Doctoral Dissertation
博士論文

Interferometer Locking Scheme
for Advanced Gravitational-Wave
Detectors and Beyond(現在および将来の重力波検出器の干渉計の動作点引き込み法)

A Dissertation Submitted for the Degree of Doctor of Philosophy
December 2019
令和元年 12 月博士 (理学) 申請
Department of Physics, Graduate School of Science,
The University of Tokyo東京大学大学院理学系研究科
物理学専攻

Yutaro Enomoto
榎本 雄太郎iii

Abstract
Direct observations of gravitational waves (GWs) by currently-operating groundbased interferometric detectors have brought unique astrophysical information,
particularly on compact stars and gravity theories. It is expected that theplanned detectors, namely the third generation detectors, which are about ten
times more sensitive than the current detectors, will provide further informationin these fields. Current GW detectors, such as Advanced LIGO, Advanced Virgo,
and KAGRA, require multiply-coupled optical cavities to be controlled at theirresonances. Thus the lock acquisition process, where all the cavities are brought
to their operation points against the cross-coupling of the cavities, is essential for
their operation and astrophysical observation. In particular, it is the most challenging to achieve the resonances of the arm cavities, because the kilometer-longarms result in narrow frequency linewidths; the arm cavity provides meaningfulerror signals only within its linewidth. In the third generation detectors, which
will have longer arms leading to narrower linewidths, the lock acquisition process
will be even more challenging. In lock acquisition of Advanced LIGO, a scheme
called arm length stabilization (ALS) has been used, where auxiliary lasers having diﬀerent wavelength from that of the main laser independently control thearms within their linewidths in a decoupled way. However, it is not trivial toscale up the system of Advanced LIGO to the third generation detectors dueto the configuration of the system; optical fibers having the same length to thearms are involved, and thus to scale it up would increase optical phase noise andloss in the fibers.
To address this issue, a new type of the ALS system was designed for KAGRA. The new configuration is simple with significantly shorter fiber length andthus it is compatible with the third generation detectors. Along with a design
study on the noise performance, an experimental test of the new ALS system was
performed in KAGRA. The test revealed that the residual fluctuations of the arm
cavities were evaluated to be less than 5 Hz in terms of root mean square, which
are smaller than the linewidth of the arm. With this level of the noise performance, lock acquisition of the Fabry–Perot Michelson interferometer of KAGRA
was achieved using the ALS system, which was the major milestone toward KAGRA joining the network observation. This achievement demonstrated that thesystem is ready for bringing the KAGRA full interferometer to the operation
mode as soon as the other part of the interferometer is available.
Lock acquisition of third generation detectors was discussed for the first time.
Utilizing the results obtained in KAGRA, the performance of the ALS systemin third generation detectors was extrapolated. Necessary modifications to the

iv
KAGRA ALS system were also discussed. The results indicated that lock acquisition of third generation detectors will be feasible by applying the KAGRA
ALS system. We also point out that a new scheme with a sub-carrier field will
make the lock acquisition process more reliable.

v

ACKNOWLEDGMENTS
This work has been supported by many people and groups. Here I would like toexpress my gratitude to them.
I am grateful to my supervisor Masaki Ando for his support. First of all, he
accepted me as a graduate student when I changed the laboratory. He gave me
freedom in research activities so that I could study what I want to study. Hiscomments on my presentations always inspired me with something new.
I am also grateful to my former supervisor Seiji Kawamura. Thanks to him,
I entered the field of gravitational-wave physics. He showed me how interesting
this field is. His attitude for discussions with other people has been a guide tome. Discussions with him have been always fun to me.
I thank Kiwamu Izumi for his support and advice. He suggested that I worked
in KAGRA and studied lock acquisition in the gravitational-wave detectors. He
has always supported our experiments. His visit to the site always moved theexperiments forward. It was my pleasure that I could work with him and learn
from him; it has been precious time. Talks with him via phone calls were great
help to me. He helped me improve the draft of the dissertation.
I also thank Yuta Michimura for his support and advice. It was the works
on the main-interferometer simulation with him that made me decide to workon KAGRA. He supported our experiments, in particular in designing the maininterferometer setup and preparing necessary items. His attitude on researchreminded me of importance of stopping once and thinking quantitatively. He
gave me constructive comments to improve the draft of the dissertation.
I would like to also thank the following people for their tremendous supports.
Ryosuke Sugimoto and Kazuya Yokogawa worked with me on the implementation of the arm length stabilization system in KAGRA. My work could not
be accomplished without their eﬀorts. Together with Kiwamu, the realization of
the system is the success of every single person of our team All Green. Thanks

vi

Acknowledgments
to Masayuki Nakano, I started working in KAGRA back in 2015. I am surethat I could not have continued the research without him. He has been always
my role model. His great eﬀort enabled us to lock the Fabry–Perot Michelsoninterferometer. Takahiro Yamamoto greatly helped us with the interferometer
commissioning work, which was indispensable to accomplish my work. He is literally the guardian of the control room. Takaaki Yokozawa supported me a lot.
He kindly entered the mine with me even if it was only for turning on a single
switch. I do not forget that he helped me to look for a “wanted” box throughout
the mine for half a day.
Masatake Ohashi greatly supported our research activities. He flexibly madesystems and rules to improve the working environment upon our frank requests.
Yoshio Saito and Takashi Uchiyama arranged the schedule of the site work.
Thanks to them, I was able to have time for my research. Yoshiki Moriwaki and
Kazuhiro Yamamoto supported our work, especially in University of Toyama.
Thanks to their support, I was able to perform preliminary measurements for
fixing the experimental design. Moriwaki-sensei also made significant eﬀorts on
the preparation of the optical components.
Shinji Miyoki supported me a lot. It would not be possible without him to
first lock the X arm cavity. His suggestion was always useful. I learned from
him importance of trying something quickly without worrying it. Yoichi Aso
and Osamu Miyakawa organized the commissioning team. Their suggestions
taught me where to go. Keiko Kokeyama supported the work particularly fromthe input optics system. I learned from her that it is important to change
setups for making progress without being afraid. Takayuki Tomaru supported
the work particularly from the cryogenic suspension system. I was impressedby his speediness in experiments. Takafumi Ushiba also helped us also from thecryogenic suspension system. He gave me a fruitful suggestion also on schemes ofwavelength conversion. Tomotada Akutsu supported the work from a variety of
aspects. I learned basics of experiments from him such as importance of making
a log entry and necessity of preparing items beforehand. Tatsuki Washimi gaveme new knowledges. In particular, we were able to reduce the level of 60 Hz noise
by a factor of more than ten thanks to his suggestion. Eiichi Hirose supported
the work mainly from the point of the core optics. He gave me the information of
the test mass coating for green. He also gave me comments on the power budgetof the X arm. Kentaro Somiya shared interesting ideas with us, especially on
quantum noise. Discussion with him was always interesting and stimulative tome. Ryutaro Takahashi, Ayaka Shoda, Mark A. Barton, Fabián Erasmo PeñaArellano, Lucia Trozzo, Koki Okutomi, Enzo Nicolás Tapia San Martin, NaokoAcknowledgments
Ohishi, Naoatsu Hirata, and Naohisa Sato contributed the completion of room
temperature suspensions for the optics. I could not perform any experimentsin the KAGRA interferometer without their contributions. They also helped
me with installing the steering mirror behind SR2. Thanks to the wrapper
tools developed by Chihiro Kozakai, I managed to have easy access to pastdata of KAGRA. Shoichi Oshino supported our work, especially in maintaining
and developing the digital control system. Darkhan Tuyenbayev supported thecommissioning work from calibration. Other than that, I remember well that it
was with him that we identified the issue of birefringence midnight in the mine.
Takahiro Miyamoto showed me the way of studying the KAGRA detector
in the graduate course. I was impressed that a small component he had tested
at Kashiwa was finally implemented as a part of the cryogenic suspension andit did really work at the KAGRA site. Kouseki Miyo helped me a lot. He
kindly implemented the remote control interface for the signal generators. His
imperturbable calm is always impressive to me. Yoshinori Fujii also helped me alot. He always answered questions from me about the suspensions. I recall a lotof scenes with Kouseki and Yoshinori, as we started working (partially) at thesite from the first year of the graduate course. Tomohiro Yamada has stimulatedme, as he showed good results through his original work. I spent much timewith Kohei Yamamoto in the life at the site. His vitality was always impressive
to me. Dinner with him, Masayuki, and Gui-Guo Ge was always fun and goodrest for me. Terrence Tak Lun Tsang helped me a lot through the suspension
works. He was so smart and eager that he learned many new things rapidly,
which was stimulative. Kenta Tanaka worked on the transmission monitor of thearm cavities. His preparation of the optics greatly contributed to the initial lockof the X arm.
Masahiro Kamiizumi, Tomonori Tomura, and Katsuhiko Shimode helped mevery much from the point of analogue electronics. I probably asked them too
much, but they responded with their best. Working on electronics with Shimodesan was one of the most interesting time for me. Mitsuharu Yoshimura supported
our work a lot. Particularly, he laid many long coaxial cables and power supply
cables used in our experiment. Seiji Furuta led the maintenance work of themine. He always welcomed us with a smile at the entrance of the mine before
and after the mine work. Kyoichi Takayama, Mihoko Okinaka, Yayoi Hara,
Yukari Maeda, Miho Iwamatsu, and Eri Sakamoto greatly supported our workthrough the administration work. We could not do anything without their eﬀort.
I never forget the taste of the soup Okinaka-san made for us one day.
Stefan W. Ballmer was always impressive to me. The way he strongly pushed

vii

viii

Acknowledgments

the experiment forward was amazing. We could not close the DARM control loop
without him. Koji Arai always gave me fruitful suggestions in the Interferometer
Sensing and Control meetings. The modification plan of the radio-frequency
photodetectors for the green PDH loops was entirely based on his suggestion.
He also greatly helped me with improving a manuscript of a paper.
Kentaro Komori is very impressive to me. He is so smart that discussions
with him are always fun. I was so impressed by his enthusiasm on experiments.
Koji Nagano has been the closest colleague to me. His advice was always useful.
He helped me to improve the draft of this dissertation. I was so lucky that I
could work with him. Tomofumi Shimoda is so smart that what he pointed
out often provided me with new insights. He also helped me to improve thedraft. Kunihiko Hasegawa helped me through his unique knowledge on his field.
He introduced me the chain curry restaurants, which I love now. Naoki Aritomi
told us interesting updates from the filter cavity experiment, which has been verystimulative. Hiroki Takeda helped me through his knowledge on astronomical
outputs of the gravitational-wave observations. Satoru Takano helped me when
he visited the KAGRA site, to align the main interferometer beam to the X end.
Takuya Kawasaki’s knowledge and insight on quantum physics were impressive
to me so that the discussion with him was fun. Ching Pin Ooi corrected myEnglish writing, which was very helpful and easy to understand. Yuki Miyazaki
greatly helped me with the initial experiment of the arm length stabilization lastyear, when he visited the KAGRA site. Naoki Kita also helped me with theinitial alignment of the KAGRA beam, especially on the transmission monitor.
Mayuko Niwata helped us with the oﬃce work. She kindly gave me advice when
I could not decide my future direction.
Matteo Leonardi, Eleonora Capocasa, and Yuhang Zhao supported me when
I visited their experiment, the filter cavity experiment. The experiences there
were precious to me.
My simulation work relied on Matlab modules called Noise Budget. I thankthe team led by Cristopher Wipf, who developed the modules.
Akito Kusaka, Masato Takita, Yuji Tachikawa, Takuro Ideguchi, and Hidetoshi
Katori reviewed the dissertation. Their comments helped me to improve the dissertation.
I have been financially supported by ALPS (Advanced Leading Graduate
Course for Photon Science) program in the University of Tokyo, and JSPS Fellowship DC1. This work was supported by MEXT, JSPS Leading-edge Research
Infrastructure Program, JSPS Grant-in-Aid for Specially Promoted ResearchAcknowledgments
26000005, JSPS Grant-in-Aid for Scientific Research on Innovative Areas 2905:
JP17H06358, JP17H06361 and JP17H06364, JSPS Grant-in-Aid for JSPS Fellows 17J00973, JSPS Core-to-Core Program A. Advanced Research Networks,
JSPS Grant-in-Aid for Scientific Research (S) 17H06133, the joint research program of the Institute for Cosmic Ray Research, University of Tokyo, National
Research Foundation (NRF) and Computing Infrastructure Project of KISTIGSDC in Korea, the LIGO project, and the Virgo project.
Finally, I sincerely thank my family for their understanding and continuous
support. Thanks to them, I spent a happy time pursuing what I want to study.
I do not know how I can express my gratitude.

ix

xi

CONTENTS
List of Figures

xv

List of Tables

xviii

Glossaryxix

1 Introduction
1.1 Outline of the dissertation . . . . . . . . . . . . . . . . . . . . . .

1
2

2 Gravitational Waves
3
2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Sources of gravitational waves . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Radiation of GWs . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Detecting gravitational waves . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Free mass motion under the presence of GWs . . . . . . . 8
2.3.2 Concept of GW detectors . . . . . . . . . . . . . . . . . . 10
3 Gravitational-Wave Detector
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Laser interferometric detectors . . . . . . . . . . . . . . . .
3.1.2 Resonant bar detectors . . . . . . . . . . . . . . . . . . . .
3.1.3 Torsion bar detectors . . . . . . . . . . . . . . . . . . . . .
3.1.4 Atom interferometer . . . . . . . . . . . . . . . . . . . . .
3.1.5 Pulsar timing array . . . . . . . . . . . . . . . . . . . . . .
3.2 Ground-based interferometric detectors and their noise sources . .
3.2.1 Summary of the detectors . . . . . . . . . . . . . . . . . .
3.2.2 Configuration . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Global detector network . . . . . . . . . . . . . . . . . . .

11
11
11
13
13
14
14
14
14
15
16

xii

CONTENTS
3.2.4 Fundamental noise sources . . . . . . . . . . . . . . . . . .
Second generation detectors . . . . . . . . . . . . . . . . . . . . .
3.3.1 Science target . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 KAGRA as the fourth detector in the network . . . . . . .
Third generation detectors . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Science target . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . .

17
21
21
22
25
25
26
28

4 Interferometer
4.1 Michelson interferometer . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Response . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Detection scheme . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Laser noise rejection . . . . . . . . . . . . . . . . . . . . .
4.2 Fabry–Perot cavity . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Response . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Detection scheme . . . . . . . . . . . . . . . . . . . . . . .
4.3 Dual-recycled Fabry–Perot Michelson interferometer . . . . . . . .
4.3.1 Power recycling . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Resonant sideband extraction . . . . . . . . . . . . . . . .
4.3.3 Detection scheme . . . . . . . . . . . . . . . . . . . . . . .
4.4 Advanced techniques . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Squeezed light injection . . . . . . . . . . . . . . . . . . .
4.4.2 Filter cavity . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Detuned RSE . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Challenges in the 3G detectors . . . . . . . . . . . . . . . . . . . .
4.5.1 Long arms . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Weak actuators . . . . . . . . . . . . . . . . . . . . . . . .

31
31
31
33
36
37
37
38
41
41
42
43
44
44
45
46
46
46
47

5 Lock Acquisition
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Previous techniques . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Initial LIGO scheme . . . . . . . . . . . . . . . . . . . . .
5.2.2 Third harmonic demodulation technique . . . . . . . . . .
5.2.3 Variable finesse technique . . . . . . . . . . . . . . . . . .
5.3 ALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Necessity of ALS . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Requirements . . . . . . . . . . . . . . . . . . . . . . . . .

49
49
52
52
52
53
54
55
55

3.3

3.4

CONTENTS
5.3.3

xiii
ALS in Advanced LIGO . . . . . . . . . . . . . . . . . . . 56

6 Configuration of KAGRA
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Optical configuration . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Control scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Lock acquisition scheme . . . . . . . . . . . . . . . . . . . . . . .

59
59
61
65
66

7 KAGRA-Type ALS
69
7.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.2.1 Optical and servo topology . . . . . . . . . . . . . . . . . . 71
7.2.2 Noise Design . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.3.1 Control loops . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.3.2 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.3.3 Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.4 Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.4.1 Interferometer setup . . . . . . . . . . . . . . . . . . . . . 89
7.4.2 The first scheme: direct transition from the ALS system . 89
7.4.3 The second scheme: transmission lock . . . . . . . . . . . . 91
7.5 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.5.1 Sensing noise . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.5.2 Actuator saturation . . . . . . . . . . . . . . . . . . . . . . 101
7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.6.1 Expected noise sources . . . . . . . . . . . . . . . . . . . . 102
7.6.2 Doppler noise . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.6.3 Air turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 105
8 ALS in the Third Generation
107
8.1 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.2 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.2.1 Main interferometer parameters . . . . . . . . . . . . . . . 108
8.2.2 ALS configuration . . . . . . . . . . . . . . . . . . . . . . 109
8.3 Noise estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.4 Discussion: sub-carrier CARM locking as common path ALS . . . 117

xiv

CONTENTS

9 Conclusion
123
9.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9.2 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
9.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A Spectral Analysis

127

B Classical Control Theory
129
B.1 Linear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B.2 Feedback loop and its stability . . . . . . . . . . . . . . . . . . . . 130
C Fabry–Perot Cavity

133

D Sideband Picture and Modulation

137

E Quantum State of Light139
E.1 Vacuum state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
E.2 Coherent state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
E.3 Squeezed state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
F Preliminary Measurements
143
F.1 Fiber phase noise measurement . . . . . . . . . . . . . . . . . . . 143
F.2 Measurements on the VCOs . . . . . . . . . . . . . . . . . . . . . 144
F.2.1 Actuation eﬃciency . . . . . . . . . . . . . . . . . . . . . . 144
F.2.2 Frequency noise of the output signal . . . . . . . . . . . . 146
G List of Devices

147

Bibliography151

List of Figures

xv

LIST OF FIGURES
2.1

Schematic view of how free test masses are moved by GWs . . . .

3.1
3.2
3.3

Laser interferometric detectors . . . . . . . . . . . . . . . . . . . . 12
Response functions of Michelson interferometers . . . . . . . . . . 15
Antenna pattern functions of an L-shaped detector to GWs invarious polarization modes . . . . . . . . . . . . . . . . . . . . . . 16
Schematic sensitivity curves of typical GW telescopes . . . . . . . 17
Transfer functions from seismic motion to the bottom of the pendula 18
Observation schedule and the approximate sensitivities of the 2G
detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Sensitivities of the 2G detectors . . . . . . . . . . . . . . . . . . . 24
Sensitivities of the 3G detectors . . . . . . . . . . . . . . . . . . . 28

3.4
3.5
3.6
3.7
3.8
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9

Schematic view of a Michelson interferometer . . . . . . . . . . .
Schematic view of an optical layout for the balanced homodyne
readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic diagram of a Fabry–Perot cavity . . . . . . . . . . . . .
Response functions from GW strain to phase deviation of the output optical field in various interferometer configurations . . . . . .
PDH signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic diagram of a dual-recycled Fabry–Perot cavity . . . . .
Conceptual diagram of how the ASD of strain-equivalent shotnoise changes as the interferometer configuration changes . . . . .
Schematic diagrams of how the squeezed vacuum is incorporated
with the interferometer . . . . . . . . . . . . . . . . . . . . . . . .
Response transfer function of a cavity with the FSR of 3.75 kHz
to the frequency fluctuations of input laser . . . . . . . . . . . . .

9

32
35
37
39
40
41
43
45
47

xvi

List of Figures
4.10 Design sensitivities of the 2G and the 3G detectors in low frequency, in terms of displacement . . . . . . . . . . . . . . . . . . . 48
5.1
5.2
5.3
5.4
5.5
5.6

Lock acquisition with and without ALS . . . . . . . . . . . . . . .
Schematic diagrams of the lock acquisition of Initial LIGO . . . .
Schematic diagrams of the variable finesse technique . . . . . . . .
Conceptual diagram of ALS . . . . . . . . . . . . . . . . . . . . .
Schematic view of the ALS system of LIGO . . . . . . . . . . . .
Schematic diagrams of the lock acquisition schemes with ALS . .

51
52
53
55
57
58

6.1
6.2
6.3
6.4

Schematic view of KAGRA . . . . . . . . . . . . . . . . . . . . . .
Picture and schematic view of a cryogenic payload . . . . . . . . .
Optical configuration of KAGRA . . . . . . . . . . . . . . . . . .
Schematic view of the carrier and sideband fields and the detection
ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60
61
62

Schematic view of the ALS system of KAGRA . . . . . . . . . . .
ASDs of the simulated noise spectra of the KAGRA ALS system .
Schematic diagram of the subcomponent loops of the system. . . .
Picture of the PSL table . . . . . . . . . . . . . . . . . . . . . . .
Layout of the optics for the ALS system on the PSL table . . . .
Picture of the laid optical fibers . . . . . . . . . . . . . . . . . . .
Configuration of the polarization Michelson interferometers for thecancellation of fiber noise . . . . . . . . . . . . . . . . . . . . . . .
7.8 Layout of the optics for the ALS system on the POP table . . . .
7.9 Layout of the optics for the ALS system on the POS table . . . .
7.10 Pictures of in-vacuum steering mirrors behind PR2 and SR2 . . .
7.11 Conceptual diagram of the VCO circuit . . . . . . . . . . . . . . .
7.12 Interferometer configuration for the demonstration of lock acquisition with the KAGRA ALS . . . . . . . . . . . . . . . . . . . .
7.13 Time series of the optical power at various detection ports during
the first lock acquisition scheme . . . . . . . . . . . . . . . . . . .
7.14 Time series of the optical power at various detection ports during
the second lock acquisition scheme . . . . . . . . . . . . . . . . .
7.15 ASDs of sensing noise of the ALS system . . . . . . . . . . . . . .
7.16 ASDs of sensing noise of the ALS system for the Y arm with andwithout the fiber noise cancellation loop engaged . . . . . . . . .
7.17 Noise budget plot of sensing noise of the ALS system for ALS
CARM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71
74
77
80
80
81

7.1
7.2
7.3
7.4
7.5
7.6
7.7

66

82
83
83
84
86
87
91
93
94
96
97

LIST OF FIGURES

xvii

7.18 Doppler coupling of the longitudinal motion of PR2 . . . . . . . . 99
7.19 ASD of the control signal of ALS DARM applied to the test massactuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.20 Expected ASD of sensing noise of ALS CARM in the DRFPMI . 102
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8

Possible configurations of the control of the auxiliary laser frequency111
Expected ASDs of residual noise of CARM, ∆fres (f ; κCARM ), in
the Cosmic-Explorer-like detector . . . . . . . . . . . . . . . . . . 115
Expected noise performance of the single arm ALS in the CosimicExplorer-like detector . . . . . . . . . . . . . . . . . . . . . . . . . 117
Conceptual diagram of the hand-over of the CARM control as thelast step of lock acquisition . . . . . . . . . . . . . . . . . . . . . . 118
Schematic diagram of the interferometer and control configuration
for a reliable lock acquisition scheme . . . . . . . . . . . . . . . . 120
CARM resonance of the carrier and sideband fields in terms of thecarrier frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Layout of the optics for the generation of the sub-carrier field . . 121
Noise budget plot of the sub-carrier scheme for the CARM control 122

B.1 Schematic diagram of linear systems . . . . . . . . . . . . . . . . 129
B.2 Nyquist plots and Bode plots . . . . . . . . . . . . . . . . . . . . 131
C.1 Schematic view of a Fabry–Perot cavity and associated electric fields133
C.2 Optical power in a cavity and roundtrip phase . . . . . . . . . . . 135
D.1 Modulation picture and sideband picture . . . . . . . . . . . . . . 138
E.1 Schematic diagram of quantum fluctuations of the vacuum state
and a squeezed state . . . . . . . . . . . . . . . . . . . . . . . . . 142
F.1 Picture of the experimental setup for the preliminary measurement of fiber phase noise . . . . . . . . . . . . . . . . . . . . . . . 143
F.2 ASD of fiber frequency noise in the preliminary measurement . . . 144
F.3 Measured VCO responses to the input voltage . . . . . . . . . . . 145
F.4 Setup for the measurement of the VCO frequency noise . . . . . . 145
F.5 Measured ASD of the output of the VCO . . . . . . . . . . . . . . 146
G.1 Picture of special modifications on the RF PDs for the arm PDH
sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

xviii

List of Tables
G.2 Top view of the prototype low noise VCO circuit . . . . . . . . . . 149

LIST OF TABLES
3.1

Summary of the ground-based interferometeric detectors . . . . . 14

6.1

Design and measured values of the main interferometer parametersof KAGRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Five length DoFs of a DRFPMI and their sensing signals . . . . . 65

6.3
7.1
7.2
7.3
7.4

Main diﬀerences between the ALS systems of KAGRA and Advanced LIGO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Design of the control bandwidth of each loop of the system . . . . 75
List of major noise sources and assumed magnitude of them . . . 76
Comparison of the expected and achieved noise levels . . . . . . . 103

8.1

Parameter set of the Cosmic-Explorer-like detector . . . . . . . . 108

G.1
G.2
G.3
G.4

List of devices (optics) . . . . . . . . . . . . . . . . . . . . . . . . 147
List of devices (opto-electronics) . . . . . . . . . . . . . . . . . . . 148
List of devices (opto-mechanics) . . . . . . . . . . . . . . . . . . . 149
List of devices (electronics) . . . . . . . . . . . . . . . . . . . . . . 149

xix

GLOSSARY
Symbolsi

imaginary unit

c
G
h
ℏ
kB

speed of light
Newtonian constant of gravitation
Planck’s constant
Dirac’s constant: ℏ = h/2π
Boltzmann constant

M⊙

Solar massEFfh(t), h(Ω)
L
P
λ
Ω
ω

Electric field
Cavity finesse
Fourier frequency
Strain of gravitational waves
Arm length
Optical power
Wavelength of lightSideband or Fourier (angular) frequency
Carrier (angular) frequency

xx

Glossary

Abbreviations
1G
2G
3G
ASD
ALS
AOM
BBH
BH
BNS
BS
DoF
DRFPMI
DRMI
EOM
ETM
FPMI
FSR
GW
IMC
ITM
LO
NS
OMC
PD
PDH
PLL
PMC
PR
PRFPMI
PRM
PSD
PSL
RF
RMS
RSE

First Generation
Second Generation
Third Generation
Amplitude Spectral Density
Arm Length Stabilization
Acousto-Optical Modulator
Binary Black Hole
Black Hole
Binary Neutron Star
Beam Splitter
Degree of Freedom
Dual-Recycled Fabry–Perot Michelson Interferometer
Dual-Recycled Michelson Interferometer
Electro-Optical Modulator
End Test Mass
Fabry–Perot Michelson InterferometerFree Spectral Range
Gravitational Wave
Input Mode Cleaner
Input Test Mass
Local Oscillator
Neutron Star
Output Mode Cleaner
Photodiode/Photodetector
Pound Drever Hall
Phase-Locked Loop
Pre-Mode Cleaner
Power Recycling
Power-Recycled Fabry–Perot Michelson Interferometer
Power Recycling Mirror
Power Spectral Density
Pre-Stabilized Laser
Radio Frequency
Root Mean Square
Resonant Sideband Extraction

Glossary
SHG
SR
SRM
VCO

xxi
Second Harmonic Generation
Signal Recycling
Signal Recycling Mirror
Voltage-Controlled Oscillator

Interferometer DoFs
DARM
CARM
PRCL
SRCL
MICH

Diﬀerential ARM length
Common ARM length
Power Recycling Cavity Length
Signal Recycling Cavity Length
MICHelson LengthInterferometer Sensing Ports
AS
REFL
POP
POS
TRX
TRY

Anti-Symmetric portinterferometer REFLection port
Pick-Oﬀ port of Power recycling cavity
Pick-Oﬀ port of Signal recycling cavity
TRansmission port of the X arm
TRansmission port of the Y arm

1

Chapter 1
INTRODUCTION
Gravitational waves (GWs) are space-time ripples traveling through the spacetime, which were first predicted in General Relativity [1,2]. Since an object thatis heavier and moves faster emits GWs more, it is thought that the observable
GWs are sourced from astronomical or cosmological origin [3, 4]. It is expectedthat direct observations of GWs reveal undiscovered nature of gravity theory,
astronomy and cosmology in a unique way. In fact, the direct observations of
GWs by terrestrial interferometric detectors have provided new discoveries [5,6].
Moreover, larger terrestrial detectors that are planned to be built in near future
will reveal a lot of new insights on those fields [7, 8].
The direct observations of GWs have revealed astronomical information suchas the existence of binary black hole systems [5], and the heaviest stellar mass
black hole ever [9, 10]. Simultaneous observation of GWs and gamma rays indicated that the origin of a short gamma ray burst is a binary neutron starmerger [11]. The observations have also set limits to gravity theories, such as
speed of GWs [11]. In addition to these results, it is expected that the near-future
larger GW detectors provide more; with their better sensitivities, they will be capable of, for example, identifying the equation of state of neutron stars [12, 13],
probing the mechanism of core-collapse supernovae [14, 15], and setting morestrict limits on gravity theories [16, 17].
Modern GW detectors [18–20], which are sensitive enough to fulfill the scientific targets raised above, inevitably employ kilometer-long coupled opticalcavities controlled with very sensitive but narrow-ranged interferometric sensors.
Therefore, considering that the controls do not last arbitrarily long as they areoccasionally broken by external disturbances, the process to somehow bring theinterferometer of the GW detector to the operation point is essential in realizing
the observation of GWs. Such a process is called interferometer locking or lockacquisition. Especially, deterministic and systematic lock acquisition is necessary

2Chap.1 Introduction
for stable long term observation. The most challenging part of lock acquisition
is to control kilometer-long optical cavities. A scheme called arm length stabilization (ALS) plays an essential role in deterministically controlling the cavities
in the lock acquisition sequence [21]. An ALS scheme utilizes auxiliary lasers
to sense the cavity lengths. In the ALS scheme that has been used so far, theauxiliary lasers are injected to the interferometer from end stations, which are
kilometer-away from the corner station, with a fraction of the main laser sentfrom the corner station to the end station via optical fibers [22]. Because of thisfeature, it is not clear if this scheme is scalable to longer-baseline GW detectors.
To address such an issue, a new ALS system that is scalable to such detectors
has been designed and implemented for KAGRA, a GW detector soon to join the
network observation [23] having key features important in the future detectors[24, 25]. In the dissertation, the detailed design and the implementation of thesystem are presented, and the experimental results in the performance tests of
the ALS system are reported, where the Fabry–Perot Michelson interferometerof KAGRA was successfully locked. In addition, implications of the results on
the future GW detectors are also discussed.

1.1

Outline of the dissertation

This dissertation is organized as follows. Chapter 2 summarizes the basic properties of GWs. Chapter 3 overviews GW detectors, instruments to observe GWs,
focusing on ground-based interferometric GW detectors. Science outcomes of thecurrent detectors and those of the detecters planned to be built in near future
are also summarized. Chapter 4 describes principles and techniques of interferometry involved in the modern and future GW detectors. Chapter 5 reviews
lock acquisition schemes in GW detectors. A scheme called ALS is introduced inparticular, and its importance is clarified. Chapter 6 overviews the configurationof KAGRA focusing on its interferometer as the experimental setup where the
new ALS scheme has been developed and demonstrated. Chapter 7 is the mainpart of the dissertation, which describes the design of the ALS system, its implementation, and the demonstrations and characterizations of its performance.
Chapter 8 discusses the possible configuration and noise performance of the ALS
system in the planned GW detectors, which can be inferred from the characterizations of the KAGRA ALS scheme. The necessary techniques in the future
ALS scheme are also assessed. Chapter 9 summarizes the results, provides the
future prospects, and finally concludes the dissertation.

