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Applications and Integration of Optical Frequency Combs
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Shen, Boqiang
(2021)
Applications and Integration of Optical Frequency Combs.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/p5z5-n346.
Abstract
Optical frequency combs have a wide range of applications in science and technology, including but not limited to timekeeping, optical frequency synthesis, spectroscopy, searching for exoplanets, ranging, and microwave generation. The integration of microresonator with other photonic components enables the high-volume production of wafer-scale optical frequency combs, soliton microcombs. However, it faces two considerable obstacles: optical isolation, which is challenging to integrate on-chip at acceptable performance levels, and power-hungry electronic control circuits, which are required for the generation and stabilization of soliton microcombs. In this thesis, we describe the design and early commissioning of the laser frequency comb for astronomical calibration using electro-optic modulation. We also focus on the realization of a novel and compact chip-scale optical frequency comb, soliton microcomb, including the progress made towards the visible soliton microcomb generation and the demonstration of low power operation of a soliton microcomb along contours of constant power in the phase space. We introduce a soliton spectrometer using dual-locked counter-propagating soliton microcombs to provide high-resolution frequency measurement. Finally, we look into the integration of lasers and high-Q microresonators. The self-injection locking process has been shown to create a new turnkey soliton operating point that eliminates difficult-to-integrate optical isolation as well as complex startup and feedback loops. Moreover, this technique also simplifies the access to high-efficiency dark soliton states without special dispersion engineering of microresonators.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
optical frequency comb; optical microresonator; soliton microcomb
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Vahala, Kerry J.
Thesis Committee:
Faraon, Andrei (chair)
Marandi, Alireza
Leifer, Stephanie D.
Vahala, Kerry J.
Defense Date:
20 May 2021
Record Number:
CaltechTHESIS:05292021-053844660
Persistent URL:
DOI:
10.7907/p5z5-n346
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Article adapted for Ch. 4
DOI
Article adapted for Ch. 5
DOI
Article adapted for Ch. 6
DOI
Article adapted for Ch. 7
DOI
Article adapted for Ch. 8
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ORCID
Shen, Boqiang
0000-0003-0697-508X
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Deposited By:
Boqiang Shen
Deposited On:
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12 Jun 2025 00:06
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Applications and integration of optical frequency combs

Thesis by

Boqiang Shen

In Partial Fulfillment of the Requirements for the
Degree of
Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2021
Defended May 20th, 2021

Boqiang Shen
ORCID: 0000-0003-0697-508X

iii

ACKNOWLEDGEMENTS
The time I spent at Caltech during my Ph.D. studies was exciting and rewarding. It
is a pleasure to have the opportunities to research with so many talented and friendly
people. Without their help and support, this thesis would not have been possible.
First and foremost, my sincere appreciation goes to my advisor, Prof. Kerry Vahala,
for his continuous encouragement, support, and guidance on research and life over
the past six years. More than a scientist and a pioneer in our field, Prof. Vahala is
undoubtedly the best mentor that I have ever seen. He has given us a lot of patience
and freedom to try our research interests with enormous valuable instructions. The
best decision I have ever made was choosing him as my advisor.
I would like to express my gratitude to Prof. Oskar Painter, Prof. Alireza Marandi,
Dr. Stephaine Leifer, and Prof. Andrei Faraon for their service as my graduate
committee and insightful input on my thesis. I am deeply thankful to Stephanie for
being so kind and warmhearted. We had a lot of conversations about the research
and life, which made me feel like she was my elder sister. I would also like to
thank Prof. John Bowers, Prof. Qiang Lin, Prof. Tobias Kippenberg, and Dr. Scott
Diddams for their support during my Ph.D. studies.
I also gratefully acknowledge my colleagues. I would like to offer my special thanks
to Dr. Dong Yoon Oh, Dr. Seung Hoon Lee, Dr. Yu-Hung Lai, Dr. Xinbai Li, Dr.
Ki Youl Yang, Dr. Qi-Fan Yang, and Dr. Chengying Bao for sharing their helpful
advice and ideas as well as productive collaborations. I was greatly enlightened by
them. I also acknowledge Dr. Xu Yi and Dr. Myoung-Gyun Suh for their support. I
also want to thank Heming Wang, Lue Wu, Zhiquan Yuan, Maodong Gao, Qing-Xin
Ji, and Bohan Li. I learned so much from them and enjoyed the conversations and
interactions with them. I likewise thank Dr. Mahmood Bagheri and Dr. Gautam
Vasisht from JPL, Prof. Rebecca Oppenheimer from AMNH, Dr. Junqiu Liu from
EPFL, Dr. Lin Chang, Warren Jin and Chao Xiang from UCSB, and Dr. Yang
He and Jingwei Lin from Rochester as well as the staff at Palomar Observatory for
fruitful collaborations and exciting moments.
I appreciate the company of my roommate and good friend Yu-An Chen during my
Ph.D. journey. We shared the food and space, played the game, and pursued our
dream together. My Ph.D. life wouldn’t have been enjoyable without meeting him
and other friends at Caltech: Baoyi Chen, Yunxuan Li, and Hsiao-Yi Chen.

iv
Last but not least, I thank my parents for their continuous support and love. They
let me explore the outside world without any pressure.

ABSTRACT
Optical frequency combs have a wide range of applications in science and technology, including but not limited to timekeeping, optical frequency synthesis,
spectroscopy, searching for exoplanets, ranging, and microwave generation. The
integration of microresonator with other photonic components enables the highvolume production of wafer-scale optical frequency combs, soliton microcombs.
However, it faces two considerable obstacles: optical isolation, which is challenging
to integrate on-chip at acceptable performance levels, and power-hungry electronic
control circuits, which are required for the generation and stabilization of soliton
microcombs. In this thesis, we describe the design and early commissioning of
the laser frequency comb for astronomical calibration using electro-optic modulation. We also focus on the realization of a novel and compact chip-scale optical
frequency comb, soliton microcomb, including the progress made towards the visible soliton microcomb generation and the demonstration of low power operation
of a soliton microcomb along contours of constant power in the phase space. We
introduce a soliton spectrometer using dual-locked counter-propagating soliton microcombs to provide high-resolution frequency measurement. Finally, we look into
the integration of lasers and high-Q microresonators. The self-injection locking
process has been shown to create a new turnkey soliton operating point that eliminates difficult-to-integrate optical isolation as well as complex startup and feedback
loops. Moreover, this technique also simplifies the access to high-efficiency dark
soliton states without special dispersion engineering of microresonators.

PUBLISHED CONTENT AND CONTRIBUTIONS

[1] Chengying Bao, Boqiang Shen, Myoung-Gyun Suh, Heming Wang, Kemal
Şafak, Anan Dai, Andrey B Matsko, Franz X Kärtner, and Kerry Vahala.
Oscillatory motion of a counterpropagating Kerr soliton dimer. Phys. Rev. A,
103(1):L011501, 2021.
B. S. participated in the experiment and the writing of the manuscript.
[2] Chengying Bao, Myoung-Gyun Suh, Boqiang Shen, Kemal Şafak, Anan Dai,
Heming Wang, Lue Wu, Zhiquan Yuan, Qi-Fan Yang, Andrey B Matsko, et al.
Quantum diffusion of microcavity solitons. Nat. Phys., 2021.
B. S. participated in the experiment and the writing of the manuscript.
[3] Chengying Bao*, Zhiquan Yuan*, Heming Wang, Lue Wu, Boqiang Shen,
Keeyoon Sung, Stephanie Leifer, Qiang Lin, and Kerry Vahala. Interleaved
difference-frequency generation for microcomb spectral densification in the
mid-infrared. Optica, 7(4):309–315, 2020.
B. S. participated in the experiment and the writing of the manuscript.
[4] Lin Chang*, Weiqiang Xie*, Haowen Shu*, Qi-Fan Yang, Boqiang Shen,
Andreas Boes, Jon D Peters, Warren Jin, Chao Xiang, Songtao Liu, et al. Ultraefficient frequency comb generation in AlGaAs-on-insulator microresonators.
Nat. Commun., 11(1331), 2020.
B. S. participated in the writing of the manuscript.
[5] Yang He*, Qi-Fan Yang*, Jingwei Ling, Rui Luo, Hanxiao Liang, Mingxiao
Li, Boqiang Shen, Heming Wang, Kerry Vahala, and Qiang Lin. Self-starting
bi-chromatic LiNbO3 soliton microcomb. Optica, 6(9):1138–1144, 2019.
B. S. participated in the LiNbO3 soliton microcomb generation and prepared
the data.
[6] Warren Jin*, Qi-Fan Yang*, Lin Chang*, Boqiang Shen*, Heming Wang*,
Mark A Leal, Lue Wu, Maodong Gao, Avi Feshali, Mario Paniccia, et al.
Hertz-linewidth semiconductor lasers using CMOS-ready ultra-high-Q microresonators. Nat. Photon., 2021.
B. S. conducted the photonic alignment and the linewidth measurement, analyzed the data, and participated in the writing of the manuscript.
[7] Yu-Hung Lai*, Myoung-Gyun Suh*, Yu-Kun Lu, Boqiang Shen, Qi-Fan Yang,
Heming Wang, Jiang Li, Seung Hoon Lee, Ki Youl Yang, and Kerry Vahala.
Earth rotation measured by a chip-scale ring laser gyroscope. Nature Photonics,
14(6):345–349, 2020.
B. S. participated in the experiment and the writing of the manuscript.
[8] Seung Hoon Lee*, Dong Yoon Oh*, Qi-Fan Yang*, Boqiang Shen*, Heming
Wang*, Ki Youl Yang, Yu-Hung Lai, Xu Yi, Xinbai Li, and Kerry Vahala.

vii
Towards visible soliton microcomb generation. Nat. Commun., 8(1295), 2017.
B. S. participated in the experiment of soliton microcomb generation at 780
nm and prepared the data.
[9] Xinbai Li*, Boqiang Shen*, Heming Wang*, Ki Youl Yang*, Xu Yi, Qi-Fan
Yang, Zhiping Zhou, and Kerry Vahala. Universal isocontours for dissipative
kerr solitons. Opt. Lett., 43(11):2567–2570, 2018.
B. S. participated in the experiment and the numerical simulation, prepared
the data, and participated in the writing of the manuscript.
[10] Zachary L Newman, Vincent Maurice, Tara Drake, Jordan R Stone, Travis C
Briles, Daryl T Spencer, Connor Fredrick, Qing Li, Daron Westly, Bojan R
Ilic, et al. Architecture for the photonic integration of an optical atomic clock.
Optica, 6(5):680–685, 2019.
B. S. participated in the experiment and demonstrated the 10GHz soliton
microcombs.
[11] Boqiang Shen*, Lin Chang*, Junqiu Liu*, Heming Wang*, Qi-Fan Yang*,
Chao Xiang, Rui Ning Wang, Jĳun He, Tianyi Liu, Weiqiang Xie, et al.
Integrated turnkey soliton microcombs. Nature, 582(7812):365–369, 2020.
B. S. conceived the project, packaged the chip, conducted the experiment,
prepared the data, and managed the draft of the manuscript.
[12] Heming Wang, Yu-Kun Lu, Lue Wu, Dong Yoon Oh, Boqiang Shen, Seung Hoon Lee, and Kerry Vahala. Dirac solitons in optical microresonators.
Light Sci. Appl., 9(205), 2020.
B. S. prepared the data and participated in the writing of the manuscript.
[13] Lue Wu*, Heming Wang*, Qifan Yang, Qing-Xin Ji, Boqiang Shen, Chengying
Bao, Maodong Gao, and Kerry Vahala. Greater than one billion Q factor for
on-chip microresonators. Opt. Lett., 45(18):5129–5131, 2020.
B. S. participated in the experiment and prepared the data.
[14] Ki Youl Yang*, Dong Yoon Oh*, Seung Hoon Lee*, Qi-Fan Yang, Xu Yi,
Boqiang Shen, Heming Wang, and Kerry Vahala. Bridging ultrahigh-Q devices
and photonic circuits. Nat. Photon., 12(5):297–302, 2018.
B. S. participated in the experiment and the simulation, prepared the data, and
participated in the writing of the manuscript.
[15] Qi-Fan Yang*, Qing-Xin Ji*, Lue Wu*, Boqiang Shen, Heming Wang,
Chengying Bao, Zhiquan Yuan, and Kerry Vahala. Dispersive-wave induced
noise limits in miniature soliton microwave sources. Nat. Commun., 12(1442),
2021.
B. S. participated in the experiment and the numerical simulation, and prepared
the data.
[16] Qi-Fan Yang*, Boqiang Shen*, Heming Wang*, Minh Tran, Zhewei Zhang,
Ki Youl Yang, Lue Wu, Chengying Bao, John Bowers, Amnon Yariv, et al.

viii
Vernier spectrometer using counterpropagating soliton microcombs. Science,
363(6430):965–968, 2019.
B. S. conducted the experiment of counterpropagating soliton microcomb
generation and the measurements of laser frequencies, prepared the data, and
participated in the writing of the manuscript.
*These authors contributed equally to this work.

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . vi
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Optical frequency combs . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Chapter II: Electro-optic frequency combs . . . . . . . . . . . . . . . . . . . 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Schematic of the electro-optic frequency comb . . . . . . . . . . . . 4
2.3 Dispersion compensation and pulse shaping . . . . . . . . . . . . . . 6
Chapter III: Searching for exoplanets using an electro-optic frequency comb
in the Palomar Radial Velocity Instrument (PARVI) . . . . . . . . . . . . 8
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Laser frequency comb design and setup . . . . . . . . . . . . . . . . 9
3.3 Spectral flattener design and setup . . . . . . . . . . . . . . . . . . . 11
3.4 PARVI LFC operation and interfaces . . . . . . . . . . . . . . . . . 12
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Chapter IV: Towards visible soliton frequency combs . . . . . . . . . . . . . 16
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Silica resonator geometry and dispersion design . . . . . . . . . . . 18
4.3 Soliton microcombs generation using pump at 1064 nm . . . . . . . 22
4.4 Soliton microcombs generation using pump at 778 nm . . . . . . . . 24
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Chapter V: Low power operation of soliton microcombs along iso-contours . 28
5.1 Dissipative Kerr soliton phase diagram and iso-power contours . . . 29
5.2 Measurement system and low power operation . . . . . . . . . . . . 30
5.3 Iso-contours of soliton pulse width . . . . . . . . . . . . . . . . . . 33
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Chapter VI: Vernier spectrometer using dual-locked counterpropagating soliton microcombs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.1 Concept of Vernier spectrometer . . . . . . . . . . . . . . . . . . . . 37
6.2 Measurement of a static laser . . . . . . . . . . . . . . . . . . . . . 39
6.3 Measurement of dynamic lasers . . . . . . . . . . . . . . . . . . . . 39
6.4 High-resolution spectroscopy . . . . . . . . . . . . . . . . . . . . . 41
6.5 Measurement of multi-line spectra . . . . . . . . . . . . . . . . . . . 41
6.6 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter VII: Integrated turnkey soliton microcombs . . . . . . . . . . . . . .
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Concept of turnkey soliton microcomb . . . . . . . . . . . . . . . .
7.3 New turnkey operating point . . . . . . . . . . . . . . . . . . . . . .
7.4 Demonstration of turnkey soliton generation . . . . . . . . . . . . .
7.5 Theory of turnkey soliton generation . . . . . . . . . . . . . . . . .
7.6 Additional measurements . . . . . . . . . . . . . . . . . . . . . . .
7.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter VIII: Dark soliton microcombs in CMOS-ready ultra-high-Q microresonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 CMOS-ready ultra-high-𝑄 microresonators . . . . . . . . . . . . . .
8.3 Hertz-linewidth integrated laser . . . . . . . . . . . . . . . . . . . .
8.4 Mode-locked dark soliton microcomb . . . . . . . . . . . . . . . . .
8.5 Performance comparison . . . . . . . . . . . . . . . . . . . . . . . .
8.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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LIST OF ILLUSTRATIONS

Number
Page
2.1 Schematic of a typical electro-optic frequency comb. SG: analog
radio-frequency signal generator. Amp: electrical amplifier. PS:
electrical phase shifter. VATT: variable microwave attenuator. VDC :
direct-current voltage source. CW laser: continuous-wave (CW)
laser. EDFA: erbium-doped fiber amplifier. PM: phase modulator.
AM: amplitude modulator. . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Autocorrelation measurement of the typical electric-optic (EO)
comb pulse after dispersion compensation. The red line is the
linewidth fit to the autocorrelation measurement data (blue) yielding
pulse width about 1.6 ps. Two commercial phase modulators were
used for EO modulation. . . . . . . . . . . . . . . . . . . . . . . . 6
3.1 Detailed setup of the laser frequency comb for PARVI. Bidirectional arrow marks the unit that is being remotely operated or monitored. Rb clock: rubidium clock. LO: low phase noise microwave
local oscillator. Amp: microwave amplifier. FC: frequency counter.
PS: microwave phase shifter. EDFA: erbium-doped fiber amplifier.
VATT: variable microwave attenuator. VOA: variable optical attenuator. CW laser: line-referenced continuous-wave laser. PM: phase
modulator. AM: amplitude modulator. PD: photodetector. VDC :
direct-current voltage port. Vm : microwave modulation port. DCU:
dispersion compensation unit. HNLF: highly nonlinear fiber. BPF:
bandpass filter. SLM: spatial light modulator. . . . . . . . . . . . . . 10
3.2 Detailed setup of the spectral flattener. The spectral transmission
is controlled by adjusting the voltage on the liquid crystal display
(LCD) pixels of the SLM, altering the polarization reflected by the
grating. SLM: spatial light modulator . . . . . . . . . . . . . . . . . 11
3.3 False color image of the laser frequency comb on the PARVI
spectrometer. The broad and flat comb lines cover a range from 1350
nm to 1750nm, with a continuum dispersive wave peaked at 1250nm.
Two input fiber channels of the spectrometer are illuminated by the
LFC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

xii
3.4 Screenshot of the graphic user interface of the laser frequency
comb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.1 Soliton frequency comb generation in dispersion-engineered silica resonators. (a) A rendering of a silica resonator with the calculated TM1 mode profile superimposed. (b) Regions of normal and
anomalous dispersion are shown versus silica resonator thickness (𝑡)
and pump wavelength. The zero dispersion wavelength (𝜆ZDW ) for
the TM1 mode appears as a blue curve. The dark green band shows
the 10-dB bandwidth of anomalous dispersion created by TM1-TE2
mode hybridization. The plot is made for a 3.2-mm-diameter silica
resonator with a 40◦ wedge angle. Three different device types I,
II, and III (corresponding to 𝑡 = 7.9 µm, 3.4 µm and 1.5 µm) are
indicated for soliton generation at 1550 nm, 1064 nm and 778 nm.
(c) Measured 𝑄 factors and parametric oscillation threshold powers
versus thickness and pump wavelength for the three device types.
Powers are measured in the tapered fiber coupler under critical coupling. Effective mode area (𝐴eff ) of the TM1 mode family is also
plotted as a function of wavelength and thickness. (d) A photograph
of a silica resonator (Type III device pumped at 778 nm) while generating a soliton stream. The pump light is coupled via a tapered
fiber from the left side of the resonator. The red light along the
circumference of the resonator and at the right side of the taper is
believed to result from short wavelength components of the soliton
comb. (e) Soliton frequency comb spectra measured from the devices. The red, green and blue soliton spectra correspond to device
types I, II, and III designed for pump wavelengths 1550 nm, 1064 nm
and 778 nm, respectively. Pump frequency location is indicated by a
dashed vertical line. The soliton pulse repetition rate of all devices
is about 20 GHz. Differences in signal-to-noise ratio (SNR) of the
spectra originate from the resolution of the optical spectrum analyser (OSA). In particular, the 778 nm comb spectrum was measured
using the second-order diffracted spectrum of the OSA, while other
comb spectra were measured as first-order diffracted spectra. Insets:
cross-sectional scanning electron microscope (SEM) images of the
fabricated resonators. White scale bar is 5 µm. . . . . . . . . . . . . 20

xiii
4.2 Microresonator dispersion engineering and soliton generation at
1064 nm. (a) Simulated group velocity dispersion (GVD) of TM
mode families versus resonator thickness. The angle of the wedge
ranges from 30◦ to 40◦ in the colored regions. Measured data points
are indicated and agree well with the simulation. The error bars depict
standard deviations obtained from measurement of 8 samples having
the same thickness. (b) Measured relative mode frequencies (blue
points) plotted versus relative mode number of a soliton-forming TM1
mode family in a 3.4 µm thick resonator. The red curve is a parabolic
fit yielding 𝐷 2 /2𝜋 = 3.3 kHz. (c) Experimental setup for soliton
generation. A continuous-wave (CW) fiber laser is modulated by an
electro-optic phase modulator (PM) before coupling to a ytterbiumdoped fiber amplifier (YDFA). The pump light is then coupled to the
resonator using a tapered fiber. Part of the comb power is used to
servo-control the pump laser frequency. FBG: fiber Bragg grating.
PD: photodetector. PC: polarization controller. (d) Optical spectra
of solitons at 1064 nm generated from the mode family shown in b.
The two soliton spectra correspond to different power levels with the
blue spectrum being a higher power and wider bandwidth soliton.
The dashed vertical line shows the location of the pump frequency.
The solid curves are sech2 fittings. Inset: typical detected electrical
beatnote showing soliton repetition rate. The weak sidebands are
induced by the feedback loop used to stabilize the soliton. The
resolution bandwidth is 1 kHz. . . . . . . . . . . . . . . . . . . . . . 21

xiv
4.3 Dispersion engineering and soliton generation at 778 nm. (a)
Calculated effective indices 𝑛eff for TE1, TE2, TM1, and TM2 modes
at 778 nm plotted versus thickness for a silica resonator with reflection
symmetry (i.e., 𝜃 = 90◦ ). The TM1 and TE2 modes cross each other
without hybridization. (b) Zoom-in of the dashed box in panel a.
(c) As in (b) but for a resonator with 𝜃 = 40◦ . An avoided crossing
of TM1 and TE2 occurs due to mode hybridization. Insets of b
and c show simulated mode profiles (normalized electric field) in
resonators with 𝜃 = 90◦ and 𝜃 = 40◦ , respectively. The color bar is
shown to the right. (d) Calculated group velocity dispersion (GVD)
of the two modes. For the 𝜃 = 40◦ case, hybridization causes a
transition in the dispersion around the thickness 1.48 µm. The points
are the measured dispersion values. (e) (f) Measured relative mode
frequencies of the TM1 and TE2 mode families versus relative mode
number 𝜇 for devices with 𝑡 = 1.47 µm and 𝑡 = 1.49 µm. (g)
Calculated total second-order dispersion versus frequency (below)
and wavelength (above) at four different oxide thicknesses (number
in lower left of each panel). Red and blue curves correspond to the
two hybridized mode families. Anomalous dispersion is negative
and shifts progressively to bluer wavelengths as thickness decreases.
Background color gives the approximate corresponding color spectrum. 23

xv
4.4 Soliton generation at 778 nm. (a) Experimental setup for soliton
generation. A 1557 nm tunable laser is sent to a quadrature phaseshift keying modulator (QPSK) to utilize frequency-kicking and is
then amplified by an erbium-doped fiber amplifier (EDFA). Then,
a periodically-poled lithium niobate (PPLN) waveguide frequencydoubles the 1557 nm input into 778 nm output. The 778 nm pump
light is coupled to the resonator for soliton generation. A servo
loop is used to maintain pump locking. (b) Measured relative mode
frequencies of the TM1 mode family versus wavelength for devices
with 𝑡 = 1.47 µm. A number of crossing mode families are visible.
The red curve is a numerical fit using 𝐷 2 /2𝜋 = 49.8 kHz and 𝐷 3 /2𝜋 =
340 Hz. (c) Optical spectrum of a 778 nm soliton generated using
the device measured in b with pump line indicated by the dashed
vertical line. The red curve is a spectral fitting which reveals a pulse
width of 145 fs. Most of the spurs in the spectrum correspond to
the mode crossings visible in b. Inset shows the electrical spectrum
of the detected soliton pulse stream. The resolution bandwidth is
1 kHz. (d) Measured relative mode frequencies of the TE2 mode
family versus wavelength for devices with 𝑡 = 1.53 µm. The red
curve is a fit with 𝐷 2 /2𝜋 = 4.70 kHz and 𝐷 3 /2𝜋 = −51.6 Hz. (e)
Optical spectrum of a soliton generated using the device measured in
d with pump line indicated as the dashed vertical line. A dispersive
wave is visible near 758 nm. Inset shows the electrical spectrum of
the detected soliton pulse stream. The resolution bandwidth is 1 kHz.

