width, Round
1.8        Width, Round

m
m

Figure 2.12: Transmission through various straight-bent-straight waveguides versus effective ra-
dius. All transmission is averaged over wavelengths of 1400 - 1700 nm. A) Total transmission
for Euler bends, normalized to input transmission. B) Fundamental mode transmission for Euler
bends, normalized to output transmission. C) Total transmission for both Euler and round bends,
normalized to input transmission. D) Fundamental mode transmission for both Euler and round
bends, normalized to output transmission.

54



2.4 Splitting Ratio and Directional Couplers

Perhaps the most important parameter to consider when designing topological photonic

lattices is J , the hopping strength, which is a measure of the coupling between adjacent site-

rings. From coupled mode theory, J or κex can be expressed:

J =
k2

2

vg
Lring

, (2.4)

where k is the coupling constant, vg is the group velocity of the relevant mode, and Lring is the

circumference of the ring [31]. k2 here gives the dropped transmission ratio through the coupling

region, assuming negligible coupling losses. We note here that k is the same k used in Equation

1.3. This relationship is depicted in Figure 2.13, where four Euler bend waveguides and two

coupling waveguides are joined to create a directional coupler.

The directional coupler has been extensively studied both theoretically and experimentally

and is a standard component in modern photonics. For a symmetric coupler like the one depicted

in Figure 2.13, the dropped power depends sinusoidally on both the coupling constant k and the

coupling length L [84].

PDrop = sin2(k̄L) = k2, (2.5)

when normalized to input power and assuming that the two coupled modes propagate with the

same phase constant. Here, k̄ is an alternative but common definition of the coupling constant that

55



Input

DropThrough

L

Gap

{

Figure 2.13: A schematic of a directional coupler made of four Euler bends and two straight
coupling regions. Input, through, and drop ports are indicated as well as the coupling gap and
coupling length.

56



Input

Drop

L = 12 

Gap

{

Reff = 20

Gap (nm)
200 250 300 350 400 450 500

0.8

0.2

0.4

Transmission vs Coupling Gap

m
m

Tr
an

sm
is

si
on

0.6

Through Drop
Through

A B

Figure 2.14: A) Schematic depicting an Euler bend directional coupler with an effective radius
of 20 µm and a coupling length of 12 µm. B) Transmission in the drop and through ports
of the directional coupler as a function of the coupling gap, normalized to input transmission.
Transmission is averaged over the wavelength band 1400 - 1700 nm.

highlights the sinusoidal dependence of the dropped power on the coupling length. Generally, k

will be used, following the definition and transfer matrix treatment in Reference [31].

In practice, numerical simulation is the easiest way to get access to the relevant quantities,

in particular due to the reliance on evanescent mode overlap. The strong dependence on coupling

gap for the above structure, again simulated with full 3D FDTD, is shown in Figure 2.14.

Here, the transmission is averaged over the wavelength band 1400 - 1700 nm. The effective

radius of the bends is 20 µm while the coupling length is 12 µm. The SiN waveguide cross

section is again 1200 by 800 nm. As the gap is increased, the transmission in the drop port

falls off exponentially due to the source of the coupling being overlap with the evanescent tail

of the waveguide mode. As previously, transmission is normalized to the input power. Notably,

57



0.7

0.4

0.5

Tr
an

sm
is

si
on 0.6

0.3

10 12 14 16 18 20
Coupling Length (     )m

Drop
Through

Transmission vs Coupling Length

Figure 2.15: Transmission in the drop and through ports of the directional coupler as a function
of the coupling length, normalized to input transmission. The effective radius is fixed at 20 µm
and the coupling gap is 300 nm. Transmission is averaged over the wavelength band 1400 - 1700
nm.

controlling the gap gives access to both strong and weak couplings with gap sizes achievable

with modern fabrication techniques. As a result, when fabricating devices, it is a practical choice

to use a single value of the coupling length L and vary the gap as necessary, since the coupling

length also impacts the FSR, as will be explored in detail in the next section.

For completeness, Figure 2.15 shows the normalized and averaged through and drop port

transmission of the same directional coupler as a function of coupling length. In this case, the

58



coupling gap is fixed at 300 nm. For these parameters, the drop transmission exhibits a largely

linear dependence on the coupling length. In actuality, as indicated in Equation 2.5, the depen-

dence is actually oscillatory, however the characteristic length to observe the oscillations for these

parameters is much larger than is reasonable to consider for the design of topological lattices.

Both Figures 2.14 and 2.15 displayed the through and drop transmissions of directional

couplers averaged over the band 1400-1700 nm. For lattices designed for linear physics, this

or calculating the coupling for a single wavelength of interest (such as 1550 nm) is typically

sufficient. However, for lattices intended for nonlinear physics where broadband light may be

generated, further analysis is required. Figure 2.16 shows the results that produced the data for

Figure 2.14, without averaging over the simulation bandwidth.