3

Chapter 2
GRAVITATIONAL WAVES
Gravitational waves (GWs) are space-time ripples traveling through the spacetime. The existence of GWs was first predicted by General Relativity of A.
Einstein in 1916 [1,2], and was proven indirectly by the observation of the energy
loss in a binary pulsar PSR B1913+16 by R. A. Hulse and J. H. Taylor [26,27]. In 2015, two Advanced LIGO detectors detected a GW from a binary
black hole merger [5], which marked the beginning of an era of GW astronomy.
In this chapter, first the physics of GW is overviewed, then sources of GWs
are summarized, and the method of the observation of GWs are briefly andconceptually explained.

2.1

Derivation

In this section, let us derive the expression of GWs, and how a test mass moves
with GWs in the framework of General Relativity in four-dimensional space-time.
In General Relativity, our space-time is considered as a four-dimensional manifold with distance defined in such a way that the distance between two points
xµ and xµ + dxµ can be expressed as
ds2 = gµν (x)dxµ dxν .

(2.1)

Here and hereafter, Greek indices run from 0 to 3, and Einstein’s convention isused. gµν (x) is the metric tensor of space-time. In General Relativity, the metrictensor gµν completely characterizes the geometry of the space-time, which isgravity.
The metric tensor satisfies the Einstein equation, which can be expressed asfollows:
8πG
1
(2.2)
Rµν − gµν g αβ Rαβ = 4 Tµν .
2c

4Chap.2 Gravitational Waves
Here, c is the speed of light and G is the Newtonian gravitational constant. A
tensor field Rµν is referred to as Ricci tensor, which is defined in the following
equations:Rµν := Rµαν α ,∂∂Rαβγ δ := − α Γ δβγ + β Γ δαγ − Γ δαµ Γ µβγ + Γ δβµ Γ µαγ ,
∂x (
∂x
)
∂
∂
1 γδ
∂
γ
gβδ + β gαδ − δ gαβ .
Γ αβ := g
2
∂xα
∂x
∂x

(2.3)
(2.4)
(2.5)

Here Rαβγ δ is called as Riemann curvature tensor, and Γ γαβ is called connection
coeﬃcients or Chirstoﬀel symbols. Tµν in the right-hand side of Equation (2.2) isthe energy-momentum tensor, so it can be said that the Einstein equation (2.2)
connects the energy and momentum in the space-time to the geometry of thespace-time.
In Minkowski space-time, where the gravity does not exist, the space-time
is flat in the sense that the metric of η = diag[−1, 1, 1, 1] can be used throughout the space-time. Here let us consider describing the Einstein equation in a
perturbative way using the deviation from η, hµν := gµν (x) − ηµν . If we assume
the gravity is suﬃciently weak and consider only up to the first order of h, thenEquation (2.2) can be simplified into the following:∂ 2 h̄µβ∂ 2 h̄αβ∂ 2 h̄µν∂ 2 h̄µα
16πG
−
η
+
+
=
Tαβ ,
αβ
∂xµ ∂xν∂xµ ∂xν ∂xα ∂xµ ∂xβ ∂xµc4
1h̄αβ := hαβ − ηαβ η µν hµν .
2

−η µν

(2.6)
(2.7)

Since it is well-known [3] that we can choose such a coordinate system that theLorenz gauge condition∂ µνh̄ = 0(2.8)∂xµis satisfied, let us always impose the Lorenz gauge condition hereafter. Then
Equation (2.6) turns to be− □h̄αβ := −η µν

16πG∂ 2 h̄αβ=Tαβ .
∂xµ ∂xνc4(2.9)

This equation indicates that h, the deviation of the metric tensor from η, issourced by the energy-momentum T and propagates as waves in the space-time,
which shows the presence of the GWs.

2.2. Sources of gravitational waves

5

To discuss the propagation of the GWs in a flat space-time, let us consider
Equation (2.9) in vacuum (Tαβ = 0):□h̄αβ = 0.

(2.10)

It is also known that there is a coordinate system that satisfies the transverse–
traceless (TT) gauge conditions [4]:‡1h0α = 0,(2.11)

hjk = 0,jk

(2.12)

∂h
= 0.
∂xj

(2.13)

Note hαβ = h̄αβ , since η αβ hαβ = 0 holds in the TT gauge condition. Let us set
xµ = [ct, x, y, z], and consider GWs traveling in +z direction. Then the solutionof Equation (2.10) will be expressed by using two independent functions h+ (t)and h× (t) as follows:0000 h+ (t − z ) h× (t − z )cchαβ = 
0 h× (t − z ) −h+ (t − z )
c
c
0
0
0


0
0
.
0
0

(2.14)

Here h+ (t) and h× (t) are called plus (+) mode and cross (×) mode, respectively.
We now know that GWs in General Relativity propagate at the speed of light,
and have two independent polarizations.
If General Relativity is not assumed, up to six independent polarizations of
GWs can be considered [3], or the propagation speed of GWs can be diﬀerent
from c. Such deviations from General Relativity have been constrained to someextent by observation of GWs [11, 28].

2.2

Sources of gravitational waves

First the description of the radiation of GWs is explained, and then the interesting sources of GWs are introduced.
‡1

Latin indices run from 1 to 3.

6

Chap.2 Gravitational Waves

2.2.1

Radiation of GWs

Let us consider the radiation of GWs from a source having energy-momentum
localized. Then the formal solution of Equation (2.9) is written as
4Gh̄ (t, x) = 4
c

∫

αβ

T αβ (t − |x − x′ | /c, x′ ) 3 ′d x.
|x − x′ |(2.15)

Assuming the typical speed of the motion of the source is smaller than c and the
distance between the observer and the source is larger than the wavelength of
the GWs of interest, Equation (2.15) can be approximated as∫
4Gαβh̄ (t, x) ≃ 4
T αβ (t − r/c, x′ )d3 x′ ,(2.16)
crwhere r is the distance between the observer and the center of the source. Here,using the Lorenz gauge condition ∂T µν /∂xµ = 0,‡2 the quadrupole formula
2G ¨jkI (t − r/c)c4 r

(2.17)

xj xk T 00 (t, x)d3 x.

(2.18)h̄jk (t, x) ≃is obtained, where∫jk

I (t) :=In the TT gauge condition, these expression can be rewritten as
2G ¨TTI (t − r/c),
c4 r jk
1
TT
Ijk:= Pjl Ilm Pmk − Pjk Plm Ilm ,2
Pjk := δjk − n̂j n̂k .

hTTjk (t, x) ≃(2.19)(2.20)(2.21)

Here n̂ is a unit vector pointing from the center of the source to the observer.
Here it can be said that the lowest order radiation of GWs is quadrupole. Thus
the GWs are radiated from the non-axisymmetric motions of mass. In addition,
Equation (2.19) tells us that the higher speed motion of the heavier mass radiates
the larger GWs. Therefore, the sources of the observable GWs are considered tobe limited to the extreme astronomical or cosmological phenomena.
‡2

Correct up to O(h). This corresponds to the conservation of energy-momentum.

2.2. Sources of gravitational waves

2.2.2

Candidates

Compact binary coalescence
The most promising sources of GWs are compact binary coalescences. Eleven
GW detection events have been published up to now, and all of them were fromcompact binary coalescences. The first detection event and nine other eventswere originated in mergers of two orbiting black holes (BHs) [9]. These events
have brought us information on binary synthesis [29–32]. The other publishedevent of GW detection was from merging neutron stars (NSs) [6]. This event
was also observed by follow-up observations in wide range of electromagnetic
waves [33]. This event indicated that the origin of short gamma-ray bursts isthe merger of two orbiting NSs [11,34], and also provided an implications on theorigin of heavy metals [35]. The GW event from the merger of an NS and a BH
is also expected, but have not been observed yet.
The frequency of GWs just before the merger can be approximately written
as f ∼ 4.4 kHz × (M⊙ /M ) [36], where M⊙ is solar mass, and M is the total
mass of the binary. Thus the mass range of M ∼ 1 − 100 M⊙ is the target of
the ground-based GW detectors. The typical amplitude of the GWs observedon the Earth is considered as h ∼ 10−21 . It is estimated that ∼ 1 − 50 events of
GWs can be detected in one year with the current sensitivity of the ground-based
detectors [23].‡3
Supernova
Another interesting sources of GWs are core collapse of supernovae. If there is
non-axisymmetric component in the core collapse, GWs would be emitted. The
estimated amplitude of the GWs is h ∼ 10−22 − 10−21 seen from 10 kpc awayfrom the source, and the typical frequency is considered to be ∼ 1 kHz [37]. If
GWs from supernovae are observed, it would be unique information to access
what happens inside the star when a supernova occurs.
Rotating neutron star
If there is non-axisymmetry in a rotating NS, it will emit GWs. Since the frequency of the emitted GWs is typically the double of the rotation frequencyof the NS, fast rotating NSs including millisecond pulsars are the target of theground-based detectors. The GW observations have imposed upper limits on‡3

Estimated sensitivity and period of Observation 3 (Figure 3.6).

7

8

Chap.2 Gravitational Waves
ellipticity ε of known pulsars; ε < 1 × 10−5 with 95% credibility for Crab Pulsar,
for example [38].
Merger of supermassive black holes
Merger of supermassive BHs may be regarded as an extreme case of compact
binary coalescence. Although it has been known that there is a supermassive
black hole (M ∼ 106 M⊙ ) in most galaxies, scenario of its formation has not
yet been determined. It is expected that GWs around 1 mHz from mergers ofsupermassive BHs will provide indications to the scenario [39].
Cosmic gravitational wave backgroundIf a number of GWs interfere with each other, their combination behaves as
stochastic background. Among such GWs, cosmic GW background coming fromquantum fluctuations during inflation is one of the most interesting targets [40].
Although the expected spectrum of such GWs is flat, it is key to choose anobservational frequency band where the foreground GWs are removable.

2.3

Detecting gravitational waves

After summarizing how free mass moves as GWs come, the detection methodsof GWs are conceptually explained.

2.3.1

Free mass motion under the presence of GWs

In General Relativity, the trajectory of a free mass is a geodesic, which meansthat the trajectory x(τ ) satisfies the geodesic equation,d2 xµdxα dxβ
µ
+
Γ
(x(τ
))
= 0,
αβ
dτ 2
dτ dτ

(2.22)

where τ is a parameter for the geodesic.
Hereafter, let us impose the TT gauge condition and chose time t as theparameter τ . In this condition, xµ (t) = [ct, X] satisfies Equation (2.22), where
X is a set of three arbitrary real numbers. This fact indicates that we canplace test masses whose spatial coordinates remain unchanged. Then let us
consider how the distance between such two test masses changes in time under the presence of GWs. Here, for simplicity, let us consider a plane wave

2.3. Detecting gravitational wavesy9+ modex

 modeFigure 2.1: Schematic view of how free test masses on xy plane are moved by
GWs traveling along z axis. It can be seen that the test masses tidally move.
with + polarization propagating in z direction, and test masses on xy planexµ (t; θ) = [ct, R cos θ, R sin θ, 0] and [ct, 0, 0, 0]. Such GW can be modeled by
setting h+ (t) = hp cos Ωt, h× (t) = 0 in Equation (2.14). The spatial distance
between two test masses can be calculated as follows:√√
gjk xj (t; θ)xk (t; θ) = R 1 + hp cos 2θ cos Ωt()
1= R 1 + hp cos 2θ cos Ωt ,2

(2.23)(2.24)

ignoring the second and higher order of h. Equation (2.24) shows that the test
mass moves as shown in the upper half of Figure 2.1 with the GWs in + mode.
Similarly, the motion of the test mass with GWs in × mode, h+ (t) = 0 , h× (t) =hc cos Ωt, can also be obtained:√)(1
gjk xj (t; θ)xk (t; θ) = R 1 + hc sin 2θ cos Ωt .
2

(2.25)

Equation (2.25) shows that the test mass moves as shown in the lower half of
Figure 2.1 with the GWs in × mode.

10

Chap.2 Gravitational Waves

2.3.2

Concept of GW detectors

As discussed in the previous subsection, the distance between free test masseschanges with GWs. Therefore, GWs can be detected by measuring the distance
between free test masses. One such realization of this idea is an interferometric
GW detector, where the distance between two masses along x axis and thatbetween two test masses along y axis are compared by using laser interferometry.

11

Chapter 3
GRAVITATIONAL WAVE
DETECTOR
In this chapter, GW detectors are introduced focusing on ground-based interferometric GW detectors. First, various types of GW detectors are overviewed.
Then ground-based interferometric detectors are introduced in detail. Next, theconcrete GW detectors so called the second and third generation detectors arereviewed.

3.1

Overview

In this section, various kinds of GW detectors that are built or proposed so far
are overviewed.

3.1.1

Laser interferometric detectors

In laser interferometric GW detectors, diﬀerential length change of the two arms
is measured by using interference of laser light. The concept of laser interferometric GW detectors is shown in Figure 3.1. At both ends of each arm, mirrors
are placed. They behave as test masses and move as GWs come. The laser interferometric GW detectors can be categorized into two: ground-based detectors
and space-borne detectors.
Ground-based
Ground-based laser interferometric detectors have been developed most intensively, and they are the only detectors that really detected GWs so far [9]. Since

12Chap.3 Gravitational-Wave Detector

Figure 3.1: Conceptual view of laser interferometric GW detectors. As GWs
arrive at the detector, one arm stretches while the other arm shrinks. The
diﬀerential changes in the arm lengths are detected by using laser interferometrictechniques.
it is not easy to have free test masses on the Earth, their test masses are suspended as pendula so that the test masses are regarded as free-falling for GWs
in the frequencies above the resonant frequency of the pendula. In this scheme,in spite of the intensive eﬀort to reduce the eﬀect of seismic motion on the test
masses, the target frequency of observation has been limited by f ≳ 5−10 Hz [41].
Currently, detectors categorized in the second generation (2G) detectors such as
Advanced LIGO, Advanced Virgo, and KAGRA are operating [23]. The third
generation (3G) detectors having longer arms than those of the 2G detectors arealso planned worldwide [7,8]. This scheme is described in more detail in the next
subsection.
Space-borne
In contrast to the ground-based ones, it is rather easy to prepare free test masses
in space. In addition, it is also easier to have long arm length. Utilizing thesefeatures, the target frequency band of the space-borne detectors is ∼ 1 mHz −10 Hz, which is lower than that of the ground-based ones. One major mission of3.1. Overview
space-borne GW detectors is LISA [42]. The arm length of LISA is planned to be
2.5 × 109 m, and the target frequency band is 0.1 − 100 mHz. The science targetof LISA is mainly binary BHs. Since it is diﬃcult to get laser beam back andforth in such a long arm, LISA adopts the optical transponder scheme, where
the space craft receives laser light from the other edge of an arm, and inject laserthat is phase-locked to the received laser light. Another major proposed missionof this category is DECIGO [43]. The arm length of DECIGO is planned to be
1 × 106 m, and the target frequency band is 0.1 − 10 Hz. The science target of
DECIGO is cosmic GW background from the inflation, as well as binary BHs
and binary NSs. To reach such good sensitivity, it is planned that DECIGO uses
optical configuration similar to that of ground-based ones, though it might bechallenging to do in space.

3.1.2

Resonant bar detectors

In a local inertial frame, GWs can be treated as external force to act on masses [3].
The distance between two masses that are elastically connected to each otherchanges larger at the resonant frequency of this two mass system than otherfrequency. Resonant bar detectors are the realization of this idea; they aredesigned to be sensitive to GWs around their resonant frequency. The firstexperimental approach to directly detect GWs was a resonant bar detector by J.
Weber [44]. The typical target frequency is typically a few kilohertz [45].

3.1.3

Torsion bar detectors

Due to the eﬀect of seismic motion, it is not straightforward to have detectors on
the Earth sensitive to GWs at frequency lower than ∼ 1 − 10 Hz. One approachto have good sensitivity in such frequency region is to employ a torsion bar as atest mass. In this scheme, rotational motion of a torsion pendulum is used as aprobe for GWs. Use of a torsion pendulum has an advantage that the resonant
frequency of torsional motion can be much lower than that of translational motion
with the same size of the suspension system, which means a torsion pendulum
is superior to an ordinary pendulum in the attenuation of the seismic motion.
A major experiment toward the realization of torsion bar detectors is TOBA
experiment [46].

13

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Chap.3 Gravitational-Wave Detector
Table 3.1: Summary of the ground-based interferometeric detectors.

DetectorsYear
Sensitivity√ at
100 Hz [/ Hz]

3.1.4

1G
TAMA300,
Initial LIGO,
Virgo,GEO6001990’s – 2010’s

2G

3G

Advanced LIGO,Advanced Virgo,
KAGRA

Einstein Telescope,
Cosmic Explorer

2010’s – 2020’s

2030’s – 2040’s∼ 3 × 10−23∼ 3 × 10−24∼ 3 × 10−25

Atom interferometer

Another approach to built detectors on the Earth sensitive in the low frequency
region is to use atom interferometers. In this scheme, matter-wave interferometers of free-falling cold atoms serve as test masses to GWs, between which lasers
propagate for measuring the distance [47]. There are projects to realize atominterferometric detectors such as MIGA [48] or ZAIGA [49]. There are also proposes to build atom interferometers in space such as AGIS [47] or AIGSO [50].

3.1.5

Pulsar timing array

In the pulsar timing array scheme, the strain of GWs is inferred from the changeof the interval of the arrival of the pulsar signals from many known pulsars. The
repetition rate of pulsar signals is so precise that it is expected that the repetition
rate changes of pulsar signals caused by GWs can be detected [51]. The target
frequency band of this scheme is ∼ 10−9 − 10−6 Hz [52].

3.2

Ground-based interferometric detectors andtheir noise sources

3.2.1

Summary of the detectors

Table 3.1 summarizes the past, present and future ground-based interferometricdetectors. The detectors are commonly categorized into three generations. The
first generation (1G) detectors have arm length of from several hundred meters
to several kilometers. The 2G detectors have 3 − 4 km arms. Advanced LIGO

3.2. Ground-based interferometric detectors and their noise sources1014

L = 1500km
L = 500km
L = 150km
L = 50km

Response [rad]

1013

1012

1011

1010100

101

102
Frequency [Hz]

103

104

Figure 3.2: Response functions of Michelson interferometers with various arm
lengths. The expression of the response function is given by Equation (4.9).
and Advanced Virgo utilize the infrastructure of the corresponding 1G detectors,
while KAGRA has been built upon new infrastructure. The 3G detectors areplanned to have 10 − 40 km arms in new sites.

3.2.2

Configuration

The working principle of ground-based interferometric detectors is as simple asthat of the Michelson interferometer (Figure 3.1). Each mirror forming the interferometer is suspended as a pendulum so that the mirror moves as a test mass as
GWs pass through the detector. The motion of the mirrors caused by the GWs
creates diﬀerential phase variations of the laser in the arms, and the Michelsoninterferometer converts them into variations of the laser power, which are thendetected by photodetectors.
Figure 3.2 illustrates the detector response function from the strain of GWs
to the measurable phase variations by the Michelson interferometer. It shows
the responses of the Michelson interferometers that have several diﬀerent arm
lengths. This figure indicates two things. One is that longer arm length provides

15

16

Chap.3 Gravitational-Wave Detector

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0.0

0.0

0.5

0.5

0.5

0.5

1.0
0.5

0.0
0.0

0.5 0.5(a) + mode.

0.5

1.0
0.5

0.0
0.0

0.5 0.5(b) × mode.

0.5

1.0
0.5

0.0
0.0

0.5 0.5(c) Average.

Figure 3.3: Antenna pattern functions of an L-shaped detector to GWs in various polarization modes [4]. For each polarization, the distance from the origin
measured in a certain direction depicts the magnitude of the response to GWs
coming from the direction. The color of the surface also shows the distance. The
response function is normalized so that the maximum response is equal to unity.
better response to the GWs in low frequencies. The other thing is that, for the
GWs at a certain frequency f , the increase of the detector response saturates
as the arm length reaches c/4f . This is because the phase change of the GWs
themselves during the propagation of the laser in the arms becomes significant if
the arm length is longer than c/4f . Therefore, to have a good sensitivity around
100 Hz, which is the current target band of the ground-based interferometricdetectors, the arm length of ∼ 500 km is required.
Since it is not realistic to achieve such long arm length on the Earth, thecurrent detectors employ optical cavities to enhance the eﬀective arm length [53].
Laser light propagates multiple roundtrips in the optical cavities so that theeﬀective arm length is enhanced by the average number of roundtrips. Theinterferometer configuration including advanced interferometry is described in
the next chapter.

3.2.3

Global detector network

For the ground-based GW detectors, global network observation with multiple
detectors is essential. Coincident observation by two detectors is necessary toconfidently claim the detected transients as GWs. Three detectors are needed
to localize the source of the detected GWs. Figure 3.3 shows the response of a

3.2. Ground-based interferometric detectors and their noise sources

Figure 3.4: Schematic sensitivity curves (ASDs) of typical GW detectors. Contributions from the three fundamental noise sources are shown.

detector to GWs in a certain polarization from a certain direction. This indicatesthat it is not possible to localize the source of GW transients only by a single
detector. The source is localized rather by diﬀerences in time of arrivals at
multiple detectors. The four detector observation is essential in disentangling
polarizations of GWs for the test of non-tensorial GWs [54]. In addition, thefour detector network will increase the temporal and spatial coverage of thelocalizable GW events. In particular, smooth recovery of the operation mode ofeach detector via a reliable lock acquisition scheme helps improve the temporal
coverage, namely the duty cycle.

3.2.4

Fundamental noise sources

Figure 3.4 conceptually shows typical noise spectrum of the ground-based interferometer. The magnitude of the stationary noise is described by the power
spectral density (PSD) in the frequency domain (Appendix A). In the field of
GW detection, the amplitude spectral density (ASD), which is the square rootof the PSD, is also used for the description of the noise level. Each fundamental
noise source is described in the following subsection.

17

18

Chap.3 Gravitational-Wave Detector

102

SingleDoubleTriple

100
10 1
10 2
10 3
10 4
10 5
10 610 1

1 Hz
0.1 Hz102
101
Transfer Function [m/m]Transfer Function [m/m]

101

100
10 1
10 2
10 3
10 4
10 5

100
101
Frequency [Hz]

(a)

102

10 610 2

10 1

100Frequency [Hz]

101

102

(b)

Figure 3.5: Transfer functions from seismic motion to the bottom of the pendula.
(a) How the attenuation function changes as the number of pendulum stages
increases. (b) How the attenuation function changes as the resonant frequencyof the pendulum decreases.
Seismic noise
Since the detectors are built on the Earth, vibrations of the test mass mirrors
introduced from the seismic motion are inevitable. To attenuate the eﬀect of theseismic motion on the test masses, multi-stage pendula are employed. Figure
3.5 shows the transfer functions from the ground motion to the bottom stageof a multi-stage pendulum, with diﬀerent number of stages or with diﬀerent
resonant frequencies. This figure illustrates that use of multi-stage pendula andhaving low resonant frequencies eﬀectively increases the attenuation factor. To
have pendula with low resonant frequencies with limited space, spring-antispring
systems are employed such as an inverted pendulum [55] for horizontal motion
or a geometric antispring filter [56] for vertical motion. Another approach is to
reduce the seismic motion itself; building the whole detector underground withstable bedrock also helps to improve the seismic noise [24].
Seismic noise is a dominant noise source below ∼ 10 Hz, because seismic
motion itself is larger at low frequencies and it is rather diﬃcult to have large
seismic attenuation there. Apart from the sensitivity point of view, having a goodseismic attenuation is essential for the detectors, since it helps the stability of thedetectors and the reliability of the lock acquisition process of the interferometer.

3.2. Ground-based interferometric detectors and their noise sources

19

Thermal noise
Thermal noise is thermal motion of the center of mass and/or the reflection
surface of test mass mirrors having finite temperature.‡1 According to thefluctuation-dissipation theorem, the PSD of the thermal noise Sx (f ) is expressedas
4kB T
Sx (f ) = −Im[H(2πf )],(3.1)(2πf )
where kB is the Boltzmann constant, T is the temperature of the thermal bathof interest, H is the transfer function from the force (that is conjugate to thedisplacement x) to the displacement x, and f is the temporal frequency of the
motion. In the Levin’s reciprocal approach [58], the PSD is also described in amore generic way asSx (f ) = −8kB ∑ Ti Wdiss,i,(2πf )2 i
F02

(3.2)

where F0 is the amplitude of an imaginary sinusoidal test force having the same
distribution as that of the laser beam for the sensing of the displacement, Ti isthe temperature of each thermal bath, and Wdiss,i is energy loss per unit time tothe thermal bath labeled as i under the condition where the imaginary force isapplied to the test mass. This formula is also applicable to the case where the test
mass is connected to multiple thermal baths having diﬀerent temperature [59].
Let us discuss thermal noise of a single pendulum system as an example,
where the system dynamics is characterized by the transfer function
H(2πf ) =1m(2π)2 [f02 (1 + iϕ(f )) − f 2 ]

.

(3.3)

Substituting it into Equation (3.1) leads to the the following PSD,
Sx (f ) =f02 ϕ(f )4kB T.
m(2π)3 f (f 2 − f02 )2 + f04 ϕ2 (f )(3.4)

This equation indicates that smaller ϕ(f ) leads to better thermal noise except
for the frequencies around f0 . ϕ(f ) is called loss angle, which characterizes theenergy dissipation of the oscillator. Heavier mass m also leads to better thermal
noise.
‡1
Even if the test mass mirrors and their suspensions are at zero temperature, they move at
the level of zero point fluctuations. The zero point fluctuations and the thermal noise can be
described in the same frame work [57].

20

Chap.3 Gravitational-Wave Detector
According to Equation (3.4), it is necessary to use mirrors (including opticalcoatings on them) and their suspensions with low mechanical loss, in order tohave good floor level of the thermal noise spectra. In fact, Advanced LIGO
employs monolithic fused silica for the test masses and their suspension fibersof the bottom stage, to reduce the mechanical loss. KAGRA adopts sapphire
for the test masses and their suspension fibers of the bottom stage, because
sapphire is known to have low mechanical loss at cryogenic temperatures. It isalso obvious in Equation (3.4) that lowering the temperature of the test mass
reduces thermal noise [60]. KAGRA cools its test masses down to ∼ 20 K. The
3G detectors such as Einstein Telescope and Cosmic Explorer also plan to use
cryogenic mirrors [7,8]. Thermal noise is a dominant noise source around 100 Hz,
where the detectors have the highest sensitivity to the GW strain.
Quantum noise
Since laser interferometric GW detectors utilize interference of laser light, quantum fluctuations of electromagnetic field become a noise source. Such noise iscalled quantum noise. In simple interferometer configurations,‡2 quantum noise
can be split into two kinds: shot noise and quantum radiation pressure noise.
Shot noise is sensing noise coming from photon number fluctuations at thephoton detection. Let us discuss its dependence on the input laser power tothe detector. Since laser interferometric GW detectors linearly convert GWs
into laser power fluctuations detected by photodiodes, the signal strength is
proportional to the input laser power to the interferometer,√ P0 . The standard√
deviation of photon number fluctuations is proportional to N and then P0 ,‡3
where N is the number of photons detected by photodiodes.
Therefore, signal√
to-noise ratio of the shot noise is proportional to P0 , so that using high power
laser is essential. Shot noise limits the sensitivity in the frequency region wheref ≳ 100 Hz [18, 20, 61].
Radiation pressure noise arises from fluctuating displacement of the test mass
mirrors caused by quantum fluctuations of radiation pressure of laser photons.
As mentioned above, the standard deviation of photon number fluctuations, andthus radiation pressure fluctuations, is proportional to the square root of the laser
power
the displacement of the radiation pressure noise is proportional√ so that2
to P0 /mf . Here we used the fact that the test mass response to the external‡2Unless advanced interferometric techniques that produce quantum correlation between
sensing noise and backaction noise are used.
‡3
As long as the laser light is Poissonian.

3.3. Second generation detectors
force is written as −1/m(2πf )2 if the mass can be treated as a free-falling mass,
where m is mass. Therefore lowering the input laser power P0 and/or having
heavy test masses m can reduce the radiation pressure noise. Radiation pressure
noise is one of the dominant noise sources in ∼ 10 − 100 Hz [18, 20, 61].
From the above discussions, it is clear that shot noise and radiation pressure
noise have inverse dependence on the laser power. This trade-oﬀ relation is aconsequence of the Heisenberg’s Uncertainty Principle. It leads to the standard
quantum limit of the quantum noise level [62], which cannot be beaten without
quantum nondemolition measurement schemes. Recently, experimental realization and practical implementation of measurement schemes to beat the standard
quantum limit have been intensively developed [63–68]. From the theory side,
more fundamental limit beyond the standard quantum limit is investigated fromthe perspective of quantum information [69, 70].

3.3

Second generation detectors

Advanced LIGO, Advanced Virgo and KAGRA, which are categorized in the
2G GW detectors, are currently operating. They have the following features:
their arm√length reaches 3 − 4 km, and their best strain sensitivity approaches∼ 10−24 / Hz around 100 Hz [18, 20, 61]. This sensitivity level can be describedas follows: with this sensitivity, GWs from merging binary black holes (BBHs)
and binary neutron stars (BNSs) can be detected with a reasonable rate [23]. In
this section, the science target of the 2G detectors are overviewed, and then thefeatures of each 2G detector are introduced.

3.3.1

Science target

Direct observation of GWs
The primary target of the 2G detectors was to directly detect GWs for the firsttime. It has been important not only because it would be the first direct proof ofexistence of GWs but because it opens a new window to observe astronomical objects, particularly objects in strong gravity field. In fact, the BHs of GW150914
were larger than any other stellar mass BHs that had ever been reported [10].
Direct observation of BBH mergers has enabled us to obtain the information of
BH formation and binary evolution [29,30], though accumulated statistics is notenough yet. It has also made it possible to test General Relativity using a perturbed BH, which is a remnant of a BBH merger [71, 72], though the sensitivity

21

22

Chap.3 Gravitational-Wave Detector
is not enough yet.
Multi-messenger observation of compact binary coalescence
Another target is to observe GW transients along with electromagnetic (andneutrino) observatories. GW170817, the first GW event of BNS merger, wasassociated with many electromagnetic follow-up observations at diﬀerent wavelengths [6, 33], which provided rich scientific information. Two notable results
are the following. (1) Short gamma ray burst (GRB170817A) was observed witha time delay of ∼ 1.7 s with respect to the merger time, which supports the theory that short gamma ray bursts follow binary neutron star mergers [11]. This
simultaneous detection of GWs and gamma ray also set a limit to diﬀerence ofthe propagation speed of GWs and electromagnetic waves [11]. (2) A set of optical and infrared (and also ultraviolet) telescopes observed optical and infraredtransients (kilonova) from the volume localized by the GW detection [33]. It
indicates that a kilonova comes from the radioactive decay of r -process nuclei
synthesized in the ejecta of the merger. This observation is consistent with thetheory that heavy metals in our Galaxy were synthesized at the BNS mergers [73–75]. Since optical observations identified the host galaxy (NGC4993), it
provided a new independent method to infer the Hubble constant, by combining
the luminosity distance inferred from the GWs of the merger and the redshift ofthe host galaxy [76, 77].

3.3.2

Detectors

Figure 3.6 shows past and near future schedule of the observations and the approximate sensitivity of each detectors at each observation period. Here thefeatures of them are explained.
Common features
Common features among the 2G detectors can be listed as follow:• the design sensitivity reaches as high as ≳ 100 Mpc in terms of BNS merger
detection range,• the arm length is ∼ 3 − 4 km,• input laser power is on the order of 100 W at maximum,3.3. Second generation detectors

Figure 3.6: Observation schedule and the approximate sensitivities of the 2G
detectors.
• interferometer configuration is so-called Dual-Recycled Fabry–Perot Michelson Interferometer (DRFPMI; see Chapter 4).