xvi
5.1 Dissipative Kerr soliton phase diagram and iso-power contours.
The phase diagram features normalized pump power 𝑓 2 along the
vertical axis and normalized detuning 𝜁 along the horizontal axis.
The green region contains stable soliton states. Black dotted lines
(gray dashed lines) are iso-power contours using Eq. 5.4 with Raman
term (w/o Raman). 𝑝 is incremented from 4.0 to 8.0 in steps of 0.5.
Red lines are simulated iso-power contours using Eq. 5.1. Blue dots
give the measured soliton iso-power contours at the following soliton
powers: 93, 99.5, 117.5, 125, 129, 136, 145 µW (left to right), which
correspond to 𝑝 values of 5.1, 5.5, 6.4, 6.8, 7.0, 7.4, and 7.9. For these
measurements, 𝑄 = 197 million (𝜅/2𝜋 = 0.98 MHz), 𝜅 𝐸 /𝜅 = 0.26,
and 𝛾 = 2.1 × 10−3 . Inset shows the measured iso-power contours
using another similar device, with soliton powers of 299, 320, and
335 µW (left to right), which correspond to 𝑝 values of 5.4, 5.7, and
6.0. For these measurements, 𝑄 = 115 million (𝜅/2𝜋 = 1.69 MHz),
𝜅 𝐸 /𝜅 = 0.39, and 𝛾 = 2.8 × 10−3 . Large green and blue data points
correspond to spectra in Fig. 5.2 . . . . . . . . . . . . . . . . . . . . 31
5.2 Measurement system and low power operation. (a) Measurement
setup. EOM: electro-optical phase modulator. PC: polarization controller. PM: In-line power meter. PD: Photodetector. FG: function
generator. (b) Soliton spectra at normalized detuning and pumping
power (𝜁 = 21, 𝑓 2 = 53) (blue) and (𝜁 = 8.0, 𝑓 2 = 9.0) (green). The
corresponding phase diagram locations are marked in Fig. 5.1. Red
curve: squared hyperbolic-secant fitting. . . . . . . . . . . . . . . . 32
5.3 Iso-contours of soliton pulse width. The device is unchanged from
Fig. 5.1 main panel. Red solid lines (black dotted lines) are simulated
(Eq. 5.3 theory) iso-contours of normalized pulse width 𝜏𝜃 ranging
from 0.21 to 0.135 (equidistant steps of 0.015). Blue solid lines are
the linear interpolation from measurement of iso-contours at 190,
170, 155, 145, 140 fs, which correspond to 𝜏𝜃 : 0.168, 0.150, 0.137,
0.128, 0.124. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

xvii
6.1 Spectrometer concept, experimental setup and static measurement. (a) Counter propagating soliton frequency combs (red and
blue) feature repetition rates that differ by Δ 𝑓𝑟 , phase-locking at the
comb tooth with index 𝜇 = 0 and effective locking at 𝜇 = 𝑁 thereby
setting up the Vernier spectrometer. Tunable laser and chemical
absorption lines (grey) can be measured with high precision. (b)
Experimental setup. AOM: acousto-optic modulator; CIRC: circulator; PD: photodetector. Small red circles are polarization controllers.
Inset: scanning electron microscope image of a silica resonator. (c)
Optical spectra of counter-propagating solitons. Pumps are filtered
and denoted by dashed lines. (d) Typical measured spectrum of 𝑉1𝑉2
used to determine order 𝑛. For this spectrum: Δ 𝑓𝑛1 − Δ 𝑓𝑛2 = 2.8052
MHz and Δ 𝑓𝑟 = 52 kHz giving 𝑛 = 54. (e) The spectrograph of
the dual soliton interferogram (pseudo color). Line spacing gives
Δ 𝑓𝑟 = 52 kHz. White squares correspond to the index 𝑛 = 54 in
panel c. (f) Measured wavelength of an external cavity diode laser
operated in steady state. (g) Residual deviations between ECDL laser
frequency measurement as given by the MSS and a wavemeter. Error
bars give the systematic uncertainty as limited by the reference laser
in panel b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2 Laser tuning and spectroscopy measurements. (a) Measurement
of a rapidly tuning laser showing index 𝑛 (upper), instantaneous frequency (middle), and higher resolution plot of wavelength relative to
average linear rate (lower), all plotted versus time. (b) Measurement
of a broadband step-tuned laser as for laser in panel A. Lower panel
is a zoom-in to illustrate resolution of the measurement. (c) Spectroscopy of H12 C14 N gas. A vibronic level of H12 C14 N gas at 5 Torr
is resolved using the laser in panel A. (d) Energy level diagram showing transitions between ground state and 2𝜈1 levels. The measured
(reference) transition wavenumbers are noted in red (blue). . . . . . . 40

xviii
6.3 Measurement of a fiber mode-locked laser. (a) Pulse trains generated from a fiber mode-locked laser (FMLL) are sent into an optical
spectral analyzer (OSA) and the MSS. (b) Optical spectrum of the
FMLL measured by the OSA. (c) Optical spectrum of the FMLL
measured using the MSS over a 60-GHz frequency range (indicated
by dashed line). (d) Measured (blue) and fitted (red) FMLL mode
frequencies versus index. The slope of the fitted line is set to 249.7
MHz, the measured FMLL repetition rate. (e) Residual MSS deviation between measurement and fitted value. . . . . . . . . . . . . . . 42
6.4 Multi-frequency measurements. (a) A section of 𝑉˜1,2 . Pairs of
beatnotes coming from the same laser are highlighted and the derived
𝑛 value is marked next to each pair of beatnotes. (b) Zoom-in on the
highlighted region near 858 MHz in (a). Two beatnotes are separated
by 1.0272 MHz. (c) Cross-correlation of 𝑉˜1 and 𝑉˜2 is calculated for
each 𝑛 and the maximum can be found at 𝑛 = 63. . . . . . . . . . . . 45
7.1 Integrated soliton microcomb chip. (a) Rendering of the soliton
microcomb chip that is driven by a DC power source and produces
soliton pulse signals at electronic-circuit rates. Four microcombs are
integrated on one chip, but only one is used in these measurements.
(b) Transmission signal when scanning the laser across a cavity resonance (blue). Lorentzian fitting (red) reveals 16 million intrinsic Q
factor. (c) Frequency noise spectral densities (SDs) of the DFB laser
when it is free running (blue) and feedback-locked to a high-Q Si3 N4
microresonator (red). For comparison, the frequency noise SDs of
ultra-low-noise integrated laser on silicon (grey) and a table-top external cavity diode laser (black) are also plotted. (d) Images of a
pump/microcomb in a compact butterfly package. . . . . . . . . . . . 50

xix
7.2 The turnkey operating point. (a) Conventional soliton microcomb
operation using a tunable c.w. laser. An optical isolator blocks the
back-scattered light from the microresonator. (b) Phase diagram,
hysteresis curve, and dynamics of the microresonator pumped as
shown in (a). The blue curve is the intracavity power as a function
of cavity-pump frequency detuning. Laser tuning (dashed red line)
accesses multiple equilibria. (c) Measured evolution of comb power
pumped by an isolated, frequency-scanned ECDL. The step in the
trace is a characteristic feature of soliton formation. (d) Turnkey
soliton microcomb generation. Non-isolated operation allows backscattered light to be injected into the pump laser cavity. Resonances
are red-shifted due to self-phase modulation (SPM) and cross-phase
modulation (XPM). (e) Phase diagram, hysteresis curve, and dynamics of pump/microresonator system. A modified laser tuning curve
(dashed red line) intersects the intracavity power curve (blue) to establish a new operating point from which solitons form. The feedback
phase 𝜙 is set to 0 in the plot. Simulated evolution upon turning-on
of the laser at a red detuning outside the soliton regime but within
the locking bandwidth is plotted (solid black curve). (f) Measured
comb power (upper panel) and detected soliton repetition rate signal
(lower panel) with laser turn-on indicated at 10 ms. . . . . . . . . . . 51
7.3 Optical and electrical spectra of solitons. (a) The optical spectrum
of a single soliton state with repetition rate 𝑓𝑟 = 40 GHz. The red
curve shows a sech2 fitting to the soliton spectral envelope. (b)
(c) Optical spectra of multi-soliton states at 20 GHz and 15 GHz
repetition rates. Insets: Electrical beatnotes showing the repetition
rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

xx
7.4 Demonstration of turnkey soliton generation. (a) 10 consecutive
switching-on tests are shown. The upper panel gives the measured
comb power versus time. The laser is switched on periodically as
indicated by the shaded regions. The lower panel is a spectrogram
of the soliton repetition rate signal measured during the switching
process. (b) Phase diagram of the integrated soliton system with
respect to feedback phase and pump power. The pump power is normalized to the parametric oscillation threshold. (c) Turnkey success
probability versus relative feedback phase of 20 GHz (upper panel)
and 15 GHz (lower panel) devices. Each data point is acquired from
100 switch-on attempts. See Methods for additional discussion. . . . 54
7.5 Continuous-wave states of the injection-locked nonlinear resonator. Horizontal axis is the normalized detuning 𝛼, and vertical
axis is the normalized optical energy on the pump mode |𝜌| 2 . Resonator characteristics are shown as the blue curves, with |𝐹 | 2 = 1
(lower) to 4 (upper). Laser locking characteristics are shown as the
red curves, with 𝜙 = −5𝜋/6 (upper left) to 5𝜋/6 (lower right). . . . . 59

xxi
7.6 Numerical simulations of turnkey soliton generation. (a) Conventional solitons are generated by sweeping the laser frequency.
Parameters are 𝐾 = 0 (no feedback) and |𝐹 | 2 = 4. The normalized
laser frequency is swept from 𝛼L = −2 to 𝛼L = 6 within a normalized
time interval of 400. Upper panel: soliton field power distribution
as a function of evolution time and coordinates. Middle panel: dynamics of the pump mode power (black) and comb power (blue).
Lower panel: a snapshot of the soliton field at evolution time 𝜏 = 350
(𝛼L = 5), also marked as a white dashed line in the upper panel
and a black dashed line in the middle panel. (b) Multiple solitons
are generated under conditions of nonlinear feedback. Parameters
are 𝐾 = 15, 𝜙 = 0.15𝜋, |𝐹 | 2 = 3 and 𝛼L = 5. Upper and middle
panels are the same as in (a). Lower panel: snapshots of the soliton
field at evolution time 𝜏 = 45 (gray dashed line) and 𝜏 = 70 (black
solid line), also marked as white dashed lines in the upper panel
and black dashed lines in the middle panel. (c) A single soliton is
generated under conditions of nonlinear feedback. Parameters are
𝐾 = 15, 𝜙 = 0.3𝜋, |𝐹 | 2 = 3 and 𝛼L = 5. Upper and middle panels
are the same as in (a). Lower panel: snapshots of the soliton field at
evolution time 𝜏 = 350 (gray dashed line) and 𝜏 = 380 (black solid
line), also marked as white dashed lines in the upper panel and black
dashed lines in the middle panel. . . . . . . . . . . . . . . . . . . . . 62
7.7 Optical and electrical spectra of different microcomb types. (a)(b)
Optical spectra of breather solitons and a chaotic comb. Inset: Electrical beatnote signals. (c) Optical spectrum of a soliton crystal state. 63
7.8 Tuning of turnkey soliton microcomb system. (a) Turnkey generation of a chaotic comb. Upper panel: Comb power evolution. Lower
panel: Spectrograph of RF beatnote power. (b)(c) Comb power evolution when the pump laser frequency is driven from blue to red (b)
and red to blue (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

xxii
8.1 CMOS-ready ultra-high-𝑄 Si3 N4 microresonators. (a) Cross sectional diagram of the ultra-low loss waveguide, consisting of Si3 N4
as the core material, silica as the cladding, and silicon as the substrate (not to scale). (b) Photograph of a CMOS-foundry-fabricated
200 mm diameter wafer after dicing (upper panel), and top view
showing 30 GHz 𝐹𝑆𝑅 Si3 N4 ring resonators and a 5 GHz 𝐹𝑆𝑅
racetrack resonator from a different reticle (lower panel). (c) The
𝑄 factor for each of three 30 GHZ 𝐹𝑆𝑅 ring resonators on each of
the 26 dies of the wafer shown in b was calculated as the average 𝑄
factor in the 1620 nm to 1650 nm range. A wafer map of the highest
𝑄 factor on each die (upper panel) and histogram of 𝑄 factors of
those 78 resonators (lower panel) demonstrate that ultra-high 𝑄 is
achieved across the wafer. (d) Transmission spectrum (upper panel)
of a high-𝑄 mode at 1560 nm in a 30 GHz ring resonator. Interfacial and volumetric inhomogeneities induce Rayleigh scattering,
causing resonances to appear as doublets due to coupling between
counter-propagating modes. Intrinsic 𝑄 of 220 M and loaded 𝑄 of
150 M are extracted by fitting the asymmetric mode doublet. The
ring-down trace of the mode (lower panel) shows 124 ns photon lifetime, corresponding to a 150 M loaded 𝑄. (e) Measured intrinsic
𝑄 factors plotted versus wavelength in a 30 GHz ring resonator with
8 𝜇m wide Si3 N4 core (upper panel) and a 5 GHz racetrack resonator
with 2.8 𝜇m wide Si3 N4 core (lower panel). Insets: simulated optical
mode profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

xxiii
8.2 Hybrid-integrated narrow-linewidth laser based on ultra-high-𝑄
Si3 N4 microresonator. (a) Schematic of the hybrid laser design (not
to scale) and frequency noise test setup. The red (yellow) arrow
denotes the forward (backscattered) light field. ISO: optical isolator; AOM: acousto-optic modulator; PC: polarization controller;
PD: photodetector. (b) Measurement of single-sideband frequency
noise of the free-running and self-injection locked DFB laser. The
minimum frequency-noise levels are 1 Hz2 Hz−1 , 0.8 Hz2 Hz−1 , 0.5
Hz2 Hz−1 for resonators with 30 GHz, 10 GHz and 5 GHz 𝐹𝑆𝑅,
respectively. The dashed lines give the simulated thermorefractive
noise (TRN). (c) Photograph of a 10.8 GHz 𝐹𝑆𝑅 ring resonator
fabricated with a drop port. (d) A comparison of single sideband
frequency noise measured from the through port and drop port of the
same device. The drop port enables the resonator itself to act as a
low-pass filter, yielding a white-noise floor of 0.2 Hz2 Hz−1 . . . . . . 70

xxiv
8.3 Formation of mode-locked Kerr combs. (a) Measured mode family dispersion is normal. The plot shows the integrated dispersion
defined as 𝐷 int = 𝜔 𝜇 − 𝜔 𝑜 − 𝐷 1 𝜇 where 𝜔 𝜇 is the resonant frequency of a mode with index 𝜇 and 𝐷 1 is the 𝐹𝑆𝑅 at 𝜇 = 0. The
wavelength of the central mode (𝜇 = 0) is around 1550 nm. The
dashed lines are parabolic fits (𝐷 int = 𝐷 2 𝜇2 /2) with 𝐷 2 /2𝜋 equal
to −20.3 kHz and −80.2 kHz corresponding to 5 GHz and 10 GHz
𝐹𝑆𝑅, respectively. Note: 𝐷 2 = −𝑐𝐷 21 𝛽2 /𝑛 eff where 𝛽2 is the group
velocity dispersion, 𝑐 the speed of light and 𝑛 eff the effective index of
the mode. (b) Experimental comb power (upper panel) and detected
comb repetition rate signal (lower panel) with laser turn-on indicated
at 5 ms. (c) Measured optical spectra of mode-locked Kerr combs
with 5 GHz (upper panel) and 10 GHz (lower panel) repetition rates.
The background fringes are attributed to the DFB laser. (d) Singlesideband phase noise of dark pulse repetition rates. Dark pulses with
repetition rate 10.8 GHz and 5.4 GHz are characterized. Inset: electrical beatnote showing 5.4 GHz repetition rate. (e) Phase diagram
of microresonator pumped by an isolated laser. The backscattering
is assumed weak enough to not cause mode-splittings. The detuning is normalized to one half of microresonator linewidth, while the
intracavity power is normalized to parametric oscillation threshold.
Green and red shaded areas indicate regimes corresponding to the
c.w. state and Kerr combs. The blue curve is the c.w. intracavity power, where stable (unstable) branches are indicated by solid
(dashed) lines. Simulated evolution of the unisolated laser is plotted
as the solid black curve, which first evolves towards the middle unstable branch of the c.w. intracavity power curve, and then converges to
the comb steady state (average normalized power shown) as marked
by the black dot. The initial condition is set within the self-injection
locking bandwidth, while feedback phase is set to 0. f, Simulated
intracavity field (upper panel) and optical spectrum (lower panel) of
the unisolated laser steady state in panel (e). . . . . . . . . . . . . . . 72

xxv
8.4 Coherence of integrated mode-locked Kerr combs. (a) Optical
spectrum of a mode-locked comb with 43.2 GHz repetition rate generated in a microresonator with 10.8 GHz 𝐹𝑆𝑅. (b) Single-sideband
optical frequency noise of the pump and comb lines as indicated in
panel a, selected using a tunable fiber-Bragg-grating (FBG) filter.
(Inset: the same data in log-log format) (c) Wavelength dependence
of white frequency noise linewidth of comb lines in panel a. . . . . . 73
8.5 Performance comparison of integrated microresonators and lasers.
Upper: Best-to-date integrated ultra-high-𝑄 (> 10 M) microresonators with integrated waveguides. Lower: Linewidth of best-todate integrated narrow-linewidth lasers. . . . . . . . . . . . . . . . 76
8.6 Comparison of finesse and intrinsic 𝑄 factors of state-of-the-art
integrated microresonators. . . . . . . . . . . . . . . . . . . . . . 77

Chapter 1

INTRODUCTION
1.1

Optical frequency combs

The optical frequency comb [1, 2] is a collection of phase-locked, coherent laser
lines featuring equally spaced frequencies. These frequencies are given by the
expression:
𝑓𝑛 = 𝑛 𝑓𝑟 + 𝑓0
(1.1)
where, 𝑓𝑟 is the repetition rate of the comb, which is the frequency difference
between two adjacent comb lines, whereas 𝑓0 is an offset frequency that is common
to all the lines.
Optical frequency combs have a wide range of applications in science and technology, including but not limited to, time keeping [3, 4], optical frequency synthesis
[5–9], spectroscopy [8, 10–14], ranging [15–18], astronomical calibration [19–25],
and microwave generation [26–31].
There are a few methods for optical frequency comb generation. An easy approach to
the formation of an optical frequency comb employs a few cascaded electro-optical
(EO) modulators driven by a microwave signal to impose a series of sidebands on
a continuous-wave laser. Due to the recent advances in integrated lithium niobate
waveguides, the EO comb uses off-the-shelf components from the telecommunication industry and therefore has great reliability.
Another approach to optical frequency combs generation has attracted great interest
due to its advantages of a small footprint, low power consumption, and integration capability. It offers the possibility of miniature comb systems integrated on a
semiconductor chip. They are often called microcombs which rely on the Kerr nonlinearity that is enhanced by the optical power build-up in high-Q microresonators
or ring-like resonators. Benefited from modern fabrication techniques, the ultra-low
propagation loss inside the microresonators results in the quality (Q) factors ranging
from 106 up to nearly 1011 . The milliwatt levels input power can be enhanced to
as high as kilowatts in such high-Q resonators, and the Kerr nonlinearity spontaneously generates two sidebands through parametric oscillation [32, 33] and more
cascaded sidebands through four-wave mixing process [32]. It was shown to lead
to a frequency comb in 2007 [34]. Developments related to solitons in optical fiber

cavities [35, 36] combined with a better understanding of combs in microresonators
eventually led to operational modes where solitons could be generated [37–41]. In
the time domain, a soliton is a nonlinear pulse that propagates inside a fiber cavity
or a microresonator. The pulse maintains its shape by balancing the anomalous
dispersion with its Kerr nonlinearity. The external pump compensates for the propagating loss, and the pulse can go around the cavity indefinitely. When coupled out
from the cavity, pulses from different round trips form a periodic pulse train and
become a frequency comb called a soliton microcomb. Soliton microcombs have
been demonstrated in many materials such as magnesium fluoride [37], diamond
[42], silicon nitride [39, 40, 43–45], silica [38, 46], aluminum nitride [47], and recently, lithium niobate [48], and III-V semiconductors [49]. By modern lithographic
control technique, the geometric dispersion engineering also enables the operation
of octave-spanning combs of THz-repetition rate [44, 45] and towards visible band
combs [50, 51].
1.2

Thesis outline

In this thesis, we focus on the applications of the optical frequency combs, including
usage of the electro-optic combs on the search of exoplanets (Chapter 3) and the
impact of the soliton microcombs on spectroscopic applications (Chapter 6). We
also demonstrated soliton microcombs towards visible band (Chapter 4) and low
power operation of microcomb along the iso-contours for dissipative Kerr soliton
systems (Chapter 5). At last, we look into the possibility of fully integrated chipbased soliton microcombs. The self-injection locking process has been shown to
create a new turnkey operating point that eliminates complex startup and feedback
loops (Chapter 7). Moreover, this technique simplifies access to dark soliton states
as well (Chapter 8).
The thesis is organized as follows:
Chapter 2 introduces the background of the electro-optic frequency comb. The
reliable and easy-to-operate electro-optic comb is a powerful wavelength calibration
tool for precision radial velocity (PRV) measurement used in chapter 3.
Chapter 3 describes the design and early commissioning of the laser frequency
comb (LFC) for astronomy. Detailed setups of the electric-optic frequency comb
and the spectral flattener are discussed. The user interfaces further facilitate the
data acquisition during the Radial Velocity (RV) observation on the sky in search
of exoplanets.

Chapter 4 focuses on the progress made towards the visible soliton microcomb
generation. A simple method of engineering the dispersion of the cavity is used to
broaden the window of available pump wavelengths for bright soliton generation.
20 GHz soliton microcombs are generated using the pump at 1 µm and 778 nm. The
shortest wavelength soliton-to-date is demonstrated covering the 755-790 nm band.
Chapter 5 investigates contours of constant power and constant pulse width for
dissipative Kerr solitons. Measurements were shown to be in good agreement with
the Lugiato-Lefever equation numerical model augmented by Raman interactions,
as well as the prediction of closed-form expressions with Raman interaction. Stable
soliton operation for pump powers as low as 10.8 mW was also demonstrated in the
course of this work.
Chapter 6 introduces soliton spectrometer using dual-locked counter-propagating
soliton microcombs to provide high-resolution frequency measurements of rapid
continuously and step tuned lasers as well as complex multi-line spectra. In combination with a tunable laser, the spectrometer also enables precise measurement of
absorption spectra including random spectral access (as opposed to only continuous
spectral scanning).
Chapter 7 demonstrates an integrated electronics-rate soliton microcomb in a butterfly package. The unisolated laser-microresonator system creates a new turnkey
soliton operating point, which eliminates the optical isolator, as well as the electronic control circuit. Given these compelling features combined with its compact
footprint, the application of this method represents a milestone of integration and
high-volume production of optical frequency combs. Due to the simplification of
the soliton microcomb system, this approach could be applied in other integrated
high-Q microresonator platforms to attain soliton microcombs across a wide range
of applications.
Chapter 8 presents a new regime of Kerr comb operation in normal dispersion
microresonators supported by ultra-low-loss silicon nitride waveguides, fabricated
in a high-volume complementary metal-oxide-semiconductor (CMOS) foundry. By
self-injection locking a conventional semiconductor distributed-feedback (DFB)
laser to these ultra-high-𝑄 microresonators, we reduce noise by five orders of
magnitude, yielding a previously unattainable frequency level for integrated lasers.
Within the same configuration, the dark soliton microcomb both operates turnkey
and attains coherent comb operation under conditions of normal dispersion without
any special dispersion engineering.

Chapter 2

ELECTRO-OPTIC FREQUENCY COMBS
2.1

Introduction

In this chapter, we will first introduce a versatile type of optical frequency comb:
electro-optics (EO) frequency comb [52–56]. It employs a few cascaded EO modulators driven by a single microwave signal to create a grid of sidebands on a continuouswave laser. This results in a spectrally flat optical frequency comb that can be further
compressed into a shorter pulse by additional nonlinear spectral broadening. Compared to the conventional mode-lock laser comb and microresonator-based frequency
comb, the EO frequency comb features a simple structure, deterministic nature, and
thus great reliability. The EO frequency comb described in this chapter will be used
in the field for wavelength calibration in search of the exoplanets.
2.2

Schematic of the electro-optic frequency comb

The schematic layout for a typical electro-optic frequency comb is illustrated in
Fig. 2.1. The continuous-wave (CW) laser is coupled into cascaded lithium niobate
(LiNbO3 ) phase modulators (PMs), and then an amplitude modulator (AM), also
known as an intensity modulator (IM). The optical field after the modulators can be
described as:

𝐸 (𝑡) = 𝐴𝐸 0 𝑒 −𝑖2𝜋 𝑓0 𝑡 × 𝑒𝑖

𝑘 𝛽PM,𝑘 sin(2𝜋 𝑓 𝑚 𝑡+𝜑 𝑘 )

× [𝑒𝑖𝜑DC + 𝑒𝑖𝛽AM sin(2𝜋 𝑓𝑚 𝑡+𝜑AM ) ] (2.1)

where 𝐴 is the attenuation factor to account for the insertion loss of modulators.
𝛽PM,𝑘 , 𝜑 𝑘 are the modulation depth and relative phase of the 𝑘-th phase modulator.
𝛽AM , 𝜑AM , 𝜑DC are the modulation depth, relative phase, and the direct current
(DC) bias phase of the amplitude modulator.
We can expand the phase modulation term using the Jacobi–Anger identity:

𝑒𝑖𝛽 sin 𝜑 =

𝐽𝑛 (𝛽)𝑒𝑖𝑛𝜑

(2.2)

𝑛=−∞

where 𝐽𝑛 is the n-th order Bessel function of the first kind. Therefore, the field
after phase modulation can be seen as a collection of laser lines with equally spaced

SG
VATT

Amp
PS

VDC

CW laser

EDFA

Figure 2.1: Schematic of a typical electro-optic frequency comb. SG: analog
radio-frequency signal generator. Amp: electrical amplifier. PS: electrical phase
shifter. VATT: variable microwave attenuator. VDC : direct-current voltage source.
CW laser: continuous-wave (CW) laser. EDFA: erbium-doped fiber amplifier. PM:
phase modulator. AM: amplitude modulator.
sidebands at the modulation frequency 𝑓𝑚 ( 𝑓𝑛 = 𝑓0 + 𝑛 𝑓𝑚 ) and the number of
comb lines is roughly twice the modulation depth 𝛽PM,𝑘 according to the numerical
simulation.
The pump laser frequency 𝑓0 can be stabilized to a molecular or atomic reference,
e.g. 𝑓0 = 𝑓𝑎𝑡𝑜𝑚 , to provide an anchor for the optical frequency. The phase/amplitude
modulators are driven by an amplified radio-frequency (RF) signal. The RF modulation frequency 𝑓𝑚 can be tuned from MHz to tens of GHz and can be synchronized
with a compact Rb clock which is GPS-disciplined to provide sub-Hertz accuracy.
To maximize the number of sidebands, the phase of PMs should be synchronized
via RF phase shifters (𝜑 𝑘 = 0, for any 𝑘) and therefore the number of comb lines
are determined by the sum of independent PMs, which is approximately 𝑘 2𝛽PM,𝑘 .
By setting the modulation phase 𝜑AM to 0, the modulation depth 𝛽AM to 𝜋, and the
bias phase 𝜑DC to 0 or 𝜋, a linear chirp of modulation sidebands can be achieved by
time gating of the intensity modulator [57]. The intensity modulator also improves
the flatness of the EO comb [58] because some of the comb lines generated solely
by PMs can have lower intensities due to the zeros of Bessel functions. In the
experiment, the modulation phase can be tuned by phase shifter: the optical spectrum
of EO comb will be symmetric with respect to the pump laser line when 𝜑AM = 0
or 𝜑AM = 𝜋. By measuring the average optical output power before and after the

modulation, the bias phase 𝜑DC and modulation depth 𝛽AM can also be set by the
bias voltage and the variable microwave attenuator, respectively.
2.3

Dispersion compensation and pulse shaping

Autocorrelation signal (a.u.)