Without the averaging, it is immediately clear that even over a bandwidth of a few hundred

nanometers, the coupling depends very strongly on wavelength. This property is a well known

feature of directional couplers, and has been the subject of significant study [84]. In the context

of nonlinear topological lattices, this and the resulting wavelength dependence of J is a potential

concern, especially for devices intended for broadband comb generation. While this value does

not affect the single ring FSR, J directly affects the FSR of the super-ring formed by the edge

states of the lattice. If the super-ring FSR is changing with wavelength / frequency, it could

impact comb formation dynamics in a way comparable to an additional dispersion term. As a

result, broadband lattice design is an area of interest for future study.

Finally, it is important to briefly consider mode mixing at this stage as well. Even using

the waveguide cross-section and bend parameters that appear to be robust to mode mixing ef-

59



1400 1500 1600 1700

0.8

0.6

0.4

0.2

0.0

Wavelength (nm)

D
ro

p 
Tr

an
sm

is
si

on 200
250
300
350
400
450
500

Drop Transmission vs Wavelength

Gap (nm)

Figure 2.16: Drop transmission of the directional coupler over the wavelength band 1400 - 1700
nm for various coupling gaps. Transmission is normalized to input transmission.

60



1.000

0.980

0.990

0.985

0.995

1400 1500 1600 1700
Wavelength (nm)

Drop
Through

Fundamental Mode Transmission vs Wavelength

Tr
an

sm
is

si
on

Figure 2.17: Fundamental mode transmission, normalized to total transmission, is displayed for
drop and through ports of an Euler bend directional coupler with effective bending radius of 20
µm, coupling gap of 300 nm, and a coupling length of 12 µm. The waveguide cross section is
1200 by 800 nm.

61



fects, the directional coupler geometry introduces additional possible points of failure. First, and

most obviously, there are additional straight-bent junctions. Second, and more importantly, the

evanescent coupling from one waveguide to the other is achieved via mode overlap. Since it is

known that the waveguide cross-sections required to achieve anomalous dispersion also support

higher-order modes, it is useful to check that the evanescent mode overlap occurs only between

the modes of interest (the fundamental TE modes of each waveguide). For this reason, these

simulations are also performed in full 3D FDTD.

Figure 2.17 shows that the fundamental mode transmission normalized to the total trans-

mission at both the drop and through ports remains above 99% for the vast majority of the 1400

- 1700 nm band. As will be seen in Chapter 3, this suppression of mode mixing is adequate for

the purposes of designing nonlinear topological lattices, however further optimization may be re-

quired for future designs that incorporate rings with higher Q factors. The reason for this can be

understood with Equation 1.8, which shows that a higher Q also results in a higher finesse, which

directly captures the expected number of round trips in the ring. By extension, more round trips

means the circulating light will pass through more straight-bent junctions, potentially causing

enhanced mode mixing.

2.5 Free Spectral Range and Add-Drop Filters

The final geometry to simulate in order to design a topological lattice for nonlinear pho-

tonics is the full add-drop filter. An add-drop filter is still relatively simple, being just a single

62



ring-resonator with two coupling waveguides, as described in Chapter 1. The layout and four

ports are depicted schematically in Figure 2.18A.

At this stage, it is prudent to move away from resource intensive 3D FDTD simulations

for two reasons. The first reason is simply that the geometries here are getting quite large, and

could be difficult to mesh accurately (especially with small coupling gaps). The second reason

is that resonant structures can be time consuming to simulate using time-domain techniques.

In particular, FDTD simulations generally require the energy in the system to decay below a

threshold level before any results that require Fourier transformation (such as a transmission

spectrum) can be calculated reliably. Depending on the Q, a resonant structure may trap energy

for very long timescales. As a result, even the small resonators that are the subject of this work

may require impractical computational resources.

In the following, the ADF is simulated using a similar technique, Lumerical’s varFDTD

which calculates a series of effective refractive indices to map the 3D geometry to a more com-

putationally tractable 2D geometry [81]. As mentioned previously, this kind of technique can

struggle to accurately capture mode mixing effects, and as a result only parameters found to have

minimal mode mixing will be used. In particular, the SiN waveguide cross-section is chosen to

be 1200 by 800 nm, and the ring is made up of 12 µm straight regions and Euler bends with a 20

µm effective radius. The coupling gap is chosen to be 300 nm.

Transmission spectra for the through and drop ports, normalized to the input port intensity,

are shown in Figure 2.18B and C, respectively. As anticipated following the analytical treatment

in Chapter 1, the spectra exhibit a series of dips and corresponding peaks that occur at the phase

63



Input Drop

Through

Gap = 300 nm

{ Add

L = 12 m

Reff = 20 m

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0.4

0.2

0.0
1520 1540 1560 1580 1520 1540 156