Advanced LIGOAdvanced LIGO has two identical detectors at Hanford in Washington and Livingston in Louisiana in the U.S. [18]. They were built on the infrastructure of
Initial LIGO [81,82]. The third LIGO detector, LIGO India, is also planned [83]and approved [84]. The design sensitivity of Advanced LIGO is shown in Figure
3.7. Two Advanced LIGO detectors detected the first GW event, GW150914, tomark the beginning of an era of GW astronomy in the period of Observation 1(O1). The arm length is approximately 4 km. Each test mass and its suspension
wire are made of fused silica and the weight of the mass is 40 kg. The world’s
best sensitivity so far in terms of the BNS merger detection range was recordedby the Livingston Observatory, which is ∼ 140 Mpc [85].
Advanced Virgo
Advanced Virgo detector is built at Pisa in Italy [61], utilizing the infrastructureof the initial version of Virgo detector [86]. The design sensitivity of Advanced
Virgo is shown in Figure 3.7. Advanced Virgo joined the network observation
at the late stage of Observation 2 (O2), and detected several compact binary
coalescence events including the first BNS merger, GW170817 [9]. It is notablethat localization of the BNS with the three detector network made it possible for

23

24

Chap.3 Gravitational-Wave Detector

Sensitivity [/sqrt(Hz)]10 21

Advanced LIGOAdvanced Virgo
KAGRA

10 22

10 23

10 24101

102
103
Frequency [Hz]Figure 3.7: Sensitivities of the 2G detectors [23, 78–80].
optical telescopes to survey almost all the potential transients in the localized
sky region [87]. The arm length is 3 km. Each test mass and its suspension wire
are made of fused silica and the weight of the mass is 42 kg.
KAGRA
KAGRA is built in the underground tunnel at Kamioka in Japan, in a newly
built infrastructure [24]. The design sensitivity of KAGRA is shown in Figure
3.7. KAGRA plans to join the network as the fourth detector at the last stage of
Observation 3 (O3). It is expected that KAGRA joining the network enables us
to increase the network duty cycle, to improve the sky coverage of the localizablesources of GWs [23, 88], and to disentangle polarizations of GWs [54]. KAGRA
has two unique features in its detector itself. One is that the detector is built
underground [20, 24]. The other is that the key components, test masses arecooled down to cryogenic temperature [20, 25]. These features will be of great
importance for the 3G detectors [8,89]. KAGRA has its own importance not only
as one detector of the four-detector network but also as a test bed to accumulate

3.4. Third generation detectors
knowledges and experiences for the future detectors. The arm length is 3 km.
Each test mass and its suspension wire are made of crystalline sapphire and theweight of the mass is 23 kg. The test mass is cooled down to 20 K.

3.3.3

KAGRA as the fourth detector in the network

Let us here focus on the scientific importance of KAGRA joining the network
observation as the fourth detector. The fourth detector is considered to improve
temporal and spatial coverage of localizable events as explained below.
Network duty cycle can be used to quantify the contribution of the fourth
detector to the temporal coverage. Let us compare the ratio of time when threeor more detectors are operating in the following simplified cases: (i) there arethree detectors and each detector has the duty cycle of 75 %, and (ii) there
are four detectors and each detector has the duty cycle of 75 %. The ratioof time with three or more detectors online can be given by straightforward
calculations: (i) 42 % and (ii) 74 %. It is worth noting here that the duty cycleof the single detector was ∼ 70 − 80 % and the three-detector duty cycle was
44 % in the early stage of O3 [90]. These numbers are close to the outcome ofthe simple consideration. Therefore, it is expected that the temporal coverage
will be improved by ∼ 1.7, by comparing the results of (i) and (ii).
Adding the fourth detector will also improve the spatial coverage. Simulation
works revealed that KAGRA joining the network as the fourth detector improve
the sky localization by more than ∼ 15 % for half of all the compact binary
coalescence events even if the sensitivity of KAGRA is limited [91].

3.4

Third generation detectors

The 3G detectors are the planned GW detectors which will be online from 2030’s[92, 93]. The planned arm length is on the order of 10 km. They will haveapproximately 10 times better sensitivity than those of the 2G detectors. Better
sensitivity will provide benefits in two ways. One is the number of detectable
events. A naive discussion tells us that N times better strain sensitivity leads to
N 3 times more statistics. The other is the maximum signal-to-noise ratio of the
GW increases by a factor of N with an N times more sensitive detector. With
such detectors, fruitful science outcomes are expected as summarized below. In
this section, the science target of the 3G detectors are overviewed, and then theplanned detectors are also introduced.

25

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Chap.3 Gravitational-Wave Detector

3.4.1

Science target

Neutron star physics
How matter in a neutron star behaves still remains unknown; the equation ofstate, the relation between mass and radius, or the maximum mass are not yet
understood well. It is of great importance to investigate them because they convey the information of how matter behaves under the extreme condition of highpressure, high density, strong gravity, and strong magnetic field. It is predictedthat they are encoded in the GWs both from late inspiral of BNS mergers andfrom a remnant of BNS mergers. As each NS comes close to each other in thelate inspiral phase, the tidal deformation aﬀects the orbit evolution and thus
modulates the emitted GWs [13, 94]. From the waveform of GW170817, someconstraints were set to tidal deformability (a dimensionless parameter characterizing the deformation) of the NSs [95]. With the 3G detectors, it is expectedthat the tidal deformability will be clearly measurable and so strong constraints
will be set that a significant fraction of the proposed equations of state will be
ruled out [12, 13]. After the merger of a BNS, a massive NS can be formed in an
excited state, which will then emit quasiperiodic GWs. It has been reported thatthe characteristic frequency of such GWs depends on the equation of state [96].
From the numerical simulations, its dependency can be approximately expressedas()
R1.8 − 2 km
f ∼ 4.0 kHz ×
(3.5)
8 km
for a massive neutron star formed from a BNS with its total mass equal to
2.7M⊙ , where R1.8 denotes radius of an NS for given equations of state. With
3G detectors operating for a year, radius of an NS may be determined within≲ 100 m, which is smaller than systematic errors in theoretical modeling [97].
Supernovae mechanism
The mechanism of supernovae remains unknown mainly because of two reasons;
it is diﬃcult to fully simulate the explosion and the core cannot be accessed withelectromagnetic waves. It is expected that core-collapse supernovae may produce
detectable GWs [14]. It is also expected that the 2G detectors could detect GWs
from core-collapse supernovae, but the expected rate is very low. Thus the 3G
detectors may be needed to detect such GWs for the first time [15]. With 3G
detectors, we can expect one event in a few years, and we can also expect that,from the detected signals, g-mode excitation of proto-neutron star formed at the3.4. Third generation detectors
core may be probed and signs of the convection or the standing accretion shock
instability may be caught resulting in constraining the supernova models [15].
Test of General Relativity
General Relativity has been one of the fundamental laws of the Universe formore than one hundred years as no deviations from it has been found. One ofthe direct ways to observe such deviations in strong gravity regime is to probe BH
perturbations with GWs. A remnant of a merged BBH is damped by emitting
GWs, which can be modeled by quasi-normal modes of a Kerr BH. According
to General Relativity, a BH is characterized only by the mass and spin,‡4 andso are the frequency and damping rate of quasi-normal modes. GWs from thequasi-normal modes were detected from the post-merger phase of GW150914 [72].
With the 3G detectors, the deviations in the frequency and the damping rate will
be detectable within a few percent, to set constraints on gravity theories [16].
Another way to test General Relativity is to search for extra polarizations inpropagating GWs, since General Relativity predicts only two tensor polarizations
while general metric theories allow six polarizations [98, 99]. It is also expectedthat the existence of nontensorial polarization modes of GW may be tested withthe 3G detectors [17].
Standard siren measurement
Since there has been discrepancy in the reported values of the Hubble constant,
one is from spatial spectrum of the CMB temperature fluctuations [100] andthe other is from the observation of type Ia supernovae [101], it is importantto have another independent method to obtain the Hubble constant. A GW
standard siren measurement is one such candidate [102]; the luminosity distanceof compact binary coalescence can be inferred from its inspiral signals and thus
the Hubble constant can be obtained by combining the redshift of the source.
In fact, the Hubble constant was measured to be 70.0+12.0−8.0 km/c/Mpc by using
GW170817 and the redshift of the identified host galaxy [76], though the precision
was not high enough to resolve the discrepancy. With the 3G detectors, it isexpected not only that the Hubble constant can be determined with enoughprecision but also that the evolution history of the Universe can be precisely
measured [103–105] since the 3G detectors can detect binary coalescence at highredshift as mentioned below.
‡4

Electric charge is ignored here since astronomical BHs are considered to be neutral.

27

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Chap.3 Gravitational-Wave Detector
Population of compact binary systems
Since the targeted sensitivity of the 3G detectors is good even at low frequencies(∼ 10 Hz), it is expected that they are capable of detecting GWs from compact
binary coalescence with high signal-to-noise ratio all across the history of the
Universe (redshift z ≳ 10) [8, 89, 93]. With this ability of the detectors, thecomplete population of stellar-mass BHs from the era of first stars to the present
time will be revealed. They will also be able to probe how the BBH merger rate
and the underlying star formation rate have evolved from the early Universe,
through the observations of BBH mergers providing the inference of their masses
and spins.

Sensitivity [/sqrt(Hz)]10 22

Einstein Telescope
Cosmic Explorer
Advanced LIGO (For reference)

10 23

10 24

10 25

101

102
103
Frequency [Hz]104

Figure 3.8: Sensitivities of the 3G detectors [7, 8, 106].

3.4.2

Detectors

There are two major projects of the 3G detectors, Einstein Telescope [7] and√Cosmic Explorer [8, 93]. The sensitivity of each detector reaches below 10−24 / Hz.
The sensitivity curves are shown in Figure 3.8.

3.4. Third generation detectors
The characteristic of each detector can be overviewed as follows. The arm
length of Einstein Telescope is 10 km. A unique feature of Einstein Telescope is
its xylophone design, which employs one detector specialized for low frequency
GWs and the other for high frequency GWs. The test masses of the low frequency
detectors will be cooled down to cryogenic temperature. The detectors will bebuilt in a triangular-shape underground tunnel that houses three pairs of thexylophones [89].
Cosmic Explorer will be an L-shaped detector with the arm length of 40 km.
The project will be split into two stages. The first stage adopts room temperature
fused silica test masses. In the second one, cryogenic silicon will be used as test
masses for better thermal noise performance and better capability of storing highpower laser in arm cavities [93].

29

31

Chapter 4
INTERFEROMETERIn this chapter, basic interferometric methods and techniques that are or will be
implemented in the 2G and/or 3G detectors are summarized. They are indispensable to reach the targeted sensitivity of each detector. Since they make theinterferometer of the GW detector complicated, it becomes important to study
lock acquisition processes of the interferometer. In the last section, expected
technical challenges in the 3G interferometers are overviewed.

4.1

Michelson interferometer

4.1.1

Response

Let us consider the response of a Michelson interferometer as shown in Figure
4.1 to GWs, h(t), propagating along z axis. Let us assume each mirror moves asa free mass, and the beam splitter (BS) has split ratio of 50:50. Let us denotethe distance between the BS and each end mirror in the x or y direction by Lx
or Ly , respectively. Let us define L by L = (Lx + Ly )/2. According to Equation(2.1), the relation()
dx
1≃ ±c 1 − h(t)
(4.1)
dt
2
holds. Therefore, the duration Tx (t) it takes for the laser light to propagate in
the x arm in a roundtrip way can be described by‡1
‡1

Assuming |Lx − Ly |/L ≪ 1, L is used for the interval instead of Lx .

32

Chap.4 Interferometer

Figure 4.1: Schematic view of a Michelson interferometer.
(∫
Tx (t) =roundtrip

2Lx 1
=
+
c
2

dx
dt

)−1

∫ t

dx

(4.2)

h(t′ )dt′ .

(4.3)

t−2L/cSimilarly, the duration Ty (t) it takes for the laser light to propagate in the y arm
can be written by∫2Ly 1 t
(4.4)
Ty (t) =−h(t′ )dt′ .
c2 t−2L/c
Therefore, defining a laser electric field incident to the BS by Ein (t) = E0 eiω0 t ,the electric field at the output port Eout (t) is expressed as
1[Ein (t − Tx (t)) − Ein (t − Ty (t))]
2[]
1= E0 eiω0 (t−Tx (t)) 1 − ei(ϕ+ϕGW (t)) ,2
2ω0 (Lx − Ly )
ϕ=
,
∫ c

Eout (t) =t

ϕGW (t) = ω0
t−2L/ch(t′ )dt′ .

(4.5)
(4.6)
(4.7)
(4.8)

4.1. Michelson interferometer

33

By detecting phase variations ϕGW (t) at the output port, the GWs are detected.
The Fourier transform of Equation (4.8) leads to ϕGW (Ω) = CMI (Ω)h(Ω), defining the response function of a Michelson interferometer CMI (Ω):
CMI (Ω) =4.1.2

2ω0 L sin(ΩL/c) −iΩL/c
e
.
c
ΩL/c

(4.9)

Detection scheme

Because we detect power of the laser field by photodetectors, we have to useinterferometric techniques to readout the phase of the output laser field, whichconvert it into optical power varations. Here, such techniques are summarized.
Mid-fringe detection
The most primitive way to detect the phase variations of the electromagnetic
field of the output laser is so-called Mid-fringe detection. Substituting π/2 into
ϕ in Equation (4.6), the laser power at the output port Pout (t) turns to be‡2
Pout (t) ≃P0 P0+ ϕGW (t),22

(4.10)

where P0 = |E0 |2 . In this way, Pout (t) has a linear term of ϕGW (t) and so the
GWs can be detected. This scheme is called mid-fringe detection because thehalf of the input power goes to the output port.
Homodyne readout
One commonly used scheme is homodyne readout. If ϕ is kept to be zero, theoutput field becomes−iEout (t) =E1 eiω0 t ϕGW (t),(4.11)
2
where E1 = E0 e−iω0 Tx (t) . This condition is called dark fringe condition, since theoutput port is kept dark in the sense that no constant field is coming out there.
A merit of adopting the dark fringe condition is that all of the constant fields
are reflected back to the input side, which enables us to use a technique called
power recycling (see Section 4.3).
‡2
Here the higher order terms of ϕGW are ignored. In the field of precision interferometry,such time-varying phase terms are actively controlled so that this approximation is valid. The
same approximation is also used hereafter.

34

Chap.4 Interferometer
Let us assume that there is a reference field denoted by Eref = −iAE1 , where
A is a real number. The interference of these two fields are written as()
ϕGW (t)iω0 t
Ehomo (t) := Eref (t) + Eout (t) = −iE1 e
A+.
(4.12)
2
The optical power Phomo (t) of the interference field Ehomo (t) is[]
Phomo (t) ≃ P0 A2 + AϕGW (t) ,(4.13)

where linear signals of the GWs are obtained.
One implementation of the homodyne readout scheme is DC readout [107].
If ϕ is kept to be a certain small number instead of zero,()
ϕ ϕGW (t)iω0 t,(4.14)
Eout (t) = −iE1 e+22where the first term can be regarded as a reference field. One of the merits of
DC readout is there is no need to prepare an additional reference light field.
DC readout is employed in the sensing of the length degree of freedom (DoF)
corresponding to GWs‡3 in the 2G detectors.
Another implementation is balanced homodyne readout (Figure 4.2). The
diﬀerence of the outputs of these two photodetector, namely PBHD , is expressed
as[]
PBHD (t) ≃ P0 A2 + AϕGW (t) ,(4.15)
where some symbols are defined as some ways. These two are the major benefitsof balanced homodyne readout. One is noise associated with the reference field
can be cancelled out by the subtraction. The other is the relative phase between
the output field Eout (t) and the reference field Eref (t) can be controlled at anarbitrary point, which is crucial in realizing quantum backaction evasion [108].
It is planned that the scheme of balanced homodyne readout will be used in the
upgrade of Advanced LIGO, so called A+ [109, 110].
Heterodyne readout (RF readout)
In order to read out the output field in the dark fringe condition, we can prepare
another kind of reference fields. Let us consider a reference filed denoted by
Eref = −iAE1 sin Ωm t. Usually, Ωm /2π is chosen from the range of ∼ 10 −‡3This DoF is called DARM (Figure 4.6).

4.1. Michelson interferometer

35

Figure 4.2: Schematic view of an optical layout for the balanced homodyne
readout. The output field from the main interferometer interferes with a reference
field at a 50:50 beam splitter. The signals from two photodetectors are subtractedwith each other so that noise associated with the reference field will be cancelledout.
200 MHz.‡4 This is because this scheme is called radio frequency (RF) readout.
The interference of the reference and output fields are written as
Ehetero (t) := Eref (t) + Eout (t) = −iE1 e

iω0 t

(
)
ϕGW (t)A sin Ωm t +.
2(4.16)The optical power Phetero (t) of the interference field Ehetero (t) is[]
Phetero (t) ≃ P0 A2 sin2 Ωm t + A sin Ωm t · ϕGW (t) ,‡4(4.17)In the field of the GW detectors, these frequencies are commonly used for the following
reasons. One is that the frequency band of the phase variations we want to detect (including
auxiliary DoFs) is, at most, up to 1 MHz so that filtering of Ωm components is feasible withoutaﬀecting the phase variation signals very much. Another reason is that, up to a few hundredsof MHz, it is rather easy to handle circuits associated with the photodetectors as a lumpedelement model. Eﬀective capacitance of a photodiode with reasonable diameter (∼ 1 mm) isalso tolerable.

36

Chap.4 Interferometer
which is detected by an RF photodetector. The output of the RF photodetector
is then demodulated by sin Ωm t, which yields[
Vdemod := Phetero (t) sin Ωm t = P0]
A
ϕGW (t) + (RF terms) .
2(4.18)The RF terms consist of terms proportional to sin nΩm t or cos nΩm t, where n isany non-zero integer. Since the RF terms can be easily low-passed out by passive
electrical components, a linear signal of ϕGW (t) is obtained in this way.
A commonly used realization of heterodyne readout of a Michelson interferometer is the use of Schnupp asymmetry [111]. Let us set Lx − Ly = las , wherelas is macroscopic length diﬀerence (on the order of centimeters to meters). If
we inject a phase-modulated laser field Ein (t) = E0 exp[iω0 t + iβ cos Ωm t] to theinterferometer, the output field turns to be[
Eout (t) = −iJ0 (β)E1 eiω0 t

ϕGW (t)(2J1 (β)/J0 (β)) sin α sin Ωm t +2](4.19)

in a neighborhood of the dark fringe condition, where α = Ωm las /c and Jn is theBessel function. Here, terms proportional to sin nΩm t or cos nΩm t with n > 1
are ignored. In this way, the reference field is provided at the output port and soheterodyne readout can be realized. This scheme is used for central Michelsoninterferometer of the GW detectors (Section 4.3).

4.1.3

Laser noise rejection

As pointed out in the first proposal [53], a Michelson interferometer can rejectlaser noise so that it does not couple to the output signal of diﬀerential phaseshift. Let us consider frequency noise coupling by expressing the input field by
Ein = E0 ei(ω0 t+δψ(t)) . Then the output of the interferometer can be written as−iEout (t) =E1 eiω0 t
2

(

)
2las
δf (t) + ϕGW (t) ,c(4.20)

where δf = d(δψ)/dt is frequency fluctuations of the input laser. This equationshows that the frequency noise coupling can be arbitrarily small by making two
arms symmetrical.

4.2. Fabry–Perot cavity

Figure 4.3: Schematic diagram of a Fabry–Perot cavity. The amplitude reflectivity of the input mirror is denoted by r.

4.2

Fabry–Perot cavity

4.2.1

Response

As briefly mentioned in Section 3.2.2, a Fabry–Perot cavity‡5 is employed in eacharm of the Michelson interferometer of the GW detectors (Figure 4.6), in orderto make the eﬀective arm length longer. Such an interferometer is called Fabry–
Perot Michelson interferometer (FPMI). Let us consider an optical cavity shownin Figure 4.3. Let Ein (t) = E0 eiω0 t be the incident field to the cavity. Let us thenexpress a roundtrip phase of the laser field in the cavity by ϕ + δϕ(t), where ϕ isconstant part and δϕ(t) is time-varying small deviations of the phase. Hereafter,
let us use Fourier domain to discuss the phase deviations. If ϕ is set to be an
integer multiple of 2π, meaning that the laser is kept at the resonance of thecavity, the reflection Eout (t) can be described as‡5

See Appendix C for more detail.

37

38

Chap.4 Interferometer
Eout (t) = E0 eiω0 t · e−iCFP δϕ(f ) ,[CFP =2 sinhr

1/2(1 + r)(4.21)](4.22)|ln r|(1 + iΩ/Ωcav )
2

1
2F
≃
π 1 + iΩ/Ωcav()Ω
1 − r ≪ 1,
≪ fFSR ,2π(4.23)

whereΩcav = |ln r| · fFSR ≃ 2π

fFSR
,
2F

2π
,
1 − r2cfFSR =.
2L
F≃(4.24)(4.25)(4.26)

Here, Ωcav /2π is called cavity pole frequency or half linewidth of the cavity, theinverse of which represents average storage time of the cavity. fFSR is free spectral
range (FSR), which corresponds to the separation of two nearby resonances interms of frequency. F is called finesse, which is a (dimensionless) ratio of the FSR
to the full linewidth (2Ωcav /2π). Finesse is proportional to average roundtrip
numbers in the cavity. Equation (4.23) shows how a Fabry–Perot cavity amplifies
the phase deviations inside the cavity. The GW signal is amplified by a factorof 2F/π and the amplification gets reduced with a low-pass response with thecorner frequency equal to the cavity pole frequency. Figure 4.4 compares the
responses of a FPMI and a MI with long arms. Each cavity mirror closer to the
BS is called input test mass (ITM), and the farther one is called end test mass(ETM) (Figure 4.6).

4.2.2

Detection scheme

The most commonly used scheme to detect the phase deviations of the reflected
laser field is the Pound–Drever–Hall (PDH) technique [112]. This method applies heterodyne readout to the sensing of a cavity. Let us consider the casewhere a phase-modulated laser field is injected to an optical cavity; Ein (t) =E0 exp[iω0 t + iβ cos Ωm t]. If the modulation frequency is chosen in such a waythat the modulation sidebands‡6 are far from resonance, the reflection of the‡6

Please refer to Appendix D for sideband picture of modulations.

4.2. Fabry–Perot cavity

39

1013

Response [rad]

1012
1011
1010
109
8
1010
0

L = 500km
L = 3km
L = 3km with FP cavities
102
103
101
Frequency [Hz]

104

105

Figure 4.4: Response functions from GW strain to phase deviation of the output
optical field in various interferometer configurations. The gray dashed line shows
the response of a MI with long arms. The blue line shows the response of a MIwith short arms. The red line shows the response of a FPMI having short arms
but long eﬀective arm length (F = 262).
cavity Eout (t) can be written as[]
2J1 (β)iω0 tEout (t) = J0 (β)E0 e· 1−i
cos Ωm t − iCFP δϕ(f ) .
J0 (β)(4.27)

Therefore, the optical power of the reflection Pout (t) is
Pout (t) = 4J0 (β)J1 (β) cos Ωm t · CFP δϕ(f )+ (constant term) + (2Ωm or higher order terms).

(4.28)

This shows that the phase variations can be detected by demodulating the photodetector output with cos Ωm t.

40

Chap.4 Interferometer
It is worth noting that the PDH signal is obtained by a demodulation withcos Ωm t while the Michelson signal with Schnupp asymmetry is obtained by ademodulation with sin Ωm t, comparing Equations (4.19) and (4.28). The phase of
the demodulation sinusoid that gives the PDH signal is called in-phase (I-phase)
while the phase that gives the Michelson signal is called quadrature phase (Qphase).
Figure 4.5 shows how the PDH signals behave as ϕ changes across FSRs.
This illustrates that the linear range of the PDH signal is very narrow. In orderto have the amplification eﬀect of the cavity, it is necessary to keep controlling
the phase, ϕ, within the narrow range very close to the resonance.

1.00

1.0
0.5

0.75

0.0
0.5

PDH signal [a.u.]

0.50

1.0

4

2

0

2

4
1e 3

0.25
0.00
0.25
0.50
0.75
1.00

1.00

0.75

0.50

0.25

0.00

Phase/2

0.25

0.50

0.75

1.00

Figure 4.5: How typical PDH signal behaves (F = 1500). The inset shows theenlarged view around the resonance.

4.3. Dual-recycled Fabry–Perot Michelson interferometer

4.3

Dual-recycled Fabry–Perot Michelson interferometer

A Michelson interferometer having Fabry–Perot cavities in both arms is called a
Fabry–Perot Michelson interferometer (FPMI). The current GW detectors combine with a FPMI two techniques, which are power recycling and resonant sideband extraction.‡7 A FPMI with these techniques incorporated is called a dualrecycled Fabry–Perot Michelson interferometer (DRFPMI). Figure 4.6 shows aschematic view of a DRFPMI.
The main interferometer
degrees of freedom

ETM

YARM

Laser
Source

PRC
PRM

ITM
ly

DARM: (Lx – Ly)/2
Ly CARM: (Lx + Ly)/2
PRCL: lpr + (lx + ly)/2
SRCL: lsr + (lx + ly)/2
MI MICH: (lx – ly)/2
Lx
BS
lx

lpr

lsr
SRM

ITM

ETM

XARM
SRC

Figure 4.6: Schematic diagram of a dual-recycled Fabry–Perot cavity. The definition of each length DoF is also described.

4.3.1

Power recycling

As the FPMI is operated in dark fringe condition, most of the laser power comesback to the input side. The power recycling mirror (PRM), which is placed at the
input side of the FPMI, resonantly reflects back such a return beam to the FPMI
‡7

Resonant sideband extraction is also known as signal recycling.

41

42

Chap.4 Interferometerpart so as to enhance the laser power at the FPMI part. Higher power leads tobetter signal-to-noise ratio of shot noise as mentioned in Section 3.2.4. This
technique is called power recycling (PR) [113]. The PRM has to be controlled sothat the carrier field satisfies the anti-resonant‡8 conditions of the cavity formedby the PRM and ITMs, which is called the PR cavity.

4.3.2

Resonant sideband extraction

A mirror placed at the output side of the FPMI is called the signal recycling
mirror (SRM). Although the laser field coming from the input side does not go
to the output side, diﬀerential modulations of the laser field introduced in thearms, which include GW signals, leak to the output side. If such modulation
sidebands‡9 resonate between the SRM and the ITMs, average storage time ofthe sidebands in the arm cavities becomes smaller, or more specifically, eﬀectivefinesse of the arm cavities for the sidebands becomes smaller, so that the cutoﬀ frequency in the response to GW signals is pushed up. This technique iscalled resonant sideband extraction (RSE) [113, 114]. The eﬀective finesse canbe written by√
1 − Rs√ F,(4.29)
Feﬀ =1 + Rswhere Rs is the reflectivity of the SRM. The SRM has to be controlled so thatthe carrier field satisfies the resonant conditions of the cavity formed by the SRM
and ITMs, which is called the SR cavity.
Figure 4.7 schematically shows how the ASD of (strain-equivalent) shot noiseis reshaped by PR and RSE. This figure shows that RSE improves shot noisein high frequencies at the cost of worse shot noise in low frequencies. It also
shows that increasing the power at the BS by a factor of N with a help of PR isequivalent to increasing the arm finesse by N and broadening the bandwidth by
N with RSE at the same time. Since the latter does not increase the laser powerat the BS, combining PR and RSE can provide the optimized shot noise curve
while the power at the BS and the power inside the arm cavities are balancedwith a desired ratio.
‡8
When the roundtrip phase in a cavity is a half-integer multiple of 2π, such a condition iscalled anti-resonant.
‡9
Please refer to Appendix D for sideband picture of modulations.

4.3. Dual-recycled Fabry–Perot Michelson interferometer

Figure 4.7: Conceptual diagram of how the ASD of strain-equivalent shot noisechanges as the interferometer configuration changes.

4.3.3

Detection scheme

A DRFPMI is a coupled interferometer consisting of five length DoFs. The nameof each DoF is defined in Figure 4.6. Since all of them are coupled with eachother, it is not straightforward to separately detect each DoF. Here the common
sensing scheme employed in the 2G detectors are explained. The injection laser
field is phase-modulated at two diﬀerent frequencies. The Schnupp asymmetry
and the modulation frequencies are chosen so that one pair of sidebands (f1 )‡10leaks to the output side while the other (f2 ) does not. The length of recycling
cavities are set so that the carrier field satisfies the anti-resonant or resonantconditions of the PR or SR cavities, respectively, when f2 sidebands resonate in
the PR cavity and f1 sidebands resonate in the PR-SR coupled cavity. In thisway, three DoFs of central interferometer (PRCL, SRCL, and MICH) are sensed
with heterodyne readout by using the modulation sideband fields as signal fields
and the carrier field as a reference field. DARM is sensed with DC readoutat the output port of the DRFPMI. As the output mode-cleaner (OMC)‡11 is‡10‡11

See also Figure 6.4.
Advanced Virgo uses two OMCs in series.

43

44

Chap.4 Interferometer
employed to reject undesired RF sidebands and higher order transverse modes,
the transmission of the OMC, namely OMC TRANS, is used for DARM sensing.
Before the DARM error signal is switched to OMC TRANS, the PDH signal at
the transmission of the SRM (AS port; Figure 6.3) is used. CARM is sensed by
the PDH signal at the reflection of the main interferometer (REFL port; Figure6.3) as the f1 or f2 sidebands are used as the reference field. Although eitherof the PDH signals at REFL port and the pick-oﬀ port of the PR cavity (POPports; Figure 6.3) is linear combination of CARM and PRCL, they are linearly
independent so that CARM and PRCL can be sensed independently. A similar
thing holds for SRCL. In a later section, Table 6.3 summarizes the error signal
for each DoF in the case of KAGRA.

4.4

Advanced techniques

To improve the sensitivity further, other advanced techniques are implemented
or to be implemented in the 2G and the 3G detectors, exploiting quantum natureof light.

4.4.1

Squeezed light injection

Unless light fields are prepared in a certain quantum mechanical state, shot noiseof the interferometer output follows the Poisson distribution, which means thatthe√standard deviation of the number of received photons in a certain period
is N if the expectation of the photon number is N . If the light field is in asqueezed state, the fluctuations of the number of received photons can be madebelow the level of the Poisson distribution. Please refer to Appendices D and E
for more detail. If a light field is in a squeezed state, fluctuations of electric fieldalong a certain direction in the complex plane are reduced while the fluctuations
along the perpendicular direction are increased [115].
In the context of the GW detectors, use of squeezed light can reduce shotnoise at the cost of the increase of radiation pressure noise. Figure 4.8 shows
how squeezed light can be incorporated in an interferometer of the GW detectors.
Since the interferometer reflects the laser field back to where it is injected from,the laser field at the detection port is determined by vacuum fields entering to
the interferometer. By injecting squeezed vacuum there and properly controllingthe squeezing angle, the reduction of shot noise can be realized.
Advanced LIGO and Advanced Virgo are currently operating with a help ofsqueezed light injection, without introducing additional excess noise [116, 117].

4.4. Advanced techniques(a) Squeezed light injection.

45(b) How a filter cavity rotates the squeezing
ellipse.

Figure 4.8: Schematic diagrams of how the squeezed vacuum is incorporated
with the interferometer.

KAGRA also has a plan to implement squeezing.

4.4.2

Filter cavity

Since radiation pressure noise limits the sensitivity only in low frequency region,
the overall quantum noise curve can be improved by controlling the squeezing
angle in a frequency-dependent way; the angle is controlled so that radiationpressure noise is reduced in low frequency region while shot noise is reducedin high frequency region. Such a rotation of squeezing angle can be realized
by reflecting a squeezed field with a fixed squeezing angle oﬀ a detuned opticalcavity, namely a filter cavity (Figure 4.8) [108].
The principle of this technique has been demonstrated [63]. Worldwide eﬀorts
are being made toward its implementation to the GW detectors [63,64,118]. It isplanned that a filter cavity will be employed in the upgrade of each 2G detector
and the 3G detectors [8, 89, 109].