Any dispersive components can be used to compensate for the linear chirp of modulation sidebands. Usually, line-by-line waveshaping is convenient for compensating
arbitrary dispersion from electro-optic combs with different configurations, such as
repetition rate, pump laser frequency, and fiber loop length. Single-mode fibers or
dispersion compensation Bragg grating can also be used if the resulted linear chirp
of EO comb lines is fixed. After dispersion compensation, the phase of electro-optic
comb teeth will be flattened and the comb will form a temporal pulse, which can be
verified by autocorrelation measurement. In Fig.2.2, the autocorrelation of an EO
comb pulse with around 1.6 ps temporal width is measured.

Measured
Fitting

-10

-5

Delay (ps)

Figure 2.2: Autocorrelation measurement of the typical electric-optic (EO)
comb pulse after dispersion compensation. The red line is the linewidth fit to the
autocorrelation measurement data (blue) yielding pulse width about 1.6 ps. Two
commercial phase modulators were used for EO modulation.
If we amplify the pulse first by the optical amplifier and then send it through a
highly nonlinear fiber (HNLF) or a nonlinear waveguide with specially designed
dispersion, the self-phase modulation of optical pulses will continue to narrow
down its temporal pulse width. In the frequency domain, the new comb lines will

be created via the four-wave mixing process. After further line-by-line attenuation,
the relatively broad and flat electro-optic comb is a powerful wavelength calibration
tool for precision radial velocity (RV) measurement which we will discuss in the
next chapter.

Chapter 3

SEARCHING FOR EXOPLANETS USING AN ELECTRO-OPTIC
FREQUENCY COMB IN THE PALOMAR RADIAL VELOCITY
INSTRUMENT (PARVI)
3.1

Introduction

1The periodic Doppler shifts in the stellar spectrum of the host star can be measured
to infer the presence of an orbiting exoplanet. Planet detection via the Doppler shifts
or Precision Radial Velocity (PRV) technique resulted in the first detection in 1995
of a planet orbiting a mature solar star, 51 Pegasi [59], and was recognized with
the 2019 Nobel Prize in Physics. Since that initial discovery, over 800 planets have
been found using this technique.
The measurement relies on a highly stable and precisely calibrated spectrometer.
Almost all of the calibrations are provided by some combination of arc lamps,
etalons, 𝐼2 gas cells, or laser frequency combs. These instruments have detected
planets with a broad range of masses and orbital separations with a floor of PRV
precision around 1 m s−1 , although a new suite of instruments, including ESPRESSO
[60], EXPRES [61], and NEID [62] promises to break this barrier. A recent National
Academy study has described the need to make Extreme Precision Radial Velocity
observations (EPRV) to measure the Doppler reflex motion of an Earth-mass planet
in a one-year orbit around a solar-type star, 9 cm s−1 .
The laser frequency comb (LFC) generated by electro-optic modulation (EOM)
offers an extremely stable wavelength calibration at the few cm s−1 RV level [63]. In
this chapter, we describe the design and early commissioning of the LFC with great
potential at infrared wavelengths. Compared to commercial visible band Menlo
astrocombs, as most of the stars in our galaxy are M-dwarf stars which are smaller
and cooler and primarily emit light in the near-infrared, NIR astrocombs produce
larger RV signals. And they also provide RV measurement capability in the spectral
range where there are less stellar jitters than the signals in the visible band.
The implementation of the spectrometer, which is precisely calibrated by the LFC,
1 Work presented in this chapter remains a work in progress.

The LFC system is assembled,
installed, and tested at Palomar Observatory by Boqiang Shen and Stephanie Leifer. The flattener was
first assembled by a team at CalStateLA, and was then reconfigured by Mahmood Bagheri, Stephanie
Leifer, and Boqiang Shen. Boqiang Shen is the copyright owner of the LFC control software.

together with the advanced Adaptive Optics (AO) system for the 5 m Hale Telescope
on Palomar Mountain is called the Palomar Radial Velocity Instrument (PARVI).
3.2

Laser frequency comb design and setup

The laser frequency comb design in PARVI is similar to the design described in
Metcalf et al. [22], Obrzud et al. [64], and Yi et al. [20]. The proof of concept has
been demonstrated at the NASA Infrared Telescope Facility (IRTF) and the W. M.
Keck observatory 10 m telescope. Three EOM combs are operating at observatories
for PRV spectrograph wavelength calibration besides this, including the one at the
Hobby Eberly Telescope’s Habitable Planet Finder (HPF) [65], and the one in Subaru
Observatory’s Infrared Doppler Instrument (IRD) [21].
The detailed experimental setup of the PARVI comb is provided in Fig. 3.1. The
pump laser is stabilized to an HCN absorption feature at 1559.914 nm. Three
cascaded phase modulators and one amplitude modulator are driven by a single 10
GHz ultra-low noise phase noise microwave generator which is locked to a GPSreferenced rubidium clock. We recycle and amplify the RF power from the external
termination ports of the first two PMs and synchronize all the modulators via phase
shifters. Variable microwave attenuators are used to optimize the input RF power
of the RF amplifier and the modulation depth of the amplitude modulator. The
remaining microwave power of the third PM is sent to a frequency counter for
monitoring and diagnosis purpose. The amplified pump light goes through these
modulators, becomes a 10 nm wide seed comb, and is then intensity stabilized to the
optimal operating point by a servo box that provides the DC bias and compensates
for the charging and drifting in the sytem. The seed comb is then sent through a
Waveshaper (Finisar model 1000A) to remove the linear chirp, amplified by the highpower optical amplifier (Pritel), and broadened by a specially designed HNLF spliced
by NIST. The nonlinear fiber package consists of three segments: 5 m of normal
dispersion HNLF, 60 cm of single-mode fiber, and 2 m of anomalous dispersion
HNLF. The high-intensity pump lines of the broadened comb are attenuated down
by a bandpass filter. After the dynamic spectral flattener, the broadened comb lines
are capped at a certain intensity level and formed a flat-top spectral envelope. The
flat and broad comb is then combined with the pump line and injected into the
PARVI spectrograph for wavelength calibration.
The laser frequency comb assembly is comprised entirely of polarization-maintaining
(PM) parts except for the single-mode fiber segment in the HNLF package. The

10
HCN absorption

CW laser

GPS

LFC Output

VOA
EDFA

Rb clock
Amp
LO

VATT
PM

φ PS φ

Servo

VDC

VOA

DCU

Spectral flattener
Spectrometer

EDFA
HNLF
BPF

SLM
Remotely controlled via software

Figure 3.1: Detailed setup of the laser frequency comb for PARVI. Bidirectional
arrow marks the unit that is being remotely operated or monitored. Rb clock:
rubidium clock. LO: low phase noise microwave local oscillator. Amp: microwave
amplifier. FC: frequency counter. PS: microwave phase shifter. EDFA: erbiumdoped fiber amplifier. VATT: variable microwave attenuator. VOA: variable optical
attenuator. CW laser: line-referenced continuous-wave laser. PM: phase modulator.
AM: amplitude modulator. PD: photodetector. VDC : direct-current voltage port.
Vm : microwave modulation port. DCU: dispersion compensation unit. HNLF:
highly nonlinear fiber. BPF: bandpass filter. SLM: spatial light modulator.

comb generator assembly, together with the control rack-mount computer, is contained in a temperature-controlled steel instrument rack. The temperature can be
well controlled within 0.1 °C by the thermal control system, while the facility environmental temperature can vary between -5 °C to 25 °C seasonally. The frequency
stability of the LFC is dominated by the reference laser stability, which has been
characterized by Gabe Ycas at NIST and reported in Yi et al. [20].

45° polarizer

Grating

SLM
Output collimator

Input collimator
Computer

Spectrometer

Figure 3.2: Detailed setup of the spectral flattener. The spectral transmission
is controlled by adjusting the voltage on the liquid crystal display (LCD) pixels
of the SLM, altering the polarization reflected by the grating. SLM: spatial light
modulator
3.3

Spectral flattener design and setup

The spectral flattener design is similar to the design described by Probst et al. [66].
It is primarily a free-space setup and housed in a thermally controlled enclosure on
the optical bench next to the comb instrument rack. The key part of the flattener is a
reflective spatial light modulator (SLM) from Meadowlark Optics. The broadened
optical frequency comb after the HNLF package is sent into free space by a fiber
optic collimator and passed through a 45°polarizer. The polarized light hits the
grating where it gets spectrally dispersed and then is projected onto the SLM via
the lens. 1920 × 1152 liquid crystal pixels on the SLM have been grouped into
480 channels which covers the entire wavelength range that is of our interest, with
a resolution of 1.6 nm per channel. Each channel can apply voltages independently

12
to piece-wise adjust the phase difference between the electric field components on
two optical axes of the liquid crystal pixels and thus change the polarization of the
return light. The light is then recombined at the grating and passed the polarizer
again, resulting in wavelength-dependent attenuation levels. The attenuated light is
then collected by a fiber collimator and part of the signal is sent to a spectrometer
for feedback.
Before using the flattener, two calibration steps are required for proper performance:
The voltage response of SLM was calibrated by measuring the transmission when
equal voltages are applied on the SLM pixels. The relation between the channel
number and the corresponding wavelength is calibrated by setting the SLM as a
bandpass filter on each channel.
Based on the difference between the current comb intensity readouts on the spectrometer and the desired intensity level, the control software calculates the voltage
increments or decrements on each channel to achieve the corresponding attenuation.
The steady-state is reached after a few iterations and a flat-top spectrum can be
obtained. Note that, in our design, only one polarizer is used. The attenuation level
at each channel is dynamic via the feedback loop instead of a static configuration.
By truncating the spectrum at around 20dB below its maximum, the LFC provides
uniform illumination on the PARVI spectrometer, see Fig. 3.3.
3.4

PARVI LFC operation and interfaces

The entire laser frequency comb system can be remotely monitored and controlled
using a National Instruments (NI) LabView graphical user interface (GUI), see Fig.
3.4.
There are two buttons on the GUI to switch the comb operating mode between "on",
"idle", and "off". When the comb is not being used for observations, the highpower EDFA that amplifies the seed comb is turned off. The LFC usually switches
between "on" and "idle" mode, where the mini comb is running continuously. The
full system, including the power supplies of microwave amplifiers and oscillator,
EDFA, and pump laser, can also be turned off in orders when a complete shutdown
is required.
The GUI also displays the attenuation controls for both the comb and pump overlay.
The amount of attenuation applied to the comb spectrum that is injected into the
PARVI spectrograph can be controlled by the user to match the intensity of the
target star under observation. The 1559.914 pump line can also be superimposed

Figure 3.3: False color image of the laser frequency comb on the PARVI spectrometer. The broad and flat comb lines cover a range from 1350 nm to 1750nm,
with a continuum dispersive wave peaked at 1250nm. Two input fiber channels of
the spectrometer are illuminated by the LFC.

on the spectrum with an independent attenuation control and serves as a wavelength
marker and indicate the location of the line-referenced pump frequency on the
PARVI spectrograph. The spectrum data from the spectrometer in the dynamic
spectral flattener is displayed at the center of the GUI. It also provides an option for
dynamic flattening which adjusts the voltages applied to each channel on the SLM
at a frequency set by the user. The routines of the look-up table (LUT) generation
and wavelength-to-pixel calibration for SLM are also built into the interface. It also
displays the status of whether the rubidium clock is locked to a GPS signal at the
observatory, whether the laser is locked to the gas cell, and the measured repetition
rate of the frequency comb by the frequency counter. Temperature is monitored and
reported in three locations within the comb instrument rack as well as in the spectral
flattener enclosure. In addition, the dispersion compensation of Waveshaper and
EDFA pump current settings can also be set in the advanced setting section.

Figure 3.4: Screenshot of the graphic user interface of the laser frequency comb.

Moreover, the GUI features the User Datagram Protocol (UDP) server which continuously broadcasts all the status information of the LFC with timestamps in JavaScript
Object Notation (JSON) strings on the local network, and the Internet protocol suite
(TCP/IP) server which listens and accepts the commands from clients to change the
LFC state without logging into the computer. Therefore, the LFC system could be
integrated into the PARVI data acquisition pipeline.
3.5

Summary

The Palomar Radial Velocity Instrument (PARVI) remains a work in progress, its
final integration and testing are postponed due to the COVID-related closure of
Palomar Observatory where the instrument is located.
Future upgrades of the laser frequency comb will incorporate a filter cavity to further
reduce the phase noise of comb lines away from the pump and the use of a selfreferenced pump laser to reference the offset frequency. The current HNLF is also
going to be replaced by a PM HNLF package or a packaged nonlinear waveguide
with a better spectral broadening design. With the LFC fully operational, we will
provide a calibration source with improved frequency stability. The LFC works with
a compact, diffraction-limited, single-mode fiber-fed spectrometer for a wide range
of exoplanet searching and other projects requiring extreme absolute frequency
stability.
A PARVI-like instrument operating across near-IR and into the visible, from 350
nm to 2400 nm, could revolutionize EPRV observations of exoplanets on a space

15
platform where small weight and volume are critical. The generation of a visible
band optical frequency comb and the integration of an optical frequency comb is
on demand. In the following chapters, we will focus on another type of optical
frequency combs called soliton microcombs, which are based on the nonlinearity
inside ultra-high-Q microresonators.

16
Chapter 4

TOWARDS VISIBLE SOLITON FREQUENCY COMBS
1Frequency combs have applications that extend from the ultra-violet into the midinfrared bands. Microcombs, a miniature and often semiconductor-chip-based device, can potentially access most of these applications, but are currently more limited
in spectral reach. Here, we demonstrate mode-locked silica microcombs with emission near the edge of the visible spectrum. By using both geometrical and modehybridization dispersion control, devices are engineered for soliton generation while
also maintaining optical 𝑄 factors as high as 80 million. Electronics-bandwidthcompatible (20 GHz) soliton mode locking is achieved with low pumping powers
(parametric oscillation threshold powers as low as 5.4 mW). These are the shortest
wavelength soliton microcombs demonstrated to date and could be used in miniature
optical clocks. The results should also extend to visible and potentially ultra-violet
bands.
4.1

Introduction

Soliton mode locking [37–40, 43] in frequency microcombs [67] provides a pathway
to miniaturize many conventional comb applications. It has also opened investigations into new nonlinear physics associated with dissipative Kerr solitons [37]
and Stokes solitons [68]. In contrast to early microcombs [67], soliton microcombs
eliminate instabilities, provide stable (low-phase-noise) mode locking, and feature
a highly reproducible spectral envelope. Many applications of these devices are
being studied, including chip-based optical frequency synthesis [9], secondary time
standards [4], and dual-comb spectroscopy [11, 12]. Also, a range of operating
wavelengths is opening up by use of several low-optical-loss dielectric materials for
resonator fabrication. In the near-infrared (IR), microcombs based on magnesium
fluoride [37], silica [38, 46], and silicon nitride [39, 40, 43–45] are being studied
for frequency metrology and frequency synthesis. In the mid-IR spectral region,
silicon nitride [69], crystalline [70], and silicon-based [71] Kerr microcombs as well
as quantum-cascade microcombs [72] are being studied for application to molecular
fingerprinting.
1 Work presented in this chapter has been published in [50] “Towards visible soliton micro-

comb generation”, Nature Communications 8, 1295 (2017). The 780nm soliton microcomb was
demonstrated by Dongyoon Oh and Boqiang Shen.

17
At shorter wavelengths below 1 µm, microcomb technology would benefit optical
atomic clock technology [73], particularly efforts to miniaturize these clocks. For
example, microcomb optical clocks based on the D 1 transition (795 nm) and the
two-photon clock transition (798 nm) in rubidium have been proposed [4]. Also, a
microcomb clock using two-point locking to rubidium D 1 and D 2 lines has been
demonstrated by frequency doubling from the near-IR. More generally, microcomb
sources in the visible and ultra-violet bands could provide a miniature alternative
to larger mode-locked systems such as titanium sapphire lasers in cases where high
power is not required. It is also possible that these shorter wavelength systems
could be applied in optical coherence tomography systems [74, 75]. Efforts directed
towards short wavelength microcomb operation include 1 µm microcombs in silicon
nitride microresonators [76] as well as harmonically-generated combs. The latter
have successfully converted near-IR comb light to shorter wavelength bands and
even into the visible band [77, 78] within the same resonator used to create the
initial comb of near-IR frequencies. Also, crystalline resonators [79] and silica
microbubble resonators [80] have been dispersion-engineered for comb generation
in the 700 nm band. Finally, diamond-based microcombs afford the possibility of
broad wavelength coverage [42]. However, none of the short wavelength microcomb
systems have so far been able to generate stable mode-locked microcombs as required
in all comb applications.
A key impediment to mode-locked microcomb operation at short wavelengths is
material dispersion associated with the various dielectric materials used for microresonator fabrication. At shorter wavelengths, these materials feature large normal dispersion that dramatically increases into the visible and ultra-violet bands.
While dark soliton pulses can be generated in a regime of normal dispersion [81],
bright solitons require anomalous dispersion. Dispersion engineering by proper
design of the resonator geometry [79, 80, 82] offers a possible way to offset the
normal dispersion. Typically, by compressing the waveguide dimension of a resonator, geometrical dispersion will ultimately compensate a large normal material
dispersion component to produce overall anomalous dispersion. For example, in
silica, strong confinement in bubble resonators [80] and straight waveguides [83]
has been used to push the anomalous dispersion transition wavelength from the nearIR into the visible band. Phase matching to ultra-violet dispersive waves has also
been demonstrated using this technique [83]. However, to compensate the rising
material dispersion, this compression must increase as the operational wavelength is
decreased, and as a side effect, highly-confined waveguides tend to suffer increased

18
optical losses. This happens because mode overlap with the dielectric waveguide interface is greater with reduced waveguide cross section. Consequently, the residual
fabrication-induced roughness of that interface degrades the resonator 𝑄 factor and
increases pumping power (e.g., comb threshold power varies inverse quadratically
with 𝑄 factor [32]).
Minimizing material dispersion provides one way to ease the impact of these constraints. In this sense, silica offers an excellent material for short wavelength
operation, because it has the lowest dispersion among all on-chip integrable materials. For example, at 778 nm, silica has a group velocity dispersion (GVD) equal to
38 ps2 km−1 , which is over 5 times smaller than the GVD of silicon nitride at this
wavelength (> 200 ps2 km−1 )[84]. Other integrable materials that are also transparent in the visible, such as diamond [42] and aluminium nitride [85], have dispersion
that is similar to or higher than silicon nitride. Silica also features a spectrally-broad
low-optical-loss window so that optical 𝑄 factors can be high at short wavelengths.
Here we demonstrate soliton microcombs with pump wavelengths of 1064 nm and
778 nm. These are the shortest soliton microcomb wavelengths demonstrated to
date. By engineering geometrical dispersion and by employing mode hybridization,
a net anomalous dispersion is achieved at these wavelengths while also maintaining
high optical 𝑄 factors (80 million at 778 nm, 90 million at 1064 nm). The devices
have large (millimetre-scale) diameters and produce single soliton pulse streams at
rates that are both detectable and processable by low-cost electronic circuits. Besides illustrating the flexibility of silica for soliton microcomb generation across a
range of short wavelengths, these results are relevant to potential secondary time
standards based on transitions in rubidium [4]. Using dispersive-wave engineering
in silica, it might also be possible to extend the emission of these combs into the
ultra-violet as recently demonstrated in compact silica waveguides [83].
4.2

Silica resonator geometry and dispersion design

The silica resonator used in this work is shown schematically in Fig.4.1a. A fundamental mode profile is overlaid onto the cross-sectional rendering. The resonator
design is a variation on the wedge resonator [86], and its geometry can be fully
characterized by its resonator diameter, silica thickness (𝑡), and wedge angle (𝜃)
(see Fig. 4.1a). The diameter of all resonators in this work (and the assumed
diameter in all simulations) is 3.2 mm, which corresponds to a free spectral range
(FSR) of approximately 20 GHz, and the resonator thickness is controlled to obtain
net anomalous dispersion at the design wavelengths, as described in detail below.

19
Further details on fabrication are given elsewhere [86]. As an aside, we note that a
waveguide-integrated version of this design is also possible [87]. Adaptation of that
device using the methods described here would enable full integration with other
photonic elements on the silicon chip.
Fig. 4.1(b) illustrates how the geometrical dispersion induced by varying resonator
thickness 𝑡 offsets the material dispersion. Regions of anomalous and normal
dispersion are shown for the TM1 mode family of a resonator having a wedge angle
of 40◦ . The plots show that thinner resonators enable shorter wavelength solitons.
Accordingly, three device types (I, II, and III shown as the colored dots in Fig.
4.1(b)) are selected for soliton frequency comb operation at three different pump
wavelengths. At a pump wavelength of 1550 nm, the anomalous dispersion window
is wide because bulk silica possesses anomalous dispersion at wavelengths above
1270 nm. For this type I device, a 7.9-µm thickness was used. Devices of type II and
III have thicknesses near 3.4 µm and 1.5 µm for operation with pump wavelengths of
1064 nm and 778 nm, respectively. Beyond geometrical control of dispersion, the
type III design also uses mode hybridization to substantially boost the anomalous
dispersion. This hybridization occurs within a relatively narrow wavelength band
which tunes with 𝑡 (darker green region in Fig. 4.1(b)), and is discussed in detail
below. Measured 𝑄 factors for the three device types are plotted in the upper panel
of Fig. 4.1(c). Maximum 𝑄 factors at thicknesses which also produce anomalous
dispersion were: 280 million (Type I, 1550 nm), 90 million (Type II, 1064 nm), and
80 million (Type III, 778 nm).
Using these three designs, soliton frequency combs were successfully generated
with low threshold pump power. Shown in Fig. 4.1(d) is a photograph of a type
III device under conditions where it is generating solitons. Fig. 4.1(e) shows
optical spectra of the soliton microcombs generated for each device type. A slight
Raman-induced soliton self-frequency-shift is observable in the type I and type II
devices [38, 88–90]. The pulse width of the type III device is longer and has a
relatively smaller Raman shift, which is consistent with theory [90]. The presence
of a dispersive wave in this spectrum also somewhat offsets the smaller Raman shift
[43]. Scanning electron microscope (SEM) images appear as insets in Fig. 4.1(e)
and provide cross-sectional views of the three device types. It is worthwhile to
note that microcomb threshold power, expressed as 𝑃th ∼ 𝐴eff /𝜆 P 𝑄 2 (𝜆 P is pump
wavelength and 𝐴eff is effective mode area) remains within a close range of powers
for all devices (lower panel of Fig. 4.1c). This can be understood to result from

20
λZDW(TM1)

Q (M)

pump
III

2 hybrid
TM1-TE
1300
1000
Wavelength (nm)

700

band

200
20
100

Pth (mW)

Normal

t (µm)

300

Bulk silica

t (µm)

10
778

1600

Aeff (µm2)

Anomalous

1064
1550
Wavelength (nm)

pump at 1550 nm

pump at 1064 nm

pump at 778 nm

Power

20 dB

185

190

195

260

III

280
Frequency (THz)

380

385

390

Figure 4.1: Soliton frequency comb generation in dispersion-engineered silica
resonators. (a) A rendering of a silica resonator with the calculated TM1 mode
profile superimposed. (b) Regions of normal and anomalous dispersion are shown
versus silica resonator thickness (𝑡) and pump wavelength. The zero dispersion
wavelength (𝜆 ZDW ) for the TM1 mode appears as a blue curve. The dark green
band shows the 10-dB bandwidth of anomalous dispersion created by TM1-TE2
mode hybridization. The plot is made for a 3.2-mm-diameter silica resonator with
a 40◦ wedge angle. Three different device types I, II, and III (corresponding to
𝑡 = 7.9 µm, 3.4 µm and 1.5 µm) are indicated for soliton generation at 1550 nm,
1064 nm and 778 nm. (c) Measured 𝑄 factors and parametric oscillation threshold
powers versus thickness and pump wavelength for the three device types. Powers
are measured in the tapered fiber coupler under critical coupling. Effective mode
area (𝐴eff ) of the TM1 mode family is also plotted as a function of wavelength and
thickness. (d) A photograph of a silica resonator (Type III device pumped at 778
nm) while generating a soliton stream. The pump light is coupled via a tapered
fiber from the left side of the resonator. The red light along the circumference
of the resonator and at the right side of the taper is believed to result from short
wavelength components of the soliton comb. (e) Soliton frequency comb spectra
measured from the devices. The red, green and blue soliton spectra correspond to
device types I, II, and III designed for pump wavelengths 1550 nm, 1064 nm and
778 nm, respectively. Pump frequency location is indicated by a dashed vertical
line. The soliton pulse repetition rate of all devices is about 20 GHz. Differences in
signal-to-noise ratio (SNR) of the spectra originate from the resolution of the optical
spectrum analyser (OSA). In particular, the 778 nm comb spectrum was measured
using the second-order diffracted spectrum of the OSA, while other comb spectra
were measured as first-order diffracted spectra. Insets: cross-sectional scanning
electron microscope (SEM) images of the fabricated resonators. White scale bar is
5 µm.

a 10

(ωµ −ω0 −µD1 )/2π (MHz)

β2 (ps2 km-1)

Normal

Anomalous

100

-5

-20

3.5
Thickness (µm)

90/10
CW laser

YDFA

FBG
PC

Servo
Feedback Loop

µ Disk

Power (dBm)

-10

50
-50
Relative mode number µ

-100

TM1
TM2
TM3

-30

Soliton A
Soliton B
Sech2 Fitting (44 fs)

100

pump

20 dB

Sech Fitting (52 fs)

-40
-1

-50

Frequency (MHz
+ 20.28535 GHz)

-60
-70
1000

1050

1100

1150

Wavelength (nm)

Figure 4.2: Microresonator dispersion engineering and soliton generation at
1064 nm. (a) Simulated group velocity dispersion (GVD) of TM mode families
versus resonator thickness. The angle of the wedge ranges from 30◦ to 40◦ in
the colored regions. Measured data points are indicated and agree well with the
simulation. The error bars depict standard deviations obtained from measurement of
8 samples having the same thickness. (b) Measured relative mode frequencies (blue
points) plotted versus relative mode number of a soliton-forming TM1 mode family
in a 3.4 µm thick resonator. The red curve is a parabolic fit yielding 𝐷 2 /2𝜋 = 3.3
kHz. (c) Experimental setup for soliton generation. A continuous-wave (CW) fiber
laser is modulated by an electro-optic phase modulator (PM) before coupling to
a ytterbium-doped fiber amplifier (YDFA). The pump light is then coupled to the
resonator using a tapered fiber. Part of the comb power is used to servo-control
the pump laser frequency. FBG: fiber Bragg grating. PD: photodetector. PC:
polarization controller. (d) Optical spectra of solitons at 1064 nm generated from
the mode family shown in b. The two soliton spectra correspond to different power
levels with the blue spectrum being a higher power and wider bandwidth soliton. The
dashed vertical line shows the location of the pump frequency. The solid curves are
sech2 fittings. Inset: typical detected electrical beatnote showing soliton repetition
rate. The weak sidebands are induced by the feedback loop used to stabilize the
soliton. The resolution bandwidth is 1 kHz.