46

Chap.4 Interferometer

4.4.3

Detuned RSE

As described in Section 4.3, a SR cavity can reshape the response function of
the detector to GWs. If the SR cavity is detuned by certain amount, radiationpressure noise can be cancelled in a narrow frequency range thanks to quantum
correlation between shot noise and radiation pressure noise, as well as GW signals
are amplified at the frequencies according to the amount of detuning [119, 120].
This technique is called detuned RSE. It is adopted in the design configurationsof KAGRA and the low frequency detectors of Einstein Telescope [20, 89].

4.5

Challenges in the 3G detectors

4.5.1

Long arms

The 3G detectors will have 3 − 10 times longer arms than those of the 2G. The
arm length of Einstein Telescope will be 10 km, and that of Cosmic Explorer willbe 40 km. A possible challenge arising from long arm length is diﬃculty in thefrequency stabilization control using CARM.
Assuming that the finesse of a virtual cavity corresponding to CARM is aconstant regardless of the arm length,‡12 the linewidth of the CARM cavity
becomes narrower in terms of frequency if the arm cavities get longer. This
indicates that requirements on frequency stabilization control becomes stringent
in the 3G detectors. In particular, lock acquisition of CARM becomes even morechallenging. In addition, the error signal of CARM will be more easily distortedby the ringing eﬀect due to the dynamical evolution of the electromagnetic fieldin the cavity [121, 122]; laser frequency change in a timescale shorter than the
inverse of the linewidth of the cavity causes the eﬀect. The other is that therequirements on the CARM control by the ALS system become stringent. Pleaserefer to Chapter 8 for more detail.
Another eﬀect of long arms is that the frequency of the FSR is no longer
above the interested frequency range. In the case of Cosmic Explorer, the FSR
is 3.75 kHz, which is not only within the observation frequency band but also
‡12

The fluctuations of the injected laser and common displacement in two cavity lengths
couple with each other. This DoF can be treated as a cavity. The PRM makes its eﬀectivefinesse larger than that of a single arm cavity. The assumption that the finesse is independentof the arm length is reasonable because the finesse is determined purely by optical loss in thearm cavities if the power recycling is in the optimal condition, which has little dependence on
the cavity length.

180
90
0
90
180

Gain

100
10 1
10 2
10 3
10 4
10 5
6
10 10
0

47

101

102
103
Frequency [Hz]104
Phase [deg]Phase [deg]Gain4.5. Challenges in the 3G detectors(a) Overview

100
10 1
10 2
10 3
10 4
10 5
6
10 3740
180
90
0
90
180

3745
3750
Frequency [Hz]3755(b) Enlarged view around the first
FSR

Figure 4.9: Response transfer function of a cavity with the FSR of 3.75 kHz andthe finesse of 500 to the frequency fluctuations of input laser.
likely to be within the control frequency band of CARM. Figure 4.9 shows the
response transfer function of a cavity to the frequency fluctuations of input laser.
It behaves in a complicated way like a notch filter around the frequencies of theinteger multiple of the FSR [123]. In order to have a stable control loop ‡13of CARM with the bandwidth up to ∼ 10 kHz, it is necessary to flatten such
structures by the servo filter. This will possibly be a new challenge in the 3G
detectors.

4.5.2

Weak actuators

As an example, Figure 4.10 compares low frequency noise spectra of Advanced
LIGO and Cosmic Explorer in terms of displacement, showing that the requirement on displacement noise of a test mass will be ∼ 10 times more stringent in
Cosmic Explorer. This implies that ∼ 10 times weaker actuators with inevitably
smaller range have to be used for each test mass. This will make the DARM
control even more diﬃcult, which relies on test mass actuators. In particular,
lock acquisition of DARM will be challenging due to saturation of the actuators.
‡13

See Appendix B for stability conditions of a control loop.

48

Chap.4 Interferometer

Sensitivity [m/sqrt(Hz)]10 16

Cosmic Explorer
Advanced LIGO (For reference)

10 17
10 18
10 19
10 20
10 21

6 × 100

101
Frequency [Hz]2 × 101

3 × 101

Figure 4.10: Design sensitivities of the 2G and the 3G detectors in low frequency,
in terms of displacement.

49

Chapter 5
LOCK ACQUISITION
Lock acquisition or interferometer locking is a process to bring the interferometerfrom a state where each DoF is moving in a stochastic way to the observation
state, where each DoF is controlled with a low noise sensor that generally has anarrow linear range. Such sensors show highly nonlinear behaviors outside theirlinear range and each DoF is cross-coupled with each other meaning that it isusually not possible to lock each DoF one by one. Therefore, lock acquisition
process is not straightforward and thus it requires intensive development. In thischapter, the overview of lock acquisition and the summary of the lock acquisition
schemes of the present and past GW detectors are given.

5.1

Overview

Each mirror of the interferometer swings in longitudinal direction by ∼ 1 µm in
0.1 − 1 Hz due to seismic motion, in spite of passive and active seismic attenuation. In these conditions, each length DoF of the interferometer (Figure 4.6)
stochastically fluctuates and stays only for a short period in the linear range of
its sensor around the operating point. In order to bring the interferometer tothe operation mode, every DoF has to be somehow confined within their linear
ranges.
A process of interferometer locking has three major diﬃculties. The first one
is that each mirror freely swings at the beginning. The second one is that each
DoF of the interferometer is cross-coupled with each other. The third one isthat the error signal of an single DoF shows nonlinear behaviors. Locking theinterferometer against these diﬃculties has been considered to be one of the mostchallenging processes in realizing the observation of GW detectors. In fact, it
typically took about one year to achieve the first lock of the interferometer inthe case of the current detectors. Since the process is challenging, many eﬀorts

50

Chap.5 Lock Acquisition
has been made on this topic as follows.
One obvious way to lock the interferometer is to engage the control loops forthe length DoFs during the short period when the sensor signals are in their linear
ranges. Such schemes were used in TAMA300 [124, 125] and Initial LIGO [126].
If it is possible to lock the interferometer in this way, this approach has a great
advantage that the lock acquisition process can be finished in short time. However, it also has the following drawbacks. The first one is that the error signals
for DoFs of the central interferometer can be largely disturbed or even vanished,depending on a dynamical state of the arm cavities. The second one is that it
is not straightforward to diagnose the lock acquisition process in case it is not
successful. It is not possible to measure the responses of the interferometer in
an intermediate state of lock acquisition for diagnostic purposes. The third oneis that it is necessary to stop a swinging mirror during a short period when thecorresponding DoF is in the linear range of its sensor. This condition imposes requirements of the maximum force actuators can exert. Typical force F necessary
to catch the resonance of an cavity can be written as
2
mvRMS
mvRMS
F
F =
≃
,
∆t
λ

(5.1)

where vRMS is the root mean square (RMS) velocity of a test mass, ∆t is theaverage time it takes for a test mass to move across the linear range of the PDH
signal around the resonance, F is the finesse of the cavity, and λ is the wavelengthof the main laser. This limitation to the maximum force of the actuators and/or
the RMS velocity is called kinetic limit (Figure 5.1). It is of great relevance inthe 2G or 3G detectors, since their low frequency sensitivity imposes stringent
requirements on actuator noise.
Another approach for interferometer locking is to sequentially bring the DoFs
into the linear ranges by using auxiliary sensors. Variable finesse technique in
Virgo [127] can be regarded as a pioneering work of this approach. This technique utilizes the PDH signals obtained from the transmission of arm cavities tocontrol them in intermediate steps of lock acquisition, which are not disturbed
by dynamical states of the central interferometer. In addition, it starts fromkeeping the Michelson interferometer at mid-fringe to reduce the finesse of the
PR cavity so that the central part and the arm part are even more decoupled.
A more dedicated approach is employed by a technique called arm lengthstabilization (ALS) [128]. This technique utilizes auxiliary lasers having diﬀerent
wavelength than that of the main laser as auxiliary sensors for arm cavities. It
is adopted in Advanced LIGO [22] and KAGRA [129], and planned to be used

5.1. Overview(a) Lock acquisition without ALS51(b) Lock acquisition with ALS

Figure 5.1: Schematic picture of lock acquisition of a cavity with and without
ALS. The red line shows the optical power inside a cavity against the mirror
position, which exhibits resonant feature. The blue rectangles depict a mirror.
(a) Without the help of ALS, the motion of a free swinging mirror has to be
stopped within a short period when the cavity stays around the resonance. This
leads to the kinetic limit. (b) Using the control by an ALS system, the motion
can be gradually damped, and the cavity length can be eventually set wherever
one wants, including at the resonance.

52

Chap.5 Lock Acquisition

(a)

(b)

(c)

Figure 5.2: Schematic diagrams of the lock acquisition of Initial LIGO. Red
arrows represent the carrier fields of the laser beam while blue ones represent
the sideband fields. In the step (a), the central PRMI is controlled using the
sideband fields. In (b), the resonance of one arm cavity is caught. In (c), thethe resonance of the other arm is finally caught.
in Advanced Virgo in its upgrade. The technique ALS is the main topic of thisdissertation.

5.2

Previous techniques

5.2.1

Initial LIGO scheme

The power-recycled Fabry–Perot Michelson interferometer (PRFPMI) of Initial
LIGO was locked in the following sequence [126]. First, the central interferometeris controlled so that the modulation sidebands field resonate there (Figure 5.2).
In this step, the demodulated signals of the reflection PD output are used forthe control of PRCL and MICH. Next, the resonance of one of the arm cavities
is caught when the arm goes across it resonance. Finally, the resonance of theother arm is also caught to end the lock acquisition process.

5.2.2

Third harmonic demodulation technique

The above scheme has an issue that the error signals for PRCL and MICH canbe largely disturbed or even vanished due to the resonance of the arm cavities.
In order to mitigate this issue, a robust signal extraction scheme called third
harmonic demodulation was developed in TAMA300 [125]. This scheme does not5.2. Previous techniques

(a)

53

(b)

(c)

Figure 5.3: Schematic diagrams of the variable finesse technique. In the step(a), both arm cavities are controlled at their resonances using the transmission
PDH signals while the central interferometer is controlled with the Michelsoninterferometer being at mid-fringe. In (b), the condition of MICH is graduallychanged to dark fringe. In (c), the Michelson interferometer reaches dark fringe
and then the error signal for each DoF is switched to that in the final state.

rely on the carrier field of the laser for the extraction of the PRCL and MICH‡1
so that the arm cavities aﬀect much less when the carrier field is close to theresonance of an arm. Instead of the carrier filed, the second order modulation
fields are used as reference fields of PRCL and MICH sensing. This scheme isstill used in Advanced LIGO and KAGRA.

5.2.3

Variable finesse technique

In Virgo detector, the PRFPMI was locked using the variable finesse technique[127,130] (Figure 5.3). First, each arm cavity is kept at resonance using the PDH
signals from the transmission of the arm, and MICH is controlled at mid-fringe.
PRCL is also controlled. At this stage, the coupling between the arms and thecentral interferometer is small since MICH is at mid-fringe so that the finesse of
the PR cavity is very small. Next, the condition of MICH is gradually changed
to dark fringe. Finally, the error signal for each DoF is switched to that in the
final state. This technique has been used in Advanced Virgo because the opticalconfiguration is a PRFPMI until now, at the time of O3.
‡1

And also SRCL in a DRFPMI.

54

Chap.5 Lock Acquisition

5.3

ALS

A technique called ALS uses one auxiliary laser for each arm cavity so as toindependently sense the arm cavity length, or equivalently the resonant frequencyof the cavity with respect to the main laser frequency [128]. In the first stageof lock acquisition, the arm cavities and central interferometer are controlled
separately. Once CARM and DARM are confined within the linear range of
their final sensing signals coming from the main laser, the CARM and DARM
controls are handed over to these signals and, eventually, all DoFs are controlledusing the main laser signals.
The features and advantages of use of ALS can be summarized as follows:(i) Use of diﬀerent wavelength (from that of the main laser) enables us to
independently sense the length fluctuations of one arm.‡2 With the helpof dichroic coating of mirrors, the optical configurations are designed sothat each auxiliary laser field is aﬀected only from the corresponding arm
cavity. In addition, the finesse of the arms for the auxiliary laser can be
set relatively low for the ease of locking the cavity (In the case of KAGRA,
the finesse was set to about 50, as noted in Table 6.1).
(ii) By utilizing frequency conversion of lasers, e.g. second harmonic generation(SHG), optical interference between the main laser and the auxiliary lasers
are realized so that the diﬀerence among the frequencies of these lasers arealways monitored. Thanks to this, the resonant frequencies of the arms arecontrolled with respect to the main laser frequency and vise versa.
(iii) The frequency of each auxiliary laser can be actuated so that the laser are
kept resonating in the arm. Adding such actuators enables us to sense thediﬀerence between the resonant frequency of the arm and the main laser
frequency even if the PDH signal of the arm associated with the main laser
is far away from its linear range. The signals from the auxiliary lasers arealways kept within their linear range.
The above items are conceptually illustrated in Figure 5.4. It is worth notingthat the advantage (i) inevitably comes at a cost of a potential disadvantage;‡2
There is a scheme to independently measure displacement of mirrors by utilizing digitalinterferometry [131, 132], without introducing diﬀerent-colored lasers. Such a scheme may beutilized for ALS instead of using auxiliary lasers.

5.3. ALS

55

the optical paths are diﬀerent for the main and auxiliary lasers‡3 so that phasenoise in the diﬀerential paths biases the ALS sensing.

Figure 5.4: Conceptual diagram of ALS.

5.3.1

Necessity of ALS

It is considered that the ALS scheme is necessary in the lock acquisition process
in the 2G and 3G detectors with the DRFPMI configuration. This is because it
is not clear how to extend lock acquisition schemes without ALS developed so
far for the PRFPMI configuration to the DRFPMI configuration.‡4 In addition,with ALS, the kinetic limit is not imposed on the arm cavities thanks to thethird feature described above (also by Figure 5.1). Therefore, adopting a lockacquisition scheme with ALS reduces the requirements on the test mass suspensions and actuators. For these reasons, schemes with ALS are so far the only
solutions for lock acquisition of a kilometer-scale DRFPMI.

5.3.2

Requirements

Here the requirements on the ALS system are described. The noise requirement
can be given primarily by the following condition (primary noise requirement).
The frequency fluctuations of the main laser with respect to the resonance ofthe single arm cavity are suppressed within the linewidth of the arm cavity interms of the RMS. If this condition is satisfied, it is possible to hand over thecontrols from the ALS system to the main laser PDH signals, by utilizing thesignal derived from the optical power in transmission in intermediate steps oflock acquisition, as is done in Advanced LIGO [22].
‡3
Dispersion of the main and auxiliary lasers is not negligible. More practically, it is technically diﬃcult to make the input mode cleaner and the input Faraday isolator dichroic.
‡4
There is an ongoing simulation work to extend variable finesse technique to a DRFPMI[133].

56

Chap.5 Lock Acquisition
A more ambitious noise target can be set by the CARM linewidth.‡5 It has tobe noted here that the linewidth of the CARM cavity is narrower than that of asingle arm cavity or the DARM cavity, since PR eﬀectively enhances the finesseof the arm while RSE eﬀectively reduces the finesse. If the RMS of the residual
frequency fluctuations is suppressed below the level of the CARM linewidth, theCARM control can be directly handed over from the ALS CARM to the main
laser PDH signal. This would greatly simplify the lock acquisition process andcontribute in reducing the down time of the detector. The CARM linewidth canbe expressed as√()
1 − RPRM
Tloss√
κCARM ≃
κARM ,(5.2)+
1 + RPRM TITM
where κARM and κCARM are the linewidths of the single arm and CARM, respectively, RPRM is the reflectivity of the PRM, TITM is the transmissivity of theITMs, and Tloss is the roundtrip loss in the arm cavity. As TPRM is usually froma few to ten percent, the CARM linewidth is ∼ 10 − 50 times narrower than the
single arm linewidth.
Another requirement on the ALS system is the range of frequency actuationof the auxiliary lasers injected to the arms (feature (iii)). The range has to belarge enough to follow the fluctuations of the resonant frequency of the arm.

5.3.3

ALS in Advanced LIGO

Figure 5.5 is the schematic view of the ALS system of Advanced LIGO [22].
Small fraction of the pre-stabilized laser (PSL) are picked oﬀ and sent from thecorner station (area that houses ITMs, BS, and so on) to both end stations (areathat houses an ETM) via optical fibers. The picked-oﬀ PSL is used to stabilize
the frequency of an auxiliary laser placed behind each ETM, by taking beat noteof the PSL and the infrared output of the auxiliary laser in the end station. The
laser head of the auxiliary laser has built-in SHG setup. Its green laser output‡6
is injected to the corresponding arm cavity. The resonant frequency of the arm
is sensed with the PDH signal of the green laser. The PDH signal is fed back to
the reference frequency of the beat note. Lock of one arm cavity in green canbe achieved easily because the finesse of the arm is set relatively low. The green
laser transmitting through each arm is extracted on an optical table in the corner
station. On the table, beat note of the two transmitted beams and beat note of‡5

CARM serves as a single optical cavity with the full interferometer locked.
In Advanced LIGO the main laser has wavelength of 1064 nm and thus the frequencydoubled laser has wavelength of 532 nm, which is green.
‡65.3. ALS

57

AUX Y

Phase-Frequency Discriminator
Voltage Controlled Oscillator
Servo

ETMY

~~

From PSL

Y arm

ITMY

IMC
PRM
PR3

ITMX

From From
X arm Y arm

From PSL

Optical Fibers

From PSL

ETMX

AUX X

SR2

To X end

To Y end

X arm

BS

PR2

~~

PSL

ALS
CARM

ALS
DARM

SR3
SRM

Figure 5.5: Schematic view of the ALS system of Advanced LIGO.
the transmission of the X arm and the second harmonic field of the picked-oﬀ
PSL are obtained. Each beat note is phase-locked and, by doing so, the DARM
signal and the CARM‡7 signal are extracted from the former and the latter beatnotes, respectively.
The lock acquisition scheme of Advanced LIGO with the ALS system is thefollowing (Figure 5.6). First, the resonance of the green lasers in the arms are
acquired. Second, CARM and DARM are controlled so that the main laser is
kept oﬀ-resonant of each arm by certain amount. Third, lock of the dual-recycled
Michelson interferometer (DRMI) in the central interferometer is acquired as
three DoFs of the DRMI are controlled using the third harmonic demodulation
technique. Fourth, the control of CARM and DARM is handed over from the
ALS system to the transmission power of the main laser of the arms and the
PDH signal of the anti-symmetric port (transmission of the SRM), respectively.
Finally, the oﬀset of the main laser frequency with respect to the arm cavity
resonances is gradually reduced to switch all the controls to the main laser PDH
signals. They achieved the full operation of the DRFPMI with this scheme.
‡7

In fact, this is not a pure CARM.

58

Chap.5 Lock Acquisition

(a)

(b)

(c)

Figure 5.6: Schematic diagrams of the lock acquisition schemes with ALS. In thestep (a), each green laser keeps resonating in the corresponding arm cavity sothat the displacement of the arm cavity length can be sensed with respect to themain laser. In (b), the central DRMI is controlled by using the sideband fields
while the main laser is kept oﬀ-resonant by ALS. In (c), both cavities are broughtaround their resonances and then the error signal for each DoF is switched tothat in the final state.
Putting a focus on the performance of their ALS system, it has been reported that
the RMS of the residual frequency fluctuations reached 6 Hz [134] or 8 Hz [135] inthe single arm operation, which were significantly smaller than the full linewidthof the single arm, 84 Hz.
Issues
The following issues of the lock acquisition and ALS scheme of Advanced LIGO
can be raised.
• The scheme requires the optical fibers have to be laid along the arms. Thus
the longer arms in the 3G detectors would force us to use the longer fibers.
If a similar ALS scheme is used in the 3G detectors, it is expected thatfiber phase noise and optical loss increase, which might harm the overall
performance.
• The lock acquisition sequence of CARM consists of multiple steps that
use the arm transmission as an error signal. This complicates the lockacquisition process.
In the following chapters, a new ALS scheme that addresses these issues is described after introducing the interferometer of KAGRA, where the new ALS
scheme has been implemented and demonstrated.

59

Chapter 6
CONFIGURATION OF KAGRA
KAGRA is one of the 2G ground-based interferometric GW detectors. Whileits design sensitivity and size of the arms or test masses are on the same orderof those of other 2G detectors, it has unique features that will be keys in thenext generation GW detectors, such as underground facility and cryogenic test
masses. In this chapter, the configuration of KAGRA is overviewed with a focus
on its interferometer, where the new ALS system has been implemented andtested.

6.1

Overview

KAGRA is constructed in the Kamioka mine, Gifu Prefecture, Japan [24]. The
detector of KAGRA is built in the L-shaped underground tunnel in the mine.
Each point of the tunnel is at least 200 m below the surface of the ground. The
purpose of building the detector in underground facility is the following. One isthat the level of seismic motion is lower, which is beneficial for both sensitivity
and stability. Another is that it is expected that physical conditions such astemperature, humidity, and air pressure are more stable.
Figure 6.1 shows the top view of the whole system of the KAGRA detector. Two arms, namely the X and Y arms, are 3-km long for each. The longvacuum tube is installed in each arm of the L-shaped tunnel. At the both edgeof each arm, large cryostats are placed to house the cryogenic payloads for thetest masses. The cryostat is, with proper thermal insulation, connected to avertical vacuum tube reaching the second floor, which is the base of the test
mass suspension. At the bottom of five horizontal and vertical seismic isolation
stages at room temperature [136], the cryogenic payload is suspended, whichis cooled down to approximately 20 K. These test mass suspensions are called
Type-A. Since the longitudinal motion of the test masses directly becomes dis-

60

Chap.6 Configuration of KAGRA

ETMY

3 km
IMC

~
ITMY

IMMT2

PR2

PRM

BS

~~

LASER
Room
Enclosure

IMMT1

PR3

ITMX
SR2

3 km

ETMX

SR3

SRM
OMMT2
OSTM

OMMT1

OMC

Figure 6.1: Schematic view of the whole system of KAGRA. The suspension system of the core optics are also shown. The gray-colored area represents vacuumtanks and tubes. Each test mass and its payload is surrounded by a cryostat.

placement noise of the detector, the highest seismic isolation is applied to them.
The BS and SR mirrors are suspended with Type-B suspensions [137]. Comparedwith Type-A, a Type-B suspension has fewer number of isolation stages and theroom temperature payload. The PR mirrors are suspended with Type-Bp suspensions [138], which are the simplified version of Type-B. For Type-A and -B,
the suspension chain starts from an inverted pendulum while, for Type-A, -B,
and -Bp, geometric antispring filters are employed for vertical isolation. Other
mirrors labeled as IMC, IMMT1, IMMT2, OMMT1, or OMMT2 are suspended
with Type-C suspensions, which are simple double pendula that were also usedin TAMA300. Most stages including the mirror stage of each suspension havecoil-magnet actuators.‡1
Figure 6.2 shows the cryogenic suspension of the test mass of KAGRA. Test
‡1
A magnet is attached to the object one wants to actuate (e.g. a mirror), and a coil isplaced correspondingly on the reaction mass. By introducing electrical current in the coil, theforce between the magnet and coil can be varied linearly to the applied current.

6.2. Optical configuration(a) Picture of a cryogenic payload.

61(b) Schematic view of a cryogenic payload.

Figure 6.2: Picture and schematic view of a cryogenic payload. (a) Picture ofa cryogenic payload viewed from side. The test mass chain is surrounded by
its reaction mass chain, which is further surrounded by a security frame. (b)Schematic view of a cryogenic payload only showing the test mass chain. Eachstage of the marionette, intermediate mass, and test mass is surrounded by its
reaction mass. The reaction mass chain is suspended from the platform.
mass is made of sapphire crystal with a weight of 23 kg. The test mass is suspended by four sapphire fibers. The upper edge of each fiber is connected toa sapphire blade, which is attached to the intermediate mass. The intermediate mass is suspended from the marionette stage, which is mainly used for thelarge-range actuation of longitudinal and angular motion of the test mass.

6.2

Optical configuration

Figure 6.3 shows the optical configuration of KAGRA. KAGRA adopts a DRFPMI as the main interferometer. Tables 6.1 and 6.2 summarizes the parameters

62

Chap.6 Configuration of KAGRA

TRY
Port

~~ Y arm

3 km

PSL

ITMY

IMC
PRM

Reference
PMC
cavityModulation
System

AUX X
AUX Y

IMC
REFL
Port

IFI
REFL Port

BS

PR2

PR3

ITMX
POP Port

TRX
Port

X arm

SR2

~~

Laser
Source

ETMY

ETMX

3 km

Optical Fibers
SR3

Laser Room
EnclosureSRM
AS PortOMC
OMC Trans. Port
GW Signals

Figure 6.3: Optical configuration of KAGRA.

associated with the main interferometer. The wavelength of the main laser is1064 nm. The output of the main laser amplifier passes through the pre-modecleaner (PMC), which is a bow-tie cavity that rejects the junk higher order opticalmodes and RF amplitude fluctuations of the laser [139]. A fraction of the leakageof the PMC is guided to the reference cavity, which is a stable 10-cm long Fabry–
Perot cavity for the frequency reference. The main output of the PMC is sent to
the modulation system [140], where phase and amplitude modulation is applied
to the main laser. These are the breakdown of the PSL system of KAGRA. Aftercoming out of the laser room enclosure and entering the vacuum tube, the PSL
is injected to a triangular optical cavity composed of three suspended mirrors,
called the input mode cleaner (IMC) [139]. The IMC has mainly three purposes.

6.2. Optical configuration
The first is to be a frequency reference of the second stage of the frequency
stabilization. The second is to reduce beam jitter of the main laser incident to
the main interferometer. The third is to reject higher order optical modes thatmight have been generated in the modulation system. The transmission of the
IMC subsequently passes through a Faraday isolator called the input Faraday
isolator (IFI), where the reflection of the main interferometer is extracted. The
transmitted beam is then steered toward the main interferometer by a pair ofcurved mirrors, called input mode matching telescope 1 and 2 (IMMT1 and 2).
The power or signal recycling mirrors are composed of three mirrors. The PRM
and SRM perpendicularly reflect the beam from the main interferometer, whilepairs of PR2 and PR3, and SR2 and SR3 serve as telescopes. They help to makethe recycling cavities stable in terms of transverse modes. The transmission of
the SRM, which is the output of the main interferometer, is steered by a pair ofcurved mirrors, called output mode matching telescope 1 and 2 (OMMT1 and2), and a flat mirror called output steering mirror (OSTM). The beam subsequently propagates through a bow-tie cavity on a suspended breadboard, called
the output mode cleaner (OMC). Variations of the transmitted optical powerfrom the OMC corresponds to GW signals detected by KAGRA. The name ofeach detection port of the optical signals is also described in the figure.
For the purpose of lock acquisition, two auxiliary lasers are placed in the laserroom beside the main laser. Each auxiliary laser provides two laser outputs:
infrared and green. The infrared beam is the primary output with a wavelengthof 1064 nm, which is same as that of the main laser. The secondary output is thegreen laser, which is produced in the laser head by SHG of the primary beam.
Thus it has a wavelength of 532 nm. The two green lasers are delivered via opticalfibers to the optical tables placed nearby PR2 or SR2. The fibers have the lengthof approximately 60 m. The green lasers are injected to the main interferometerfrom the back side of PR2 or SR2. The BS, PR2 and SR2 have dichroic coatingon their front side so that they are transmissive in green at 532 nm. Thanks tothese dichroic properties, the lasers from PR2 and SR2 are incident to the X
and Y arms, respectively. The transmitted beam of each arm cavity is separated
into infrared and green beams by a dichroic mirror, and their optical power isseparately detected at TRX or TRY ports.
It is worth mentioning that the finesse of the arm cavities of KAGRA ishigher than those of Advanced LIGO and Advanced Virgo approximately bya factor of three. This is mainly because only KAGRA cools its test masses
down to cryogenic temperature; less optical power in the central interferometer ispreferred given the optical absorption of the substrate of the test mass mirror and

63

64

Chap.6 Configuration of KAGRA
thus the combination of higher finesse of the arm cavities and higher reflectivityof the SRM is preferred.‡2 This fact makes lock acquisition of KAGRA evenmore challenging and the arm cavities of KAGRA a suitable test bed of lockacquisition schemes.
Table 6.1: Design and measured values of the main interferometer parameters of
KAGRA.
Parameter

Designed

Arm cavity length [m]3000 [141]

Arm FSR [kHz](49.96541)

Roundtrip Gouy
phase [deg]250.75 [141]Arm finesse for 1064 nm

1530 [141]Arm finesse for 532 nm

51 [145]

Arm roundtrip loss
for 1064 nm [ppm]PR cavity length [m]SR cavity length [m]Roundtrip Gouy phaseof PR cavity [deg]Roundtrip Gouy phaseof SR cavity [deg]Schnupp assymmetry [m]PRM transmissivity [%]SRM transmissivity [%]IMC length [m]IMC finesse
Modulation 1 (f1 ) [MHz]Modulation 2 (f2 ) [MHz]Modulation 3 (f3 ) [MHz]66.591 [141]66.591 [141]Measured
X: 2999.990(2) [129, 142]
Y: 3000.015(1) [143]
X: 49.96557(4) [129, 142]
Y: 49.96517(2) [143]
X: 250.7(3) [129, 142]
Y: 250.9(3) [143]
X: 1410(30) [129]
Y: 1320(30) [144]
X: 41(2) [146]
Y: 53(3) [146]
X: 86(3) [129]
Y: –
–
–

33.0 [141]31(1) [147]35.0 [141]48(2) [148]3.3299 [141, 145]
10 [145]
15 [145]
26.639 [141]520 [150]
16.880962 [141, 145]
45.015898 [141, 145]
56.269873 [145]–10.35 [145]–26.65095(2) [149]540(70) [151]
16.8732705 [149]44.995388 [149]–

100 [141]‡2
Please refer to Section 4.3.2 for how RSE balances the optical power in the central interferometer.

6.3. Control scheme

65

Table 6.2: Linewidths of the arm cavities of KAGRA that are derived from thevalues in Table 6.1.
Parameter
Single arm full linewidthfor 1064 nm [Hz]CARM full linewidth [Hz]DARM full linewidth [Hz]Designed
32.7
1.6
783

Measured
X: 35.4(7)
Y: 37.9(8)––Table 6.3: Five length DoFs of a DRFPMI and their sensing signals. The namesof the demodulation signals are defined in the following way: X-phase demodulation at k (integer) MHz of the photodetector outputs at NAME port is called
NAMEkX.
DoF name
DARM
CARM
PRCL
SRCL
MICH

6.3

Error signal
OMC TRANS (or AS17Q)
REFL45I
POP45I
POP17I
POP17Q

Control scheme

In this section, the designed control scheme of the main interferometer in theobservation mode is briefly described. For the sensing of length DoFs of the
DRFPMI, the main laser is phase-modulated at two diﬀerent frequencies, namely
f1 and f2 (Figure 6.4). Defining the FSR of the IMC as fIMC , f1 and f2 can be
written by f1 = 3fIMC and f2 = 8fIMC . f1 and f2 are integer multiple of fIMC
so that the modulation sidebands transmit through the IMC together with thecarrier field. These frequencies are listed in Table 6.1. As shown also in Table 6.1,
the Schnupp asymmetry of the central Michelson interferometer is about 3.33 m.
With this asymmetry, f1 sidebands leak to the SR cavity while f2 sidebands are
perfectly reflected by the Michelson interferometer when the carrier field is at
the dark fringe. DARM is controlled with an oﬀset of about 1 − 10 pm so thata small fraction of carrier field leaks out of the SR cavity, which enables us tosense DARM with the DC readout scheme.
The error signal for each length DoF is listed in Table 6.3. At REFL and POP

66

Chap.6 Configuration of KAGRA

Figure 6.4: Schematic view of the carrier and sideband fields and the detection
ports in the main interferometer. The carrier field resonates in the arm cavities
and the PR cavity. Due to the DARM oﬀset, a small fraction of the carrier
leaks out of the Michelson interferometer and the SR cavity. The f1 sidebands
resonate in the PR-SR coupled cavity while the f2 sidebands resonate in the PR
cavity. At REFL and POP port, the beat notes between the carrier field and thesideband fields are detected.
port, the beat notes between the carrier field and the sideband fields are detectedwith photodetectors that have a resonant circuit at f1 and f2 frequencies so as
to have high signal-to-noise ratio. The DARM signal is fed back to the positionof one of ETMs with the bandwidth of about 50 Hz. The CARM signal is fed
back to the frequency of the main laser by the combination of changing the IMC
length and directly actuating the laser frequency with the bandwidth of about
10 kHz. The PRCL, SRCL, and MICH signal are fed back to the position of the
PRM, the SRM, and the BS, respectively, with the bandwidth of about 30 Hz.

6.4

Lock acquisition scheme

In this section, the lock acquisition scheme of the KAGRA interferometer, or
how to reach the control configuration described in the previous section, is described. The scheme is similar to that of Advanced LIGO. The main diﬀerences
are the following two. One is that the scheme of the ALS is diﬀerent. The ALS

6.4. Lock acquisition scheme
scheme of KAGRA is described in the next chapter in detail. The other is that,in the default plan, the amplitude modulation sidebands at the frequency of f3
are employed in the intermediate steps of lock acquisition [141]. f3 is equal to
10fIMC . This value is chosen so that the sidebands do not resonate in any cavitiesof the main interferometer. The use of these non-resonant sidebands of amplitude modulation enables us to realize an improved version of the third harmonic
demodulation technique. Let us recall that, in the third harmonic demodulation
technique, the variations in the DoFs of the central interferometer are carried by
the phase modulation sidebands of f1 or f2 , and they are sensed with heterodyne
readout by using the second order sidebands, which have sideband frequency of
2f1 and 2f2 , as reference fields; the signals in the photodetector outputs areconsequently around 3f1 or 3f2 (Specifically, REFL51I/Q and REFL135I). Since
the 2f1 and 2f2 sidebands are amplitude modulation,‡3 it is possible to sense thesame signals by alternatively preparing new amplitude modulation sidebands.
This is the working principle of this scheme. The signals of the length DoFs
appear around f3 ± f1 or f3 ± f2 instead. The advantage of this scheme is thatthe signals are further separated from the dynamics of arm cavities comparedwith the third harmonic demodulation technique; the beat note between the carrier field, which carries the signals from the arm cavities, and the third order
sidebands pollutes the signals in the third harmonic demodulation technique.
KAGRA employs a pair of Mach–Zehnder interferometers to generate amplitude
modulation only from electro-optic modulators (EOMs), each of which is a phasemodulator [140].

‡3Phase modulation at f is inevitably accompanied with amplitude modulation at 2f .

67

69

Chapter 7
KAGRA TYPE ALS
In this chapter, a newly developed ALS scheme that is used in KAGRA is described in detail. First, the design of the scheme is explained. Second, theimplemented optics and circuits associated with the ALS system are overviewed.
Third, the demonstration of the lock acquisition scheme of KAGRA using the
ALS system is described. Finally, the characterized performance of the ALS
system is explained. The works covered by the present chapter were primarily
done by the author of the dissertation.

7.1

Requirements

As mentioned in Section 5.3.2, for smooth handing over from the ALS system
to the main laser signals, the ALS system has to be capable of suppressing thefluctuations of the main laser frequency below the full linewidth of a single arm
cavity. This requirement can be quantified by the condition that the RMS of∆fres (f ; κARM ) is below the full linewidth of the arm, where∆fres (f ; κARM ) = [∆fmain (f ) − ∆fARM (f )]L(f ; κARM ),1
L(f ; κc ) =.
1 + if /κc

(7.1)
(7.2)

Here κARM is the (half) linewidth of the arm, ∆fmain (f ) and ∆fARM (f ) are thefrequency fluctuations of the main laser entering the main interferometer andthe resonance of the arm, respectively, and L(f ; κc ) is the transfer function of alow-pass filter with a corner frequency of κc . We here consider the low-passed
fluctuations because the error signal of the arm cavity senses these low-passed
fluctuations‡1 and thus its RMS has to be compared with the linewidth of the‡1

Please refer to Equation (4.23).

70

Chap.7 KAGRA-Type ALS
Table 7.1: Main diﬀerences between the ALS systems of KAGRA and Advanced
LIGO.
KAGRA
Green laser injection site Central area
Optical fibers
Within the central area
Signals of the arm DoFs
Summations in electronics
Number of optical sensors 4 (or 6 a )Number of SHG setups
2aAdvanced LIGO
End stations
Along the arms
Optical beat notes
63

If the fiber noise cancellation loops are involved.

arm. In the case of KAGRA, the full linewidth of the single arm cavity is 33 Hz,‡2
and thus the requirement on ∆fres (f ; κARM ) is 33 Hz (Table 6.2). This value issmaller than that of Advanced LIGO, which is 84 Hz [22].
While the above requirement is the primary target of the frequency stabilityby the ALS system, the more ambitious target can be given by the CARM
linewidth. This target can be quantified by the condition that the RMS of∆fres (f ; κCARM ) is below the full linewidth of CARM, where∆fres (f ; κCARM ) = [∆fmain (f ) − ∆fCARM (f )]L(f ; κCARM ).

(7.3)

Here κCARM is the (half) linewidth of CARM, and ∆fCARM (f ) is the fluctuationsof the resonant frequency of CARM. In the case of KAGRA, the full linewidth of
CARM is 1.6 Hz (Table 6.2). If the condition is satisfied, it is possible to directly
hand over the CARM control from the ALS system to the main laser PDH signal
obtained at REFL port, which will greatly simplify the whole lock acquisition
process.
As is also mentioned, an requirement is imposed on the range of the frequency
actuation of the auxiliary lasers. Since the RMS of the arm length fluctuation
due to seismic motion is about 2 µm on a noisy day at the KAGRA site [152],
the range has to be larger than 0.2 MHz.

7.2

DesignThe ALS scheme of KAGRA is diﬀerent from that of Advanced LIGO in theirconcepts, designs, and control strategy. As explained in the previous chapter,‡2

This corresponds to 0.35 nm in terms of the cavity length displacement.

7.2. Design71

Laser Source

Demodulator

Acousto-Optic Modulator

Phase-Frequency Discriminator

Electro-Optic Modulator
Faraday Isolator
Fiber Coupler

−1

ETMY

RF Signal Generator
Voltage Controlled Oscillator

Optical Fiber~~ Y arm

Servo

Photodetector

Slow Servo
Fast (2 inputs / 2 Outputs)

CARM Slow

DARM Error
ALS DARM

DARM servo

Digital System

IMC

CARM Fast

PRM

AUX Y

REFL
PD
PLL
PD X

CARM Error

PR3

ITMX
PDH
PD Y

+
+
+
−

ETMX

SR2
SR3

PLL
PD Y

X arm

BS

PR2
PDH
PD X

Digital System

~~

PSL

AUX X

ITMY

+
+

CARM servo
CARM Error
Slow
Fast
ALS CARM

SRM
CARM Slow
CARM Fast

ALS DARM Digital System

Figure 7.1: Schematic view of the ALS system of KAGRA.
KAGRA employs the green laser beams injected from the central area of the interferometer, instead of the other ends of the arm cavities. This made the control
configuration simple by eliminating the long optical fibers along the kilometerlong arms. The main diﬀerences of them are summarized in Table 7.1. Thissection covers the design of the optical and control configuration of the ALS
system and the noise budget design of the system.

7.2.1

Optical and servo topology

Figure 7.1 shows the optical and control topology of the ALS system. Let us firstsummarize the working principle before going into the detail. First, phase-locked
loops (PLLs) make the frequencies of the infrared outputs of the auxiliary lasers
follow the PSL frequency with certain oﬀsets. The frequency of the green laserfrom each auxiliary laser follows the resonant frequency of the correspondingarm by using the combination of a voltage controlled oscillator (VCO) and a

72

Chap.7 KAGRA-Type ALS
double-path acousto-optic modulator (AOM) as a frequency actuator. At thispoint, on the assumption that the auxiliary laser is tightly locked to the PSL
via the PLL, each control signal applied to the frequency actuator is equivalent
to the diﬀerence between the PSL frequency and the arm resonant frequency.
Therefore, CARM can be sensed by the sum of the two control signals while
DARM by the diﬀerence between the two. Such error signals are called ALSCARM and ALS DARM. ALS CARM is fed back to the PSL frequency while
ALS DARM is to (diﬀerential) arm cavity lengths, so as to achieve the controlof the two length DoFs of the arm cavities. By this control, the main laser
frequency is fixed to certain points with respect to the resonances of the arm
cavities. The positions of the resonances are adjusted by the frequency oﬀsets in
the PLLs of the auxiliary lasers.
The detail of the system can be described as follows. For the PLLs of theauxiliary lasers, beat notes between the PSL and the infrared output of theauxiliary laser are detected by photodetectors (denoted by PLL PD X and Y
in the figure). At phase-frequency discriminators, the signals from the beatnote photodetectors are demodulated by mixing them with local oscillator (LO)signals from RF signal generators. One can adjust the auxiliary laser frequency
with respect to the PSL frequency by changing the LO frequency. To close
the PLL, each demodulated beat note signal is filtered by an analogue servo
circuit and then fed back to the frequency of the auxiliary laser. The frequency
actuation is achieved by the piezoelectric transducer attached to the laser cavity,
accompanied with the temperature control for drift compensation. The green
laser output from the laser head of each auxiliary laser is frequency-shifted by adouble path AOM. Successively, phase modulation is applied by an EOM in thelaser room enclosure for signal extraction via the PDH technique. After delivered
via the optical fibers, they are injected to the X and Y arms from the rear sideof PR2 and SR2, respectively. The reflection from each arm cavity is extracted
by a Faraday isolator and detected by a photodetector (PDH PD X or Y) on
the optical table (POP or POS table, respectively) to provide the PDH error
signal. Each PDH error signal is filtered by another analogue servo circuit andthen fed back to a low-noise VCO that drives the AOM. The two control signalsof the PDH loops are combined to obtain the ALS CARM and ALS DARM
error signals. The CARM loop partially involves analogue control filters, while
the DARM loop is entirely realized by a digital control, because the CARM loop
requires wide control bandwidth up to about 1−10 kHz. Then an extra analogue
circuit is used for adding the two control signals and successively applying a lowpass filter to the summation, in order to to synthesize the ALS CARM error

7.2. Design
signal in such a way that its frequency response is matched with the main laser
CARM signal (REFL CARM). The REFL CARM and ALS CARM are sent tothe same analogue servo board named the CARM servo so that the two input
signals can be gradually switched to accomplish smooth hand-over from the ALS
system to the main laser PDH signal. The CARM servo has two output ports
named slow and fast outputs. The slow output is fed back to the IMC length
while the fast output is to the error point of the IMC control loop. Combining two
feedback paths enables us to form a dual loop CARM control [153]. Meanwhile,for the DARM control, the two control signals of the arm PDH loops are sampled
and digitally processed by real-time digital controllers. The output of the digital
controller is fed back diﬀerentially to the ETM motion. Once the ALS CARM
and ALS DARM loops are fully engaged, the resonant frequency of each arm
cavity can be tuned to a desired position with respect to the PSL frequency bychanging the LO frequency for the corresponding PLL.
In addition to the main control loops of the ALS system, optics and servo
for the cancellation of fiber phase noise are prepared; what is called fiber phasenoise here is phase fluctuations the lasers pick up as they propagate through theoptical fibers. For the purpose of this cancellation technique, a small fraction ofeach green laser is picked oﬀ and immediately reflected back between the EOM
and the fiber coupler in the laser room enclosure. Another small fraction is also
picked oﬀ and reflected back from POP or POS table to the laser room. Theinterference of the two reflection is detected in the laser room, and the fringeof it can be controlled to mid-fringe by feeding back the detected fringe signal
to the optical path length; a piezoelectric transducer is attached to the mirror
immediately before the fiber coupler so that its position can be actuated. This
control is secondary in the sense that it can improve the noise performance of
the ALS system if fiber noise limits the noise level but it is not indispensable forthe control topology of the ALS system.
The features of the KAGRA ALS can be summarized as follows. The first
feature is that auxiliary green lasers are injected from the corner station, as
opposed to the end stations. Thanks to this feature, the optical fiber lengthis not on the same order to the arm length but can be significantly shortened
to the level of approximately 60 m, which makes the system scalable to the 3G
detectors [8,89]; fiber phase noise and optical loss do not increase even if the arm
cavity length gets long. Moreover, it is also realistic to even eliminate opticalfibers in the configuration of the KAGRA-type ALS. The second is that theconfiguration is simple in the sense that the number of SHG setups is less. If
the fiber noise cancellation is not used, the number of the involved sensors is

73

74

Chap.7 KAGRA-Type ALS

Frequency noise [Hz/sqrt(Hz)]also less. The third feature is that the ALS CARM and ALS DARM signals areproduced by summing the two signals from two arms in electronics or real-time
controllers, respectively. This increases the flexibility of the ALS control loops,
compared to the case where these signals are obtained with optical beat notes.‡3

100

Total
Main-laser related
Seismic
PLL related
Optical fiber
Green lock VCO
Cumulative RMS

CARM linewidth: 1.6Hz

RMS=0.64Hz

10 1
10 2
10 3
10 4
10 5
6
10 10
1

100

101

102

Frequency [Hz]103

Figure 7.2: ASDs of the simulated noise spectra of the ALS system in design. The
blue dashed line shows total noise: quadrature sum of the contributions of all thenoise sources considered in the design study. The magenta line shows the total
contributions of noise related to the PSL and its frequency stabilization. The
brown line shows the estimated residual contribution of arm length fluctuations
caused by seismic motion. The solid cyan line shows the contribution fromthe PLLs of the auxiliary lasers including free-run frequency fluctuations of theauxiliary lasers. The green line shows the contribution of phase noise caused by
the optical fibers for the green lasers. The light green line shows the contributionof phase noises of the VCOs. The light blue dotted line shows the cumulative
RMS of the blue dashed line integrated from the high frequency side. The RMS
integration is bounded by 0.1 Hz because we assumed that the control hand-overlasts for 10 s at most.

‡3

In the case of Advanced LIGO, the true CARM cannot be sensed by their ALS system.

7.2. Design

75

Table 7.2: Design of the control bandwidth of each loop of the system. The unity
gain frequencies (UGF) are shown.
Loop name
UGF
PLL
20 kHzArm PDH loop 20 kHz
ALS CARM loop 10 kHz
ALS DARM loop 25 Hz

7.2.2

Noise Design

In order to verify the design of the whole ALS system, we numerically simulated
the noise performance of the ALS. The noise design of the KAGRA ALS system
is described here.
We numerically simulated the noise contributions in the KAGRA ALS systemin the frequency domain. The servo loops were also simulated and designed.
The conditions and assumptions of the simulation are summarized in Tables 7.2
and 7.3. The simulation results of the noise contributions to residual frequency
fluctuations of the main laser with respect to the CARM resonance are shown in
Figure 7.2. The spectra of ∆fres (f ; κCARM ) are computed by using a commercial
software, Simulink. The constructed Simulink model is available here [154]. The
calculation results show that, in the designed system, the RMS of the residual
frequency fluctuations with the ALS control can be as low as 0.64 Hz, whichis smaller than the CARM linewidth. The dominant noise contribution above
about 1 Hz was expected to come from frequency noise of the VCOs. Below 1 Hz,
it was expected that phase noise of the green lasers introduced by the opticalfibers was dominant. We also confirmed that the primary noise requirement
can be achieved; the RMS of ∆fres (f ; κARM ) is 2.0 Hz and so smaller than the
linewidth of the single arm cavity, 33 Hz.
Fiber phase noise had been estimated based on a measurement of phase noiseof an optical fiber of length of 5 m placed on an optical table (Appendix F for thedetail). We scaled the measured level of phase noise in such a way that the PSD
of fiber phase noise is proportional to the fiber length. Since it was expected thatthe contribution of fiber phase noise was not critical in achieving the requirement
based on the preliminary measurement, the fiber noise cancellation loop was notincluded in the simulation work.

76

Chap.7 KAGRA-Type ALS
Table 7.3: List of major noise sources and design magnitude of them.
Noise source
Optical fiber
VCO
Arm length fluctuations
Free-run frequency noiseof the auxiliary laser
PSL frequency noise

7.3

ASD at 1√Hz
0.01 Hz/ Hz

Note
Appendix F
Extrapolation from√LIGO VCO [155] with
0.5 Hz/ Hza safety factor of 10√−9
3 × 10 m/ Hz [152]√
1 × 105 Hz/ Hz 1/f dependency [156]√
1 × 104 Hz/ Hz Calculated from [152]

Implementation

In this section, first, the configuration of each control loop in the ALS systemthat has been installed in KAGRA is overviewed. Successively, optical and electrical components in the actual system are described in detail. Comprehensive
information of the components such as the product numbers can be found in
Appendix G.

7.3.1

Control loops

Here, the control loops that are the building blocks of the whole system are
explained; they are the PLLs between the main and auxiliary lasers, the arm
cavity loops of the auxiliary lasers, fiber noise cancellation loops, and ALS CARM
and DARM loops. Sensors, servo filters, and actuators of them are described
below (Figure 7.3).
Phase-locked loop
The sensor of the PLL is a photodetector. It receives the interference fields
between the main laser and one of the auxiliary lasers. Its photodiode converts
the received optical power into electrical current, and the circuit part of thephotodetector then converts the current into voltage signal. The output of thephotodetector is typically at several tens of megahertz.
The raw output of the photodetector is then demodulated at the phasefrequency discriminator by mixing it with local oscillator signal. The demodulated signal, which is typically down-converted below about 1 MHz, is regarded

7.3. Implementation

77

Auxiliary
laser(built-in)

Piezoelectric
actuator

Main
laser

To green
PDH loop

Effective sensor (Phase diff. à Voltage)
Photodetector(broadband)

PFD

Optical power Demodulationà Voltage

Voltageà Frequency

Filter circuit(a) Phase-locked loop.
Fiber noise
cancellation loop To arm
Fromlaser module
VCO

Effective sensor (Frequency diff. à Voltage)
Photodetector(resonant)
Optical power Demodulator
à Voltage

AOM
Actuator
Voltageà Frequency

To ALS CARM/DARM loops

Filter circuit(b) Green PDH loop.

Figure 7.3: Schematic diagram of the subcomponent loops of the system. The
configuration of the fiber noise cancellation loops is shown in Figure 7.7.

as the error signal of the PLL; it is proportional to the relative phase deviations
between the main and auxiliary lasers.
The demodulated voltage signal is then sent to a filter circuit (“universal analogue circuit” described below). The circuit is a series of variable gain amplifier
stages and filter stages with zeros and poles, which are realized with operational
amplifiers.
The output voltage signal of the filer circuit is sent to the laser module of
the auxiliary laser, which eventually drives piezoelectric actuator of the laser
cavity to actuate the laser frequency. The eﬀective frequency bandwidth of
the frequency actuation is about 100 kHz, since there is a bunch of mechanicalresonances above ∼ 200 kHz.

78

Chap.7 KAGRA-Type ALS
Green PDH loop
The sensor of the arm cavity loop of the green laser is also a photodetector. It
receives the reflection fields of the arm cavity, which are the interference between
the carrier and the phase-modulation sideband fields. The photocurrent generated at the photodiode is converted into voltage signal via a resonant circuit. Its
resonant frequency is tuned at the frequency of the phase modulation.
The raw output of the photodetector is demodulated at a mixer circuit toproduce the PDH signal. The frequency band of the signal is below about 1 MHz.
The PDH signal is proportional to the diﬀerence between the auxiliary laser
frequency and the resonant frequency of the arm. The PDH signal is then sent
to a similar filter circuit.
The output voltage signal of the filter circuit then drives a VCO. The VCO
drives an AOM, which shifts the frequency of the green output of the auxiliary
laser. As a whole, the amount of the frequency shift is proportional to the output
voltage of the filter circuit.
Fiber noise cancellation loop
The sensor of the fiber noise cancellation loop is a subtraction‡4 of the outputsof two photodetectors, which sense DC optical power of two ports of a Michelsoninterferometer. The Michelson interferometer is designed to be sensitive to theoptical length variations in the optical fiber for the green laser. The sensor output
is then filtered in a similar filter circuit.
The output voltage signal of the filter circuit is amplified by a high-voltage
amplifier with the output range of 0 − 150 V. The amplified signal drives apiezoelectric actuator attached to a mirror so that the motion of the mirror
compensates the optical length variations.
ALS CARM and DARM loops
The eﬀective sensors of the ALS CARM and ALS DARM loops are the control
signals of the green PDH loops, which are the voltage signals applied to the
VCOs. As for the ALS CARM loop, the error signal is filtered by a similar
analogue filter circuit. As for the ALS DARM loop, the filtering is performed indigital real-time controllers.
There are two actuators in the ALS CARM loop. One changes the length of
the IMC. The length of the IMC is actuated by pushing one mirror of the IMC‡4

The subtraction is actually done in the filter circuit downstream.

7.3. Implementation(MCe; a suspended mirror placed at the far end of the IMC) with coil-magnet
actuators. Since the control system of the IMC forces the PSL frequency tofollow the IMC resonance, changing the IMC length actuates the PSL frequency
via the IMC control. The other injects the voltage signal into the error point of
the IMC control. This injection biases the IMC control and thus actuates the
PSL frequency through the control.
The actuator of the ALS DARM loop changes the lengths of the arm cavities.
The length of the arms are actuated by pushing one or more ETMs with coilmagnet actuators. The test mass stage is used for the higher frequencies while
the marionette stage is used for the lower frequencies.

7.3.2

OpticsLaser room
In the laser room enclosure, a large optical table with the dimension of 1500 mm
by 4800 mm is placed, which is called the PSL table (Figure 7.4). One cornerof the table (in the +x and +y side in the interferometer frame) is assigned
for the optics associated with the ALS system. Figure 7.5 shows the layout ofsuch optics. Important optical components among them are introduced in thefollowing.
Two free-space laser modules are fixed on the table. They are commercial
products named Prometheus from Coherent Inc., which are based on non-planarring oscillator configurations and have a SHG setup inside so that each module
has infrared and green laser outputs. The power of the infrared output is about
1 W while that of the green output is about 100 mW. For detection of beat notes
between the PSL and the auxiliary lasers, we employed commercial broadband
photodetector units with silicon photodiode. The reason why we chose silicon
photodiode, which has fast response but relatively low responsivity for 1064 nm(compared to InGaAs, for example) is that there is a lot of optical power on
the photodiodes thanks to the new configuration of ALS.‡5 About the use of theAOMs, it is worth noting that the “minus” first order diﬀraction is chosen for
the X arm beam while the “plus” first order diﬀraction is used for the Y arm
beam; here we defined the terminologies of the “plus” and “minus” first orderdiﬀractions in such a way that the “plus” one has higher frequency than that‡5

In the ALS of Advanced LIGO, the optical power of the green beat notes are limited for
two reasons. One is that the power of the green lasers produced by the SHG setup is not high.
The other is that the auxiliary green lasers for the beat notes are the transmission of one of
the arm cavities

79

80

Chap.7 KAGRA-Type ALS

Figure 7.4: Picture of the PSL table focusing on the ALS part. Yellow-coloredjacketed fibers can be found in the middle.
15

Pick-off from the PSL
20

25

30

35

Laser module (X)40

PLL PD X
8-32

M4

45

PLL
PD Y

Laser module (Y)

50

8-32

M4

55

130

135

140

145

150

155

160

165

170

175

180

185

190

Figure 7.5: Layout of the optics for the ALS system on the PSL table.

60

7.3. Implementation

81

Optical Fibers
with Protections

Laser Room

Cable Rack

Figure 7.6: Picture of how the optical fibers are laid outside the laser room
enclosure. The tubes that have the fibers in them are laid on cable racks andthe walls of the laser room.

of the input beam while “minus” one has lower frequency. This selection leads
to opposite eﬃciencies of the frequency actuations. Thanks to this feature, thepositive bias voltages applied to the VCOs are compensated with each other atthe very first stage of the analogue ALS CARM servo. We laid a pair of opticalfibers for green laser for each arm.‡6 Each fiber has length of about 60 m. Bothends of the fiber are end-capped to avoid damage of the surfaces. The bare fiber
is protected by protective outer jacket, which is further wrapped in corrugatedtubes together with shock absorbers outside the laser room or the areas aroundPOP or POS table (Figures 7.4 and 7.6). The jacketed optical fibers are laid
vertically along rigid pillars standing on the optical tables from the tables to ceiling cable racks. The interferometer configuration of the fiber noise cancellation
system is described in Figure 7.7. It employs a polarization Michelson interferometer controlled at mid-fringe. This interferometer locks the fluctuations of the
propagation phase of the laser in the longer arm, which includes the fiber, withrespect to that in the shorter arm.

82

Chap.7 KAGRA-Type ALS

Laser Room Enclosure

POP/POS Table
10% trans.
BS

Piezo Actuator

PBS

HWP

QWP

Optical Fiber
From
Laser Output
PBS

QWP
To
PR2/SR2

HWP

HWP: Half wave plate
QWP: Quarter wave plate
PBS: Polarizing BS

Figure 7.7: Configuration of the polarization Michelson interferometers for thecancellation of fiber noise.

POP/POS table
Figures 7.8 and 7.9 show the layout of optics on POP and POS tables, respectively. Photodetectors on the tables for the PDH signals of green are resonanttype circuits. For this purpose, we modified standard RF photodetectors of
KAGRA [157] so that the resonant frequencies are 33 MHz and 32 MHz for the
X and Y arm, respectively. We also swapped the photodiodes from InGaAs
ones to silicon ones. The last (horizontal) steering mirrors on the tables havedichroic coatings so that they reflect green beams (532 nm) but transmit infrared(1064 nm). Their mirror mounts have two piezo-motor actuators for each for theirremote alignment adjustment. On POP table, the transmission of this mirror
is used for POP port of the main interferometer. There is a pair of dichroic
mirrors to form a periscope at the edge of each table. After the periscopes,
the green beams are injected to vacuum chambers through glass viewports with
anti-reflective coatings. As shown in Figure 7.1, we have one steering mirror in
vacuum immediately behind PR2 or SR2 (Figure 7.10). These mirrors also have
two piezo-motor actuators.

‡6

Four fibers for green are laid in total.

7.3. Implementation

83

To the steering mirror
behind PR2
14

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96

Periscope
Gr L3(f=300mm)
Fiber coupler
Gr L1(f=150mm)

Gr M3

Gr M1

Harmonic Splitter
Gr BS(90% trans)
Gr DCPD

Gr M2
Gr F.I.

Gr HWP1
Gr QWP

Gr M4

Gr L2(f=100mm)

Gr HWP2

Main laser
POP port

Gr RFPD
Gr L4(f=200mm)
Gr M5

OUTPUT (BNC): 0-10 V max.

POWER CONNECTOR

0.13in 0.01

2.2mm
0.09inSM1 (1.035-40)
EXTERNAL
THREADSERIES
1.09in52.5mm2.07in1.96in[3.3mm
DETECTORAREA
SURFACE0.3]ACTIVE
OFFSET

49.9mm

27.7mmPOWER SWITCH

70.9mm2.79in2.33inDETECTOR
9.8mm
0.39in59.3mmPOWER INDICATOR LED

0.42in0.98in24.9mm0.98in8 POSITION ROTARY GAIN SWITCH

SM05 (0.535-40)
INTERNAL
THREADSERIESSM1 ADAPTER and
INCLUDED:

0.51in25.0mmSM1RR RETAINING RING
0.80in12.8mm0.98in20.3mmAND M4 x 0.7 x 6mm

1.97in50.0mm25.0mmFOR INFORMATION ONLYNOT FOR MANUFACTURING PURPOSES

22.5mm0.89inPROJECTIONDRAWINGNAME

DATE

w wAdjustableSi
w . t hMETRICo r l a b sUNIVERSAL. Gainc o m Detector

N/A

A
B
REV

24.9mm0.98in10.7mmTHREAD MOUNTS ACCEPT:
8-32 x 0.25in

UPH30/M
PDA100A2

0.05 kg
0.10

REDRAWN ©WKC
APPROVALDRAWNCOPYRIGHT
AV
CG
RH
2011 BY THORLABS
2017
15/JUN/15ITEM
08/JUN/15
08/DEC/17
MATERIAL
MATERIAL#
POST HOLDER 30.0mm
APPROX WEIGHTVALUESAND
MAY
IN CONTAIN
PARENTHESISROUNDOFFARE CALCULATEDERRORS

Figure 7.8: Layout of the optics for the ALS system on the POP table.

13

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96

Gr M3
15

Gr HWP1

Harmonic Splitter
Gr L1(f=400mm)

20

Gr L1(f=100mm)
25

Faraday Isolator
Gr BS(90% transmission)
BeamDumper for IR

30

Gr HWP2
Gr L3(f=200mm)

Gr M6
Gr QWP

Gr M2

Gr M4
Gr M5

35




Gr RFPD

40

Figure 7.9: Layout of the optics for the ALS system on the POS table.

11

Gr M1
10

10

Fiber coupler

9

Periscope (hight=98.4mm)
58

5To the steering mirror
behind SR2

7

POS

84

Chap.7 KAGRA-Type ALS(a) Behind PR2.

(b) Behind SR2.

(c) Another view behind SR2.

Figure 7.10: Pictures of in-vacuum steering mirrors behind PR2 and SR2.

7.3.3

Circuits

Phase-frequency discriminator
For demodulations of the beat notes of the PLLs, we made and used the circuit
modules of phase-frequency discriminator. This circuit outputs the voltage signal
proportional to the phase diﬀerence between the beat note and the LO if theirfrequencies are identical. The output is positive or negative constant voltage ata certain value depending on whether the frequency of the beat note is lower
or higher than the LO frequency, respectively, if their frequencies are diﬀerent.
This functionality enables us to easily bring the frequency of the auxiliary laser
close to that of the PSL when it gets far away.
RF signal generator
In the ALS system, diﬀerent commercial RF signal generators are employed forthree diﬀerent purposes. The first purpose is the LO of the PLLs. Stable signal
generators, E8663D from Keysight Technologies, are used for this purpose. Sincephase noise of the LO directly becomes sensing noise of the ALS system, so it isdesirable that the noise level of the LO is low. They
√ provide a stable RF signal
with the phase noise level as low as 1.6 × 10−4 rad/ Hz in the condition that thecarrier frequency is between 250 kHz and 250 MHz and the sideband frequency
is 1 Hz [158]. They can also sweep the output frequency in a phase-continuous

7.3. Implementation
way upon remote control commands so that the operation point of the ALS(the PSL frequency with respect to the arm cavity resonance) can be smoothly
changed without disturbing the whole control loops. The LO frequencies are set
around 40 MHz and 46 MHz for the X and Y arms, respectively. The second
purpose is the phase modulation and demodulation for the arm PDH signals.
The frequencies for the X and Y arm are chosen to be 33 MHz and 32 MHz,
respectively. The third purpose is the frequency reference for the VCOs, which
is described in the section of VCO.

Demodulator
The demodulator circuit modules employed for the green PDH signals are standard ones in KAGRA. These ones have the smaller phase delay in the highfrequency region among the two types of demodulator modules that KAGRA
has [159]. Amplification circuits of LO signals, frequency mixers, and low-passfilters for the demodulated signals are built in a single chassis. This module
outputs I- and Q-phase demodulations and the relative phase between the signal
applied to the EOM and the LO signal was adjusted so that the PDH error signal
did not leak in the Q-phase signal. The relative phase can be remotely adjusted
by a voltage-controlled phase shifter.

VCO
Phase noise of the output of the VCO circuit also directly leads to sensing noiseof the ALS system. Thus we made our own VCO circuits by simplifying thedesign of those in Advanced LIGO. Figure 7.11 shows the conceptual diagram of
the VCO. The frequency division by ten reduces phase noise of the seed VCO bya factor of ten at the cost of the reduction of the actuation range by the same
factor. For the constant frequency reference, another stable RF generator is used
for both arms in common. The output of the reference is set to 70.35 MHz so thatthe output of the VCO unit is around 80 MHz, which is the nominal operating
frequency of the AOMs. Their actuation eﬃciency and range of frequency werecharacterized. The eﬃciency is 0.42 MHz/V – 0.43 MHz/V and the range is about
1 MHz, which satisfies the requirement on the actuation range of the whole ALS
system. Phase noise of the output was also characterized. The details of thesepreliminary measurements of the VCOs are described in Appendix F.