22
a partial compensation of reduced 𝑄 factor in the shorter wavelength devices by
reduced optical mode area (see plot in Fig. 4.1(c)). For example, from 1550 nm to
778 nm the mode area is reduced by roughly a factor of 9, and this helps to offset a
3-times decrease in 𝑄 factor. The resulting 𝑃th increase (5.4 mW at 778 nm versus
approximately 2.5 mW at 1550 nm) is therefore caused primarily by the decrease
in pump wavelength 𝜆 P . In the following sections, additional details on the device
design, dispersion, and experimental techniques used to generate these solitons are
presented.
4.3

Soliton microcombs generation using pump at 1064 nm

Dispersion simulations for TM modes near 1064 nm are presented in Fig. 4.2(a)
and show that TM modes with anomalous dispersion occur in silica resonators
having oxide thicknesses less than 3.7 µm. Aside from the thickness control, a
secondary method to manipulate dispersion is by changing the wedge angle (see Fig.
4.2(a)). Both thickness and wedge angle are well controlled in the fabrication process
[82]. Precise thickness control is possible because this layer is formed through
calibrated oxidation of the silicon wafer. Wedge angles between 30 and 40 degrees
were chosen in order to maximize the 𝑄 factors [86]. The resonator dispersion is
characterized by measuring mode frequencies using a scanning external-cavity diode
laser (ECDL) whose frequency is calibrated using a Mach-Zehnder interferometer.
As described elsewhere [37, 38], the mode frequencies, 𝜔 𝜇 , are Taylor expanded as
𝜔 𝜇 = 𝜔0 + 𝜇𝐷 1 + 𝜇2 𝐷 2 /2 + 𝜇3 𝐷 3 /6, where 𝜔0 denotes the pumped mode frequency,
𝐷 1 /2𝜋 is the FSR, and 𝐷 2 is proportional to the GVD, 𝛽2 (𝐷 2 = −𝑐𝐷 21 𝛽2 /𝑛0 where
𝑐 and 𝑛0 are the speed of light and material refractive index). 𝐷 3 is a thirdorder expansion term that is sometimes necessary to adequately fit the spectra (see
discussion of 778 nm soliton below). The measured frequency spectrum of the
TM1 mode family in a 3.4 µm thick resonator is plotted in Fig. 4.2b. The plot
gives the frequency as relative frequency (i.e., 𝜔 𝜇 − 𝜔0 − 𝜇𝐷 1 ) to make clear the
second-order dispersion contribution. The frequencies are measured using a radiofrequency calibrated Mach-Zehnder interferometer having a FSR of approximately
40 MHz. Also shown is a fitted parabola (red curve) revealing 𝐷 2 /2𝜋 = 3.3 kHz
(positive parabolic curvature indicates anomalous dispersion). Some avoided mode
crossings are observed in the spectrum. The dispersion measured in resonators of
different thicknesses, marked as solid dots in Fig. 4.2a, is in good agreement with
numerical simulations.
The experimental setup for generation of 1064 nm pumped solitons is shown in Fig.

1.436

neff

1.431
1.45

1.5

1.427
1.4

1.45
1.5
t (µm)

1.432 θ = 40˚

1.55 1.431
1.45

TM1-TE2
hybrid
t (µm)

1.5 0

θ = 40˚
θ = 90˚

Normal
TE2

TM1
TE2

TM1
Anomalous

TM1

-10
TE2
-20

t = 1.47 µm

TE2

-10

TM1

-20

200
Relative mode number µ

β2 (ps2 km-1)

TM2

neff

TM1

β2 (ps2 km-1)

t (µm)

1.430

-200

TM1

(ωµ −ω0 −µD1 )/2π (GHz)

TE2

neff

1.433

(ωµ −ω0 −µD1 )/2π (GHz)

1.432 θ = 90˚

TE1

200

800

750

t = 1.49 µm
200
Relative mode number µ

Wavelength (nm)
660 640 620
540

520

450 440 430

200
100
-100
-200
1.47 µm

1.4

1.45

1.5
t (µm)

1.55

380

1.27 µm

400

460

1.10 µm

480
540 560
Frequency (THz)

0.96 µm

580

660

680

700

Figure 4.3: Dispersion engineering and soliton generation at 778 nm. (a)
Calculated effective indices 𝑛eff for TE1, TE2, TM1, and TM2 modes at 778 nm
plotted versus thickness for a silica resonator with reflection symmetry (i.e., 𝜃 =
90◦ ). The TM1 and TE2 modes cross each other without hybridization. (b) Zoomin of the dashed box in panel a. (c) As in (b) but for a resonator with 𝜃 = 40◦ .
An avoided crossing of TM1 and TE2 occurs due to mode hybridization. Insets
of b and c show simulated mode profiles (normalized electric field) in resonators
with 𝜃 = 90◦ and 𝜃 = 40◦ , respectively. The color bar is shown to the right. (d)
Calculated group velocity dispersion (GVD) of the two modes. For the 𝜃 = 40◦
case, hybridization causes a transition in the dispersion around the thickness 1.48
µm. The points are the measured dispersion values. (e) (f) Measured relative mode
frequencies of the TM1 and TE2 mode families versus relative mode number 𝜇
for devices with 𝑡 = 1.47 µm and 𝑡 = 1.49 µm. (g) Calculated total second-order
dispersion versus frequency (below) and wavelength (above) at four different oxide
thicknesses (number in lower left of each panel). Red and blue curves correspond
to the two hybridized mode families. Anomalous dispersion is negative and shifts
progressively to bluer wavelengths as thickness decreases. Background color gives
the approximate corresponding color spectrum.

4.2c. The microresonator is pumped by a continuous wave (CW) laser amplified
by a ytterbium-doped fiber amplifier (YDFA). The pump light and comb power
are coupled to and from the resonator by a tapered fiber. Typical pumping power
is around 100 mW. Solitons are generated while scanning the laser from higher
frequencies to lower frequencies across the pump mode [37, 38, 43]. The pump
light is modulated by an electro-optic PM to overcome the thermal transient during
soliton generation [38, 43, 91]. A servo control referenced to the soliton power is
employed to capture and stabilize the solitons [91]. Shown in Fig. 4.2d are the

24
optical spectra of solitons pumped at 1064 nm. These solitons are generated using
the mode family whose dispersion is characterized in Fig. 4.2b. Due to the relatively
low dispersion (small 𝐷 2 ), these solitons have a short temporal pulse width. Using
the hyperbolic-secant-squared fitting method [38] (see orange and green curves in
Fig. 4.2d), a soliton pulse width of 52 fs is estimated for the red spectrum. By
increasing the soliton power (blue spectrum), the soliton can be further compressed
to 44 fs, which corresponds to a duty cycle of 0.09% at the 20 GHz repetition rate.
Finally, the inset in Fig. 4.2d shows the electrical spectrum of the photo-detected
soliton pulse stream. Besides confirming the repetition frequency, the spectrum is
very stable with excellent signal-to-noise ratio (SNR) greater than 70 dB at 1 kHz
resolution bandwidth.
4.4

Soliton microcombs generation using pump at 778 nm

As the operational wavelength shifts further towards the visible band, normal material dispersion increases. To generate solitons at 778 nm, an additional dispersion
engineering method, TM1-TE2 mode hybridization, is therefore added to supplement the geometrical dispersion control. The green band region in Fig. 4.1b gives
the oxide thicknesses and wavelengths where this hybridization is prominent. Polarization mode hybridization is a form of mode coupling induced dispersion control.
The coupling of the TM1 and TE2 modes creates two hybrid mode families, one
of which features strong anomalous dispersion. This hybridization is caused when
a degeneracy in the TM1 and TE2 effective indices is lifted by a broken reflection
symmetry of the resonator [92]. The wavelength at which the degeneracy occurs is
controlled by the oxide thickness and determines the soliton operation wavelength.
Finite element method (FEM) simulation in Fig. 4.3a shows that at 778 nm, the TM1
and TE2 modes are expected to have the same effective index at the oxide thickness
1.48 µm when the resonator features a reflection symmetry through a plane that is
both parallel to the resonator surface and that lies at the centre of the resonator. Such
a symmetry exists when the resonator has vertical sidewalls or equivalently a wedge
angle 𝜃 = 90◦ (note: the wet-etch process used to fabricate the wedge resonators
does not support a vertical side wall). A zoom-in of the effective index crossing
is provided in Fig. 4.3b. In this reflection symmetric case, the two modes cross
in the effective-index plot without hybridization. However, in the case of 𝜃 = 40◦
(Fig. 4.3c), the symmetry is broken and the effective index degeneracy is lifted. The
resulting avoided crossing causes a sudden transition in the GVD as shown in Fig.
4.3d, and one of the hybrid modes experiences enhanced anomalous dispersion.

25
1557 nm
CW laser
Servo
Feedback Loop

QPSK

EDFA

t = 1.47 µm

770

775
780
785
Wavelength (nm)
pump

PPLN

790

(ωµ −ω0 −µD1 )/2π (GHz)

0.2
0.1
t = 1.53 µm
-0.1
750

20 dB

-20

770
780
Wavelength (nm)

790

pump

20 dB

800

-20

µ Disk

-1
Frequency (MHz
+ 19.98 GHz)

-40

Power (dBm)

PD
Power (dBm)

778 nm

760

-1
Frequency (MHz
+ 19.76 GHz)

-40

FBG
-60

-60

90/10
770

775
780
785
Wavelength (nm)

790

750

760

770
780
790
Wavelength (nm)

800

Figure 4.4: Soliton generation at 778 nm. (a) Experimental setup for soliton
generation. A 1557 nm tunable laser is sent to a quadrature phase-shift keying
modulator (QPSK) to utilize frequency-kicking and is then amplified by an erbiumdoped fiber amplifier (EDFA). Then, a periodically-poled lithium niobate (PPLN)
waveguide frequency-doubles the 1557 nm input into 778 nm output. The 778 nm
pump light is coupled to the resonator for soliton generation. A servo loop is used
to maintain pump locking. (b) Measured relative mode frequencies of the TM1
mode family versus wavelength for devices with 𝑡 = 1.47 µm. A number of crossing
mode families are visible. The red curve is a numerical fit using 𝐷 2 /2𝜋 = 49.8 kHz
and 𝐷 3 /2𝜋 = 340 Hz. (c) Optical spectrum of a 778 nm soliton generated using
the device measured in b with pump line indicated by the dashed vertical line. The
red curve is a spectral fitting which reveals a pulse width of 145 fs. Most of the
spurs in the spectrum correspond to the mode crossings visible in b. Inset shows the
electrical spectrum of the detected soliton pulse stream. The resolution bandwidth
is 1 kHz. (d) Measured relative mode frequencies of the TE2 mode family versus
wavelength for devices with 𝑡 = 1.53 µm. The red curve is a fit with 𝐷 2 /2𝜋 = 4.70
kHz and 𝐷 3 /2𝜋 = −51.6 Hz. (e) Optical spectrum of a soliton generated using
the device measured in d with pump line indicated as the dashed vertical line. A
dispersive wave is visible near 758 nm. Inset shows the electrical spectrum of the
detected soliton pulse stream. The resolution bandwidth is 1 kHz.
To verify this effect, resonators having four different thicknesses (𝜃 = 40◦ ) were
fabricated and their dispersion was characterized using the same method as for the
1064 nm soliton device. The measured second-order dispersion values are plotted
as solid circles in Fig. 4.3d and agree with the calculated values given by the solid
curves. Fig. 4.3e and Fig. 4.3f show the measured relative mode frequencies versus
mode number of the two modes for devices with 𝑡 = 1.47 µm and 𝑡 = 1.49 µm. As

26
before, upward curvature in the data indicates anomalous dispersion. The dominant
polarization component of the hybrid mode is also indicated on both mode-family
branches. The polarization mode hybridization produces a strong anomalous dispersion component that can compensate normal material dispersion over the entire
band. Moreover, the tuning of this component occurs over a range of larger oxide
thicknesses for which it would be impossible to compensate material dispersion
using geometrical control alone. To project the application of this hybridization
method to yet shorter soliton wavelengths, Fig. 4.3g summarizes calculations of
second order dispersion at a series of oxide thicknesses. At a thickness close to
1 micron, it should be possible to generate solitons at the blue end of the visible
spectrum. Moreover, wedge resonators having these oxide film thicknesses have
been fabricated during the course of this work. They are mechanically stable with
respect to stress-induced buckling [93] at silicon undercut values that are sufficient
for high-𝑄 operation.
For soliton generation, the microresonator is pumped at 778 nm by frequencydoubling a continuous wave (CW) ECDL operating at 1557 nm (see Fig. 4.4a). The
1557 nm laser is modulated by a quadrature phase-shift keying (QPSK) modulator for
frequency-kicking, and then amplified by an erbium-doped fiber amplifier (EDFA).
The amplified light is sent into a periodically-poled lithium niobate (PPLN) device
for second-harmonic generation. The frequency-doubled output pump power at 778
nm is coupled to the microresonator using a tapered fiber. The pump power is
typically about 135 mW. The soliton capture and locking method was again used
to stabilize the solitons [91]. A zoom-in of the TM1 mode spectrum for 𝑡 = 1.47
µm with a fit that includes third-order dispersion (red curve) is shown in Fig. 4.4b.
The impact of higher order dispersion on dissipative soliton formation has been
studied [37, 94]. In the present case, the dispersion curve is well suited for soliton
formation. The optical spectrum of a 778 nm pumped soliton formed on this mode
family is shown in Fig. 4.4c. It features a temporal pulse width of 145 fs as derived
from a sech2 fit (red curve). The electrical spectrum of the photo-detected soliton
stream is provided in the inset in Fig. 4.4c and exhibits high stability.
Fig. 4.4d gives the measured mode spectrum and fitting under conditions of slightly
thicker oxide (𝑡 = 1.53 µm). In this case, the polarization of the hybrid mode more
strongly resembles the TE2 mode family. The overall magnitude of second-order
dispersion is also much lower than for the more strongly hybridized soliton in Fig.
4.4b and Fig. 4.4c. The corresponding measured soliton spectrum is shown in

27
Fig. 4.4e and features a dispersive wave near 758 nm. The location of the wave
is predicted from the fitting in Fig. 4.4d (see dashed vertical and horizontal lines).
The dispersive wave exists in a spectral region of overall normal dispersion, thereby
illustrating that dispersion engineering can provide a way to further extend the
soliton spectrum towards the visible band. As an aside, the plot in Fig. 4.4d has
incorporated a correction to the FSR (𝐷 1 ) so that the soliton line is given as the
horizontal dashed black line. This correction results from the soliton red spectral
shift relative to the pump that is apparent in Fig. 4.4e. This shift results from a
combination of the Raman self shift [89, 90] and some additional dispersive wave
recoil [43]. Finally, the detected beat note of the soliton and dispersive wave is
shown as the inset in Fig. 4.4e. It is overall somewhat broader than the beatnote of
the other solitons, but is nonetheless quite stable.
4.5

Discussion

We have demonstrated soliton microcombs at 778 nm and 1064 nm using on-chip
high-𝑄 silica resonators. Material-limited normal dispersion, which is dominant
at these wavelengths, was compensated by using geometrical dispersion through
control of the resonator thickness and wedge angle. At the shortest wavelength,
778 nm, mode hybridization was also utilized to achieve anomalous dispersion
while maintaining high optical 𝑄. These results are the shortest wavelength soliton
microcombs demonstrated to date. Moreover, the hybridization method can be
readily extended so as to produce solitons over the entire visible band. The generated
solitons have pulse repetition rates of 20 GHz at both wavelengths. Such detectable
and electronics-compatible repetition rate soliton microcombs at short wavelengths
have direct applications in the development of miniature optical clocks [4] and
potentially optical coherence tomography [74, 75]. Also, any application requiring
low-power near-visible mode-locked laser sources will benefit. The same dispersion
control methods used here should be transferable to silica ridge resonator designs
that contain silicon nitride waveguides for on-chip coupling to other photonic devices
[87]. Dispersive wave generation at 758 nm was also demonstrated. It could be
possible to design devices that use solitons formed at either 778 nm or 1064 nm for
dispersive-wave generation into the visible and potentially into the ultra-violet as
has been recently demonstrated using straight silica waveguides [83].

28
Chapter 5

LOW POWER OPERATION OF SOLITON MICROCOMBS
ALONG ISO-CONTOURS
1Dissipative Kerr solitons can be generated within an existence region defined on a
space of normalized pumping power versus cavity-pump detuning frequency. The
contours of constant soliton power and constant pulse width in this region are
studied through measurement and simulation. Such iso-contours impart structure to
the existence region and improve understanding of soliton locking and stabilization
methods. As part of the study, dimensionless, closed-form expressions for soliton
power and pulse width are developed (including Raman contributions). They provide
iso-contours in close agreement with those from the full simulation, and, as universal
expressions, can simplify the estimation of soliton properties across a wide range of
systems.
Temporal optical solitons resulting from the balance of dispersion with the Kerr
nonlinearity have long been studied in optical fiber systems [96, 97]. In addition
to their many remarkable properties, these nonlinear waves are important in mode
locking [98], continuum generation [99], and were once considered as a means to
send information over great distances [100, 101]. Recently, a new type of dissipative temporal soliton [102] was observed in optical fiber resonators [36]. These
coherently driven cavity solitons (CSs) were previously considered a theoretical possibility [35], and related soliton phenomena including breather solitons and Raman
interactions have also been reported in this system [103–105]. While leveraging the
Kerr effect to balance dispersion, this soliton also regenerates using Kerr-induced
parametric amplification [32]. Their recent demonstration in microcavity systems
[37–40, 43, 44, 71] has made possible highly stable frequency microcombs [34, 67].
Referred to as dissipative Kerr solitons (DKs) in the microcavity system, soliton phenomena including the Raman self-shift [89, 90, 106], optical Cherenkov
radiation [43, 106–109], multi-soliton systems [110–112], and the co-generation
of new types of solitons [68] have been reported. Moreover, the compact soliton microcomb devices are being studied for systems-on-a-chip applications such
1 Work presented in this chapter has been published in [95] “Universal iso-contours for dissipative

Kerr solitons”, Optics Letters 43 11, 2567-2570 (2018). Boqiang Shen conducted the experiment,
did the numerical simulation, prepared the data, and participated in the writing of the manuscript.

29
as dual-comb spectroscopy [11], precision distance measurement [16, 17], optical
communications [113], and optical frequency synthesis [9].
5.1

Dissipative Kerr soliton phase diagram and iso-power contours

Regions of stability and existence are well known in driven soliton systems [114].
These properties of DKs and CSs have been studied using the Lugiato-Lefever (LL)
equation [35, 115] in a space of normalized pumping power and cavity-pump frequency detuning [37, 116–118]. In analogy with thermodynamic phase diagrams,
this soliton existence diagram also contains other regions of existence including
those for breather solitons as well as more complex dynamical phenomena [119–
121]. Fig. 5.1 is a typical diagram showing only the stable soliton region. In
thermodynamic phase diagrams, another useful construct is the iso-contour for processes performed with a state variable held constant (e.g., isochors and isotherms).
These contours not only provide a way to understand processes within the framework of the phase diagram, but impart structure to the phase diagram that improves
intuition of thermodynamical processes. In this work, contours of constant soliton
power and constant pulse width are measured and compared with theory. Closedform expressions for normalized power and pulsewidth are also developed including
the Raman process.
The normalized LL equation is shown below as Eq. 5.1 [37]. The slowly-varying
field envelope 𝜓 is defined such that |𝜓| 2 = (2𝑔/𝜅)𝑁 where 𝑁 is photon number, 𝜅 is
the cavity mode power damping rate, and 𝑔 = ℏ𝜔2c 𝑐𝑛2 /(𝑛2𝑉0 ) is the Kerr coefficient
with material refractive index 𝑛, Kerr nonlinear index 𝑛2 , optical mode volume 𝑉0 ,
cavity resonant frequency 𝜔𝑐 , Planck’s constant ℏ, and speed of light 𝑐. 𝜏 = 𝜅𝑡/2
and 𝜃 ≡ 𝜙 𝜅/2𝐷 2 are the normalized time and cavity polar coordinate (𝜙) where
𝐷 2 is the second-order dispersion parameter [37, 38]. 𝑓 2 ≡ 𝑃/𝑃th is the ratio of the
input pump power and parametric threshold power [32, 38] and 𝜁 ≡ (𝜔c − 𝜔p )(2/𝜅)
is the normalized frequency detuning between cavity resonant frequency 𝜔𝑐 , and
pump frequency 𝜔 𝑝 . 𝛾 ≡ 𝐷 1 𝜏R 𝜅/2𝐷 2 is the normalized Raman coefficient where
𝜏R is the material Raman constant [90] and 𝐷 1 /2𝜋 is the cavity free-spectral-range
[37, 38].
𝜕𝜓
1 𝜕2𝜓
𝜕|𝜓| 2
= 𝑗
|𝜓|
(1
𝜁)𝜓
𝜓 + 𝑓.
(5.1)
𝜕𝜏
2 𝜕𝜃 2
𝜕𝜃
Iso-power contours found by solving Eq. 5.1 are shown in Fig. 5.1 as red contours.
The analysis is performed for a high-Q silica resonator and parameters used in the
calculation are provided below and in the Fig. 5.1 caption. Numerical simulation is

30
based on propagating the LL equation from an initial soliton seed until steady-state
is achieved [90].
The following simplified analytical solution is also used to study soliton behavior
[37]: 𝜓 = 𝐴 + 𝐵 sech(𝜃/𝜏𝜃 )𝑒 𝜄𝜙0 , where 𝐴 is the soliton background field, 𝐵 is the
amplitude, 𝜙0 is the soliton phase, and 𝜏𝜃 ≡ q
𝜏s /𝜏0 is the normalized soliton pulse
width (𝜏𝑠 is the physical pulse width and 𝜏0 ≡ 2𝐷 2 /(𝜅𝐷 21 )). By Fourier transform,
the soliton spectrum in optical frequency 𝜈 varies as sech(𝜈/𝜈s ) where 𝜈s 𝜏s = 𝜋 −2 .
Approximate expressions giving the Raman-free dependence of amplitude and pulse
width on detuning and pump power have been developed [37, 118]. By including
high-order corrections and Raman corrections, the following improved expressions
result as well in an expression for soliton average power,
p ©
5 𝜋 2 𝑓 2 − 8𝜁 ª
64 2 3
(5.2)
𝛾 𝜁
𝐵(𝜁, 𝑓 , 𝛾) ≈ 2𝜁 1 +
® 1−
225
2𝜁 3
1 ©
1 𝜋 2 𝑓 2 − 8𝜁 ª
64 2 3
𝜏𝜃 (𝜁, 𝑓 , 𝛾) ≈ p 1 −
(5.3)
𝛾 𝜁
® 1+
225
2𝜁 3
2𝜁
p ©
𝑝(𝜁, 𝑓 , 𝛾) ≈ 2𝜁 1 +

𝜋 2 𝑓 2 − 8𝜁 ª
64 2 3
𝛾 𝜁
® 1−
225
2𝜁 3

(5.4)

where 𝑝 ≡ 𝑃sol /𝑃0 is the normalized time-averaged soliton power 𝑃sol and 𝑃0 ≡
(𝜅 𝐸 ℏ𝜔c /𝜋𝑔) 𝜅𝐷 2 /2 with 𝜅 𝐸 the optical loss rate from waveguide-resonator coupling [38]. As an aside, the requirement of the square root to be real in these
expressions (𝜁 < 𝜋 2 𝑓 2 /8) gives the approximate upper bound of detuning for
soliton existence in the phase diagram [37, 94, 114]. Dotted lines in Fig. 5.1
are the iso-power contours using Eq. 5.4 (equivalently 𝑝(𝜁, 𝑓 , 𝛾) = Constant for
𝛾 = 2.1 × 10−3 ), and they are in excellent agreement with the simulation contours.
Raman contributions become especially important at larger detuning values where
the soliton spectrum increases in width [89, 90]. To illustrate this point, the dashed
curves in Fig. 5.1 result by using Eq. 5.4 except with 𝛾 = 0.
5.2

Measurement system and low power operation

The experimental setup is shown in Fig. 5.2a. The resonator was an ultra-high-Q
silica wedge resonator having a diameter of approximately 3 mm (free spectral range
𝐷 1 /2𝜋 = 21.9 GHz). Further details on its fabrication are presented elsewhere [86].