85

86

Chap.7 KAGRA-Type ALS

96-102MHz

Vtune
Seed VCO 10Frequency
Divider

Mixer
70MHz
Reference

VCO Output
80MHz
Band-pass Filter

Figure 7.11: Conceptual diagram of the VCO circuit. The output frequency ofthe seed VCO is adjusted linearly by the input voltage labeled as Vtune . Afterthe frequency division, the frequency of the signal is then raised to the desired
value by the combination of frequency mixing and band-passing.
Universal analogue circuits
Universal analogue circuits are employed in the servos of the PLLs, PDH loops,
fiber noise cancellation loops, and CARM loop. Use of analogue circuits enables
us to have wide control bandwidths in these loops up to tens of kilohertz. The
features of them can be summarized as follows. The input and output ranges√ of−7
the circuit are ∼ ±13 V. The input equivalent noise level is ∼ 1 × 10 V/ Hz.
The targeted frequency band is below ∼ 1 MHz. The circuit has two input ports
and two output ports. Each input port has a variable gain amplifier stage anda switchable inverter. One output port is DC-coupled, namely the slow path,
while the other is AC-coupled, namely the fast path, which we utilize only for the
CARM loop. There are several switchable integrator stages in the common pathfor the purpose of gain boost in the low frequency region. The common path isthen split into the slow and fast path, where we also have a few switchable filters.
The variable gain stages and the switchable filters can be controlled remotely via
binary outputs of the real-time controllers.
Digital real-time controller
KAGRA uses digital real-time controllers that were originally developed in LIGO.
The sampling frequency of this system is 16384 Hz.‡7 The voltage signals acquiredby analogue-digital convertors are processed in real-time controllers and output
by digital-analogue converters. How the signals are processed in the controllers
is programmed by users before the programs are run. The important feature‡7
The sampling frequency can be set as high as 65536 Hz though it is usually set to 16384 Hz
or less.

7.4. Demonstration

87

here is that parameters of the programmed process such as the parameters offinite impulse response filters and matrix elements can be changed while they
are running. This feature enables us to dynamically and flexibly change theconfiguration of the control loops as the lock acquisition process proceeds. The
states of analogue circuits such as the switches and gains are also governed by
the digital controllers.
TR Y

Laser Source

Demodulator

Acousto-Optic Modulator

Phase-Frequency Discriminator

Electro-Optic Modulator
Faraday Isolator
Fiber Coupler

ETMY DARM Error

RF Signal Generator
Voltage Controlled Oscillator

Optical Fiber
Photodetector

Slow Servo
Fast (2 inputs / 2 Outputs)

CARM MCL

TR Y

~~

Servo

Y arm

CARM Fast

ITMY

+
+

PRM
REFL
PD
PLL
PD X

CARM Error

PR3
POP
PD
PDH
PD Y

SR3

PLL
PD Y

+
+
+
−

X arm

BS

PR2
PDH
PD X

TR X
Digital System

ITMX

TR X

ETMX

SR2

SRM
CARM servo
Slow
Fast

CARM Error
ALS CARM
ALS DARM Digital System

~~

AUX Y

CARM MCLCARM Error

PSL

AUX X

CARM servo

CARM Slow

Digital System

IMC

DARM servo

ALS DARMASPD

CARM Slow
CARM Fast

DARM Error

Figure 7.12: Interferometer configuration for the demonstration of lock acquisition with the KAGRA ALS. The control diagram is focused on the CARM andDARM control. The PRM and SRM were misaligned.

7.4

Demonstration

Using the ALS system that has been implemented in the main interferometer,
we have achieved lock acquisition of the FPMI of KAGRA. Figure 7.12 shows
the interferometer configuration when the demonstration of the lock acquisition

88

Chap.7 KAGRA-Type ALS
scheme with the KAGRA ALS was performed. We chose the FPMI configuration instead of the DRFPMI because the control of the DRMI (the central
interferometer of the DRFPMI) was unstable due to the failure of polarization
property of the ITMs [160, 161]; the birefringence of the ITMs due to inhomogeneity of the sapphire crystals was too large for the control of the DRMI to bestably held. Thus we simplified the central interferometer to the short Michelsoninterferometer for the demonstration purpose. The DARM control was closed
by actuating only ETMY in order to simplify the control topology; actuating
both ETMs might introduce instabilities in the control loop due to undesirable
competition of the motion of two ETMs. The fiber noise cancellation loops were
engaged to improve the noise performance of the ALS system.
In order to demonstrate, with the FPMI configuration, that full interferometer‡8 lock is achievable by utilizing the present ALS system, we performed lockacquisition of the FPMI using the following two schemes. In the first scheme,
the CARM and DARM controls were directly handed over from the ALS signals
to the main laser PDH signals. In the second schemes, the CARM (or the PSL
frequency) is controlled using the optical power of the transmitted beam of thearm cavities as the error signal in the intermediate steps as Advanced LIGO
does. The purposes of the demonstration of each scheme are the following.
The demonstration of first scheme shows that the the ALS CARM and ALSDARM controls satisfied the requirement that the fluctuations of the main laser
frequency with respect to the resonant frequency of the arm cavities are suppressed within the linewidth of the arms. It is worth mentioning that the
linewidths of the CARM and DARM of our demonstration setup are, to ourbest knowledge, the narrowest ever in the arm cavities whose lock acquisition isachieved purely with ALS.
The achievement of lock with the second scheme demonstrates the processof the CARM hand-over from the ALS signal to the transmission signal. As the
final step of the CARM lock process of the DRFPMI, which is the hand-over fromthe transmission signal to the PDH signal at REFL port, has been establishedin Advanced LIGO [22], the demonstration of the second scheme shows that
the ALS system is ready for locking the DRFPMI. Once the hand-over fromthe ALS signal to the transmission signal is demonstrated, the successive locking
procedure of the DRFPMI in Advanced LIGO, which is irrelevant to ALS, should
be applicable also to KAGRA. For DARM locking, since the SR cavity widens the
linewidth of DARM in contrast, lock acquisition of DARM should be achievablealso in the DRFPMI.
‡8DRFPMI.

7.4. Demonstration

7.4.1

Interferometer setup

The interferometer setup when the demonstrations were performed can be summarized as follows. The main laser power after the IMC was approximately 4 W.
The power after the laser transmitted through the PRM was approximately
0.4 W since the transmission of the PRM was approximately 10 %. The powerof each green laser injected to the arm was about 10 mW. The temperature of
ETMX, ITMX, ETMY, and ITMY was 179 K, 174 K, 259 K, and 293 K, sinceeach cryostats housing the test mass was in preparation status for the cool-downof the mirror. The Mach–Zehnder interferometers for the modulation system of
the PSL was not employed at that time since they could not satisfy the noise
requirement. Thus non-resonant amplitude modulation sidebands of f3 could
not be used, and the third harmonic demodulation technique was used instead.
Since the OMC was not ready, the lock acquisition sequences were demonstrated
up to a stage where DARM was sensed by RF readout with the PDH signal at
AS port instead of DC readout with the OMC transmission signal. However, thetransition from RF readout to DC readout is an independent process that has
been established, thus it is of no relevance with demonstrating the performanceof the ALS system in the lock acquisition processes.

7.4.2

The first scheme: direct transition from the ALS
systemFigure 7.13 shows how the optical power of the main infrared or auxiliary green
lasers at various detection ports evolved in time along with the steps taken
throughout the lock acquisition process. Here, each step of the first lock acquisition scheme is explained.
(I) The resonances of both arm cavities only for the green auxiliary lasers wereachieved by feeding back the PDH error signals to the VCOs.
(II) The ALS CARM control was engaged with the full frequency bandwidth
using both slow and fast outputs of the CARM servo. The crossover frequency of the slow and fast loops was set to around 80 Hz. The overall
control bandwidth was about 2 kHz at this point.
(III) In this period, the ALS DARM control was gradually engaged by feedingback the ALS DARM signal to the longitudinal motion of ETMY. The
feedback force was applied to the test mass stage in the high frequency
region and the marionette stage in the low frequency region. The cross

89

90

Chap.7 KAGRA-Type ALS
over frequency of the two actuation loops was set to around 1 Hz. The
overall control bandwidth was about 30 Hz at this stage. From this step,
the ALS system controlled the diﬀerence between the main laser frequency
and the resonant frequency of each arm so that the main laser no longer
stochastically resonated in the arms.
(IV) In this period, the resonance points of both arms were searched by sweeping
the LO frequencies of the PLLs. At the end of this period, the two arms
were held at the resonance points of the main laser.
(V) The control loop of the DoF of the central interferometer, MICH in thiscase, was engaged by using the third harmonic demodulation signal of f1 at
REFL port in Q-phase (REFL51Q) as the error signal. The MICH control
signal was fed back to the BS.
(VI) In this period, the DARM control was handed over from the ALS system
to the main laser PDH signal. The error signal used here was the PDH
signal of f1 at AS port in Q-phase normalized by the transmitted power of
the arm (AS17Q/TRY), which have broader linear range than the ordinary
PDH signal by a factor of a few. The use of the normalized one helped
to significantly reduce the rate of hand-over failures due to the associatedtransients. After the hand-over, the control bandwidth was successively
increased by a factor of 1.5 and two stages of a low frequency boost filter
were enabled.
(VII) The CARM control was handed over from the ALS system to the main
laser PDH signal of f2 at REFPl port in I-phase (REFL45I), by enabling
the first input and disabling the second input of the CARM servo circuit.
The overall control bandwidth was about 5 kHz from this point.

(VIII) The error signal of the DARM control was switched from AS17Q/TRY to
the ordinary PDH signal (AS17Q).
(IX) The error signal of the MICH control was switched from REFL51Q to the
1f signal at REFL port (REFL17Q), to finish the lock acquisition sequence.
The whole process listed above was automated. We performed four times thedemonstration in the procedure described above, and succeeded every time.

7.4. Demonstration
I II

91
III

IV

V

VI VII

VIII

IX

Figure 7.13: Time series of the optical power at various detection ports as thesteps of the first lock acquisition scheme proceed. The top panel shows the main
laser power at the transmission of the arm cavities. The middle shows the green
laser power at the transmission of the arms. The bottom shows the main laser
power at AS port. The Roman numerals from I to IX denote the steps of thelock acquisition sequence.

7.4.3

The second scheme: transmission lockFigure 7.14 shows how the optical power of the main infrared or auxiliary green
lasers at various detection ports evolved in time along with the steps taken
throughout the lock acquisition process. Here, each step of the first lock acquisition scheme is explained.
(I) The resonance of the X arm cavity only for the green auxiliary laser wasachieved by feeding the PDH error signal back to the corresponding VCO.

92

Chap.7 KAGRA-Type ALS(II) The ALS CARM (frequency) control was engaged using only the IMC
length as the actuator of the PSL frequency via the digital control system.
The control bandwidth was about 100 Hz at this point.
(III) In this period, the main laser resonance of the X arm was searched bysweeping the LO frequency of the X arm PLL.
(IV) The PSL frequency control was handed over from the ALS system to thesignals from the transmission power of the X arm, which is called TR
CARM. This signal is the square root of the transmission power, which isrealized by the real-time controllers.
(V) The resonance of the Y arm cavity only for the green auxiliary laser was
similarly achieved.
(VI) In this period, the ALS DARM control was gradually engaged in the same
way as described in the previous section. In addition, the MICH control
was engaged by using REFL51Q as the error signal at the end of thisperiod.
(VII) In this period, the main laser resonance of the Y arm was searched bysweeping the LO frequency of the Y arm PLL.

(VIII) In this period, the DARM control was handed over from the ALS system tothe normalized PDH signal at AS port as described in the previous section.
(IX) In this period, the error signal of the CARM control was gradually switchedfrom TR CARM to the normalized PDH signal at REFL port, which isnamed as REFL17I/(TRX+TRY). In this process, the locking point of the
X arm gradually shifted from the middle to the top of the resonance as theerror signals were changed.
(X) In this period, the CARM error signal was switched to the ordinary PDH
signal (REFL45I) and successively the control bandwidth of CARM wasincreased to about 5 kHz by engaging the fast analogue feedback path toform the dual-loop CARM control.
(XI) The error signal of the DARM control was switched from AS17Q/TRY to
the ordinary PDH signal (AS17Q).
(XII) The error signal of the MICH control was switched from REFL51Q to the
1f signal at REFL port (REFL17Q), to finish the lock acquisition sequence.

7.4. Demonstration

93

The whole process listed above was automated. We performed three times thedemonstration in the procedure described above, and succeeded every time.

I II

III

IV V

VI

VII

VIII IXX

XI XII

Figure 7.14: Time series of the optical power at various detection ports as thesteps of the first lock acquisition scheme proceed. The top panel shows the main
laser power at the transmission of the arm cavities. The middle shows the green
laser power at the transmission of the arms. The bottom shows the main laser
power at AS port. The Roman numerals from I to XII denote the steps of thelock acquisition sequence.

Chap.7 KAGRA-Type ALS

Frequency noise [Hz/sqrt(Hz)]

94

101

RMS=4.4Hz (X arm)
RMS=4.2Hz (Y arm)

100
10 1 RMS=2.7Hz (CARM)

RMS=2.8Hz (DARM)

10 2
10 3

X arm
X arm (Cumulative RMS)
Y arm
Y arm (Cumulative RMS)
CARM
CARM (Cumulative RMS)DARM
DARM (Cumulative RMS)

10 1

100

101

102

Frequency [Hz]103

Figure 7.15: ASDs of sensing noise of the ALS system in the basis of the X andY arms, and that of ALS CARM and DARM. The red solid trace shows the ASDof ∆fres (f ; κARM ) of the X arm. The red dotted line shows its cumulative RMS.
The magenta solid trace shows the ASD of the Y arm. The magenta dotted line
shows its cumulative RMS. The black solid trace shows the ASD of ALS CARM.
The black dotted line shows its cumulative RMS. The gray solid trace shows the
ASD of ALS DARM. The gray dotted line shows its cumulative RMS.

7.5

Characterization

In order to quantitatively characterize the performance of the KAGRA ALS
system, we have conducted the following measurements.

7.5. Characterization

7.5.1

Sensing noise

The most important characteristic of ALS systems is the residual fluctuations of
the arm DoFs, ∆fres (f ; κARM ). If the control bandwidths of CARM and DARM
are suﬃciently large, which is usually the case, sensing noise of the ALS system
limits ∆fres (f ; κARM ). Therefore, we evaluated the sensing noise level of the
ALS system in two sets of bases, (X arm, Y arm) and (CARM, DARM). The
evaluation method was the following. For the evaluation on the basis of (X arm,
Y arm), the PSL frequency was locked to the resonance of one arm cavity ofinterest by using REFL45I as the error signal while the mirrors of the other arm
cavity was misaligned not to disturb the interested arm. At the same time, thefrequency of the auxiliary green laser of the interested arm was locked to theresonance of the same arm without feeding any signals back to the main laser
or test masses from the ALS system. In this configuration, sensing noise of the
ALS system of the interested arm can be inferred by measuring the control signalof the PDH loop under the assumption that residual frequency fluctuations of
the main laser with respect to the arm is small enough to be ignored for thenoise estimation. For the evaluation on the basis of (CARM, DARM), the FPMI
was locked only with the main laser signals as described in the previous section
while the frequency of each auxiliary green laser was locked to the resonance ofthe corresponding arm. In the similar way, sensing noise of the ALS system of
CARM and DARM can be obtained by measuring the ALS CARM and ALSDARM error signals, respectively. The data for the bases of X and Y arms and
ALS CARM were acquired by digital controller after they were amplified by ananalogue circuit. The measured signals were calibrated into frequency based on
the eﬃciencies of the VCOs (Appendix F). From the measurement results of theeﬃciencies, calibration error of ∼ 3 % can be expected in the estimation.
We adopted this evaluation method to suppress the nonlinear eﬀect in themain laser PDH signals, instead of a straightforward method where one measures
the ASDs of the PDH signals while the interested DoFs are controlled by the
ALS system so that the PDH signals are kept around the zero points, which
are the center of their linear range. This is because the straightforward method
may suﬀer from the nonlinear eﬀect in the PDH signals if they are not confinedsuﬃciently close to the zero points.
Figure 7.15 shows the ASDs of sensing noise of the ALS system measured in
the two diﬀerent bases, calibrated to ∆fres (f ; κARM ). The sensing noise levelsof the X arm and Y arm, and CARM and DARM are shown along with thecorresponding cumulative RMS curves. They show that the RMSs of sensing

95

96

Chap.7 KAGRA-Type ALS
noise of the X and Y arms were 4.4 Hz and 4.2 Hz, respectively. Those of ALS
CARM and ALS DARM were 2.7 Hz and 2.8 Hz, respectively. All of these RMS
values were smaller than the linewidth of the arm cavities of KAGRA, 33 Hz,
which supported the ALS system satisfied the noise requirements of KAGRA.

Frequency noise [Hz/sqrt(Hz)]RMS=14.9Hz
101
100 RMS=4.2Hz
10 1
10 2
10 3

With the fiber noise cancellation
With the fiber noise cancellation (Cumulative RMS)
Without the fiber noise cancellation
Without the fiber noise cancellation (Cumulative RMS)
Contribution from fiber phase noise

10 1

100

101

102

Frequency [Hz]103

Figure 7.16: ASDs of sensing noise of the ALS system for the Y arm with andwithout the fiber noise cancellation loop engaged. The red solid trace shows
the ASD of ∆fres (f ; κARM ) with the fiber noise cancellation loop engaged. The
red dotted line shows its cumulative RMS. The blue solid trace shows the ASD
with the fiber noise cancellation loop disabled. The blue dotted line shows its
cumulative RMS. The black solid line shows the estimated contribution fromfiber phase noise to the blue trace. The sharp peak at 17 Hz came from vibrationof the clean booths caused by air filter fan units, which then shook the fibers.

7.5. Characterization

Frequency noise [Hz/sqrt(Hz)]

101

97

RMS=2.7Hz

100

Measured
Cumulative RMS
Total
PSL related
Seismic
PLL related

Fiber phase noise
PDH loop related
Doppler
Measurement electronics
Air turbulence

10 1
10 2
10 3
10 4
10 5 1
10

100

101

102

Frequency [Hz]103

Figure 7.17: Noise budget plot of sensing noise of the ALS system for ALS
CARM. The black trace is the measured ASD of sensing noise of ALS CARM
∆fres (f ; κARM ), which is also plotted in Figure 7.15. The gray dotted trace is
its cumulative RMS. The blue dashed line shows total noise, which is quadrature sum of the contributions of all the evaluated noise sources described in thefollowing. The magenta trace shows sum of the contributions of noise sources
related to the PSL and its frequency stabilization. The brown one shows the estimated residual contribution of arm length fluctuations due to seismic motion.
The cyan one is the contribution of the noise sources associated with the PLLsof the auxiliary lasers including the free-run frequency noise of the lasers. The
green one shows the contribution of fiber phase noise including electrical noisein the cancellation loops. The light green one is the contribution of noise fromelectronics involved in the PDH loops. The orange one is the contribution of the
Doppler shifts caused by the BS, PR2, PR3, and SR3. The yellow one shows
the noise level of the electronics involved in the sensing noise measurement. The
above contributions are included in total noise. The red shaded belt shows thepossible eﬀect of air turbulence around the POP and POS table or the PSL table.

98

Chap.7 KAGRA-Type ALS
Figure 7.16 shows the ASDs of the sensing noise spectra of the Y arm ALS
with and without the fiber noise cancellation loops engaged as an example. It
also shows the estimated contributions from fiber phase noise to the sensing noise
spectrum with the cancellation loop disabled. According to these plots, the fiber
noise cancellation helped to improve the noise level in a broad frequency regionof ∼ 10 − 2000 Hz. The suppression of the noise level recorded its maximum of∼ 500 at 17 Hz and, as a result, the RMS was improved by a factor of ∼ 3.5. It
can be also said that the estimated contribution from fiber phase noise explains
well the excess in the ASD without the fiber noise cancellation. The spectrumof fiber phase noise were measured in a separate preliminary measurement byclosing the fiber noise cancellation loop. The situation was similar in the X arm.
In order to understand the sources of measured sensing noise of the ALS system, we performed a number of supplementary measurements. Figure 7.17 shows
the contributions of the various noise sources on sensing noise in the ALS CARM
signal, evaluated in such measurements. The black solid trace shows measured
sensing noise of ALS CARM, while the blue dashed line is quadrature sum of thecontributions of all the noise sources considered so far. In the frequency region
above about 0.5 Hz, there is discrepancy between the measured noise level andthe total noise level, which is discussed in Section 7.6. It is worth noting that
this discrepancy is not a large issue because the RMS of sensing noise satisfied
the requirement and noise in this frequency region did not limit the RMS. In
contrast, below about 0.5 Hz, the Doppler shift caused by the BS, PR2, PR3,
and SR3 can explain the measured noise level.
Doppler noise
When a laser beam gets reflected by a moving mirror, a Doppler shift is introduced to the laser frequency, which can be approximated in the Fourier domain
as follows.
2π2v(f )
λi4πf=x(f ),
λ∆fDoppler (f ) =

(7.4)
(7.5)

where ∆fDoppler (f ) is the amount of the Doppler shift, λ is the wavelength ofthe laser, and v(f ) and x(f ) are the velocity and displacement of the mirror,
respectively. If a mirror reflects only the main laser or the auxiliary laser, theDoppler shift caused by the mirror motion becomes a source of sensing noise of
the ALS system. In contrast, if a mirror reflects both the main laser and the7.5. Characterization

99

auxiliary laser such as PR3 in the X arm ALS, the motion of the mirror is not anoise source; according to the equation, the auxiliary laser gets a Doppler shift
exactly twice larger than that of the main laser.
As an example, we measured the transfer function from the displacementof PR2 measured by its local sensors [138] to sensing noise of the Y arm ALS,
as shown in Figure 7.18. It shows the agreement between the measured transfer
function and the theoretical line. We also performed another measurement where
the Doppler couplings from the displacement of PR3 to sensing noise of the X
arm ALS and Y arm ALS were measured for comparison. Measurement wasdone at a single frequency of 4.1 Hz, and it was observed that the coupling to
the X arm ALS was suppressed by 20.3 dB than that to the Y arm ALS, which
supports the expectation that the Doppler shift is not a noise source for a mirrorthat reflects both the main and auxiliary lasers.

Doppler coupling [Hz/m]

108

107

100

Frequency [Hz]MeasuredModel
101

Figure 7.18: Doppler coupling of the longitudinal motion of PR2 to sensing noiseof the Y arm ALS. The red plus marks show the measured transfer function
from longitudinal motion of PR2 to sensing noise of the Y arm ALS. The data
points above 1 Hz were obtained by an active measurement where the longitudinalmotion of PR2 was excited by sinusoidal signals, while the other data points wereobtained in a passive measurement where there were no intentional excitations.
The blue line shows the theoretical Doppler frequency shift imposed on the main
laser (Equation 7.5).

Chap.7 KAGRA-Type ALS

Actuator output of ETMY TM [count/sqrt(Hz)]

100

105

Saturation

104

103

102

101
10 1

With fiber noise cancellation
With fiber noise cancellation (Cumulative RMS)
Without fiber noise cancellation (expected)
Without fibr noise cancellation (Cumulative RMS)

100

101

Frequency [Hz]

102

103

Figure 7.19: ASD of the control signal of ALS DARM applied to the actuators
at TM stage of ETMY. The red solid trace shows the measured ASD of theoutput of the ETMY TM actuators from the digital system. The red dottedline is its cumulative RMS. The blue solid trace is the expected ASD if the fiber
cancellation loops were disabled. The blue dotted line is its cumulative RMS.
The black dashed line shows the level at which the actuators saturate.

Assuming the coupling mechanism described above, the contributions of thelongitudinal motions of the BS, PR2, PR3 and SR3 were evaluated by utilizing
their local sensors at the mirror stages. The evaluated contributions are shownin the noise budget plot, Figure 7.17.
This coupling mechanism was first mentioned and discussed in the literature[22, 162]. They treated the coupling mechanism as nonlinear,‡9 but it should
be noted here that this mechanism is linear as long as the associated sensing islinear, which is the case either for KAGRA or Advanced LIGO.
‡9
The fact that they called this noise fringe wrapping also indicates that they considered
this mechanism as nonlinear.

7.6. Discussion

7.5.2

101

Actuator saturation

Figure 7.19 shows the ASD of the control signal of ALS DARM applied to theactuators at the TM stage of ETMY in the step (III) of the first lock acquisition
scheme. It can be seen that the ASD was limited by sensing noise of the ALS
system above about 4 Hz and its RMS was also limited by it. The RMS of
the control signal was smaller than the full range of the actuators by a factorof ∼ 8, which was small enough to avoid saturation of the actuators during
the lock acquisition process. It also shows the estimated ASD of the control
signal if the fiber noise cancellation loops are disabled. The estimation was
done by comparing the sensing noise levels of ALS DARM with and without
the cancellation loops. It can be seen that the RMS would be close to thesaturation level without the cancellation loops. This comparison indicates thatthe cancellation helped to have the DARM control bandwidth of about 30 Hz
without saturating the actuators of the test mass.

7.6

Discussion

The demonstration and characterization of the ALS system were performed in
the FPMI configuration instead of the DRFPMI configuration, as described in
the previous section. Here let us estimate the noise performance in the DRFPMI
of KAGRA in near future, which makes the CARM linewidth even narrower
due to the power recycling while that of DARM is widened. Thus the following
discussion is focused on CARM.
Figure 7.20 shows the measured ASD of sensing noise of the ALS systemcalibrated to ∆fres (f ; κCARM ) along with the designed ASD. The RMS expectedfrom the measured ASD is 2.2 Hz, which is larger than the full linewidth of
CARM, 1.6 Hz. Therefore, improvement in the noise performance is required tosatisfy the ambitious target so that the direct hand-over of the CARM control
is realized also in the DRFPMI.
Here, let us discuss the discrepancy between the designed and measured noise
performance, with a focus on the ambitious noise target. After the expected noise
sources are overviewed (Section 7.6.1), then unexpected noise sources, which are
Doppler noise (Section 7.6.2) and air turbulence (Section 7.6.3), are discussed.

102

Chap.7 KAGRA-Type ALS

Frequency noise [Hz/sqrt(Hz)]

101
100

RMS=2.2Hz (CARM)

10 1
10 2
10 3
10 4
10 1

CARM
CARM (Cumulative RMS)
Design

100

101

102

Frequency [Hz]103

Figure 7.20: Expected ASD of sensing noise of the ALS system calibrated to∆fres (f ; κCARM ) in the DRFPMI, along with the designed noise of the system.
The black solid trace shows the measured ASD of the sensing noise. The black
dotted line is its cumulative RMS. The blue dashed line is the simulated noiselevel in the design study, which is also shown in Figure 7.2.

7.6.1

Expected noise sources

Let us first review the noise sources that was expected in advance. Table 7.4
summarizes the comparison between the noise levels of each component expected
at the time of the design study and those evaluated after the implementation of
the components.
For phase noise from the optical fibers, it turned out that the noise level
had been underestimated. However, the contribution from fiber phase noise was
mitigated by engaging the fiber noise cancellation loops. The fiber phase noise
is discussed below in more detail. For VCO noise, the achieved noise level was
better than the conservative expectation. For arm length fluctuations, though
we could not measure the fluctuations separately from the frequency fluctuations
around 1 Hz, the upper bound shown in the table is suﬃciently low for CARM;
the noise design expects that the contribution from arm length fluctuations ismore than 1000 times smaller than the total noise level at 1 Hz. For the frequency
fluctuations of the auxiliary lasers, though the achieved level was slightly higher

7.6. Discussion

103

Table 7.4: Comparison of the expected and achieved noise levels. The ASDs at
1 Hz are shown.
Noise source
Optical fiber
VCO
Arm length fluctuations
Free-run frequency noiseof the auxiliary laser
PSL frequency noise

Expected
Achieved√√
0.01 Hz/√ Hz
0.05 Hz/√ Hz (Figure 7.16)
0.5 Hz/ Hz
0.1 Hz/ Hz (Appendix
F)
√
√
−9
−7
3 × 10 m/ Hz ≤ 1 × 10 m/ Hz [163]√√1 × 105 Hz/ Hz
3 × 105 Hz/ Hz [164]√√1 × 104 Hz/ Hz
1 × 104 Hz/ Hz [163]than the expected level only around 1 Hz, it did not matter because their eﬀect
should be suﬃciently suppressed by the PLLs as shown in Figure 7.17. For
frequency noise of the main laser (PSL), the level was almost same as expected.
To summarize the above arguments, we can say that the expected noise
sources did not degrade the designed noise performance. The fiber phase noise
would have degraded the performance without the cancellation, but the fiber
noise cancellation loops mitigated the contribution from this noise source.
Fiber phase noise
In the design study, it was expected that the contribution of fiber phase noise
limited the overall noise level only below 0.4 Hz without the cancellation technique (Figure 7.2). However, it turned out that fiber phase noise limited theoverall measured noise level in ∼ 10 − 2000 Hz without the cancellation (Figure7.16). This indicates that the the magnitude of fiber phase noise was significantly
underestimated in the design study. As is mentioned in 7.2.2 (and Appendix F),
the level of fiber phase noise in the KAGRA site was estimated based on a measurement of phase noise of an 5-meter long optical fiber that was gently placedon an optical table. Thus this discrepancy between the estimation and measurement of the fiber noise contribution is probably due to the diﬀerent environments
between the fiber testing setup and the KAGRA site. It indicates that the protection of the laid fibers from sounds or vibrations (Section 7.3.2) was not enoughin terms of the phase noise.
Although the magnitude of fiber phase noise was significantly larger thanthe expectation, it turned out that this noise was not a critical issue; the fiber
noise cancellation loops successfully mitigated the contributions of this noise. In
fact, the cancellation loops were eﬀective in terms of the residual RMS of the104

Chap.7 KAGRA-Type ALS

ALS control of the arm DoFs and the range of the actuators of the test mass,
as discussed in the previous section. With the cancellation loops engaged, thenumber of the optical sensing involved in the whole ALS system is six insteadof four (Table 7.1). Thus it should be admitted that the number of the opticalsensors is no longer an advantage of our configuration to that of Advanced LIGO.
We employed mirrors with piezoelectric transducers as actuators of the fiber
noise cancellation loops. There was a minor issue that actuation range of thetransducers limited the lock stretch of these loops (typically 30 − 60 minutes).
The drift of the fiber length caused the actuator saturation. In order to solve
this issue, a possible approach is to use the combination of AOMs and VCOs
as the actuators instead.‡10 Since it can actuate the frequency of the laser as
opposed to the phase, the actuation range at DC in terms of the optical phase
is eﬀectively infinite. The comparison and characterization of various schemes ofthe fiber noise cancellation, including the possible implementation to KAGRA
ALS in the scope, are on-going as an independent research project [165].

7.6.2

Doppler noise

It turned out that Doppler noise limits the overall performance especially in the
low frequency region. As can be seen from Figure 7.20, the measured noise level
was significantly higher than the designed one in ∼ 0.2 − 1 Hz, where the RMS
was limited. Since the limiting noise source in this frequency region was Doppler
noise caused by the suspended mirrors, it is likely that, by improving the Doppler
noise coupling by a factor of a few, the ambitious noise target can be satisfied.
In order to deal with the Doppler noise coupling, it is necessary to reduce themotion of the mirrors or to sense the motion for online noise subtraction. Since
the reduction of the motion of the mirrors is usually done by active controls, it isimportant to have good sensors for the BS, PR2, PR3, and SR3 in either case. It
would be also useful to have such sensors for the steering mirrors behind PR2 or
SR2 placed on the suspended breadboards. If we√ prepare and utilize sensors for−7them with the sensitivity level of ∼ 3×10 m/ Hz around 0.5 Hz, improvementin the Doppler noise coupling can be expected by a factor of five, which should
be suﬃcient.
‡10
In this case, the readout scheme of the interferometers for the cancellation needs to bechanged of course.

7.6. Discussion

7.6.3

105

Air turbulence

A candidate of the noise source that could explain the discrepancy in the characterization of sensing noise of the ALS system (Figure 7.17) is turbulence of air. It
has been reported that air turbulence introduces phase fluctuations in laser fields
that propagate in air [166,167]. Since the auxiliary laser and the main laser takediﬀerent paths in air, the air turbulence independently introduces phase fluctuations to the two laser beams, which leads to a noise source of the ALS system.
Let us estimate its eﬀect on the noise performance of the ALS in the following.
According to the Kolmogorov’s theory, the PSD of phase fluctuations causedby air turbulence Sϕ (f ) scales as Sϕ (f ) ∝ LV 5/3 f −8/3 [166, 167], while an experiment reported that Sϕ (f ) ∝ f −7/3 [167]. Here, L is the optical path length and
V is a constant wind speed. Therefore, the level of phase fluctuations in our casecan be estimated by knowing L and V . L was about 10 m for our case. However,
it is not obvious how we can estimate typical V in the beam paths.
It has been also known that angular jitter of the beam is caused by the air
turbulence and its PSD Sα (f ) is related to Sϕ (f ) bySα (f ) = (λ/V )2 Sϕ (f )(7.6)

in low frequency region, where λ is the wavelength of the laser beam [167, 168].
The PSD of angular jitter has a cut-oﬀ frequency of approximately 0.3V /D,
where D is the diameter of the beam [168]. Therefore, the eﬀective wind speed V
can be estimated from the spectrum of angular jitter of the beam propagating in aclean booth in the KAGRA site. In the characterization of local sensors of mirrors
called optical levers, the PSD of such angular jitter had been measured [169]. In
this characterization measurement, the cut-oﬀ frequency was about 10 Hz withD ∼ 1 mm, which leads to V ∼ 3 × 10−2 mm.‡11 Let us adopt this number for
V . The magnitude of the PSD of the angular jitter was reported to be√√
Sα (f = 1 Hz) ∼ 0.4 µrad/ Hz(7.7)
with L ∼ 0.1 m. Let us also use this number to estimate the level of phasefluctuations through Equation (7.6). By combining Equations (7.6) and (7.7)‡11

In the clean booth that housed POP and POS table, filter fan units were operating on
the roof. The total volume of air per unit time by the filter fan units were 20 m3 /s and thearea of the side of the clean booth, where the air escaped to outside, was about 240 m2 [170].
The average wind speed can also be inferred from these numbers as 8 × 10−2 m/s, whichis comparable to the number we adopt. It is reasonable that this number is larger than
3 × 10−2 m/s, because the wind probably leaks outside mainly from the area far and higher
from the tables.

106

Chap.7 KAGRA-Type ALS

and Sϕ ∝ L, Sϕ in the ALS system can be inferred as√√√S∆f (f = 1 Hz) = f 2 Sϕ (f = 1 Hz) ∼ 1 × 10−1 Hz/ Hz,(7.8)

where S∆f is the corresponding PSD in terms of frequency fluctuations.
In Figure 7.17, the red shaded belt shows the possible eﬀect of the air turbulence, which is estimated from the above discussion with allowance for estimation
error of a factor of two. The area between two lines are filled. The upper line
was calculated based onv[(
u)−8/3 ()−7/3 ]√u√ff
S∆f (f ) = 1×10−1 × t2 · f 2 max,
Hz/ Hz, (7.9)
1 Hz
1 Hz
while the lower one was onv[(
u)−8/3 ()−7/3 ]√u1√f
f
S∆f (f ) = 1 × 10−1 × t f 2 min,
Hz/ Hz. (7.10)
2
1 Hz
1 Hz
Here, the error of an overall factor of two is allowed, and both of the two diﬀerent
power laws of f −8/3 and f −7/3 in the literature are considered. From this plot,
it can be said that the contribution from the air turbulence can account for thediscrepancy in the frequency range of ∼ 2 − 100 Hz.

107

Chapter 8
ALS IN THE THIRD
GENERATIONIn this chapter, after setting a possible configuration of an ALS system for the
3G detectors, the noise performance of the ALS system is numerically simulated
based on the knowledges we have gained through the development and characterization of the KAGRA ALS. The results are discussed based on the requirementson the system. The feasibility of new techniques that will be essential in lockacquisition are also discussed. The works covered by the present chapter weredone by the author of the dissertation.

8.1

Challenges

First, let us list expected challenges in lock acquisition of the 3G interferometers.
Unless another technique is developed, ALS will be used also in the 3G detectors.
The 3G detectors will employ the similar interferometer configuration to that of
the 2G detectors, but the arm length will be about 3 − 10 times longer. In
addition, the wavelength of the main laser might be changed from 1064 nm to1550 nm or even longer, for the use of silicon as a test mass. These diﬀerences
will cause the following challenges in ALS systems in the 3G detectors.
• The linewidth of CARM will be much narrower. While the finesse of thesingle arm cavity depends on the detailed design of the interferometer,‡1‡1

The detector bandwidth (the linewidth of DARM) is determined by the combination of thefinesse of the arm cavity and the reflectivity of the SRM. Thus fixing the detector bandwidth
does not fix the arm finesse. It is determined by what the maximum signal recycling gain isand how much optical power in the central interferometer can be allowed considering the heatabsorption of the ITM substrates.

108

Chap.8 ALS in the Third Generation

Table 8.1: Parameter set of the Cosmic-Explorer-like detector. The parametersin the upper row are written in the reference [8], while those in the lower row
are set here for the following discussion.
Arm length (L)Test mass material
Wavelength (λ1 )Optical power in the arm (Parm )Detector bandwidth (κDARM )Finesse of the arm (Farm )Roundtrip loss of the arm (Tloss )Transmissivity of the PRM (TPRM )Transmissivity of the SRM (TSRM )Input laser power (PPRM )40 km
Silicon1550 nm
2 MW
400 Hz519
100 ppm
3.33 %
3.50 %
400 W

the finesse of CARM is determined by roundtrip optical loss inside the arm
cavities, which will not change drastically. Therefore, it is expected thatthe linewidth of CARM will be narrowed as the arm length gets longer.
• The bandwidth of a control loop related to the arm cavity will be limitedby the FSR. As mentioned in Section 4.5.1, the frequency response exactly
becomes zero at integer multiples of the FSR. Therefore, it is challenging
to close a stable loop with the bandwidth larger than the FSR.
• The second harmonic laser could not be used for the sensing of the arm
cavity length if the substrate of the test mass is silicon. The wavelengthsof 1550 nm or around 2.0 µm are considered to be promising candidates
as the main laser wavelength if silicon is adopted. Because it has been
known that silicon is black for the laser with the wavelength of ∼ 800 nm
or ∼ 1000 nm [171], the second harmonic field of the main laser could notpropagate through the test mass.

8.2

Configuration

8.2.1

Main interferometer parameters

As a framework of the discussion on the ALS system in the 3G detectors, let us
here set parameters of the main interferometer based on the proposal of Cosmic

8.2. Configuration

109

Explorer [8]. Table 8.1 summarizes the parameters of the main interferometer.
Here, realistic values are chosen for the parameters considering technical feasibility because the detailed parameters are not yet fixed except for those necessary
for setting the detector sensitivity [8]. However, the values will not be drastically changed unless the optical loss inside the arm is significantly reduced.
Let us assume that the property of input optics, such as the configuration of
the IMC or the frequency stability of the PSL, is same as those of KAGRA for
simplicity. From the above parameter set, the FSR is 3.75 kHz. The single arm
linewidth κARM and the CARM linewidth κCARM are calculated: κARM = 3.6 Hz
and κCARM = 6.0 × 10−2 Hz.

8.2.2

ALS configuration

From its parameters, the noise requirement of the single arm ALS of the CosmicExplorer-like detector is set by the full linewidth of the arm cavity, 2κARM =7.2 Hz, while the ambitious noise target is by the full linewidth of CARM,
2κCARM = 0.12 Hz.
As shown in the previous chapter, the performance of the KAGRA-type ALS
system was proven in two senses; the FPMI of KAGRA was locked using the
ALS system, and the RMS of its sensing noise ∆fres (f ; κARM ) was as low as
about 4 Hz, which is smaller than 7.2 Hz. In addition, the design of the systemis scalable to detectors with longer arm length such as the Cosmic-Explorerlike detector. Therefore, it is reasonable to set the configuration of the ALS
system of the Cosmic-Explorer-like detector to the KAGRA-type, with someminor modifications. As the arm length becomes longer, the finesse of the arm
for the auxiliary laser is kept at the same value and the actuator eﬃciency of
the VCO is reduced by a factor of ten. Besides, considering from the expectedchallenges listed above, the necessary modifications can be raised as follows.
• Noise level in the low frequency region (≲ 1 Hz) needs to be improved.
In particular, the Doppler noise coupling and the eﬀect of air turbulence
should be improved. For the Doppler noise coupling, it would be eﬀective
to improve the sensitivity of the local sensors of the suspended mirrors. For
air turbulence, it would be eﬀective to put the optics on the POP and POS
tables in vacuum and to reduce the length of diﬀerential optical path ofthe main and auxiliary lasers on the PSL table. The eﬀect of such changes
has to be quantitatively evaluated by a numerical simulation (Section 8.3).
• The control bandwidth of the arm PDH loops of the auxiliary lasers needs

110Chap.8 ALS in the Third Generation
to be lowered below the FSR.‡2 In the ALS system of KAGRA, the bandwidth of this loop was 20 kHz. The eﬀect of the bandwidth should be
included in the simulation.
• The wavelength of the auxiliary lasers that are injected to the arms has tobe changed from a half of that of the main laser. Here let us set the wavelength of the auxiliary lasers to 1950 nm. Thus the optical configuration
around the PLLs between the main laser and the auxiliary lasers needs tobe changed, since the SHG cannot be simply utilized to lock the frequencyof the auxiliary laser with respect to the main laser frequency.

Wavelength conversion
Here the possible configurations around the PLLs are proposed, with a focus onalternatives to the SHG for connecting two distant wavelengths.
Let us first discuss the relationship between the wavelengths of the main andauxiliary lasers (λ1 and λ2 , respectively). If two lasers with the wavelengths of
λ1 and λ2 are injected to the same optical cavity having the cavity length of L,
the phase deviation from the resonance for each laser (δj ) is expressed as()
4πL ∆λj∆Lδj =−(j = 1, 2).
(8.1)
λj
λj
L
Therefore, in order to sense δ1 from δ2 , the relation
∆λ1
∆λ2
=
λ1
λ2

(8.2)

needs to hold. In the ALS systems of KAGRA and Advanced LIGO, this relation
approximately holds thanks to the SHG. The ALS CARM control actuates thefrequency of the main laser in such a way that ∆λ1 = 2∆λ2 to keep the resonanceof the auxiliary laser, i.e. δ2 = 0. As a result,()
4π∆L 2λ2
δ1 =
−1 ,
(8.3)
λ1
λ1
which sets the lower bound in the coupling of arm length fluctuation. In the caseof KAGRA in fact, (2λ2 /λ1 − 1) ∼ 4 × 10−7 considering the frequency oﬀsets‡2
Although it is not impossible to have broader bandwidth than the FSR, it is safer torestrict the bandwidth within the FSR in the following consideration

8.2. Configuration

111
To the IMC

PSL

To the IMC

PSL

frep

Phase-Frequency
Discriminator
RF Signal Generator
Servo

Frequency fceocombsourceProbe laser
for thereference cavity

SHG

To the
X arm

AUX X

Reference cavity

To theX arm

AUX X
For AUX Y

For AUX Y

Figure 8.1: Possible configurations of the control of the auxiliary laser frequency.
The left panel shows the configuration with an optical frequency comb for converting the wavelength of the main laser to that of the auxiliary laser. The rightpanel employs a dichroic reference cavity for the wavelength conversion.
introduced in the PLLs and the AOMs are at most 120 MHz. This number issmall enough because, assuming ∆L ≲ 10 µm,
δ1 ≲ 2π × 8 × 10−6 ≪2π= 2π × 2 × 10−4 ,
Fcarm

(8.4)

which ensures the contribution of the arm length fluctuations in this mechanism
is much smaller than the linewidth of CARM.
Equation (8.2) tells us that an alternative to the SHG has to connect two
diﬀerent wavelengths λ1 and λ2 as if it is a ruler of wavelength that stretches likea rubber band. The first example of the alternative is to utilize a technique called
optical frequency comb [172]. The left panel of Figure 8.1 shows the possible
application of the optical frequency comb to the 3G ALS. The optical field ofa frequency comb consists of equidistant lines in the frequency domain. Eachfrequency can be expressed as
fk = fceo + kfrep(fceo < frep ).

(8.5)

fceo is called carrier envelope oﬀset frequency and frep is called the pulse repetition rate. In the configuration depicted in Figure 8.1, the optical beat note
between the main laser and the output of the frequency comb is detected andthe phase of the beat note is controlled by feeding the demodulated beat noteback to frep . At the same time, fceo is kept constant by the f -2f self-referencing
scheme [173,174]. In this way, fceo is fixed and frep varies in accordance with the112

Chap.8 ALS in the Third Generation

main laser frequency, and thus the frequency comb behaves like a rubber ruler.
A fraction of the comb around the wavelength of λ2 is guided so that it interfereswith each auxiliary laser. The beat note between them is controlled so that theauxiliary laser frequency follows the frequency comb. As a whole, λ2 follows λ1
with the condition (8.2) satisfied, if the frequency of each optical beat note is not
too large.‡3 With the state-of-the-art technology, an optical frequency comb thatcovers spectrum in ∼ 1050 − 2100 nm with the frequency stability of 10−16 in√1 s
is commercially available [175]. This stability corresponds to ∼ 3×10−2 Hz/ Hz
at 1 Hz, which should be stable enough; whether this stability level is enoughor not is evaluated in the next section. Therefore, it is promising to apply thistechnique to the ALS system, though the research and development of applying
it is lacking for now.
Another possible scheme to scale λ1 to λ2 is to use an optical cavity. The rightpanel of Figure 8.1 shows the configuration of such a scheme. A fraction of themain laser is sent to the reference cavity, and its length is controlled so that themain laser resonates in it. At the same time, a probe laser having wavelength of
λ2 is injected to the same cavity, with the frequency of the probe laser controlled
at the resonance of the cavity. By the same mechanism that leads to Equation(8.2), the frequency of the probe laser scales as the main laser frequency varies.
The beat note between the probe laser and the auxiliary laser is used to control
the auxiliary laser frequency in the similar way. Let us next estimate the level offrequency noise introduced by this scheme. Sensing noise of the reference cavityof the two lasers becomes the noise source.√The fundamental noise source is shotnoise, whose ASD in terms of frequency, Sshot , can be written as√
T
Sshot (f ) =
2πL

√
hc3
8P0 λ
(

= 6 × 10−5T
3 · 10−4

(8.6)
)(

0.1 m
L

)(

1.5 µmλ) 12 (

0.1 W
P0

) 21

√
Hz/ Hz,(8.7)

where T is the transmissivity of the front and end mirror, L is the cavity length,
λ is the wavelength of the laser, and P0 is the incident optical power. Using
the parameter set of T = 3 · 10−4 , L = 0.1 m, and P0 = 0.1 W, and considering
shot noise in the sensing by both lasers, the shot-noise-limited noise level is‡3
As the discussion around Equation (8.4) shows, it would not be an issue if the beat notefrequency is on the order of 100 MHz.

8.3. Noise estimation

113√∼ 1 × 10−4 Hz/ Hz. This number is low enough not to harm the overall noise
performance of the ALS system.
√ In fact, literature [176] reported the sensing−2
noise level of ≲ 1 × 10 Hz/ Hz at 1 Hz, which should also be stable enoughthough the sensing noise level was not limited by shot noise; if this level is enoughor not is assessed in the next section. Thus the use of a dichroic reference cavity
is also a promising candidate for the wavelength conversion.
In the above discussion, two configurations (optical frequency comb anddichroic reference cavity) are proposed and it seems thatis applicable.
√ either−2‡4
Let us assume the frequency noise level of 3 × 10 Hz/ Hz for the following
simulation of the overall noise performance to see if their noise performance will
be suﬃcient.

8.3

Noise estimation

8.3.1

Methods

In order to extrapolate the characterization of the KAGRA ALS (Section 7.5)
to the expected performance of the ALS system in the Cosmic-Explorer-like
detector, the following two methods were taken.
The first method is to construct a noise simulation model of the 3G ALS as
was done in the noise design of the KAGRA ALS (Section 7.2.2), by utilizing
the noise spectrum of each component that was measured for the noise budget(Figure 7.17). By constructing the noise model, the contributions of all theconsidered noise sources are correctly evaluated. This method is flexible in thesense that the spectrum of each noise source can be individually adjusted; there
is room for assuming improvement (or degradation) of each noise level. In thesimulation, the following conditions and assumptions were adopted.
• The bandwidth of the arm cavity PDH loops was set to 2 kHz, which issmaller than the FSR of 3.75 kHz. The bandwidth of the ALS CARM loop
was set to 1 kHz.
• An improvement by a factor of five in the Doppler noise coupling was
assumed. For this improvement, it is necessary to have local sensors forthe suspended mirrors √such as PR2 with the sensitivity better than thelevel of ∼ 3 × 10−7 m/ Hz around 0.5 Hz. This level is easily achievablewith proper wind shields for the sensors [169].
‡4√
The use of SHG achieved ≲ 1 × 10−4 Hz/ Hz around 1 Hz [177].

114

Chap.8 ALS in the Third Generation• It was assumed that the phase noise level due to air turbulence would be
reduced by a factor of three from the upper edge of the red shaded area in
Figure 7.17. As the ASD of the air-turbulence phase noise is proportional
to the square root of the optical path length in air (∼ 10 m was assumed
for the KAGRA setup), shortening the path length to ∼ 1 m would be
suﬃcient. It should be possible by putting the POP and POS optics invacuum and making the optics related to the ALS in the PSL room ascompact as possible.
• It was assumed that the frequency √noise level associated with the wave−2
length conversion was 3 × 10 Hz/ Hz, as is mentioned in the previous
section.

The obvious drawback of this method is that eﬀect of unconsidered noise sources
is ignored, which potentially leads to under-estimate of the overall noise level.
The second method is complementary to the first one. It is to simply recalibrate the measured sensing noise ASD of the KAGRA ALS to ∆fres (f ; κCARM )or ∆fres (f ; κARM ) using the 3G parameters. The advantage of this method is that
contributions of noise sources that are not covered in the noise budget can benaturally included. This method is conservative in the sense that no improvementin each component is assumed. Meanwhile, this method has two drawbacks. One
is that the eﬀect of the narrowed control bandwidth cannot be taken into account.
The other is that the contributions from some noise sources including the newscheme of the wavelength conversion will not be correctly included. However,these two can be assessed by the first method.

8.3.2

Results

Figure 8.2 shows the estimations of ∆fres (f ; κCARM ) by the first and second
method, with a focus on the ALS CARM performance. The black trace shows
the result of the second method; it was calculated in such a way that the sensing noise ASD of ALS CARM measured in KAGRA was simply calibrated to∆fres (f ; κCARM ) of the Cosmic-Explorer-like detector. The RMS of the estimated
ASD was 0.66 Hz, which is significantly larger than the CARM full linewidth of
0.12 Hz. This indicates that the ambitious target could not be achieved without
any improvements.
The blue dashed line is the total noise level inferred by the first method, whichis the quadrature sum of the contributions of all the considered noise sources thatwere numerically calculated using the noise model. The RMS of the total noise

8.3. Noise estimation

Frequency noise [Hz/sqrt(Hz)]

101

115

KAGRA
KAGRA (RMS)
Total
Total (RMS)
PSL related
Seismic
PLL related

RMS=0.66Hz

100
10 1

Fiber phase noise
PDH loop related
Wavelength conversion
ALS CARM Servo
Doppler
Air turbulence

10 2 RMS=0.11Hz
10 3
10 4
10 5
10 6
10 2

10 1

100

101

Frequency [Hz]

102

103

Figure 8.2: Expected ASDs of residual noise of CARM, ∆fres (f ; κCARM ), in the
Cosmic-Explorer-like detector with CARM controlled by the ALS system. The
black trace is simply calculated from the sensing noise level of ALS CARM measured in KAGRA (The second method). The gray dotted line is its cumulative
RMS. The blue dashed line shows the quadrature sum of the contributions of all
the considered noise sources (the first method). The blue dotted line shows its
cumulative RMS. The descriptions of other traces are same as those for Figure
7.17, except for the navy line that shows the contribution of noise associatedwith the wavelength conversion.

116

Chap.8 ALS in the Third Generation

spectrum was 0.11 Hz, which is slightly smaller than the CARM linewidth. Thus
the ambitious target can be barely achieved in the Cosmic-Explorer-like detector
by the ALS system. The noise spectrum is limited by three noise sources below∼ 10 Hz: (1) Doppler noise (orange line; 0.1 − 1 Hz), (2) air turbulence (red line;
1 − 10 Hz), and (3) electronics noise associated with the PDH loops (light green
line; 0.03 − 0.3 Hz), all of which contributed to the RMS. In particular, among
the noise source (3), VCO noise was dominant.
The eﬀect of the modification from the ALS system of KAGRA can also be
extracted. The solid navy line shows the noise contribution from the wavelength
conversion optics, which does not limit the total noise spectrum. It can beinferred that the eﬀect of the narrowed bandwidth of the arm PDH loops was
negligible by observing the following two points. One is that the structure in
the total noise spectrum around 1 kHz, which is the unity gain frequency of
the ALS CARM control loop, did not limit the RMS. This indicates that theoverall control system of the ALS can be stably closed. The other is that thecontributions of seismic noise (blown line) and PSL-related noise (magenta line)
did not limit the total spectrum, which indicates that the suppression by the
ALS control was suﬃcient in spite of the narrowed bandwidth of the control.
Therefore, we can conclude that the eﬀect of the modification on the overallnoise performance will be negligible.
Let us move our attention to the single arm performance ∆fres (f ; κARM ) witha focus on the primary noise requirement. Since it turned out that the eﬀect of themodification would be negligible, ∆fres (f ; κARM ) can be conservatively estimatedby the second method. By converting the measurement results of sensing noiseof the X and Y arm (shown in Figure 7.15), the RMS of ∆fres (f ; κARM ) for the
3G ALS was obtained to be 1.9 Hz and 1.7 Hz, respectively (Figure 8.3). These
numbers are smaller than the full linewidth of the single arm, 7.2 Hz. Therefore, it is expected, without any improvements assumed, that the primary noise
requirement will be satisfied by the KAGRA-type ALS system in the CosmicExplorer-like detector.
The obtained results can be summarized as follows. The primary noise requirement of the ALS system in the Cosmic-Explorer-like detector can be satisfied by simply scaling up the KAGRA ALS with the minimum modifications:to change the wavelength conversion schemes, and to narrow the control bandwidth of the arm PDH loops. On the contrary, the estimation from the simple
scaling did not reach the ambitious noise target. With modest improvement in
the Doppler noise coupling and the air turbulence assumed, the simulation told
us that the ambitious noise target can be barely achieved.

8.4. Discussion: sub-carrier CARM locking as common path ALS

117

Frequency noise [Hz/sqrt(Hz)]101 RMS=1.9Hz (X arm)
100

RMS=1.7Hz (Y arm)

10 1
10 2
10 3
4
10 10
2

X arm
Y arm
X arm (RMS)
Y arm (RMS)
100
10 1

101

102

Frequency [Hz]103

Figure 8.3: Expected noise performance of the single arm ALS in the CosimicExplorer-like detector (the second method). The measured ASDs of sensing noiseof the single arm ALS in the KAGRA were simply converted by using the 3Gparameters.

8.4

Discussion: sub-carrier CARM locking ascommon path ALS

According to the estimation results of the 3G ALS performance, the residual
noise level of the arm DoFs controlled by the KAGRA-type system will be sosmall that the PSL frequency stays within the linewidth of a single arm cavity.
Therefore, the control hand-over of DARM will be feasible, since the DARM
linewidth is larger than that of the single arm. However, the direct hand-over of
CARM might be challenging; the ambitious noise target can be slightly cleared
even if the noise improvements are assumed.
In case that the direct hand-over is not feasible, the transmitted power of thearm cavities will be used as the CARM error signal in intermediate steps of the
CARM hand-over. However, it is not trivial if the similar scheme is applicableto the 3G interferometers for the following reason. Since this scheme requires
communication between digital controllers‡5 in the end and corner stations, the‡5This scheme is most likely to be realized by digital controllers, as one needs to calculate

118

Chap.8 ALS in the Third Generation(I)

At the step (II)

~~~

~~~~

SC
f1, f2Carrier
SC ¹ f4
f1, f2
3f1, 3f2
f4

(II)

(III)

CARM: (ALS CARM à) REFL SC à REFL1f
DARM: AS 1f
PRCL: REFL 3f
SRCL: REFL 3f
MICH REFL 3f

CARM resonanceFor sub-carrier
For carrier(III)
Carrier Sub-carrier

Figure 8.4: Conceptual diagram of the hand-over of the CARM control as thelast step of lock acquisition. “SC” stands for sub-carrier.
sampling frequency of the transmission CARM will be limited by ∼ 3 kHz‡6 considering time delay due to the arm length (c.f. the FSR is 3.75 kHz). Therefore,
the control bandwidth using this error signal will be narrower than that in the
2G detectors, which might result in lack of suppression gain for CARM. Here,apart from the conventional schemes, let us discuss a reliable locking scheme for
CARM in the 3G detectors having ultra-narrow linewidth.
As mentioned in Section 5.3, an advantage of a conventional ALS scheme
is closely related to an possible disadvantage in the following sense. Use ofa diﬀerent wavelength from the main laser enables us to sense the arm DoF
independently and separately, by utilizing dichroic mirrors. However, at thesame time, this feature makes it practically impossible that the main laser andthe auxiliary laser propagate on the identical (or common) optical path. Dueto this fact, a conventional ALS scheme suﬀers from the Doppler noise coupling
and air turbulence, which will be problematic in locking CARM. Considering
the above trade-oﬀ in a conventional ALS scheme, it would be helpful to havean auxiliary laser propagating on the common path for the purpose of CARM
locking. Such a laser field can be regarded as a sub-carrier field. Researches
discussed use of a sub-carrier field in the context of the control of the central
interferometer [178–180]. Let us reconsider it as an extension of ALS.
Figure 8.4 shows a possible usage of the sub-carrier field in the DRFPMI withthe square root of the transmitted power.
‡6
In the digital system commonly used in our field, sampling frequency is limited to 2n Hz
with n being integer. With this constraint, the sampling frequency would be limited by 2048 Hz.

8.4. Discussion: sub-carrier CARM locking as common path ALS

119

the KAGRA-type ALS and sub-carrier field for lock acquisition. The outline ofthe proposed lock sequence is the following.
(I) While CARM is controlled by the ALS signals, the other DoFs are held by
the main laser PDH signals, including the third harmonic demodulation
signals. The PSL frequency is set slightly oﬀ from the CARM resonanceso that the central DoFs are not disturbed.
(II) The CARM control is handed over from the ALS signal to the sub-carrier
PDH signal (REFL f4 ). The detail is described later.
(III) The operation point of the CARM control by the sub-carrier signal is gradually shifted to the CARM resonance for the carrier, and then the control
is handed over to the carrier PDH signal at REFL port to complete CARM
locking.
Figure 8.5 shows the schematic diagram of the interferometer and control configuration of the 3G detector focusing on the proposed lock acquisition scheme.
The main laser, auxiliary lasers, and sub-carrier field are shown here, and all the
associated control loops are also shown.
The frequency of the sub-carrier field with respect to the carrier field, fsc ,should be chosen so that the sub-carrier field is close to the resonance of the
PR cavity by itself when the carrier field satisfies the anti-resonant conditionof it. By setting this condition, the CARM linewidth for the sub-carrier field
is broadened by the PR cavity,‡7 which will make the hand-over from the ALS
control significantly easier. The frequency, fsc , should also be finely tuned so that
it is close to an integer multiple of the FSR of the arm cavity. More practically,
the frequency, fsc , should also be an integer multiple of the FSR of the IMC.
It is possible to find such a frequency. In fact, assuming that the parametersof the central interferometer is same as those of KAGRA and the arm length isshortened by 174 mm from 40 km, fsc = 225.077662 MHz ∼ 5 · f2 is an example
of such a frequency oﬀset from the carrier field; if fsc ∼ (2k + 1) · f2 , where k is
an integer, the sub-carrier field satisfies the resonant condition of the PR cavity.
Figure 8.6 shows the structure of the resonances in the CARM cavity.
Figure 8.7 shows a possible optical layout on the PSL table for generating the
sub-carrier field.‡8 Phase-modulation at f4 is separately applied to the sub-carrier‡7

In the same way as the SR cavity broadens the DARM linewidth for the carrier field.
It should be beneficial to recombine the sub-carrier field before the IMC, since the IMC
reduces the beam jitter (including higher order modes) of the sub-carrier field.
‡8

120

Chap.8 ALS in the Third Generation
Laser Source

Demodulator

Acousto-Optic Modulator

Phase-Frequency Discriminator

Electro-Optic Modulator
Faraday Isolator
Fiber Coupler

−1

ETMY

RF Signal Generator
Voltage Controlled Oscillator

Optical Fiber~~ Y arm

Servo

Photodetector

Slow Servo
Fast (2 inputs / 2 Outputs)

CARM Slow

DARM Error
ALS DARM

DARM servo

Digital System

IMC

CARM Fast

Wavelength
Conversion

AUX X

AUX Y

ITMY
Carrier
Sub-carrier

PRM
REFL
PD
PLL
PD X

Slow CARM Error
Fast
PR2

PR3

PDH
PD X

+
+
+
−

X arm

BS
ITMX

PDH
PD Y

ETMX

SR2
SR3

PLL
PD Y

Digital System

~~

PSL

+
+

CARM servo
CARM Error
Slow
Fast
ALS CARM

SRM
CARM Slow
CARM Fast

ALS DARM Digital System

Figure 8.5: Schematic diagram of the interferometer and control configuration
for a reliable lock acquisition scheme. All of the main laser (red), auxiliary lasers(magenta), and sub-carrier field (purple) are incorporated here.

field for the PDH technique of CARM sensing. By setting f4 diﬀerent from f1 or
f2 (5.626946 MHz for example), the PDH signal associated with the sub-carrier
can be independently extracted at REFL port. The beat note between the carrierand sub-carrier field is detected so that its phase is kept constant. This can beunderstood as a compact and complete extension of the combination of the PLL
and the fiber noise cancellation loop in the KAGRA-type ALS system; this loop
suppresses diﬀerential phase fluctuations between the main laser (carrier field)
and the auxiliary laser (sub-carrier). Thus noise arising from the sub-carrier
generation can be naturally included in our noise simulation model. The beatnote detection can be performed on the PSL table, but it can also be done after
the IMC, which would reduce the potential diﬀerential Doppler coupling between
the carrier and sub-carrier field from the IMC.

8.4. Discussion: sub-carrier CARM locking as common path ALS
100

f1 f2

121

f2 f1

Resonance [a.u.]