31
Cavity-pump detuning (MHz)
10
12

50
40

6.0

7.9

7.0

5.7

6.8

7.4

5.4

6.4

6.0
5.5

15
5.5

7.5

5.1

7.0

6.5
6.0
5.5

10
4.0

4.5

5.0

Theory
Simulation
Experiment

15
20
25
Normalized detuning z

Pump power (mW)

Normalized Pump Power f ²

Figure 5.1: Dissipative Kerr soliton phase diagram and iso-power contours.
The phase diagram features normalized pump power 𝑓 2 along the vertical axis and
normalized detuning 𝜁 along the horizontal axis. The green region contains stable
soliton states. Black dotted lines (gray dashed lines) are iso-power contours using
Eq. 5.4 with Raman term (w/o Raman). 𝑝 is incremented from 4.0 to 8.0 in
steps of 0.5. Red lines are simulated iso-power contours using Eq. 5.1. Blue dots
give the measured soliton iso-power contours at the following soliton powers: 93,
99.5, 117.5, 125, 129, 136, 145 µW (left to right), which correspond to 𝑝 values
of 5.1, 5.5, 6.4, 6.8, 7.0, 7.4, and 7.9. For these measurements, 𝑄 = 197 million
(𝜅/2𝜋 = 0.98 MHz), 𝜅 𝐸 /𝜅 = 0.26, and 𝛾 = 2.1 × 10−3 . Inset shows the measured
iso-power contours using another similar device, with soliton powers of 299, 320,
and 335 µW (left to right), which correspond to 𝑝 values of 5.4, 5.7, and 6.0. For
these measurements, 𝑄 = 115 million (𝜅/2𝜋 = 1.69 MHz), 𝜅 𝐸 /𝜅 = 0.39, and
𝛾 = 2.8 × 10−3 . Large green and blue data points correspond to spectra in Fig. 5.2

The measurement used the TE1 mode family pumped at 1550 nm and the secondorder dispersion was measured to be 𝐷 2 /2𝜋 = 12.1 kHz at 1550 nm by a method
reported elsewhere [38]. The mode area was calculated to be 𝐴eff = 40 µm2 and
the silica Raman constant 𝜏R = 2.4 fs was also used [90], which is valid when
the soliton spectral width is below 13 THz [122]. Finally, the resonator used in
this measurement featured minimal avoided mode crossings and dispersive waves.
Their presence would interfere with the ideal power dependence predicted by the LL
equation. To measure pump detuning, weak phase-modulation of the pump light and
detection of converted amplitude modulation sidebands was performed [118, 123].
In this method, pump light reflected by the fiber Bragg grating (FBG) contains the

(a)
EOM

EDFA

Triggering &
FG
Attenuation

Servo

FG
Lock point
PD

AOM

Resonator
PM

OSA

Comb

ESA

FBG

PM
Oscilloscope

(b)
z=21 f =53
z=8.0 f =9.0
sech² fitting

Figure 5.2: Measurement system and low power operation. (a) Measurement
setup. EOM: electro-optical phase modulator. PC: polarization controller. PM:
In-line power meter. PD: Photodetector. FG: function generator. (b) Soliton spectra
at normalized detuning and pumping power (𝜁 = 21, 𝑓 2 = 53) (blue) and (𝜁 = 8.0,
𝑓 2 = 9.0) (green). The corresponding phase diagram locations are marked in Fig.
5.1. Red curve: squared hyperbolic-secant fitting.

33
Cavity-pump detuning (MHz)
10
12

0.150

0.124
0.137
0.128

0.168

60
30
0.135

0.150
0.165
0.180
0.195

0.210

Pump power (mW)

Normalized Power f ²

Theory
Simulation 20
Exp. interp.

10
15
20
25
Normalized detuning z

Figure 5.3: Iso-contours of soliton pulse width. The device is unchanged from Fig.
5.1 main panel. Red solid lines (black dotted lines) are simulated (Eq. 5.3 theory)
iso-contours of normalized pulse width 𝜏𝜃 ranging from 0.21 to 0.135 (equidistant
steps of 0.015). Blue solid lines are the linear interpolation from measurement of
iso-contours at 190, 170, 155, 145, 140 fs, which correspond to 𝜏𝜃 : 0.168, 0.150,
0.137, 0.128, 0.124.

modulation information and is analyzed by an electrical spectrum analyzer (ESA)
to retrieve the detuning frequency. The soliton spectrum transmitted past the FBG
is sent to a detector and optical spectrum analyzer (OSA) for analysis. To determine
soliton power, the FBG filtered line was manually reinserted.
5.3

Iso-contours of soliton pulse width

Triggering and locking of single soliton states used the soliton average power to
servo control the pump laser frequency [91]. Because this soliton locking method
maintains a constant soliton power, it provides a convenient way to map out the
iso-power contours. Specifically, as opposed to varying (𝜁, 𝑓 2 ) in the phase diagram
and monitoring soliton power, the iso-power measurement proceeded by varying
only the pumping power with the soliton locked at constant output power. The servo
control then compensates for these variations by adjusting the pumping frequency.
The corresponding detuning was then recorded as described above. Pump power was
varied using a combination of an acousto-optical modulator (AOM) and Erbiumdoped fiber amplifier (EDFA). Upon completion of an iso-power contour, the soliton
power setpoint was adjusted and the measurement repeated. The measured iso-power

34
data points are shown in the main panel of Fig. 5.1. Each measurement proceeded
until it was no longer possible to reliably lock the soliton state. There is overall
good agreement between measurement and theory. Errors are largest at lowest
detuning values, however, even here they are relatively small (∼10%). The ability to
measure the contours over such large ranges and their good agreement with theory
and simulations showcases the system’s robustness and quality. As an additional
test, a second loading condition was also measured. The inset to Fig. 5.1 shows
this data, which is in reasonable agreement with simulation and Eq. 5.4. Measured
soliton powers have experienced an ∼1.2 dB insertion loss between the resonator
and the detector. It is also noted that breather solitons could be stably locked near the
upper boundary in Fig. 5.1. However, the region was small, making measurement
of iso-power contours difficult. As a result, breathers were not studied in this work.
Stable generation of solitons at small detuning is of practical importance for low
pumping power operation of the soliton system. To this end, the green data point
(𝜁=8.0, 𝑓 2 =9.0) in Fig. 5.1 shows both the lowest detuning and the lowest power
soliton state observed in this study. The corresponding unnormalized quantities are
4.2 MHz and 10.8 mW. This is, to the authors’ knowledge, the lowest operating
power reported for any soliton microcomb platform. Making this result equally
important is that the repetition rate is detectable (21.9 GHz) requiring large mode
volume and hence higher pumping power levels as compared to, for example, THzrate microcombs. Corresponding soliton spectra are presented in Fig. 5.2(b). The
result was achieved by both the use of a high-quality-factor resonator sample as well
as the improved understanding gained through these measurements of the stability
regional boundaries [91]. For comparison, a soliton spectrum produced at (𝜁=21,
𝑓 2 =53) is also shown in Fig. 5.2(b). These values are plotted as the light blue data
point in Fig. 5.1 and correspond to unnormalized quantities 11.3 MHz and 63.5
mW. The cavity loading condition for these two spectra is: loaded 𝑄 = 182 million
and 𝜅 𝐸 /𝜅 = 0.44.
In parallel with the iso-power data point collection, the soliton pulse width was
also measured by fitting of the optical spectral envelope [38]. Then the data set
(𝜁, 𝑓 2 , 𝜏𝜃 ) was linearly interpolated to determine iso-contours of pulse width (blue
contours in Fig. 5.3). It was not possible to interpolate iso-pulse-width contours at
lower detuning values where there are fewer iso-power data points. For comparison,
simulated pulse width (red) and the analytical expression, Eq. 5.3 (dotted black) are
plotted. The interpolated pulse width iso-contours are less accurate than the directly

35
measured power contours but nonetheless show reasonable agreement between the
data and theory. Overall, the pulse width contours are more weakly dependent upon
normalized pumping power (i.e., more vertical) as compared to the soliton power
contours.
5.4

Summary

Contours of constant power and constant pulse width have been measured for dissipative Kerr solitons. Measurements were found to be in good agreement with the LL
equation numerical model augmented by Raman interactions. There was also good
agreement with the predictions of closed-form expressions that include the Raman
interaction. Compared with the the large-detuning approximation which predicts
that soliton power depends only upon resonator-pump detuning (i.e., vertical isopower contours), it is found that soliton power depends both upon pumping power
and detuning. The resulting tilt of iso-power contours at low detuning suggests that
soliton locking by servo control of pumping power could potentially be an option
for low-detuning ranges just as servo control of pump frequency is used at larger
detuning ranges. Stable soliton operation for pump powers as low as 10.8 mW
was also demonstrated in the course of this work. These measurements provide
structure to the phase diagram picture of soliton existence. The universal nature of
the closed-form expressions should make them suitable for use in other CS and DK
soliton platforms. Future work could consider incorporating higher order dispersion
into the analysis to include, for example, the impact of phenomena such as dispersive
waves.

36
Chapter 6

VERNIER SPECTROMETER USING DUAL-LOCKED
COUNTERPROPAGATING SOLITON MICROCOMBS
1 Frequency-agile lasers are ubiquitous in sensing, spectroscopy, and optical communications [124–126], and measurement of their optical frequency for tuning and
control is traditionally performed by grating and interferometer-based spectrometers, but more recently these measurements can make use of optical frequency
combs [1]. Frequency combs provide a remarkably stable measurement grid against
which optical signal frequencies can be determined subject to the ambiguity introduced by the equally spaced comb teeth. The ambiguity is resolved for continuously
frequency swept signals by counting comb teeth [127] relative to a known comb
tooth, and this method has enabled measurement of remarkably high chirp rates
[128]. However, many signal sources will experience intentional or unintentional
frequency jumps. Here, the ambiguity can be resolved using a second frequency
comb that has a different comb tooth spacing so as to provide a frequency Vernier
scale for comparison with the first comb [129–131]. This Vernier concept is also
used in dual comb spectroscopy [10, 11], but in measuring active signals, the method
can be enhanced to more directly (and hence quickly) identify signal frequencies
through a signal correlation technique [131]. Moreover, continuous as opposed to
discretely sampled frequencies are measured in the active approach. The power of
the Vernier-based method relies upon mapping of optical comb frequencies into a
radio-frequency grid of frequencies, the precision of which is set by the relative
line-by-line frequency stability of the two frequency combs. This stability can be
guaranteed by self-referencing each comb using a common high-stability radiofrequency source or through optical locking of each comb to reference lasers whose
relative stability is ensured by mutual locking to a common optical cavity.
In this chapter, we show that a single microresonator provides rapid and broad-band
measurement of optical frequencies with a relative frequency precision comparable
to conventional dual frequency comb systems. Dual-locked counter-propagating
(CP) solitons having slightly different repetition rates are used to implement a
1 Work presented in this chapter has been published in [14] “Vernier spectrometer using coun-

terpropagating soliton microcombs”, Science 363, 965-968 (2019). Boqiang Shen conducted the
experiment, prepared the data, and participated in the writing of the manuscript.

37
Vernier spectrometer. Laser tuning rates as high as 10 THz/s, broadly step-tuned
lasers, multi-line laser spectra, and also molecular absorption lines are characterized
using the device. Besides providing a considerable technical simplification through
the dual-locked solitons and enhanced capability for measurement of arbitrarily
tuned sources, our results reveal possibilities for chip-scale spectrometers that exceed
the performance of table-top grating and interferometer-based devices.
6.1

Concept of Vernier spectrometer

We demonstrate a broad-band, high-resolution Vernier microresonator soliton spectrometer (MSS) using a single miniature comb device that generates two mutuallyphase-locked combs. The principle of operation relies upon an optical phase locking effect observed in the generation of counter-propagating solitons within high-Q
whispering gallery resonators [110]. Soliton generation in microresonators is being
studied for miniaturization to the chip-scale of complete comb systems and these
soliton microcombs have now been demonstrated in a wide range of microresonator
systems [41]. It has been shown that counter-propagating solitons can have distinct,
controllable repetition rates and that their underlying comb spectra can be readily phase locked at two spectral points [110]. This mutual double-locking creates
line-by-line relative frequency stability for the underlying microcomb spectra that
is more characteristic of fully self-referenced dual comb systems. The resulting
Vernier of comb frequencies in the optical domain maps to an exceptionally stable
radio frequency grid for implementation of the spectrometer.
Phase-locked CP solitons
The spectral relationship of the doubled-locked cw and ccw solitons reveals the inherent optical frequency Vernier (Figure 6.1(a)). A single laser source is modulated
(Figure 6.1(b)) to produce the two mutually-coherent pump lines at order 𝜇 = 𝑁
with frequency separation Δ𝜈 (MHz range). The distinct pump frequencies cause the
soliton repetition rates to differ by Δ 𝑓𝑟 as a result of the Raman self-frequency-shift
[88–90, 110, 132]. As detailed elsewhere, the cw and ccw combs will experience
frequency locking (induced by optical backscattering) at order 𝜇 = 0 for certain
pumping frequencies [110]. This locking requires that Δ𝜈 = 𝑁Δ 𝑓𝑟 . Also, because
the two pump frequencies are derived from radio-frequency modulation of a single
laser source, they have a high relative frequency stability (Δ𝜈 is very stable) and
are effectively locked at order 𝜇 = 𝑁. This double locking sets up a stable Vernier
in the respective soliton comb frequencies. The counter-propagating solitons are

38
c.w. laser
fL
Phase locking
Dfr

Pump
Dn=NDfr

Correlation (arb.unit)

Chemical
absorption

Dfn1
Dfn2
Optical frequency

m=N

PD
50/50
PD

50/50

AOM
PD

50/50

resonator

c.w. pump

V3
50/50 50/50

Signal processing

CIRC

Frequency (MHz)

Power (arb. unit)

3.0
n=54

Ref laser

2.5
Time (1 ms/div)

CW soliton

Wavelength (pm
+1553.93305 nm)

CCW soliton
Pump

Power (20 dB/div)

CIRC

AOM

CW soliton

Dfn1-Dfn2=nDfr

3.5

Oscilloscope

50/50
Signal laser
11 mm
mm

RBW
200 Hz

Gas cell

Frequency (MHz)

m=n

m=0

0.02

-0.02
Time (1 ms/div)

Pump

1500

1550
Wavelength (nm)

CCW soliton

1600

Residuals (pm)

0.1

-0.1
1545

1550
1555
Wavelength (nm)

1560

Figure 6.1: Spectrometer concept, experimental setup and static measurement.
(a) Counter propagating soliton frequency combs (red and blue) feature repetition
rates that differ by Δ 𝑓𝑟 , phase-locking at the comb tooth with index 𝜇 = 0 and
effective locking at 𝜇 = 𝑁 thereby setting up the Vernier spectrometer. Tunable
laser and chemical absorption lines (grey) can be measured with high precision.
(b) Experimental setup. AOM: acousto-optic modulator; CIRC: circulator; PD:
photodetector. Small red circles are polarization controllers. Inset: scanning
electron microscope image of a silica resonator. (c) Optical spectra of counterpropagating solitons. Pumps are filtered and denoted by dashed lines. (d) Typical
measured spectrum of 𝑉1𝑉2 used to determine order 𝑛. For this spectrum: Δ 𝑓𝑛1 −Δ 𝑓𝑛2
= 2.8052 MHz and Δ 𝑓𝑟 = 52 kHz giving 𝑛 = 54. (e) The spectrograph of the dual
soliton interferogram (pseudo color). Line spacing gives Δ 𝑓𝑟 = 52 kHz. White
squares correspond to the index 𝑛 = 54 in panel c. (f) Measured wavelength of an
external cavity diode laser operated in steady state. (g) Residual deviations between
ECDL laser frequency measurement as given by the MSS and a wavemeter. Error
bars give the systematic uncertainty as limited by the reference laser in panel b.

generated in a high-𝑄 silica microresonator with 3 mm diameter (22 GHz soliton repetition rate) [86]. Details on the soliton generation process can be found
elsewhere [38, 91, 110]. Typical optical spectra of cw and ccw solitons span the
telecommunication C-band (Figure 6.1(c)).

39
Operation principle
The spectrometer operates as follows. A test laser frequency 𝑓 𝐿 is measured using
either of the following expressions: 𝑓 𝐿 = 𝑛 𝑓𝑟1,2 + Δ 𝑓𝑛1,2 + 𝑓0 where 𝑛 is the comb
order nearest to the laser frequency, 𝑓𝑟1,2 are the comb repetition rates, Δ 𝑓𝑛1,2 are
the heterodyne beat frequencies of the test laser with the two frequency comb teeth
at order 𝜇 = 𝑛, and 𝑓0 is the frequency at 𝜇 = 0. 𝑓𝑟1,2 and Δ 𝑓𝑛1,2 are measured
by co-detection of the combs and the test laser to produce the electrical signals
𝑉1,2 in Fig. 6.1(b). Fast Fourier transform (FFT) of 𝑉1𝑉2 gives the spectral line
at 𝑛Δ 𝑓𝑟 (Fig. 6.1(d)) using the correlation method [131] and, in turn, the order
𝑛. The correlation method can be understood as a calculation of the frequency
difference Δ 𝑓𝑛2 − Δ 𝑓𝑛1 = 𝑛Δ 𝑓𝑟 by formation of 𝑉1𝑉2 followed by fast Fourier
transform (FFT). The FFT spectrum of 𝑉1𝑉2 gives the spectral line at 𝑛Δ 𝑓𝑟 (Fig.
6.1(d)). To determine 𝑛, it requires Δ 𝑓𝑟 = 𝑓𝑟2 − 𝑓𝑟1 which is measured by heterodyne
of the solitons to produce electrical signal 𝑉3 . Figure 6.1(e) is a narrow frequency
span of the FFT of 𝑉3 and shows how the optical frequency Vernier is mapped into
a stable radio-frequency grid with line spacing Δ 𝑓𝑟 . The order corresponding to
the FFT of the 𝑉1𝑉2 signal (Fig. 6.1(d) spectrum) is also indicated. These steps
are performed automatically to provide a real time measurement of 𝑓 𝐿 relative to
𝑓0 . 𝑓0 is determined by applying this procedure to the reference laser frequency
𝑓ref (stabilized using an internal molecular reference). All of these data inputs are
automatically processed in real time to measure 𝑓 𝐿 .
6.2

Measurement of a static laser

As a preliminary test, the frequency of an external-cavity-diode-laser is measured
and compared against a wavemeter. Figures 6.1(d) and (e) (𝑛 = 54) are from
this measurement. The real-time measured wavelength of the laser (Figure 6.1(f))
fluctuates within ±0.02 pm over a 5 ms time interval. The measurement was repeated
from 1545 to 1560 nm with residual deviations less than 0.1 pm versus the wavemeter
measurement (Figure 6.1(g)). These deviations are believed to be limited by the
wavemeter resolution (±0.1 pm). The systematic uncertainty of absolute wavelength
in the current setup is set by the reference laser to around ±4 MHz (±0.03 pm).
6.3

Measurement of dynamic lasers

The large, microwave-rate free-spectral range of the MSS enables tracking of lasers
undergoing fast-chirping or discontinuous broadband tuning. Although correlation
is performed with a time interval 𝑇𝑊 = 1/Δ 𝑓𝑟 , the instantaneous frequency of

40
Scanning laser

Index n

-12.4

THz/

-0.5

Residuals
(pm)

Time (2 ms/div)
-12.39 THz/s

-5
Time (2 ms/div)

0.5

Wavelength (nm
+1555 nm)

-5
Time (2 ms/div)

Wavelength
(5 pm/div)

Wavelength (nm
+1553 nm)

6.1

Step-tuned laser

1557.613 nm

Time (0.2 ms/div)

High-resolution Spectroscopy
0.12
Absorbance (dB)

D H12CN

0.10
0.08
0.06
0.02

J=25

2.6 GHz

0.04

Ground
Measurement
Fitting

-0.1
0.1
Wavenumber (cm-1 +6433.2090 cm-1)

-1

J=26
J=27

cm -1
2090
433. 095 cm
433.
-1
R: 6
94 c -1
cm
429
M: 6 29.2214
-1
R: 6
cm
1897 cm-1
M: 6 25.1931
R: 6

J=24
J=25

2v1

J=26

Figure 6.2: Laser tuning and spectroscopy measurements. (a) Measurement of
a rapidly tuning laser showing index 𝑛 (upper), instantaneous frequency (middle),
and higher resolution plot of wavelength relative to average linear rate (lower), all
plotted versus time. (b) Measurement of a broadband step-tuned laser as for laser in
panel A. Lower panel is a zoom-in to illustrate resolution of the measurement. (c)
Spectroscopy of H12 C14 N gas. A vibronic level of H12 C14 N gas at 5 Torr is resolved
using the laser in panel A. (d) Energy level diagram showing transitions between
ground state and 2𝜈1 levels. The measured (reference) transition wavenumbers are
noted in red (blue).

the laser relative to the combs can be acquired at a much faster rate set by the
desired time-bandwidth-limited resolution. To avoid aliasing of the correlation
measurement (i.e., to determine 𝑛 uniquely), the amount of frequency-chirping
should not exceed the repetition rate 𝑓𝑟 within the measurement window 𝑇𝑊 , which
imposes a maximum resolvable chirping-rate of 𝑓𝑟 × Δ 𝑓𝑟 . This theoretical limit is 1
PHz/s for the MSS and represents a boost of 100× compared with previous Vernier
spectrometers [131].
Measurement of rapid continuous-tuning of an external cavity diode laser is shown

41
in Figure 6.3(a). The correlation measurement evolves as the laser is tuned over
multiple FSRs of the comb and thereby determines the index 𝑛 as a function of
time (Figure 6.3(a) upper panel). Measurement of the linear frequency chirp (12.4 THz/s) as well as the frequency versus time at high resolution (by subtracting
the average linear frequency ramp) are shown in the Figure 6.3(a) middle and
lower panels, respectively. The discontinuities in the measurement are caused
by electrical frequency dividers used to reduce the detected signal frequency for
processing by a low-bandwidth oscilloscope. The dividers can be eliminated by
using a faster oscilloscope. In Figure 6.3(b), measurement of broadband step tuning
(mode hopping) of an integrated-ring-resonator tunable III-V/Silicon laser diode
[133] is presented. Fast step tuning between 1551.427 nm and 1557.613 nm every 1
ms with the corresponding index 𝑛 stepping between 𝑛 = 64 and 𝑛 = 29 is observed.
The lower panel in Figure 6.3(b) gives a higher resolution zoom-in of one of the
step regions. The data points in these measurements are acquired over 1𝜇s so the
resolution is approximately 1 MHz.
6.4

High-resolution spectroscopy

This combination of speed and precision is also useful for spectroscopic measurements of gas-phase chemicals using tunable, single-frequency lasers. To demonstrate, an absorption line of H12 C14 N at 5 Torr is obtained by a scanning laser
calibrated using the MSS (Figure 6.3(c)). The linewidth is around 2.6 GHz and the
absorbance is as weak as 0.12 dB. Separate measurements on vibronic transitions
between the ground state and 2𝜈1 states were performed. The corresponding transition wavenumbers obtained by pseudo-Voigt fitting are in excellent agreement with
the HITRAN database (Figure 6.3(d)) [134].
6.5

Measurement of multi-line spectra

To illustrate a measurement of more complex multi-line spectra, a fiber mode-locked
laser (FMLL) is characterized (Figure 6.3(a)). The FMLL full spectrum (Figure
6.3(b)) was first bandpass filtered to prevent detector saturation. Also, the frequency
extraction procedure is modified to enable unique identification of many frequencies.
The reconstructed FMLL spectrum measured using the MSS is plotted in Figure
6.3(c). In an additional study of the FMLL, the MSS is used to measure 6 closelyspaced-in-frequency groups of lines located at various spectral locations spanning
2500 free-spectral-ranges of the mode-locked laser (Figure 6.3(d)). A linear fitting
defined as 𝑓𝑚 = 𝑓𝑜 + 𝑚 𝑓rep is plotted for comparison by using the photodetector-

OSA

4 ns
Mode-locked
laser

OSA

1500

Wavelength (nm)

1600
MSS

92 X
249.7 MHz

650

Measurement
Fitting

Frequency (GHz
+192.79 THz)

82 X
249.7 MHz

26
27
Frequency (GHz+192.79 THz)

550
100

100

200 2300 2400 2500
Relative index m

Residuals (MHz)

Power (arb. unit)

MSS

Power (20 dB/div)

0.5

-0.5

100 200 2300 2400 2500
Relative index m

Figure 6.3: Measurement of a fiber mode-locked laser. (a) Pulse trains generated
from a fiber mode-locked laser (FMLL) are sent into an optical spectral analyzer
(OSA) and the MSS. (b) Optical spectrum of the FMLL measured by the OSA. (c)
Optical spectrum of the FMLL measured using the MSS over a 60-GHz frequency
range (indicated by dashed line). (d) Measured (blue) and fitted (red) FMLL mode
frequencies versus index. The slope of the fitted line is set to 249.7 MHz, the
measured FMLL repetition rate. (e) Residual MSS deviation between measurement
and fitted value.

measured FMLL repetition rate 𝑓rep = 249.7 MHz, where 𝑚 and 𝑓𝑜 represent the
relative mode index and fitted offset frequency at 𝑚 = 0, respectively. The residual
deviation between the measurement and linear fitting is shown in Figure 6.3(e) and
gives good agreement. The slight tilt observed in Figure 6.3(e) is believed to result
from drifting of soliton repetition rates which were not monitored real-time. Also,
variance of residuals within each group comes from the 300 kHz linewidth of each
FMLL line. Drifting of the reference laser and FMLL carrier-envelope offset also
contribute to the observed residuals across different measurements.
6.6

Signal processing

In this section, the algorithms used to extract the absolute frequencies of lasers are
presented.

43
Single frequency laser
Through heterodyne of the test laser with the nearest comb teeth, the phase 𝜓 of the
test laser is related to the electrical signals 𝑉1,2 by
𝑉1,2 ∝ cos(𝜓 − 2𝜋𝜈𝑛1,2 𝑡),

(6.1)

where 𝜈𝑛1,2 represent the frequencies of nearest comb teeth and have order 𝑛. We
also have 𝜈𝑛2 − 𝜈𝑛1 = 𝑛Δ 𝑓𝑟 as a result of the CP soliton locking. A Hilbert transform
is used to extract the time-dependent phase 𝜓 − 2𝜋𝜈𝑛1,2 𝑡 from 𝑉1,2 which thereby
gives the heterodyne frequencies via
Δ 𝑓𝑛1,2 = 𝜓/2𝜋
− 𝜈𝑛1,2 .

(6.2)

Each data point of Δ 𝑓𝑛1,2 is obtained by linear fitting of the phase over a specified
time interval that sets the frequency resolution. Similarly, the heterodyne frequency
between the reference laser and the soliton comb can be retrieved to determine the
frequency 𝑓0 (see discussion in main text).
The Fourier transform of the product 𝑉1𝑉2 is given by
∫ 𝑇𝑊 𝑖(𝜓−2𝜋𝜈𝑛1 𝑡)
+ 𝑒 −𝑖(𝜓−2𝜋𝜈𝑛1 𝑡) 𝑒𝑖(𝜓−2𝜋𝜈𝑛2 𝑡) + 𝑒 −𝑖(𝜓−2𝜋𝜈𝑛2 𝑡) −2𝜋𝑖 𝑓 𝑡
d𝑡
𝑉g
1 2
∝ 𝛿(| 𝑓 | − 𝑛Δ 𝑓𝑟 ),
(6.3)
where sum frequency terms in the integral are assumed to be filtered out and are
therefore discarded. To accurately extract the above spectral signal, the acquisition
time window 𝑇𝑊 should be an integer multiple of 1/Δ 𝑓𝑟 , which is also related
to the pump frequency offset Δ𝜈 by 𝑇𝑊 = 𝑁𝑊 𝑁/Δ𝜈 where 𝑁 is the pump order
and 𝑁𝑊 is an integer. Moreover, the number of sampled points, which equals the
product of oscilloscope sampling rate 𝑓samp and 𝑇𝑊 , should also be an integer (i.e.,
𝑓samp 𝑁𝑊 𝑁/Δ𝜈 is an integer). Here, 𝑓samp is usually set to 2.5 or 5 GHz/s and it is
found that simple adjustment of Δ𝜈 is sufficient to satisfy this condition. As a result,
it is not necessary to synchronize the oscilloscope to external sources. It is noted
that this method is simpler than the asynchronous detection used in previous work
[131].
On account of the limited bandwidth of the oscilloscope used in the work, it was
necessary to apply electrical frequency division to the detected signals for processing
by the oscilloscope. When frequency dividers are used (division ratio 𝑟 = 8), the

44
divided electrical signals (indicated by superscript d) yield
𝑉1,2
∝ cos((𝜓 − 2𝜋𝜈𝑛1,2 𝑡)/𝑟).

(6.4)

d = Δ𝑓
As a result, the divided frequencies also satisfy Δ 𝑓𝑛1,2
𝑛1,2 /𝑟 and the correlation
between the divided signals scales proportionally by
Δ 𝑓𝑛1
− Δ 𝑓𝑛2
= 𝑛Δ 𝑓𝑟 /𝑟.

(6.5)

Therefore the required resolution bandwidth to resolve the ambiguity 𝑛 from the
measured correlation is Δ 𝑓𝑟 /𝑟 which increases the minimal acquisition time to
𝑇𝑊d = 𝑟𝑇𝑊 .
Multi-line spectra
The algorithm used here to extract a large number of frequencies simultaneously
using the MSS is different from the previous single-frequency measurements. Rather
than multiplying the signals 𝑉1 and 𝑉2 followed by Fast Fourier Transform (FFT)
in order to determine the microcomb order, we directly FFT the signals 𝑉1 and 𝑉2
followed by filtering and then frequency correlation. This avoids the generation
of ambiguities. To explain the approach, first consider an implementation similar
to that reported in the main text. There, a fiber mode-locked laser (FMLL) comb
with free-spectral-range (FSR) of about 250 MHz was optically filtered to create a
narrower frequency range of FMLL laser lines extending over only a few microcomb
teeth. The signals 𝑉1 and 𝑉2 upon FFT therefore produce a large set of frequencies
representing the individual beats of each FMLL laser line (index 𝑚) with microcomb
modes (index 𝑛). Figure 6.3(a) gives a narrow frequency span of a typical FFT
generated this way for both the 𝑉1 and 𝑉2 signals. A zoom-in of one pair of 𝑉1
and 𝑉2 signals is provided in Figure 6.4(b) and a remarkably precise frequency
separation between the beats (in view of the spectral breadth of each beat) can be
determined by correlating the upper (blue) and lower (red) spectrum (see Figure
6.4(c)). This precision results from the underlying high relative frequency stability
of the cw and ccw microcomb frequencies. As described in the main text, this
frequency separation is a multiple of Δ 𝑓𝑟 and plot of the correlation versus the
frequency separation (in units of Δ 𝑓𝑟 ) is provided in Figure 6.4(c) where the peak of
the correlation gives the index 𝑛 = 63 for this pair of beat frequencies. Proceeding
this way for each pair of peaks in Figure 6.4(a) allows determination of 𝑛 from which
the frequency of the corresponding FMLL line can be determined. It is interesting
to note that in Figure 6.4(a), there are two sets of peaks that give 𝑛=63, 64, and 65.

45
Intensity (arb. u.)

n = 65

n = 64

n = 63
850

800

n = 63

n = 64

Frequency (MHz) 900

950

Dfm1 = 858.4 MHz
Correlation (arb. u.)

Intensity (arb. u.)

n = 65

V"
857

n = 63

Df = 1.0272 MHz
858
Frequency (MHz)

859

100

Figure 6.4: Multi-frequency measurements. (a) A section of 𝑉˜1,2 . Pairs of
beatnotes coming from the same laser are highlighted and the derived 𝑛 value is
marked next to each pair of beatnotes. (b) Zoom-in on the highlighted region near
858 MHz in (a). Two beatnotes are separated by 1.0272 MHz. (c) Cross-correlation
of 𝑉˜1 and 𝑉˜2 is calculated for each 𝑛 and the maximum can be found at 𝑛 = 63.
These correspond to FMLL lines that are higher and lower in frequency relative to
the microcomb modes with indices 𝑛=63, 64, and 65. The relative alignment of
the blue and red peaks which switches sign for these sets of beat frequencies allows
determination of which FMLL line is lower and higher in frequency relative to the
microcomb teeth.
To provide more rigor to this explanation, the electrical signals consist of multiple
beat components given by,
𝑉1,2 =
𝑉𝑚1,2 , 𝑉𝑚1,2 ∝ cos(𝜓𝑚 − 2𝜋𝜈 𝜇(𝑚)1,2 𝑡),
(6.6)

where 𝜓𝑚 and 𝜈 𝜇(𝑚)1,2 represent the phase of the 𝑚-th FMLL mode and the frequencies of the microcomb order nearest to this FMLL mode, respectively, and where
𝜇(𝑚) denotes the comb order nearest the 𝑚-th FMLL mode. As described in the
main text the frequencies 𝜈 𝜇(𝑚)1,2 are related to the repetition rate difference by
𝜈 𝜇(𝑚)2 − 𝜈 𝜇(𝑚)1 = 𝜇(𝑚)Δ 𝑓𝑟 . The FFT of 𝑉1,2 is denoted by 𝑉˜1,2 and the correlation
given in Figure 6.4(c) (and used to determine the comb order 𝑛 of each spectral

46
component) is given by
∫ Δ 𝑓𝑚1 +𝜅/2
𝑉˜1 ( 𝑓 )𝑉˜2∗ ( 𝑓 + 𝑛Δ 𝑓𝑟 )d 𝑓
Δ 𝑓 −𝜅/2
∫ ∞𝑚1 ∫
2𝜋𝑖 𝑓 𝑡
d𝑓
𝑉𝑚1 (𝑡)𝑒
d𝑡
𝑉𝑚2 (𝑡 0)𝑒 −2𝜋𝑖( 𝑓 +𝑛Δ 𝑓𝑟 )𝑡 d𝑡 0
∫−∞
(6.7)
= 𝑉𝑚1 (𝑡)𝑉𝑚2 (𝑡)𝑒 −2𝜋𝑖𝑛Δ 𝑓𝑟 𝑡 d𝑡
∫ 𝑖(𝜓𝑚 −2𝜋𝜈 𝜇1 𝑡)
+ 𝑒 −𝑖(𝜓𝑚 −2𝜋𝜈 𝜇1 𝑡) 𝑒𝑖(𝜓𝑚 −2𝜋𝜈 𝜇2 𝑡) + 𝑒 −𝑖(𝜓𝑚 −2𝜋𝜈 𝜇2 𝑡) −2𝜋𝑖𝑛Δ 𝑓𝑟 𝑡
d𝑡
∝𝛿(𝜇(𝑚) − 𝑛),
where Δ 𝑓𝑚1 denotes the peak frequency of the beatnote, and 𝜅 is a predetermined
range of integration to cover the linewidth of the beatnote (here 𝜅 = 2 MHz), and
where sum frequency terms in the integral have been discarded. Therefore for
each spectral component 𝑚, its associated microcomb order number 𝜇(𝑚) can be
determined by varying 𝑛 in the above correlation until it reaches maximum (see
Figure 4.4(c)). The 𝑛 value with the maximum correlation will be assigned to the
peak as the tooth number 𝜇(𝑚) and then the absolute frequency can be recovered.
The limit of this process to accommodate more FMLL frequencies is much higher
than that given by the filter bandwidth studied in this work. It is instead set by
the spectral density of FMLL-microcomb beat lines that can be reasonably resolved
within the microcomb FSR spectral span.
6.7

Conclusion

Our soliton spectrometer uses dual-locked counter-propagating soliton microcombs
to provide high resolution frequency measurement of rapid continuously and step
tuned lasers as well as complex multi-line spectra. In combination with a tunable
laser, the spectrometer also enables precise measurement of absorption spectra,
including random spectral access (as opposed to only continuous spectral scanning).
Further optimization of this system could include generation of solitons from distinct
mode families, thereby allowing tens-of-MHz repetition rate offset to be possible
[135]. If such solitons can be dual-locked, the increased acquisition speed would
enable measurement of chirping-rates much higher than PHz/s. Operation beyond
the telecommunications band would also clearly be useful and could use internal
[43] or on-chip spectral broadeners [136], and methods for generation of soliton
microcombs into the visible band are possible [50]. Besides the performance
enhancement realized with the soliton microcombs, the use of dual-locked counter-

47
propagating solitons provides a considerable technical simplification by eliminating
the need for a second mutually phase-locked comb. Finally, chip integrable versions
of the current device employing silicon nitride waveguides are possible [87]. These
and other recently demonstrated compact and low-power soliton systems [137, 138]
point towards the possibility of compact microresonator soliton spectrometers.

48
Chapter 7

INTEGRATED TURNKEY SOLITON MICROCOMBS
1 Optical frequency combs have a wide range of applications in science and technology. An important development for miniature and integrated comb systems is
the formation of dissipative Kerr solitons in coherently pumped high-quality-factor
optical microresonators. Such soliton microcombs [41] have been applied to spectroscopy, the search for exoplanets [23, 24], optical frequency synthesis [140], time
keeping [4] and other areas [41]. In addition, the recent integration of microresonators with lasers has revealed the viability of fully chip-based soliton microcombs.
However, the operation of microcombs requires complex startup and feedback protocols that necessitate difficult-to-integrate optical and electrical components, and
microcombs operating at rates that are compatible with electronic circuits—as is
required in nearly all comb systems—have not yet been integrated with pump lasers
because of their high power requirements. Here we experimentally demonstrate
and theoretically describe a turnkey operation regime for soliton microcombs cointegrated with a pump laser. We show the appearance of an operating point at
which solitons are immediately generated by turning the pump laser on, thereby
eliminating the need for photonic and electronic control circuitry. These features
are combined with high-quality-factor 𝑆𝑖3 𝑁4 resonators to provide microcombs
with repetition frequencies as low as 15 gigahertz that are fully integrated into an
industry standard (butterfly) package, thereby offering compelling advantages for
high-volume production.
7.1

Introduction

The integration of microcomb systems faces two considerable obstacles. First, complex tuning schemes and feedback loops are required for generation and stabilization
of solitons [37, 39, 91]. These not only introduce redundant and power-hungry electronic components [137, 141], but also require optical isolation, a function that
has so far been challenging to integrate at acceptable performance levels. Second,
repetition frequencies that are both detectable and readily processed by integrated
1 Work presented in this chapter has been published in [139] “Integrated turnkey soliton microcombs”, Nature 582, 365-369 (2020). Boqiang Shen conceived the experiment, packaged the
chip and performed the measurements. Boqiang Shen also analyzed the data and participated in the
writing of the manuscript.

49
electronic circuits, such as complementary metal–oxide–semiconductor (CMOS)
circuits, are essential for comb self-referencing, the key process that underlies many
comb applications [1]. And while ultra-high-Q silica resonators [38, 87] and Damascene Si3 N4 resonators [138] can attain these rates, their integration with pumps
has not been possible. Here, we show that the nonlinear dynamics of an unisolated
laser-microcomb system creates a new operating point from which the pump laser
is simply turned-on to initiate the soliton mode-locking process. Theory and experimental demonstration of the existence and substantial benefits of this new turnkey
operating point are demonstrated. Moreover, the resulting microcomb system features Q factor performance that enables electronic-circuit rate operation using an
integrated pump.
7.2

Concept of turnkey soliton microcomb

In the experiment, integrated soliton microcombs whose fabrication and repetition
rate (40 GHz down to 15 GHz) are compatible with CMOS circuits [142] are buttcoupled to a commercial distributed-feedback (DFB) laser via inverse tapers (Fig.
7.1(a)). The microresonators are fabricated using the photonic Damascene reflow
process [138, 143] and feature Q factors exceeding 16 million (Fig. 7.1(b)), resulting
in a low milliwatt-level parametric oscillation threshold, despite the larger required
mode volumes of the GHz-rate microcombs. This enables chip-to-chip pumping of
microcombs for the first time at these challenging repetition rates. Up to 30 mW
of optical power is launched into the microresonator. Feedback from the resonator
suppresses frequency noise by around 30 dB compared with that of a free-running
DFB laser (Fig. 7.1(c)) so that the laser noise performance surpasses state-of-theart monolithically integrated lasers [144] and table-top external-cavity-diode-lasers
(ECDL). Given its compact footprint and the absence of control electronics, the
pump-laser/microcomb chipset was mounted into a butterfly package (Fig. 7.1(d))
to facilitate measurements and also enable portability. This level of integration
and packaging combined with turnkey operation makes this a completely functional
device suitable for use in any system-level demonstration.
7.3

New turnkey operating point

In conventional pumping of microcombs, the laser is optically-isolated from the
downstream optical path so as to prevent feedback-induced interference (Fig. 7.2(a)).
And on account of strong high-Q-induced resonant build-up and the Kerr nonlinearity, the intracavity power as a function of pump-cavity detuning features bista-

DC power supply

20 GHz

Laser

CMOS

Microresonator

Transmission (a.u.)

Measurement
Lorentzian fitting

Qo=16 M
-250
Frequency (MHz)

1010

DFB, free running
ULN integrated laser
External cavity laser
DFB, injection locked

Frequency noise (Hz2/Hz)

105

250

100

Lensed fiber

TEC

103

104

105
106
Frequency offset (Hz)

107

Figure 7.1: Integrated soliton microcomb chip. (a) Rendering of the soliton
microcomb chip that is driven by a DC power source and produces soliton pulse
signals at electronic-circuit rates. Four microcombs are integrated on one chip, but
only one is used in these measurements. (b) Transmission signal when scanning the
laser across a cavity resonance (blue). Lorentzian fitting (red) reveals 16 million
intrinsic Q factor. (c) Frequency noise spectral densities (SDs) of the DFB laser
when it is free running (blue) and feedback-locked to a high-Q Si3 N4 microresonator
(red). For comparison, the frequency noise SDs of ultra-low-noise integrated laser
on silicon (grey) and a table-top external cavity diode laser (black) are also plotted.
(d) Images of a pump/microcomb in a compact butterfly package.

bility. The resulting dynamics can be described using a phase diagram comprising
continuous-wave (c.w.), modulation instability (MI) combs and soliton regimes that
are accessed as the pump frequency is tuned across a cavity resonance. A typical
plot (Fig. 7.2(b)) is made versus 𝛿𝜔, the difference of cavity resonance and pump
laser frequency (i.e., 𝛿𝜔 > 0 indicates red detuning of the pump frequency relative
to the cavity frequency) [37]. The tuning through the MI regime functions to seed
the formation of soliton pulses. On account of the thermal hysteresis [145] and the
abrupt intracavity power discontinuity upon transition to the soliton regime (Fig.
7.2(c)), delicate tuning waveforms [37, 39] or active capturing techniques [91] are
essential to compensate thermal transients, except in cases of materials featuring
effectively negative thermo-optic response [48].
Now consider removing the optical isolation as shown in Fig. 7.2(d) so that backscatter feedback occurs from pumping a resonator mode. In prior work semiconductor
laser locking to the resonator mode as well as laser line narrowing have been shown

51
Conventional pumping

Laser cavity

XPM

Pump laser
Backscattered light
Soliton

SPM

Nonlinear
microresonator

Isolator

Direct pumping

Forward

Backward

Cold
resonance

Detuning

Laser cavity

+φ

Nonlinear
microresonator

c.w. state

Frequency detuning, δω

MI comb

Soliton

Frequency decreasing

Time (200 µs/div)

Frequency
(MHz)

Intracavity power

Soliton

Comb power (arb.unit)

MI comb

Intracavity power

Comb power
(arb.unit)

Injection

Frequency detuning, δω

Laser on

fr = 19.787 GHz

0 Power (a.u.) 1

Time (20 ms/div)

Figure 7.2: The turnkey operating point. (a) Conventional soliton microcomb
operation using a tunable c.w. laser. An optical isolator blocks the back-scattered
light from the microresonator. (b) Phase diagram, hysteresis curve, and dynamics
of the microresonator pumped as shown in (a). The blue curve is the intracavity
power as a function of cavity-pump frequency detuning. Laser tuning (dashed red
line) accesses multiple equilibria. (c) Measured evolution of comb power pumped
by an isolated, frequency-scanned ECDL. The step in the trace is a characteristic
feature of soliton formation. (d) Turnkey soliton microcomb generation. Nonisolated operation allows back-scattered light to be injected into the pump laser
cavity. Resonances are red-shifted due to self-phase modulation (SPM) and crossphase modulation (XPM). (e) Phase diagram, hysteresis curve, and dynamics of
pump/microresonator system. A modified laser tuning curve (dashed red line)
intersects the intracavity power curve (blue) to establish a new operating point from
which solitons form. The feedback phase 𝜙 is set to 0 in the plot. Simulated
evolution upon turning-on of the laser at a red detuning outside the soliton regime
but within the locking bandwidth is plotted (solid black curve). (f) Measured comb
power (upper panel) and detected soliton repetition rate signal (lower panel) with
laser turn-on indicated at 10 ms.
to result from backscattering of the intracavity optical field [146]. These attributes
as well as mode selection when using a broadband pump have also been profitably
applied to operate microcomb systems without isolation [137, 141, 147, 148]. However, these prior studies of feedback effects have considered the resonator to be linear
so that the detuning between the feedback-locked laser and the cavity resonance is
determined solely by the phase 𝜙 accumulated in the feedback path [149]. In contrast, here the nonlinear behavior of the microresonator is included and is shown to
have a dramatic effect on the system operating point. The nonlinear behavior causes
the resonances to be red-shifted by intensity-dependent self- and cross-phase modulation. As a result, the relationship between frequency detuning and intracavity

52
power of the pump mode 𝑃0 can be shown (see section 7.5) to be approximately
given by,
𝛿𝜔
𝜙 3 𝑃0
= tan +
(7.1)
𝜅/2
2 2 𝑃th
where 𝜅 is the power decay rate of the resonance and 𝑃th is the parametric oscillation
threshold for intracavity power. This dependence of detuning on intracavity power
gives rise to a single operating point at the intersection of Eq. (7.1) and the hysteresis
as shown in Fig. 7.2(e). Control of the feedback phase shifts the x-intercept of Eq.
(7.1) and thereby adjusts the operating point.
In the section 7.5 it is shown that the system converges to this operating point once
the laser frequency is within a locking bandwidth (estimated to be 5 GHz in the
present case). As verified both numerically (Fig. 7.2(e)) and experimentally (Fig.
7.2(f)), this behavior enables soliton mode-locking by simple power-on of the pump
laser (i.e., no triggering or complex tuning schemes). A simulated trajectory is
shown in Fig. 7.2e wherein a laser is initially started to the red of the high-Q cavity
resonant frequency and well outside its linewidth. The system is attracted towards the
resonance through a process that at first resembles injection locking of the III-V laser.
However, as the laser frequency moves towards the resonant frequency, the resonator
power rises and the Kerr nonlinearity induces evolution towards the operating point.
The system transiently exceeds the threshold for parametric oscillation and, as shown
in section 7.5, Turing rolls form that ultimately evolve into the solitons as the system
achieves steady state. An experimental trace of the comb power shows that a steady
soliton power plateau is reached immediately after turn-on of the laser. And the
stable soliton emission is further confirmed by monitoring the real-time evolution
of the soliton repetition rate signal (Fig. 7.2(f)). Numerical simulation of this
startup process is provided in section 7.5. The turnkey operation demonstrated here
is automatic, such that the entire soliton initiation and stabilization is described and
realized by the physical dynamics of laser self-injection locking in combination with
the nonlinear resonator response. Consistent with this point, the system is observed
to be highly robust with respect to temperature and environmental disturbances.
Indeed, soliton generation without any external feedback control was possible for
several hours in the laboratory.
7.4

Demonstration of turnkey soliton generation

Figure 7.3 shows the optical spectra of a single-soliton state with 40 GHz repetition
rate and multi-soliton states with 20 GHz and 15 GHz repetition rates. The deviation

53
from the theoretical sech2 spectral envelope is believed to result from a combination
of mode crossing induced dispersion and the dispersion of the waveguide-resonator
coupling strength. The pump laser at 1556 nm is attenuated at the output by
a fiber-Bragg-grating notch filter in these spectra. The coherent nature of these
soliton microcombs is confirmed by photodetection of the soliton pulse streams,
and reveals high-contrast, single-tone electrical signals at the indicated repetition
rates. Numerical simulations have confirmed the tendency of turnkey soliton states
consisting of multiple solitons, which is a direct consequence of the high intracavity
power and its associated MI gain dynamics (see section 7.5 for details). However,
single-soliton operation is accessible for a certain combination of pump power and
feedback phase.

Power (10 dB/div)

fr=40 GHz

Se
ch 2

1530

-0.5

1540

1560
Wavelength (nm)

1580
RBW
100 Hz
40 dB
0.5
Frequency (MHz
+19.787 GHz)

1580

fr=15 GHz

1600

RF power

Power (10 dB/div)

1570

fr=20 GHz

1520

1550
1560
Wavelength (nm)

RF power

Power (10 dB/div)

1540

RBW
100 Hz
40 dB

-0.5

1520

1540

1560
Wavelength (nm)

1580

0.5
Frequency (MHz
+15.056 GHz)

1600

Figure 7.3: Optical and electrical spectra of solitons. (a) The optical spectrum
of a single soliton state with repetition rate 𝑓𝑟 = 40 GHz. The red curve shows a
sech2 fitting to the soliton spectral envelope. (b) (c) Optical spectra of multi-soliton
states at 20 GHz and 15 GHz repetition rates. Insets: Electrical beatnotes showing
the repetition rates.

0 Power (arb. unit) 1

MI
comb

Soliton

−π

Parametric oscillation threshold

Time (100 ms/div)

Feedback phase

Turnkey success probability

Normalized pump power

Beatnote frequency
(MHz + 19.785 GHz)

Comb power
(arb. unit)

To demonstrate the repeatable turnkey operation, the laser current is modulated to a
preset current by a square wave to simulate the turn-on process. Soliton microcomb
operation is reliably achieved as confirmed by monitoring soliton power and the
single-tone beating signal (Fig. 7.4(a)). More insight into the nature of the turnkey
operation is provided by the phase diagram near the equilibrium point for different
feedback phase and pump power (Fig. 7.4(b)). The turnkey regime occurs above a
threshold power within a specific range of feedback phases. Moreover, the regime
recurs at 2𝜋 increments of feedback phase, which is verified experimentally (Fig.
7.4(c)). Consistent with the phase diagram, a binary-like behavior of turn-on success
is observed as the feedback phase is varied. In the measurement the feedback phase
was adjusted by control of the gap between the facets of the laser and the microcomb
bus waveguide. A narrowing of the turn-on success window with an increased
feedback phase is believed to result from the reduction of the pump power in the bus
waveguide with increasing tuning gap (consistent with Fig. 7.4(b)).

2π

3π

2π
3π
Derived phase

4π

Figure 7.4: Demonstration of turnkey soliton generation. (a) 10 consecutive
switching-on tests are shown. The upper panel gives the measured comb power
versus time. The laser is switched on periodically as indicated by the shaded
regions. The lower panel is a spectrogram of the soliton repetition rate signal
measured during the switching process. (b) Phase diagram of the integrated soliton
system with respect to feedback phase and pump power. The pump power is
normalized to the parametric oscillation threshold. (c) Turnkey success probability
versus relative feedback phase of 20 GHz (upper panel) and 15 GHz (lower panel)
devices. Each data point is acquired from 100 switch-on attempts. See Methods for
additional discussion.