10 1
10 2
10 3

Carrier
Sub-carrier (II)Sub-carrier (III)

10 4
10 5

0.2

0.0

0.2

0.4

0.6

0.8

1.0

(Carrier frequency deviation)/FSR_arm

1.2

Figure 8.6: CARM resonance of the carrier and sideband fields in terms of thecarrier frequency. The red line shows the resonance curve of the carrier field.
The purple solid line shows that of the sub-carrier field in step (II), with fsc =225.077862 MHz. The purple dashed line shows that of the sub-carrier field in
step (III), with fsc shifted by −200 Hz. The positions of the resonances of f1 andf2 sidebands are also shown for verifying that these resonances do not overlap
with any other resonances.

f4

Phase-Frequency
Discriminator
RF Signal Generator

EOM EOM

Servo

Voltage Controlled Oscillator
AOM

Laser
room
From
the PMCIMC

To the maininterferometerf 1, f 2

fsc/2

Figure 8.7: Layout of the optics for the generation of the sub-carrier field. A small
fraction of the main laser is picked oﬀ, frequency-shifted, and then recombined.

122

Chap.8 ALS in the Third Generation

Total
Total (RMS)
PSL related
Seismic
Wavelength conversion
Sub-carrier loop related

Frequency noise [Hz/sqrt(Hz)]10 1 RMS=0.012Hz
10 2
10 3
10 4
10 5
10 6
10 2

10 1

100

101

Frequency [Hz]

102

103

Figure 8.8: Noise budget plot of the sub-carrier scheme for the CARM control.
Wavelength conversion corresponds to “PLL related” in the noise budget plot of
the conventional ALS scheme. The Trace labeled as “Sub-carrier loop related”
include shot noise, photodetector noise, and electronics noise associated with the
sub-carrier sensing.
Let us estimate the noise level of the sub-carrier sensing for CARM. Since thesub-carrier sensing can be regarded as an extension of the KAGRA-type ALS,
its noise performance can be estimated by using the similar noise simulation
model. The diﬀerences from the noise budget (Figure 8.2) are that there will
be no contributions from Doppler noise, air turbulence, and VCO noise, all ofwhich are supposed to limit the overall noise performance of the 3G ALS system.
Figure 8.8 show the result of the simulation for the sub-carrier control scheme.
From the simulation result, the RMS is expected to be 1.2 × 10−2 Hz, which issignificantly smaller than the CARM linewidth. Here, the optical power of the
sub-carrier is assumed to be 100 mW.
In summary, we can expect that the use of sub-carrier will provide us with asmooth hand-over of the CARM control, since this scheme does not suﬀer from
Doppler noise or air turbulence thanks to its common path feature. In addition,
the sub-carrier CARM control can be realized with analogue circuits within thecorner station. Thus it will be helpful for realization of the 3G interferometers.

123

Chapter 9
CONCLUSION
9.1

Summary of results

A new type of the ALS system was designed for interferometer locking of KAGRA. The new ALS scheme is scalable to the 3G GW detectors, thanks to thesimplified configuration compared with the previous scheme. The study on thenoise design told us that it can satisfy the noise requirement for lock acquisitionof KAGRA that the RMS of residual frequency fluctuations of the main laseris smaller than the single arm linewidth. It was also expected that it can also
potentially satisfy the ambitious noise target that the RMS of the residual frequency fluctuations is smaller than the CARM linewidth, which will simplify thelock acquisition sequence from what has been ever achieved before.
The new ALS system was implemented in KAGRA. By using the system,
lock acquisition of the FPMI of KAGRA was achieved. Lock of the FPMI wasdemonstrated using the two diﬀerent sequences; the CARM control was directly
handed over from the ALS signal to the main laser PDH signal in the first one
while the transmission power of the arm cavity was used as the CARM error
signal in the intermediate step in the second one. The demonstration of locking
the FPMI indicated that the performance of the ALS system was good enoughfor the full operation of the KAGRA interferometer, where the CARM linewidthis even narrower.
The noise performance of the ALS system was characterized in the KAGRA
interferometer. The ASDs of sensing noise of the system were measured in thebasis of the X and Y arm, and that of CARM and DARM. Their RMSs were
4.4 Hz for the X arm, 4.2 Hz for the Y arm, 2.7 Hz for CARM, and 2.8 Hz for
DARM. All of them were smaller than the full linewidth of the single arm cavity,
33 Hz. The sensing noise ASD for CARM was intensively studied; the contributions of the possible noise sources on the overall noise performance of ALS

124

Chap.9 Conclusion

CARM were assessed, and the assessment showed that the improvement in the
Doppler noise coupling is necessary for the system to satisfy the ambitious noise
target. The eﬀect of fiber phase noise was also studied, where the fiber noise
cancellation helped the performance of the system, especially to broaden the
DARM control bandwidth without the test mass actuators saturated.
Utilizing the results of the characterization of the KAGRA ALS system, theperformance of the ALS system of a GW detector having as long arms as the
3G detectors was estimated for the first time. It was also indicated that a newscheme for converting wavelengths of lasers needs to be employed in the 3G ALS
system. The possible candidates of the scheme, use of an optical frequency comb
or a dichroic optical cavity, are raised and assessed. The estimation revealed thatthe primary noise requirement would be achieved by scaling up the KAGRA ALS
system. It was also shown that it might be possible but challenging to achieve
the ambitious noise target that the RMS of CARM fluctuations is smaller than
0.12 Hz. Thus the estimation results indicated that the lock acquisition processof 3G detectors can be achieved utilizing the KAGRA-type ALS system, but the
CARM locking process might need a help from the transmission power of thearms as is done in Advanced LIGO if the ambitious target is not satisfied. To
reinforce the CARM locking process, the use of a sub-carrier field was proposedand discussed from the viewpoint of ALS systems; this scheme can be regarded
as an extension of ALS systems that has the auxiliary laser share exactly thesame (common) path with the main laser. The noise estimation revealed thatthe proposed scheme can control the main laser frequency within the CARM
linewidth.

9.2

Future prospects

As for the ALS system installed in KAGRA, the next thing to be done is tolock the DRFPMI, which is the final configuration of KAGRA, as soon as ITMs
with good quality are ready. From the results shown in Chapter 7, it is feasible
to lock the DRFPMI with the current performance, with a help of transmission
signals as the CARM error signal. Meanwhile, by improving sensitivity of thelocal sensors of the BS, PR2, PR3, and SR3 by a factor of a few and fully utilizing
them, it can be expected that reduction of the Doppler noise coupling leads tothe improvement of the ALS system. This improvement will enable us to lockthe DRFPMI in a simplified process, where the CARM control is directly handed
over from the ALS signal to the main laser PDH signal even with the narrowed

9.3. Conclusion

125

linewidth of CARM.
Toward the realization of the 3G detectors from the viewpoint of lock acquisition, the following two things should be studied and developed. In preparation
for the use of silicon as a test mass, the first thing is to develop schemes for the
wavelength conversion, and to experimentally characterize the noise performanceof them. The use of an optical frequency comb and the use of a dichroic opticalcavity are raised as the candidates in Chapter 8. The second thing is to develop
a sub-carrier lock scheme for CARM. The existing 2G detectors can probably beutilized for the development. After a system for sub-carrier generation is developed and evaluated, it can be implemented in the 2G detectors so that the lockacquisition scheme with sub-carrier sensing will be demonstrated and evaluated,in a similar way as the ALS system was characterized in KAGRA.

9.3

Conclusion

This work demonstrated that the KAGRA-type ALS system, where the auxiliary
lasers are injected from the corner station, is an eﬀective lock acquisition schemeof arm cavities in the 2G detectors. In particular, the performance of the system was better than the requirement of a 2G detector KAGRA, which showedthat this scheme is capable of bringing the interferometer of the detector to theastrophysical observation mode. How the current performance can be improved
was also clarified.
Utilizing the experimental results on the characterization of the system, thiswork also showed that the following issues, which had not been fully considered,will be critical in lock acquisition of the 3G detectors. The issues include Doppler
noise, air turbulence, and alternatives to the SHG. By taking these issues into
account, we proposed a realistic and reliable lock acquisition scheme applicableto the 3G detectors. Thus this work has paved the way for realizing the operationof the 3G GW detectors in the astrophysical observation mode.

127

Appendix A
SPECTRAL ANALYSIS
Let us prepare a formalism to treat probabilistic fluctuations of a time-varying
physical quantity in frequency domain. Functions called power spectral density(PSD) and amplitude spectral density (ASD) are widely used. A PSD characterizes the contribution to the variance of the quantity at each frequency bin.
An ASD is given by the square root of A PSD. The unit of a PSD for a physicalquantity
whose unit is [m], for example, is [m2 /Hz], while that of the ASD is√[m/ Hz].
Let us consider the PSD of a quantity x(t).‡1 Let us assume that statisticalproperty of x does not change in time, and the average of x is zero. By using
the finite-time Fourier transform of x(t), which is given by‡2
∫ T /2xT (Ω) =dt x(t)e−iΩt ,

(A.1)

−T /2

the PSD‡3 is defined by the following equation:⟩1⟨ ††Sx (f ) := lim
xT (2πf )xT (2πf ) + xT (2πf )xT (2πf ) .
T →∞ T(A.2)

Here, ⟨·⟩ denotes the expectation value of a quantity. A PSD has the following
relation with the auto-correlation function Cx (τ ) [181]:‡1
x can either be a classical probabilistic variable or a quantum observable in the Heisenberg
picture.
‡2
If the sampling frequency is finite, the integral is replaced by a summatoin.
‡3
This is called the one-sided (single-sided) PSD, more specifically, because the interval of
integration in Equation (A.6) is one side of the frequency space.

128

Appx.A Spectral Analysis
1
Cx (τ ) = ⟨x(t)x(t + τ ) + x(t + τ )x(t)⟩,2∫∞Sx (f ) = 2dτ Cx (τ )e−2πif τ ,
∫−∞
∫ ∞
1 ∞
2πif τCx (τ ) =df Sx (f )e=df Sx (f ) cos(2πf τ ).
2 −∞
0

(A.3)
(A.4)
(A.5)

Substituting τ = 0, the relation between the variance ⟨x2 (t)⟩ and the PSD isobtained:∫ ∞2⟨x (t)⟩ =df Sx (f ).
(A.6)
0This relation shows that the PSD is the contribution to the variance of thequantity at each frequency bin.
Similarly, a cross spectral density of x(t) and y(t) is defined by⟩1⟨ †Sxy (f ) := lim
xT (2πf )yT (2πf ) + yT (2πf )x†T (2πf ) ,(A.7)
T →∞ T
which includes the definition of a PSD. The similar relation between the crossspectral density and the cross-correlation function Cxy (τ ) holds [181]:
1
Cxy (τ ) = ⟨x(t)y(t + τ ) + y(t + τ )x(t)⟩,2∫∞
Sxy (f ) = 2dτ Cxy (τ )e−2πif τ ,
∫−∞
∫ ∞
1 ∞
2πif τ
Cxy (τ ) =df Sxy (f )e=df Sxy (f ) cos(2πf τ ).
2 −∞
0

(A.8)
(A.9)
(A.10)

Equations∗
Sxy (f ) = Syx(f ),(A.11)Cxy (τ ) = Cyx (−τ )(A.12)

also hold. Equation (A.7) can be formally expressed as⟨⟩
δ(f − f ′ )Sxy (f ) = x† (2πf )y(2πf ′ ) + y(2πf ′ )x† (2πf ) ,(A.13)

wherex(Ω) := lim xT (Ω).
T →∞(A.14)

129

Appendix B
CLASSICAL CONTROLTHEORY
B.1

Linear system

Let us consider a system (Figure B.1 (a)) that has an input x(t) and an output
y(t). Let us assume that y(t) is written as∫ ∞y(t) =dτ T (τ )x(t − τ ),(B.1)
0which makes the system linear. By utilizing the Laplace transformation, which
is defined by∫∞x(s) =

dt x(t)e−st ,

(B.2)

0

Equation (B.1) leads toy(s) = T (s)x(x).
xext
x(t)

T (τ)

(B.3)
x1

+

-

y(t)
A

(a)

S
F

(b)

Figure B.1: (a) Linear system with a transfer function T . (b) Feedback control
system.

130

Appx.B Classical Control Theory

Here T (s) is called transfer function of the system. We often use angular frequency Ω (= −is) as the argument instead of s.
When T (s) is written in a form of rational function,
T (s) = kN (s)/D(s),∏N (s) =(s − zl ),

(B.4)
(B.5)

l

D(s) =∏(s − pl ),

(B.6)

m

where k is a numerical factor, zl and pl are called zeros and poles of the system.
For example, a system with a single pole at p (< 0) shows characteristic of alow-pass filter with a corner frequency of |p|. If the system has a pole whose
real part is positive, the system is unstable; arbitrary small input would lead toinfinite output [182]. This is because the inverse-Laplace transform of 1/(s − a)is eat .

B.2

Feedback loop and its stability

Let us consider a series of systems, which forms a loop (Figure B.1 (b)). From astraightforward calculation, we obtain1
xext (s),1 + G(s)G(s) = A(s)F (s)S(s),x1 (s) =

(B.7)
(B.8)

where G(s) is called the open loop transfer function. In particular, |G(s)| iscalled the open loop gain.
This system is a primitive example of a feedback control system. Let us interpret xext as external disturbances to the system, and S as the transfer function of
a sensor for x. By taking the open loop gain arbitrary large, x1 can be arbitrarysmall, which is how a feedback control works. Even if the linear range of thesensor is smaller than the range of xext , the feedback control makes it possible
for the sensor to always sense it.
Although large gain is desirable for suppressing the disturbed motion of x,
there is a limitation to the open loop transfer function coming from the stability
condition of the whole system. On the assumption that each of S, F , and A isstable, if and only if 1/(1+G) does not have a pole whose real part is positive, theB.2. Feedback loop and its stability

131

whole system is stable. It is known that there is a useful criterion to topologicallyjudge the stability, which is called the Nyquist stability criterion. Let C be an
infinitely large semicircular arc on the right side of the complex plane, and D
be the trajectory of G(s) with s ∈ C. The condition can be described as: thesystem is unstable if and if only D rotates around the point −1 on the complex
plane more than once [182] (Figure B.2 (a)). If the shape of G is simple, thestability can be judged from a Bode plot of G by looking at the phase at unity
gain frequency (Figure B.2 (b)).‡1

StableUntableStableUnstable104|Ω| = ∞Ω=0

Gain|z| = 1102
100

Phase [deg]

2
10 10
0

(a)

180
90
0
90
180

101

102
103
Frequency [Hz]

104

(b)

Figure B.2: (a) Nyquist diagrams of the open loop transfer functions G in stableand unstable cases. The blue line rotates around the point −1, while the red one
does not. (b) Bode plot of G in stable and unstable cases. The blue one crosses
the line of |G| = 1 with the phase being lower than −180 deg, while the red one
crosses with the phase being higher than −180 deg.

‡1

Unity gain frequency is the frequency at which the open loop gain is unity.

133

Appendix C
FABRY–PEROT CAVITYA Fabry–Perot cavity is an optical cavity composed of two parallel reflecting
mirrors. A light field propagates in the cavity many times and then it is amplified
inside. Here let us summarize the equations describing how a light field behavesin a Fabry–Perot cavity.
Let us calculate static field associated with a cavity shown in Figure C.1.
The field in the cavity Ecav can be calculated by an infinite series:

Ecav (t) =

∞
∑

(
)n
t1 Ei (t) r1 r2 e−iϕ

(C.1)

n=0

=

t1
Ei (t),
1 − re−iϕ(C.2)

where r = r1 r2 , and ϕ is roundtrip phase of the cavity. The reflection and thetransmission can also be calculated, by using this, as
ITM / front mirror

Ei

ETM / end mirror

Et

loss: AEcav

Er
r1, t1

r2, t2

Figure C.1: Schematic view of a Fabry–Perot cavity and associated electric fields.

134

Appx.C Fabry–Perot Cavity
t1 t2
Ei (t),
1 − re−iϕ][t21 r2 e−iϕEr (t) = −r1 +Ei (t).
1 − re−iϕEt (t) =

(C.3)
(C.4)

It is worth noting that, if we set ϕ ≡ 0 (mod 2π), r1 = r2 , and A = 0, it follows
that Et (t) = Ei (t), and Et (r) = 0. This regime is utilized for cavies called modecleaners.
From Equation (C.2), the optical power in the cavity can be calculated as
T1P0(1 − r)2 + 4r sin2 (ϕ/2)
4T1≃
P0 , (1 − R ≪ 1, |δϕ| ≪ 1),(1 − R)2 + 4δϕ2

Pin =

(C.5)
(C.6)

where R = r2 , T1 = t21 , and δϕ ≡ ϕ (mod 2π), −π < δϕ ≤ π. Figure C.2 describes
how the optical power in the cavity changes as ϕ changes. The half width at thehalf maximum of the resonance of the cavity is called cavity pole frequency fcav ,which can be expressed as
fcav :=c1−rcarcsin √ ≃(1 − R).
2πL
8πL
2 r

(C.7)

The finesse F of the cavity is defined by a ratio of the FSR (= c/2L) to the full
width at the half maximum (= 2fcav ), which is
F :=

2π
2π
√ ≃
.
1−R
4 arcsin[(1 − r)/2 r](C.8)

An average number of roundtrips in the cavity Navg can be expressed using thefinesse:Navg =∞∑n · (1 − R)Rn

(C.9)

n=0

=

R
F
≃
.
1−R
2π

(C.10)

Similarly, the average storage time of the cavity τs can be expressed as
τs ≃

2L F
.
c 2π

(C.11)

Relative power [a.u]

135

φ/2π
Figure C.2: Optical power in a cavity and roundtrip phase.

These equations show the physical meanings of finesse.
Let us then consider the dynamical property of a cavity around the resonance.
For simplicity, let us set r2 = 1 and A = 0. For time-dependent variations ofroundtrip phase around ϕ ≡ 0, the reflection field can be written as

Er (t)/Ei (t) = − [1 − iCFP δϕcav (f )] ,[CFP (Ω) =2 sinhr1/2 (1 + r)]

|ln r|(1 + iΩ/Ωcav )
2

2F
1
≃
π 1 + iΩ/Ωcav()Ω
1 − r ≪ 1,
≪ fFSR .
2π(4.22)(4.23)

Similarly, in the presence of dynamical change of input laser phase (or equivalently frequency), the reflection field can be written as

136

Appx.C Fabry–Perot Cavity
Er (t)/Ei (t) = − [1 + iDFP δϕi (f )] ,(C.12)−2iLΩ/c(1 + r)(e− 1)−2iLΩ/c
1 − re2F
12LiΩ≃−
π 1 + iΩ/Ωcav c

DFP (Ω) =()Ω
1 − r ≪ 1,
≪ fFSR .
2π(C.13)(C.14)

Here δϕcav is the deviation of the roundtrip phase inside the cavity, δϕi is phasefluctuations of the incident laser, and Ωcav = 2πfcav . Note that iΩδϕi corresponds
to (angular) frequency fluctuations of the incident laser. From these equations,a Fabry–Perot cavity has a low-pass response with the corner frequency of fcav .
Figure 4.9 shows the typical shape of the transfer function of Equation (C.13).

137

Appendix D
SIDEBAND PICTURE AND
MODULATIONLet us consider a electrical field with phase and amplitude modulations applied
to it, which can be written as
E = A(1 + βam cos(Ωm t + ϕam )) exp [iω0 t + iβpm cos(Ωm t + ϕpm )] .

(D.1)

Here, ω0 is the angular frequency of the field without modulations, Ωm is modulation frequency, and β is modulation amplitude, which we assumed to be significantly smaller than one. The expression of the field can be written in thefollowing form:
E = Aeiω0 t + Au ei(ω0 +Ωm )t + Al ei(ω0 −Ωm )t ,
)
A(
Au =
βam eiϕam + iβpm eiϕpm ,
2
)
A(
Al =
βam e−iϕam + iβpm e−iϕpm .
2

(D.2)
(D.3)
(D.4)

This expression indicates that the modulated field is equivalent to the combination of three monochromatic fields at frequencies of ω0 and ω0 ± Ωm (FigureD.1). The field having the frequency of ω0 is called carrier field, and the fieldwith ω0 ± Ωm are called upper and lower sidebands, respectively. Similarly, acombination of the upper and lower sidebands can be interpreted as a certain
combination of amplitude and phase modulations.
By employing the sideband picture for modulations, we can understand thatthe modulation frequencies for the main interferometer sensing should be set tointeger multiples of the FSR of the IMC; otherwise no modulations appear after
the IMC. The term resonant sideband extraction can also be understood with

138

Appx.D Sideband Picture and Modulation
A

Im

βpm
eiω t
0

βam

Al

Au

A
ω

Re
ω0 - Ωm

ω0

ω0 + Ωm

Figure D.1: Modulation picture and sideband picture.
the sideband picture. The GWs modulate the laser field in the arm cavities, andthis modulation can be expressed as audio-band sidebands. The SR cavity tunes
the finesse of the arm cavity for these GW sidebands. A sub-carrier field is asingle sideband so it can also be understood as a combination of amplitude andphase modulation to the carrier field.

139

Appendix E
QUANTUM STATE OF LIGHT
Let us summarize commonly used quantum states of light field for describing
laser fields in the interferometer [183, 184].

E.1

Vacuum state

Let âω be an annihilation operator of electromagnetic field in the fundamental
Hermite-Gauss mode. Quadrature operators âj (z, t), (j = 1, 2) are defined by)1 (â1 (Ω) = √ âω0 +Ω + â†ω0 −Ω ,2)1 (â2 (Ω) = √ âω0 +Ω − â†ω0 −Ω ,
∫i ∞2
)
dΩ (
âj (z, t) =âj (Ω)eiΩ(z/c−t) + â†j (Ω)e−iΩ(z/c−t) ,
2π
0

(E.1)
(E.2)
(j = 1, 2).

(E.3)

Here, ω0 can be understood as the carrier frequency while Ω as the sideband
frequency. The interpretations of the quadrature operators are given later.
The vacuum state |0⟩ is defined by the equation

This leads toâω |0⟩ = 0.

(E.4)

⟨
⟩
â†j (Ω)âk (Ω ′ ) + âk (Ω ′ )â†j (Ω) = 2πδjk δ(Ω − Ω ′ ).

(E.5)

Therefore, according to Equation (A.13), the PSDs and cross spectral densitiesof âj (z, t) in the vacuum state are written as
Sjk (f ) = δjk ,(j, k = 1, 2).

(E.6)

140

E.2

Appx.E Quantum State of Light

Coherent state

A coherent state |α⟩ is defined by applying a displacement operator D̂[α] to thevacuum state:
[∫ ∞
]
dω
†
†
|α⟩ = D̂[α] |0⟩ =: exp
α(ω)âω − α (ω)âω |0⟩ .
(E.7)
2π
0
A coherent state is an eigenstate of the annihilation operator satisfyingâω |α⟩ = α(ω)|α⟩.

(E.8)

When calculating expectation values in a coherent state, instead of applyingD̂[α] to the vacuum state, we can displace the operator Ô byÔ → D̂† [α]ÔD̂[α].

(E.9)

For example, the annihilation operator can be transformed byâω → α(ω) + âω .

(E.10)

Because this form is so simple, let us employ this convention of displacing theoperator instead of the vacuum state.
If we set α(ω) specifically byA
α(ω) = √ 2πδ(ω − ω0 ),2(E.11)

the coherent state describes the following electric field:({(
ω0 z )}
ω0 z )+ â2 (z, t) sin ω0 t −,Ê(x, t) ∝ u00 (x) [A + â1 (z, t)] cos ω0 t −cc(E.12)
where u00 (x) describes the spatial distribution of the fundamental Hermite-Gaussmode. This expression of the electric field provides the physical interpretationof the coherent state and quadrature operators. The coherent state describes a
monochromatic electric field of laser light. â1 (z, t) corresponds to quantum fluctuations of amplitude modulation to the monochromatic light, while â2 (z, t) corresponds to quantum fluctuations of phase modulation. According to Equation(E.6), the variances of quantum fluctuations of amplitude and phase modulation
are same, and they are uncorrelated.

E.3. Squeezed state

E.3

141

Squeezed state

In a similar way to a coherent state, a squeezed state is given by applying asqueezing operator Ŝ[χ] to the vacuum state:
[∫ ∞
]
dΩ
†
†
†
|χ⟩ = Ŝ[χ] |0⟩ =: exp
χ(Ω)âω0 +Ω âω0 −Ω − χ (Ω)âω0 +Ω âω0 −Ω |0⟩ .
2π
0(E.13)
Here again, let us apply Ŝ[χ] to operators instead. By setting χ(Ω) = ξΩ e−2iϕΩ ,the quadrature operators are transformed as
S † â1 S = â1 (cosh ξ + sinh ξ cos 2ϕ) − â2 sinh ξ sin 2ϕ,†

S â2 S = −â1 sinh ξ sin 2ϕ + â2 (cosh ξ − sinh ξ cos 2ϕ).

(E.14)(E.15)

This transformation can be re-arranged, by usingâζ = â1 cos ζ + â2 sin ζ,(E.16)

S † â−ϕ S = â−ϕ eξ ,(E.17)

into

†

−ξ

S â π2 −ϕ S = â π2 −ϕ e .

(E.18)

Quantum fluctuations in the direction of −ϕ are amplified by a factor of eξ ,while those in the direction of π/2 − ϕ are squeezed by a factor of eξ . Figure E.1
illustrates quantum fluctuations of quadrature operators âj (z, t) in the vacuum
or a squeezed state. Note that ϕ and ξ can be dependent on Ω. Such a case iscalled frequency-dependent squeezing.

142

Appx.E Quantum State of Light

Figure E.1: Schematic diagram of quantum fluctuations of the vacuum state (leftpanel) and a squeezed state (right).

143

Appendix F
PRELIMINARY
MEASUREMENTS
F.1

Fiber phase noise measurement

Figure F.1: Picture of the experimental setup for the preliminary measurementof fiber phase noise.
Figure F.1 shows the experimental setup for the preliminary measurementof fiber phase noise. This measurement was done on an optical table in the

Appx.F Preliminary Measurements

Freqency noise [Hz/sqrt(Hz)]144

Measured noiseElectric noise
Model
Model+Electric

100
10 1
10 2
10 3 10 1

100

102
101
Frequency [Hz]103

Figure F.2: ASD of fiber frequency noise in the preliminary measurement.

University of Toyama. A Mach–Zehnder interferometer was formed in such a
way that one path is free space, while the other path has a 5-m long opticalfiber. The Mach–Zehnder interferometer was kept at the mid-fringe simply byusing the output optical power as the error signal. The error signal was fed
back to the laser frequency to close the control. Figure F.2 shows the obtained
ASD of fiber frequency noise. Because we considered that, in the high frequency
region (f > 40 Hz), the measured ASD was limited by noise from measurement
electronics and vibrations of free-space
optics, we adopted the noise model of√
2 · 10−3 + 3 · 10−2 · (1 Hz/f )2 Hz/ Hz for the ASD of fiber frequency noise.

F.2

Measurements on the VCOs

F.2.1

Actuation eﬃciency

Figure F.3 shows the measured relations between the input voltage to the VCO
and the output frequency. When used in the ALS system, the operation pointof the VCOs were set to around 2 V. Actuation eﬃciencies around there was
0.42(1) MHz/V and 0.43(1) MHz/V for the X and Y arms, respectively.

F.2. Measurements on the VCOs

145

For X
For Y

Frequency [MHz]

81.2

81.0

80.8

80.6

80.4
0

1

2

3

Vtune [Volts]

4

5

Figure F.3: Measured VCO responses to the input voltage. The red line shows
the response of the VCO for the X arm PDH loop, while the blue one is for the
Y arm PDH loop.

VCO

Stable oscillator~ 80MHz
E8663DPFDMEASURE

VoffsetServo circuit

Figure F.4: Setup for the measurement of the VCO frequency noise.

Appx.F Preliminary Measurements

Frequency noise [Hz/sqrt(Hz)]

146

101
100
10 1
10 2 2
10

10 1

100
102
101
Frequency [Hz]

103

104

Figure F.5: Measured ASD of the output of the VCO.

F.2.2

Frequency noise of the output signal

Figure F.4 shows the experimental setup for the measurement of the output
frequency noise of the VCO. The output of the VCO was demodulated at aphase-frequency discriminator by a stable sinusoidal signal at ∼ 80 MHz. After
processed by an analogue filter, the demodulated signal was fed back to the inputof the VCO to close the PLL. From the feedback signal, the frequency fluctuationsof the VCO output were measured, with respect to the stable oscillator output.
This measurement was done for the VCO for the X arm as a representative.
Figure F.5 shows the result.

147

Appendix G
LIST OF DEVICES
Table G.1: List of devices (optics) [164].
DeviceMirror (1064 nm)
Mirror (532 nm)
HWPa (1064 nm)
HWPa (532 nm)
QWPb (532 nm)
PBSc (1064 nm)
PBSc (532 nm)
BS (532 nm)
Dichroic mirror

HBSelens (1064 nm)
lens (532 nm)
FIf (1064 nm)
FIf (532 nm)
Optical fiber
Fiber couplera

Name
PYD-10
10D20DM.11
05D20DM.11
WPH05M-1064
ZOWP-532-15mm-2-mntd
WPH10M-532
ZOWP-532-15mm-4-mntd
WPQ10M-532
PBS-20-10640
PBS-20-5320
PBSW-532
BSX10
DRX-1064/532-45P-UF-2038
TFVMQ-50.8C10-20-W1D-532/1064-ARS-30-45D
LWP-45-R532-T1064-UF-2038
N-BK7 Plano-Convex C-series
N-BK7 Plano-Convex A-series
IO-5-1064-HP
IO-5-532-HP
PM-S405-XP
PAF-X-5-A

Note
Autex
Newport
NewportThorlabs
Autex
Thorlabs
Autex
Thorlabs
Opto Sigma
Cube type, Opto Sigma
Plate type, Thorlabs
Plate type, Thorlabs
In-air, LEOd
In-vacuum, Opto Sigma
LEOd
Thorlabs
Thorlabs
Thorlabs
Thorlabs
NufernThorlabs

half-wave plate, b quarter-wave plate, c polarizing beam splitter, d LatticeElectro Optics Inc., e harmonic beam splitter, f Faraday isolator

148

Appx.G List of Devices
Table G.2: List of devices (opto-electronics) [164].
DeviceEOMg (532 nm)
AOMh (532 nm)
PDi (PLL, 1064 nm)
PDi (PDH, 532 nm)
PDi (DC, 1064/532 nm)gName
EO-T38M3-VIS
3080-125
818-BB-21A
KAGRA RFPD [157]
PDA100A
PDA100A-2

Note
CUBIC
Crystal Technology
NewportSpecial made (Figure G.1)
For fiber noise loops, Thorlabs
For fiber noise loops, Thorlabs

electro-optical modulator, h acousto-optical modulator, i photodetector

Figure G.1: Picture of special modifications on the RF PDs for the arm PDH
sensing. The modifications were needed because the pin assignment of the photodetector used in the ordinary version is diﬀerent from that in this special
version.

149
Table G.3: List of devices (opto-mechanics).
DevicePicomotor-actuated mirror mount

Name
8822
8822-UHV
PC4QR

Piezoelectric transducerNote
In air, Newport
In vacuum, Newport
Thorlabs

Table G.4: List of devices (Electronics).
DevicePFDj
Demodulator
Servo circuit
VCOk
Signal generator

Piezo driverj

NameSpecial made [185]I&Q demodulator [186]Common mode servo [187]Special madeE8663DN5181B
33611A-OCX
MDT693B

Note
High-bandwidth version
Prototype version (Figure G.2)
For PLL, Keysight
For PDH, Keysight
For VCO reference, KeysightThorlabs

phase-frequency discriminator, k voltage-controlled oscillator

GND

+5V

WenzelRF divider
LNFD2-10-100-10-1-13-15

GND

+15V

WenzelMixer
LNMX-X-X-X

Panasonic
Zener Diode 5.1V
DZ2405100L

Bypass capacitors: 10n
Murata
RCE5C1H103J1K1H03B

WenzelRF ampLNBDA-2-26-AC-AE-100-15

2 kOhmMini-Circuits
Seed VCO
ROS-102-919+ on TB-10

WenzelWenzelRF ampRF amp
LNBA-10-22-AC-100-15
LNBA-3-12-BL-100-15
Lark Engineering
Band-pass filter
MC80-10-4AA

Figure G.2: Top view of the prototype low noise VCO circuit.

151

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