7.5

Theory of turnkey soliton generation

Equations of motion
The injection locking system consists of three parts: the soliton optical field 𝐴S , the
backscattering field 𝐴B , and the laser field 𝐴L . The complete equations of motions
are [117, 149]:

𝜅 | 𝐴S | 2 + 2| 𝐴B | 2
𝜕 𝐴S
𝐷 2 𝜕 2 𝐴S
= − 𝐴S − 𝑖𝛿𝜔𝐴S + 𝑖
𝑖𝛽
𝜅R 𝜅 L 𝑒𝑖𝜙B 𝐴L
𝜕𝑡
2 𝜕𝜃 2
𝐸 th
2 + 2 2𝜋 | 𝐴 | 2 𝑑𝜃/(2𝜋)
𝜅 B
𝑑𝐴B
= − 𝐴B − 𝑖𝛿𝜔𝐴B + 𝑖
𝐴B + 𝑖𝛽 𝐴S
𝑑𝑡
𝐸 th
𝑔(| 𝐴L | )
𝑑𝐴L
= 𝑖(𝛿𝜔L − 𝛿𝜔) 𝐴L − 𝐴L +
(1 + 𝑖𝛼𝑔 ) 𝐴L − 𝜅 R 𝜅L 𝑒𝑖𝜙B 𝐴B
𝑑𝑡
(7.2)
∫ 2𝜋
where the field amplitudes are normalized so that 0 | 𝐴S | 2 𝑑𝜃/(2𝜋), | 𝐴B | 2 and
| 𝐴L | 2 are the optical energies of their respective fields, 𝑡 is the evolution time, 𝜃 is
the resonator angular coordinate, 𝜅 is the resonator mode loss rate (assumed to be
equal for 𝐴S and 𝐴B ), 𝛿𝜔 is the detuning of the cold-cavity resonance compared
to injection-locked laser (𝛿𝜔 > 0 indicates red detuning of the pump frequency
relative to the cavity frequency), 𝐷 2 is the second-order dispersion parameter, 𝐸 th
is the parametric oscillation threshold for intracavity energy, 𝛽 is the dimensionless
backscattering coefficient (normalized to 𝜅/2), 𝜙B is the propagation phase delay
between the resonator and the laser, 𝜅 R and 𝜅L are the external coupling rates for the
resonator and laser, respectively, 𝛿𝜔L is the detuning of the cold-cavity resonance
relative to the cold laser frequency, 𝛾 is the laser mode loss rate, 𝑔(| 𝐴L | 2 ) ≡
𝑔0 /(1 + | 𝐴L | 2 /| 𝐴L,sat | 2 ) is the intensity-dependent gain, 𝑔0 is the gain coefficient,
| 𝐴L,sat | 2 is the saturation energy level, and 𝛼𝑔 is the amplitude-phase coupling factor.
∫ 2𝜋
The average soliton field amplitude 𝐴S = 0 𝐴S 𝑑𝜃/(2𝜋) is also the amplitude on
the pumped mode, and by using 𝐴S in the equation for 𝐴B , we have assumed that
only the mode being pumped contributes significantly in the locking process, which
can be justified if a single-frequency laser is used. We note that the equations are
effectively referenced to the frequency of the injection-locked laser instead of the
free-running laser, which will simplify the following discussions. The frequency
difference between the cold laser and the injection-locked laser is given by 𝛿𝜔−𝛿𝜔L .
We will introduce some other dimensionless quantities to facilitate the discussion.
Define: normalized soliton field amplitude as 𝜓 = 𝐴S / 𝐸 th , normalized total
∫ 2𝜋
intracavity power as 𝑃 = 0 | 𝐴S | 2 𝑑𝜃/(2𝜋𝐸 th ), normalized amplitude on the pump,
backscattering and laser mode as 𝜌 = 𝐴S / 𝐸 th , 𝜌B = 𝐴B / 𝐸 th , 𝜌L = 𝐴L / 𝐸 th ,
respectively, normalized detuning of cavity as 𝛼 = 2𝛿𝜔/𝜅, and normalized evolution
time as 𝜏 = 𝜅𝑡/2. The equation for 𝐴S and 𝐴B can then be put into the dimensionless

56
form
2 𝜅R 𝜅L 𝑖𝜙B
𝜕𝜓
𝐷 2 𝜕2𝜓
+ 𝑖(|𝜓| + 2|𝜌B | )𝜓 + 𝑖𝛽𝜌B −
= −(1 + 𝑖𝛼)𝜓 + 𝑖
𝑒 𝜌L
𝜕𝜏
𝜅 𝜕𝜃 2
𝑑𝜌B
= −(1 + 𝑖𝛼) 𝜌B + 𝑖(2𝑃 + |𝜌B | 2 ) 𝜌B + 𝑖𝛽𝜌.
𝑑𝜏
(7.3)
The laser dynamics for 𝐴L are split into amplitude and phase parts:
1 𝑑| 𝐴L |
𝛾 𝑔(| 𝐴L | 2 )
𝑖𝜙B 𝐴B
=− +
− Re 𝜅R 𝜅L 𝑒
| 𝐴L | 𝑑𝑡
𝐴L
𝑑𝜙L
𝑔(| 𝐴L | 2 )
𝑖𝜙B 𝐴B
= 𝛿𝜔L − 𝛿𝜔 +
𝛼𝑔 − Im 𝜅 R 𝜅L 𝑒
𝑑𝑡
𝐴L
where Re[·] and Im[·] are the real and imaginary part functions, respectively. The
laser power without backscatter feedback | 𝐴L(0) | satisfies 𝑔(| 𝐴L(0) | 2 ) = 𝛾. Expanding
the gain around this point gives
| 𝐴 | 2 − | 𝐴L(0) | 2
1 𝑑| 𝐴L |
0 L
𝑖𝜙B 𝐴B
= −𝑔
− Re 𝜅R 𝜅L 𝑒
| 𝐴L | 𝑑𝑡
𝐴L
| 𝐴L,sat | 2
where 𝑔0 = 𝑔0 /(1 + | 𝐴L(0) | 2 /| 𝐴L,sat | 2 ) 2 is the gain derivative that represents the
relaxation rate of the gain dynamics. Typical values for 𝑔0 are on the order of several
GHz for III-V semiconductor lasers, which is much faster compared to the resonator
dynamics. Accordingly, the laser amplitude can be adiabatically eliminated (i.e.,
assume 𝑑| 𝐴L |/𝑑𝑡 = 0) so that the laser power adiabatically tracks the feedback. The
gain can be solved as,
𝑔(| 𝐴L | 2 ) 𝛾
𝑖𝜙B 𝐴B
= + Re 𝜅R 𝜅L 𝑒
𝐴L
which, under the assumption of fast relaxation rates, becomes independent of the
specific details of gain. Substituting this equation into the phase equation and
normalizing results in an Adler-like equation:
√
2 𝜅R 𝜅L
𝑑𝜙L
𝑖𝜙B 𝜌B
= 𝛼L − 𝛼 − Im
(1 − 𝑖𝛼𝑔 )𝑒
(7.4)
𝑑𝜏
𝜌L
where 𝛼L = (2𝛿𝜔L +𝛼𝑔 𝛾)/𝜅 is the normalized detuning of the cold-cavity resonance
compared to the free-running hot laser.
To simplify the equations further, we also consider the following approximation for
the propagation phase 𝜙B , which depends on the precise frequency of the locked laser

57
and material dispersion. We assume that the feedback length is short (𝐿
𝑐/(𝑛𝜅),
where 𝑐 is the speed of light in vacuum and 𝑛 is the refractive index of the material)
so that 𝜙B can be treated as constant. This approximation is equivalent to assuming
that the FSR of a cavity equal in length to the feedback path is much larger than the
linewidth of the high-Q resonator. By defining a pump phase variable 𝑧 = −𝑒𝑖𝜙B 𝑒𝑖𝜙L ,
the equations can be written as

𝜕𝜓
𝐷 2 𝜕2𝜓
= −(1 + 𝑖𝛼)𝜓 + 𝑖
+ 𝑖(|𝜓| 2 + 2|𝜌B | 2 )𝜓 + 𝑖𝛽𝜌B + 𝑧𝐹
𝜕𝜏
𝜅 𝜕𝜃
𝑑𝜌B
= −(1 + 𝑖𝛼 − 2𝑖𝑃 − 𝑖|𝜌B | 2 ) 𝜌B + 𝑖𝛽𝜌
𝑑𝜏
1 𝑑𝑧
𝑖𝜙 𝜌B
= 𝛼L − 𝛼 + 𝐾Im 𝑒
, |𝑧| = 1
𝑖𝑧 𝑑𝜏
𝑖𝛽𝑧𝐹
where we defined: the normalized pump,
2 𝜅R 𝜅L
|𝜌L |
𝐹=

(7.5)

(7.6)

the locking bandwidth,
𝐾=

4𝜅R 𝜅L
|𝛽|
1 + 𝛼𝑔2
𝜅2

(7.7)

and the feedback phase,
𝜙 = 2𝜙B − arctan(𝛼𝑔 ) + Arg[𝛽] +

(7.8)

where Arg[·] is the argument function. The feedback phase 𝜙 contains phase contributions from the propagation length, the amplitude-phase coupling, the backscattering, as well as an extra 𝜋/2 added to the definition of 𝜙 such that the mode is
locked to the center at 𝜙 = 0 (as discussed below).
We note that the feedback fields considered in Eq. (7.5) come entirely from inside
the resonator. In experiments, defects and facets of the coupling waveguide can also
induce reflections. However, these are neglected in the injection locking dynamics
by the following arguments. Multiple reflection sources can be incorporated into
Eq. (7.5) by adding a feedback term corresponding to each source. However, the
cumulative effect of such reflections will be to produce a wavelength dependence
that is weak compared to the resonator mode, which is spectrally very narrow on
account of its high optical Q. Such a weak wavelength dependence means that these
fields do not contribute to the locking (i.e., a constant term is added to the phasor
equation in Eq. (7.5) , which can then be absorbed into the free-running laser

58
frequency). As a specific illustration of this idea, consider that the facet reflections
at the end of the waveguide form a kind of Fabry-Perot resonator. However, the Q of
its resonances will be quite low and, accordingly, the linewidth will be of order the
FSR associated with the waveguide length. Moreover, this FSR is also comparable
in scale to the FSR of the high-Q resonator, the resonances of which are many
orders narrower than the resonator FSR. As a result, any wavelength dependence
introduced by reflections in the waveguide will be spectrally slow in comparison to
those introduced by the resonator.
For stationary solutions (e.g. a stable soliton), when backscattering is weak (𝛽
1)
so that nonlinearities caused by |𝜌B | 2 in Eq. (7.3) are negligible in comparison to
the soliton driven nonlinear terms, 𝜌B can be found as
𝜌B =

𝑖𝛽𝜌
1 + 𝑖(𝛼 − 2𝑃)

(7.9)

and the laser phasor equation reduces to an algebraic equation for the detuning 𝛼,
𝑖𝜙
𝛼 = 𝛼L + 𝐾Im 𝑒
(7.10)
1 + 𝑖(𝛼 − 2𝑃) 𝑧𝐹
Finally, neglecting the small coupled amplitude from 𝜌B to 𝜓, the equation for the
soliton field reads
𝐷 2 𝜕2𝜓
+ 𝑖|𝜓| 2 𝜓 + 𝑧𝐹 = 0.
−(1 + 𝑖𝛼)𝜓 + 𝑖
𝜅 𝜕𝜃

(7.11)

We have therefore recovered a conventional Lugiato-Lefever equation, with an additional equation that describes the detuning determined by injection locking. This
shows that the spectral properties of the injection-locked solitons are not much
different from a conventional soliton. The main difference is the comb formation
dynamics.
Continuous-wave excitation and equilibrium of locking
It is known that combs and solitons will emerge from a continuous-wave (CW)
background when the input power exceeds the parametric oscillation threshold (|𝜌| >
1), and it is desirable to first study the CW excitation of the system by setting 𝐷 2 = 0.
In this case, the Lugiato-Lefever partial differential equation reduces to an ordinary
differential equation with 𝜓 = 𝜌 and 𝑃 = |𝜌| 2 . The steady state solution can be
found from
𝑧𝐹 = [1 + 𝑖(𝛼 − |𝜌| 2 )] 𝜌
(7.12)

59
and the locking equilibrium reduces to
𝑖𝜙
𝛼 = 𝛼L + 𝐾Im 𝑒
= 𝛼L + 𝐾 𝜒(𝑃, 𝛼, 𝜙)
1 + 𝑖(𝛼 − 2𝑃) 1 + 𝑖(𝛼 − 𝑃)

(7.13)

where we have defined the CW locking response function:
𝜒(𝑃, 𝛼, 𝜙) =

(3𝑃 − 2𝛼) cos 𝜙 + (1 − 2𝑃2 + 3𝑃𝛼 − 𝛼2 ) sin 𝜙
[1 + (𝛼 − 𝑃) 2 ] [1 + (𝛼 − 2𝑃) 2 ].

(7.14)

To obtain analytical results, we will also make the approximation of infinite locking
bandwidth limit (i.e. 𝐾 → ∞), which makes the detuning independent of the freerunning laser frequency. The locking condition is then equivalent to setting the
locking response function to zero:
(3𝑃 − 2𝛼) cos 𝜙 + (1 − 2𝑃2 + 3𝑃𝛼 − 𝛼2 ) sin 𝜙 = 0
−5π/6

(7.15)

−2π/3 −π/2 −π/3
−π/6

φ=0

|ρ|2

π/6
π/3
π/2
2π/3

5π/6

|F|2 = 4
-2

|F|2 = 1

Figure 7.5: Continuous-wave states of the injection-locked nonlinear resonator.
Horizontal axis is the normalized detuning 𝛼, and vertical axis is the normalized
optical energy on the pump mode |𝜌| 2 . Resonator characteristics are shown as the
blue curves, with |𝐹 | 2 = 1 (lower) to 4 (upper). Laser locking characteristics are
shown as the red curves, with 𝜙 = −5𝜋/6 (upper left) to 5𝜋/6 (lower right).
Fig. 7.5 shows a plot for Eq. (7.12) with different pumping powers |𝐹 | 2 and Eq.
(7.15) with different feedback phases 𝜙. The intersecting point of the two curves
gives the CW steady state of the cavity. Note that there are two solutions to the
quadratic equation Eq. (7.15). Only the solution branch that satisfies 𝜕 𝜒/𝜕𝛼 < 0 is
plotted, which are the stable locking solutions (stable in the sense of CW excitations;
the instability arising from modulations are considered below). The opposite case
𝜕 𝜒/𝜕𝛼 > 0 drives the frequency away from the equilibrium.

60
When a resonator is pumped conventionally, the intracavity power 𝑃 will approach
its equilibrium given by Eq. (7.12). In the case of feedback-locked pumping, such
power changes will also have an effect on the locking response function 𝜒, pulling
the detuning to the new locking equilibrium as well (Fig. 7.2(b) and 7.2(e)). A
special case is 𝜙 = 0, where the locking equilibrium can be simply described as
𝛼=

(7.16)

i.e. the detuning is pulled away from the cold cavity resonance, and the effect is
exactly 3/2 times what is expected from the self phase modulation. This is an
averaged effect of the self phase modulation on the soliton mode and the cross phase
modulation of the backscattered mode from the soliton mode. More generally, the
detuning can be solved in terms of 𝑃 as
4 + 𝑃2 sin2 𝜙
(7.17)
𝛼 = 𝑃 − cot 𝜙 +
2 sin 𝜙
where again only the solution satisfying 𝜕 𝜒/𝜕𝛼 < 0 is given. Neglecting the higherorder 𝑃2 sin2 𝜙 term inside the square root results in a lowest order approximation:
𝛼 = tan

𝜙 3
+ 𝑃
2 2

(7.18)

which splits into two additive contributions: one from the feedback phase and the
other from the averaged nonlinear shift. This is Eq. (7.1) when written using
dimensional quantities. We note that Eq. (7.1) uses power normalized to threshold
power, while in the above analysis we used energy normalized to threshold energy.
The intracavity power and energy only differ by a factor of round-trip time, and the
normalized quantity is essentially the same.
Soliton formation
When the dispersion term is considered, the CW solution is no longer stable, which
leads to the formation of modulational instability (MI) combs. These combs will
evolve into solitons if the CW state is also inside the multistability region of the
resonator dynamics. By adjusting the pump power |𝐹 | 2 and feedback phase 𝜙, we
can change the operating point of the cavity, and map the possible comb states to
a phase diagram with |𝐹 | 2 and 𝜙 as parameters (Fig. 7.4(b)). It should be noted
that as soon as combs start to form inside the resonator, the CW results after Eq.
(7.12) becomes invalid (i.e., due to power appearing in the sidebands, we have
𝑃 > |𝜌| 2 when combs are formed instead of 𝑃 = |𝜌| 2 in the CW case), and such

61
comb formation will slightly shift the operating point of the system. However, the
CW results still indicate whether and how such combs can be started. Contrary to
conventional pumping phase diagrams (with |𝐹 | 2 and 𝛼 as parameters), where soliton
existence regions only imply the possible formation of solitons due to multistability,
the soliton existence region here guarantees the generation of solitons as the system
bypasses the chaotic comb region before the onset of MI.
Numerical simulations
We have also performed numerical simulations to verify the above analyses (Fig.
7.6). The simulation numerically integrates Eq. 7.5 with a split-step Fourier
method. Noise equivalent to about one-half photon per mode is injected into 𝜓 to
provide seeding for comb generation. Parameters common to all simulation cases
are 𝐷 2 /𝜅 = 0.015 and |𝛽| = 0.5, while others are varied across different cases and
can be found in the caption of 7.6. As an aside, the magnitude of 𝛽 was estimated
from the resonant backscatter reflectivity. The resonant reflection (measured to be
in the range of 4% - 20%) was measured by detecting the reflected optical power
from the resonator while scanning a laser across the resonances. In the first case
(Fig. 7.6(a)), conventional soliton generation by sweeping the laser frequency is
presented, showing the dynamics of a random noisy comb waveform collapsing into
soliton pulses. This is in contrast to the turnkey soliton generation in the second case
(Fig. 7.6(b)), where solitons directly “grow up” from ripples in the background.
Such ripples are generated by MI in those sections of the resonator with local
intracavity power above the threshold. Each peak in the ripples corresponds to one
soliton if collisions and other events are not considered. The process of growing
solitons out of the background will continue until there is no space for new solitons
or when the background falls below the MI threshold, and such dynamics explain
the tendency of the turnkey soliton state to consist of multiple solitons. By carefully
tuning the phase and controlling the MI gain, it is still possible to obtain a turnkey
single soliton state, as shown in the third case (Fig. 7.6(c)).
7.6

Additional measurements

Different types of microcombs in the injection locking system
There are several interesting solutions other than stable solitons that can be derived
from the regular Lugiato-Lefever equation (LLE) [117]. One is the breather soliton,
which is the type of soliton whose shape oscillates in time. Another example is the
chaotic comb, which corresponds to the unstable Turing patterns or soliton state as

Conventional pumping
2π -2

Direct pumping (multi-soliton)

Normalized detuning

Direct pumping (single-soliton)

2π

Power (a.u.)

Resonator
coordinate

Normalized power

1.5

Pump mode
Comb

1.5

0.5

100

200

300

400

Pump mode
Comb

Field amplitude

Pump mode
Comb

0.5

100

200

300

400

Normalized evolution time

Normalized evolution time
τ = 350

τ = 45

100

200

300

400

Normalized evolution time

τ = 70

τ = 350

τ = 380

Resonator coordinate

2π

Resonator coordinate

2π

Resonator coordinate

2π

Figure 7.6: Numerical simulations of turnkey soliton generation. (a) Conventional solitons are generated by sweeping the laser frequency. Parameters are 𝐾 = 0
(no feedback) and |𝐹 | 2 = 4. The normalized laser frequency is swept from 𝛼L = −2
to 𝛼L = 6 within a normalized time interval of 400. Upper panel: soliton field
power distribution as a function of evolution time and coordinates. Middle panel:
dynamics of the pump mode power (black) and comb power (blue). Lower panel:
a snapshot of the soliton field at evolution time 𝜏 = 350 (𝛼L = 5), also marked
as a white dashed line in the upper panel and a black dashed line in the middle
panel. (b) Multiple solitons are generated under conditions of nonlinear feedback.
Parameters are 𝐾 = 15, 𝜙 = 0.15𝜋, |𝐹 | 2 = 3 and 𝛼L = 5. Upper and middle panels
are the same as in (a). Lower panel: snapshots of the soliton field at evolution time
𝜏 = 45 (gray dashed line) and 𝜏 = 70 (black solid line), also marked as white dashed
lines in the upper panel and black dashed lines in the middle panel. (c) A single
soliton is generated under conditions of nonlinear feedback. Parameters are 𝐾 = 15,
𝜙 = 0.3𝜋, |𝐹 | 2 = 3 and 𝛼L = 5. Upper and middle panels are the same as in (a).
Lower panel: snapshots of the soliton field at evolution time 𝜏 = 350 (gray dashed
line) and 𝜏 = 380 (black solid line), also marked as white dashed lines in the upper
panel and black dashed lines in the middle panel.

RF power

Power (20 dB/div)

63
20 dB

RBW
50 kHz

-0.2
0.2
Frequency (GHz
+19.787 GHz)

20 dB

1560
Wavelength (nm)

RBW
50 kHz

1580
Chaotic comb

-0.2
0.2
Frequency (GHz
+19.787 GHz)

1540

1560
Wavelength (nm)

Power (20 dB/div)

1520

1540

RF power

Power (20 dB/div)

1520

Breather soliton

1580
Soliton crystal

1520

1540

1560
Wavelength (nm)

1580

Figure 7.7: Optical and electrical spectra of different microcomb types. (a)(b)
Optical spectra of breather solitons and a chaotic comb. Inset: Electrical beatnote
signals. (c) Optical spectrum of a soliton crystal state.

the pump power is increased. In addition, solitons can be self-organized and form
an equidistant pulse train in the microresonator, which is called a soliton crystal.
It is also possible to operate our system in different types of microcomb states under
certain feedback phase and laser driving frequency. Fig. 7.7 shows the experimental
spectra of breather solitons, a chaotic comb, and a soliton crystal state, respectively.
The turnkey generation of the chaotic comb is further shown in Fig. 7.7(a). The
broad and noisy RF spectrum indicates that it is not mode-locked.
Tuning of turnkey soliton microcomb system
To further explore the performance of the turnkey soliton microcomb system, the
frequency of the pump laser is driven by a linear current scan (Fig. 7.8(b),(c)).
The scan speed of the driving frequency is around 0.36 GHz/ms, estimated from
the wavelength-current response when the laser is free running. When the laser
is scanned across the resonance, feedback locking occurs and pulls the pump laser
frequency towards the resonance until the driving frequency is out of the locking

40
20

Time (100 ms/div)

Frequency decreasing

Time (2 ms/div)

Comb power (a.u.)

RF power (a.u.)

Frequency (MHz
+19.750GHz)

Comb power
(a.u.)

Frequency increasing

Time (2 ms/div)

Figure 7.8: Tuning of turnkey soliton microcomb system. (a) Turnkey generation
of a chaotic comb. Upper panel: Comb power evolution. Lower panel: Spectrograph
of RF beatnote power. (b)(c) Comb power evolution when the pump laser frequency
is driven from blue to red (b) and red to blue (c).

band. As shown in Fig. 7.8(b), the power steps indicate that soliton states with
different soliton numbers can be accessed as we tune the driving current. It is worth
noting that the soliton microcombs can be powered-on when the laser is scanned from
red-detuned side (Fig. 7.8c), which seldom happens under conventional pumping,
except in cases of an effectively negative thermo-optic response system [48]. The
comb evolution during laser scanning is a useful tool to assess the robustness of the
turnkey soliton generation.
7.7

Conclusion

Besides the physical significance and practical impact of the new operating point,
our demonstration of a turnkey operating regime is an important simplification of
soliton microcomb systems. Moreover, the application of this method in an integrated CMOS-compatible system represents a milestone towards mass production of
optical frequency combs. The butterfly packaged devices will benefit several comb
applications including miniaturized frequency synthesizers [9] and optical clocks
[4]. In these applications, their CMOS rates, besides enabling integration of electrical control and processing functions, provide for simple detection and processing of
the comb pulses (i.e., without the need for millimeter-wave-rate frequency mixers).

65
Moreover, the recent demonstration of low power comb formation in III-V microresonators [49] suggests that monolithic integration of pumps and soliton microcombs
is feasible using the methods developed here. A phase section could be included
therein or in advanced versions of the current approach to electronically control the
feedback phase. The ability to create a complete system including pump laser without optical isolation is also significant. Even in cases where solitons are pumped
using amplification such as with laser cavity solitons [150], full integration would
require difficult-to-integrate optical isolators. It is also important to note that the
current turnkey approach is a soliton forming comb while other recent work reports
on non-soliton Kerr combs [151]. Finally, due to its simplicity, this approach could
be applied in other integrated high-Q microresonator platforms [47, 48, 87] to attain
soliton microcombs across a wide range of wavelengths.

66
Chapter 8

DARK SOLITON MICROCOMBS IN CMOS-READY
ULTRA-HIGH-Q MICRORESONATORS
1Driven by narrow-linewidth bench-top lasers, coherent optical systems spanning
optical communications, metrology, and sensing provide unrivalled performance.
To transfer these capabilities from the laboratory to the real world, a key missing ingredient is a mass-produced integrated laser with superior coherence. Here,
we bridge conventional semiconductor lasers and coherent optical systems using
CMOS-foundry-fabricated microresonators with record high 𝑄 factor over 260 million and finesse over 42,000. Five orders-of-magnitude noise reduction in the pump
laser is demonstrated, enabling frequency noise of 0.2 Hz2 Hz−1 to be achieved in
an electrically-pumped integrated laser, with corresponding short-term linewidth of
1.2 Hz. Moreover, the same configuration is shown to relieve dispersion requirements for microcomb generation that have handicapped certain nonlinear platforms.
The simultaneous realization of record-high 𝑄 factor, highly coherent lasers, and
frequency combs using foundry-based technologies paves the way for volume manufacturing of a wide range of coherent optical systems.
8.1

Introduction

The benefits of high coherence lasers extend to many applications. Hertz-level
linewidth is required to interrogate and manipulate atomic transitions with long coherence times, which form the basis of optical atomic clocks [4, 73]. Furthermore,
linewidth directly impacts performance in optical sensing and signal generation applications, such as laser gyroscopes [31, 153], light detection and ranging (LIDAR)
systems [16, 17], spectroscopy [11], optical frequency synthesis [9], microwave photonics [27, 143, 147, 154, 155], and coherent optical communications[156, 157]. In
considering the future transfer of such high coherence technologies to a mass manufacturable form, semiconductor laser sources represent the most compelling choice.
They are directly electrically pumped, wafer-scale manufacturable and capable of
complex integration with other photonic devices. Indeed, their considerable advan1 Work presented in this chapter has been published in [152] “Hertz-linewidth semiconductor
lasers using CMOS-ready ultra-high-Q microresonators”, Nature Photonics 15, 346-353 (2021).
Boqiang Shen conducted the photonic alignment and linewidth measurements, prepared the data,
and participated in the writing of the manuscript.

67
tages have made them into a kind of ‘photonic engine’ for nearly all modern day
optical source technology, including commercial benchtop laser sources. Nonetheless, mass manufacturable semiconductor lasers, such as used in communications
systems, have linewidths ranging from 100 kHz to a few MHz [156], which is many
orders of magnitude too large for the above mentioned applications.
A powerful method to narrow the linewidth of a laser is to apply optical feedback
through an external reflector, for which the degree of noise suppression scales
with the square of the quality (𝑄) factor of the reflector [149, 158–163]. Ultrahigh-𝑄 microresonators are excellent candidates to achieve substantial linewidth
narrowing and have been demonstrated across a wide range of materials as discrete
[86, 163] or integrated components [31, 87, 140, 164–171]. To compare laser
linewidth between different works, the short-term linewidth – indicated by the
level of the high-offset frequency white-noise floor – is a useful metric. Hereafter
throughout the manuscript ‘linewidth’ will refer to the short-term linewidth. While
sub-Hertz linewidth has been realized in semiconductor lasers that are self-injectionlocked to discrete crystalline microresonators [163], retaining ultra-high 𝑄 factor
when moving to higher levels of integration is both of paramount importance and
challenging. As a measure of the level of difficulty, current demonstrations of
narrow-linewidth integrated lasers, despite many years of effort, feature linewidths
of 40 Hz to 1 kHz, as limited by their 𝑄 factors [139, 141, 172–174].
In this work, we present critical advances in silicon nitride waveguides, fabricated
in a high-volume complementary metal-oxide-semiconductor (CMOS) foundry.
We achieve a 𝑄 factor over 260 million, which is a record among all integrated
resonators. By self-injection locking a conventional semiconductor distributedfeedback (DFB) laser to these ultra-high-𝑄 microresonators, we reduce noise by
five orders of magnitude, yielding frequency noise of 0.2 Hz2 Hz−1 at high offset frequency, with corresponding short-term linewidth of 1.2 Hz – a previously
unattainable level for integrated lasers. Within the same configuration, a new regime
of Kerr comb operation in microresonators is supported. Specifically, the comb
both operates turnkey [139] and attains coherent comb operation under conditions
of normal dispersion without any special dispersion engineering. The comb’s line
spacing is suitable for dense wavelength division multiplexed (DWDM) communications systems. Moreover, each comb line benefits from the exceptional frequency
noise performance of the disciplined pump, representing a significant advance for
DWDM source technology. The microwave phase noise performance of the comb

68
is also comparable to that of existing commercial microwave oscillators. Overall,
experiment and theory reveal an ultra-low-noise regime in integrated photonics.
8.2

CMOS-ready ultra-high-𝑄 microresonators

The ultra-high 𝑄 factor resonators use high-aspect-ratio Si3 N4 waveguides as shown
in Fig. 8.1a. The samples are fabricated in a high-volume CMOS foundry on 200 mm
wafers (Fig. 8.1b) following the process of Bauters et al. [175], but we increase the
thickness of the Si3 N4 core from 40 nm to 100 nm. Thicker Si3 N4 enables a bending
radius below 1 mm, allowing higher integration density than the centimeter-sized
resonators demonstrated previously [31, 140, 164]. Furthermore, a top cladding
thickness of 2 µm is sufficient, which obviates the need for complex chemicalmechanical polishing and bonding of additional thermal SiO2 on top [140, 175].
To minimize the residual hydrogen content of the deposited Si3 N4 and SiO2 films,
we also employ extended thermal treatment totaling over 20 hours of annealing at
1150 ◦ C. Microresonators having three different free spectral ranges (𝐹𝑆𝑅) were
fabricated. Those resonators having 30 GHz FSR were in a whispering-gallery-mode
ring geometry while single-mode racetrack resonators with 5 GHz and 10 GHz 𝐹𝑆𝑅
were fabricated to reduce footprint (Fig. 8.1b).
The capability of CMOS-foundry fabrication to produce ultra-high 𝑄 factor at the
wafer scale is exhibited in Fig 8.1c, wherein the intrinsic 𝑄 factors of 30 GHz 𝐹𝑆𝑅
ring resonators measured throughout the wafer were observed to be clustered in the
170 M to 270 M range. Moreover, a die map (Fig 8.1c) reveals that an intrinsic
𝑄 factor in the vicinity of 200 M is observed on each die, with the exception of
a single die at the center subjected to handling error during fabrication. These 𝑄
factors were measured by transmission spectra scans using a tunable external cavity
laser (calibrated by a separate interferometer) to extract resonator linewidth and to
infer loaded, coupled, and intrinsic optical 𝑄 factors. Cavity ring down was also
performed as a separate check of these 𝑄 measurements. Spectra were observed to
occur in doublets on account of both the ultra-high-Q and the presence of waveguide
backscattering (Fig. 8.1d) [176]. By fitting the doublet line shape of the 30 GHz
ring resonator, intrinsic 𝑄 of 220 M and loaded 𝑄 of 150 M are extracted at 1560
nm, which are further confirmed by measuring the ring-down trace of the resonance
as shown in Fig. 8.1d. The spectral dependences of 𝑄-factors in ring- and racetrackresonators (Fig. 8.1e) provide insight into the origins of loss. A reduction in the value
of 𝑄 around 1510 nm is due to absorptive N-H bonds in the Si3 N4 core. Beyond this
wavelength, the intrinsic 𝑄 factor increases monotonically versus wavelength, likely

300

Maximum intrinsic Q (M)

223 224 220 232
217 227 214 217 222 220
226 233 243 84 225 264
LPCVD SiO2 (2 μm)

199 179 200 206 213 217

LPCVD Si3N4 (100 nm)
189 189 191 194
Thermal SiO2 (14.5 μm)

100
Number of resonators

1 cm

Cross section

15
10

QO=220 M
QL=150 M

FSR=30 GHz

300

8 μm

Intrinsic Q
250

Loaded Q

200

150

0.5
Measurement
Fitting
-4

-2

Frequency (MHz)

Normalized amplitude (arb. unit.)

τ=124 ns

Q factor (M)

Transmission (arb. unit.)

100
150
200
250
Maximum intrinsic Q (M)

300

100
1480

1500

1520

1540

1560

1580

1600

1620

1640

150
2.8 μm

FSR=5 GHz

QL=150 M

Intrinsic Q
Loaded Q
100

50
Measurement
Fitting

0.1

100
Time (ns)

150

200

1450

1470

1490

1510

1530

1550
1570
Wavelength (nm)

1590

1610

1630

1650

Figure 8.1: CMOS-ready ultra-high-𝑄 Si3 N4 microresonators. (a) Cross sectional diagram of the ultra-low loss waveguide, consisting of Si3 N4 as the core
material, silica as the cladding, and silicon as the substrate (not to scale). (b) Photograph of a CMOS-foundry-fabricated 200 mm diameter wafer after dicing (upper
panel), and top view showing 30 GHz 𝐹𝑆𝑅 Si3 N4 ring resonators and a 5 GHz 𝐹𝑆𝑅
racetrack resonator from a different reticle (lower panel). (c) The 𝑄 factor for each
of three 30 GHZ 𝐹𝑆𝑅 ring resonators on each of the 26 dies of the wafer shown in
b was calculated as the average 𝑄 factor in the 1620 nm to 1650 nm range. A wafer
map of the highest 𝑄 factor on each die (upper panel) and histogram of 𝑄 factors of
those 78 resonators (lower panel) demonstrate that ultra-high 𝑄 is achieved across
the wafer. (d) Transmission spectrum (upper panel) of a high-𝑄 mode at 1560 nm
in a 30 GHz ring resonator. Interfacial and volumetric inhomogeneities induce
Rayleigh scattering, causing resonances to appear as doublets due to coupling between counter-propagating modes. Intrinsic 𝑄 of 220 M and loaded 𝑄 of 150 M
are extracted by fitting the asymmetric mode doublet. The ring-down trace of the
mode (lower panel) shows 124 ns photon lifetime, corresponding to a 150 M loaded
𝑄. (e) Measured intrinsic 𝑄 factors plotted versus wavelength in a 30 GHz ring
resonator with 8 𝜇m wide Si3 N4 core (upper panel) and a 5 GHz racetrack resonator
with 2.8 𝜇m wide Si3 N4 core (lower panel). Insets: simulated optical mode profile.

Through port

Ultrahigh-Q Si3N4 resonator
-55 MHz
ISO

Sν(f)

AOM

Injection-locked

Oscilloscope

PC
III-V DFB laser

Free-running
1 km
1 mm
Hybrid-integrated narrow-linewidth laser chip

1010

Free running DFB laser
30 GHz resonator
10 GHz resonator
5 GHz resonator
TRN (30 GHz resonator)
TRN (10 GHz resonator)
TRN (5 GHz resonator)

Frequency Noise (Hz2 Hz-1)

108

104

102

100

Frequency Noise (Hz2 Hz-1)

Drop port

Linewidth measurement setup

103

Through port
Drop port
f 2 trend

102

101

100

<1 Hz2 Hz-1
10

105
Frequency offset f (Hz)

106

107

10-1
104

105
106
Frequency offset f (Hz)

107

Figure 8.2: Hybrid-integrated narrow-linewidth laser based on ultra-high-𝑄
Si3 N4 microresonator. (a) Schematic of the hybrid laser design (not to scale) and
frequency noise test setup. The red (yellow) arrow denotes the forward (backscattered) light field. ISO: optical isolator; AOM: acousto-optic modulator; PC: polarization controller; PD: photodetector. (b) Measurement of single-sideband frequency noise of the free-running and self-injection locked DFB laser. The minimum
frequency-noise levels are 1 Hz2 Hz−1 , 0.8 Hz2 Hz−1 , 0.5 Hz2 Hz−1 for resonators
with 30 GHz, 10 GHz and 5 GHz 𝐹𝑆𝑅, respectively. The dashed lines give the
simulated thermorefractive noise (TRN). (c) Photograph of a 10.8 GHz 𝐹𝑆𝑅 ring
resonator fabricated with a drop port. (d) A comparison of single sideband frequency noise measured from the through port and drop port of the same device. The
drop port enables the resonator itself to act as a low-pass filter, yielding a white-noise
floor of 0.2 Hz2 Hz−1 .
limited by Rayleigh scattering. The highest 𝑄 factor is obtained using a 30 GHz FSR
resonator (mean value of 260 M and standard deviation of 13.5 M over 34 modes),
and observed in the 1630 nm to 1650 nm wavelength range. The overall lower 𝑄
factor of the 5 GHz racetrack resonator suggests excess propagation loss in its single
mode waveguides. This is possibly caused by higher scattering loss from increased
modal overlap with the waveguide sidewall as compared to the whispering-gallery
mode waveguide.
8.3

Hertz-linewidth integrated laser

The hybrid-integrated laser comprises a commercial DFB laser butt-coupled to the
bus waveguide of the Si3 N4 resonator chip (Fig. 8.2a). The laser chip, which is

71
mounted on a thermoelectric cooler to avoid long-term drift, is able to deliver
power up to 30 mW at 1556 nm into the Si3 N4 bus waveguide. Optical feedback is
provided to the laser by backward Rayleigh scattering in the microresonator, which
spontaneously aligns the laser frequency to the nearest resonator mode. As the phase
accumulated in the feedback is critical to determining the stability of injectionlocking [139, 149, 177], we precisely control the feedback phase by adjusting the
air gap between the chips. In the case of a rigidly co-packaged laser and resonator,
feedback phase control may instead be achieved by the addition of a resistive heater
to the waveguide linking the laser and resonator. The laser output is taken through
the bus waveguide of the microresonator, and directed to a self-heterodyne setup
for frequency noise characterization. Two photodetectors and a cross-correlation
technique are used to improve detection sensitivity [178] (see Methods).
The frequency noise spectra of the self-injection locked laser system using the
30 GHz ring and the 10 GHz and 5 GHz racetrack resonators (respective intrinsic
𝑄 factors of 250 M, 56 M, and 100 M) are compared in Fig. 8.2b. The ultra-high-Q
factors enable the frequency noise of the free-running DFB laser to, in principle,
be suppressed by up to 80 dB (see Methods). In practice, however, the noise
suppression over a broad range of offset frequencies (10 kHz to 2 MHz) is limited to
50 dB by the presence of thermorefractive noise [179–181] in the microresonator.
Consistent with theory, microresonators with larger mode volume, i.e. smaller
𝐹𝑆𝑅, experience a lower thermorefractive fluctuation and exhibit reduced frequency
noise (Fig. 8.2b). At low frequency offset (below 10 kHz), frequency noise is
primarily limited by temperature drift and coupling stability between chips. This
can be suppressed by improved device packaging. At high offset frequencies (above
5 MHz), frequency noise rises with the square of offset frequency, as the maximum
noise suppression bandwidth of injection locking is limited to the bandwidth of the
resonator [152, 161, 162]. Thus, minimum frequency noise below 1 Hz2 Hz−1 is
observed at about 5 MHz offset frequency, where the contributions of rising laser
noise and falling thermorefractive noise are approximately equal. To achieve an
ultra-low white frequency noise floor at high offset frequencies, the laser output is
taken from a resonator featuring a drop-port [182] (Fig. 8.2c). The drop port provides
low-pass filtering action and is studied further [152]. This configuration yields of
a white-noise floor of 0.2 Hz2 Hz−1 , corresponding to a short-term linewidth of
1.2 Hz (Fig 8.2d).

D /(2p)= -20.3 kHz

-1
-300

-200

-100

100

200

Kerr combs
Laser on

300

FSR = 10 GHz

D /(2p)= -80.2 kHz

-1

fr=10.873 GHz

FSR = 5 GHz

Power (10 dB per division)

Integrated dispersion Dint/2p (GHz)

FSR = 5 GHz

Frequency (MHz
+10.873 GHz)

Comb power (arb. unit.)

1552

1554

1556

1558

1560

1556
1558
Wavelength (nm)

1560

FSR = 10 GHz

RF Power (arb. unit.)

-1
-2

-150

-100

-50

50
Mode index m

100

150

20
30
Time (ms)

1552

1554

-40 -20 0 20
Frequency (kHz 40
+ 5437.07 MHz)

-60
-80

-100 dBc Hz-1
-100
-114 dBc Hz-1

-120

10.8 GHz
5.4 GHz

-140
10

102

Intracavity power
(arb. unit)

-129 dBc Hz-1

-p
c.w. state

Kerr combs

Steady state

-140 dBc Hz-1

103
104
Frequency offset (Hz)

105

106

Polar angle q

Mode number

Optical power
(10 dB per division)

-40

Normalized intracavity power

Phase noise (dBc Hz-1)

-20

RBW
10 Hz

Power (20 dB
per division)

10
Normalized detuning

-50

Figure 8.3: Formation of mode-locked Kerr combs. (a) Measured mode family
dispersion is normal. The plot shows the integrated dispersion defined as 𝐷 int =
𝜔 𝜇 − 𝜔 𝑜 − 𝐷 1 𝜇 where 𝜔 𝜇 is the resonant frequency of a mode with index 𝜇 and 𝐷 1
is the 𝐹𝑆𝑅 at 𝜇 = 0. The wavelength of the central mode (𝜇 = 0) is around 1550 nm.
The dashed lines are parabolic fits (𝐷 int = 𝐷 2 𝜇2 /2) with 𝐷 2 /2𝜋 equal to −20.3 kHz
and −80.2 kHz corresponding to 5 GHz and 10 GHz 𝐹𝑆𝑅, respectively. Note:
𝐷 2 = −𝑐𝐷 21 𝛽2 /𝑛 eff where 𝛽2 is the group velocity dispersion, 𝑐 the speed of light
and 𝑛 eff the effective index of the mode. (b) Experimental comb power (upper panel)
and detected comb repetition rate signal (lower panel) with laser turn-on indicated at
5 ms. (c) Measured optical spectra of mode-locked Kerr combs with 5 GHz (upper
panel) and 10 GHz (lower panel) repetition rates. The background fringes are
attributed to the DFB laser. (d) Single-sideband phase noise of dark pulse repetition
rates. Dark pulses with repetition rate 10.8 GHz and 5.4 GHz are characterized.
Inset: electrical beatnote showing 5.4 GHz repetition rate. (e) Phase diagram of
microresonator pumped by an isolated laser. The backscattering is assumed weak
enough to not cause mode-splittings. The detuning is normalized to one half of
microresonator linewidth, while the intracavity power is normalized to parametric
oscillation threshold. Green and red shaded areas indicate regimes corresponding to
the c.w. state and Kerr combs. The blue curve is the c.w. intracavity power, where
stable (unstable) branches are indicated by solid (dashed) lines. Simulated evolution
of the unisolated laser is plotted as the solid black curve, which first evolves towards
the middle unstable branch of the c.w. intracavity power curve, and then converges
to the comb steady state (average normalized power shown) as marked by the black
dot. The initial condition is set within the self-injection locking bandwidth, while
feedback phase is set to 0. f, Simulated intracavity field (upper panel) and optical
spectrum (lower panel) of the unisolated laser steady state in panel (e).

Pump

Power (dBm)

-10 dBm

-20
-40
-60
1550

1554
1556
Wavelength (nm)

1552

Frequency Noise (Hz2 Hz-1)

b 103

1558

Pump (1555.438 nm)

104

Comb (1556.143 nm)

102

Comb (1557.194 nm)
Comb (1554.732 nm)

100

Comb (1553.687 nm)

101

1560

<1 Hz2 Hz-1

10-1
Fundamental Linewidth (Hz)

c 40

Frequency offset (MHz)

30
20

Pump

10
1554

1556
1555
Wavelength (nm)

1557

Figure 8.4: Coherence of integrated mode-locked Kerr combs. (a) Optical
spectrum of a mode-locked comb with 43.2 GHz repetition rate generated in a
microresonator with 10.8 GHz 𝐹𝑆𝑅. (b) Single-sideband optical frequency noise
of the pump and comb lines as indicated in panel a, selected using a tunable fiberBragg-grating (FBG) filter. (Inset: the same data in log-log format) (c) Wavelength
dependence of white frequency noise linewidth of comb lines in panel a.

8.4

Mode-locked dark soliton microcomb

The ultra-high Q of the microresonators enables strong resonant build-up of the
circulating intensity, providing access to nonlinear optical phenomena at low input
power levels [183]. As an example, optical frequency combs have been realized in
continuously pumped high-𝑄 optical microresonators due to the Kerr nonlinearity,
and they are finding a wide range of applications [41]. To explore the nonlinear
operating regime of the hybrid-integrated laser in pursuit of highly-coherent Kerr
combs, the mode dispersion of racetrack resonators with 5 GHz and 10 GHz 𝐹𝑆𝑅
was characterized. Their mode families are measured to have normal dispersion
across the telecommunication C-band (Fig. 8.3a). Also, the dispersion curves ex-

74
hibit no avoided-mode-crossings, which is consistent with the single-mode nature
of the waveguides. As distinct from microresonators with anomalous dispersion
wherein bright soliton pulses are readily generated, comb formation is forbidden in
microresonators with normal dispersion, unless avoided-mode-crossings are introduced to alter mode family dispersion so as to allow formation of dark pulses [81].
Surprisingly, however, it was nonetheless possible to readily form coherent combs
in these devices without either of the aforementioned conditions being satisfied.
Indeed, deterministic, turnkey comb formation was experimentally observed when
the DFB laser was switched-on to a preset driving current (see Fig. 8.3b). A clean
and stable beatnote of the comb is established 5 ms after turning on the laser,
indicating that mode-locking has been achieved (see Fig. 8.3b). Plotted in Fig. 8.3c
are optical spectra of the mode-locked Kerr combs in racetrack resonators with
5 GHz and 10 GHz 𝐹𝑆𝑅, where the typical spectral shape of dark pulses is observed
[41, 81, 184, 185]. The stability of mode-locking is characterized by measurement
of the comb beat note phase noise (Fig. 8.3d). For Kerr combs with 10.8 (5.4) GHz
𝐹𝑆𝑅, the phase noise reaches -100 (-114) dBc Hz−1 at 10 kHz and -129 (-140)
dBc Hz−1 at 100 kHz offset frequencies. We note that in order to suppress noise
at high-offset frequencies, the pump is excluded in the photodetection using a fiber
Bragg grating filter, as suggested by previous works [147].
This unexpected result is studied theoretically in the Supplement of Jin et al. [152].
Here, results from that study are briefly summarized. A phase diagram of the
microcomb system is given in Fig. 8.3e, and separates resonator operation into
continuous-wave (c.w.) and Kerr comb regimes based on the viability of parametric
oscillation [117]. The intracavity power exhibits a typical bi-stable behavior as a
function of cavity-pump frequency detuning when pumped by a laser with optical
isolation [41]. In contrast, a recent study shows that the feedback from a nonlinear
microresonator to a non-isolated laser creates an operating point for the compound
laser-resonator system in the middle branch [139]. The operating point is induced
through a combination of self- and cross-phase modulation, and is associated with
turnkey operation of soliton combs operating under conditions of anomalous dispersion [139]. Here, we have validated through simulation that a similar operating
point allows access to dark pulses (normal dispersion) without the requirement for
extra dispersion engineering provided by avoided mode crossings. As has been
previously shown for bright solitons, the phase of the feedback path plays a major
role. Indeed, control of this phase through precision control of the coupling gap

75
between and laser and resonator chips enabled suppression of comb formation for
frequency noise measurements reported in Fig. 8.2 b,d. The black curve in Fig. 8.3e
gives the dynamics of the compound laser-resonator system when initialized at a
point that is within the locking bandwidth of the system. It first evolves towards the
operating point located on the middle branch of the c.w. power bi-stability curve,
where comb formation can be initiated, and then converges to a steady Kerr comb
state (average normalized power shown). The spectral and temporal profile of the
steady state solutions show that flat-top pulses are formed in the microresonator with
normal dispersion (Fig. 8.3f). Though the possible presence of dark pulse formation in microresonators pumped by a self-injection locked laser has been previously
observed [184–186], the theory of the mutually coupled system has only recently
been clarified [139, 152, 187, 188].
The combs generated in these devices exhibit several important properties. In
Fig. 8.4a, the spectrum of a 43.2 GHz repetition rate comb is presented. Curiously,
this spectrum was generated in a microresonator having a 10.8 GHz 𝐹𝑆𝑅. The
appearance of rates that are different from the 𝐹𝑆𝑅 rate has been observed for dark
pulses [81]. This line spacing is compatible with DWDM channel spacings and
10 comb teeth feature on-chip optical power over -10 dBm, which is a per channel
power that is readily usable in DWDM communication systems [189]. However,
most significant is that the white-frequency-noise-level floor for each of these optical
lines (Fig. 8.4b) is measured to be on the order of 1 Hz2 Hz−1 . We note that these
spectra are white at higher offset frequency, i.e., not rising for higher offset as
discussed above for the laser source. The corresponding linewidths of the comb
teeth are plotted in Fig. 8.4c. One of the lines exhibits degraded linewidth of
approximately 30 Hz, which is suspected to be due to its coincidence with a sublasing-threshold side-mode of the DFB laser. Notably, certain comb teeth are quieter
than the pump due to the filtering of pump noise by the ultra-high-𝑄 modes. These
results represent a two order-of-magnitude improvement as compared to previously
demonstrated integrated microcombs [137, 139, 141, 186].
8.5

Performance comparison

For devices with both integrated waveguide coupler and resonator, a few platforms
have emerged as able to provide ultra-high 𝑄 (𝑄 > 10 M). In silica ridge resonators,
a 𝑄 factor of 205 M has been demonstrated [87], while in low-confinement silicon
nitride, a 𝑄 factor of 216 M has been demonstrated [164]. However, these platforms
pose challenges to photonic integration with large scale and high density, e.g. the

76
Table 1 | Current integrated ultra-high-Q microresonators
and narrow-linewidth lasers
Microresonators
Ref #

Material

Cladding

Q (M)

FSR (GHz)

Finesse

Si3N4 (this work)

Oxide

260

42,600

Oxide

3.3

1,400

Oxide

2.7

910

Oxide

216

2.7

3,000

Si3N4 (low confinement)

Si3N4 (high confinement)

Oxide

200

38,400

Oxide

21,700

Oxide

6,200

Oxide

100

7,200

Air

100

10,340

SiO2

Air

205

15.2

15,800

Oxide

5.4

630

LiNbO3

Oxide

210

170

Phosphorous-doped silica Doped-oxide

Lasers
Ref #

Operation principle

Configuration

Linewidth (Hz)

Self-injection locking
(this work)

Hybrid III-V/Si3N4

1.2

External cavity

Hybrid III-V/Si3N4

External cavity

Heterogeneous III-V/Si

140

External cavity

Heterogeneous III-V/Si3N4

4,000

External cavity

Monolithic III-V

50,000

Figure 8.5: Performance comparison of integrated microresonators and lasers.
Upper: Best-to-date integrated ultra-high-𝑄 (> 10 M) microresonators with integrated waveguides. Lower: Linewidth of best-to-date integrated narrow-linewidth
lasers.
use of suspended structures [87] or the requirement for centimeter-level bending
radius [164]. While these limitations are not present in high-confinement silicon
nitride resonators, the highest demonstrated 𝑄 factor is lower, 67 M [165]. In Fig.
8.5, we list key figures of merit for integrated microresonators with ultra-high-𝑄
factors. In addition to the record-high Q factor, owing to their compact footprint, the
current resonators stand out among ultra-high Q resonators for having the highest
finesse as well. Fig. 8.6 provides a comparison as a plot of the 𝑄 and finesse of the
current work with the state-of-the-art.
We further compare the current hybrid-integrated laser linewidth to state-of-the-art
results in Fig.8.5. The linewidth of monolithic III-V lasers is generally limited to
the 100 kHz to 1 MHz range by passive waveguide losses well above 1 dB cm−1 ,
with best demonstrated linewidth below 100 kHz [190]. Phase and amplitude
noise scale according to the square of cavity losses [149, 162]. Thus, hybrid

77
105
This work
High-confinement
Si3N4

Finesse

Silica

Low-confinement
Si3N4

103
Silicon

LiNbO3

Phosphorous-doped Silica

102
10

Intrinsic Q (M)

100

Figure 8.6: Comparison of finesse and intrinsic 𝑄 factors of state-of-the-art
integrated microresonators.

integration, where the active III-V and passive photonic chips are assembled postfabrication, and heterogeneous integration [191], where III-V material is directly
bonded to the passive chip during fabrication, have emerged as primary technologies
to create narrow-linewidth integrated lasers. As shown in Fig. 8.5, hybrid and
heterogeneous integration can produce linewidth well below 1 kHz. In this work,
high-offset frequency noise is suppressed to a white noise floor of 0.2 Hz2 Hz−1 ,
or equivalently, a 1.2 Hz short-term linewidth, which is more than an order of
magnitude improvement over the best results to date [172].
8.6

Discussion

As single-frequency or mode-locked lasers, these hybrid-integrated devices are readily applicable to many coherent optical systems. For example, while laboratory communication experiments pursuing spectral efficiency approaching 20 bit s−1 Hz−1
rely on high performance single-frequency fiber lasers [157], narrow-linewidth integrated photonic comb lasers could accelerate the adoption of similar schemes in
practical data-center and metro links [113, 137, 139, 141, 189, 192]. Microwave
photonics [27, 143, 147, 154, 155], atomic clocks [4, 73], and quantum information [193] will also benefit greatly from the reduced size, weight, power, and cost
provided by the combination of ultra-high 𝑄 and photonic integration.
Many improvements beyond the results presented here are feasible. We infer propagation loss of 0.1 dB m−1 , however, lower loss of 0.045 dB m−1 is feasible in thinner
cores [175], suggesting that the limits of 𝑄 for this platform have not been fully

78
explored. Spiral resonators with increased modal volume can suppress low-offset
frequency noise induced by thermodynamic fluctuations [194]. Finally, heterogeneous integration of III-V lasers and ultra-high-𝑄 microresonators may eventually
unite the device onto a single chip [173, 174, 191], leading to scalable production
with high yield using foundry-based technologies.

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