Doctoral Dissertation
博士論文

Control scheme for a Fabry–Pérot typeinterferometric space gravitational wave
antenna(Fabry–Pérot 型宇宙重力波望遠鏡の制御手法に関する研究)
A Dissertation Submitted for the Degree of Doctor of Philosophy
令和元年 12 月博士（理学）申請

December 2019

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

Koji Nagano
長野 晃士

Abstract
Control scheme for a Fabry–Pérot typeinterferometric space gravitational wave antenna
by Koji Nagano
Supervisor: Takashi Uchiyama
A gravitational wave is a ripple of space time. The existence of gravitational
wave was predicted by Albert Einstein, who developed general relativity. The
first observation of gravitational waves from a black-hole binary with Advanced
Laser Interferometer Gravitational-Wave Observatory (aLIGO) started a newera of gravitational wave physics, astronomy, and cosmology.
The observation frequency of the current ground-based gravitational wave observatory, such as aLIGO, is limited between ∼ 10 Hz and a few kHz. The lower
frequency limit is mainly determined by the seismic motion. Even below 10 Hz,
some observational targets are predicted. One target is the coalescence of theintermediate-mass black holes with masses in the range between ∼ 103 M⊙ and∼ 106 M⊙ . The intermediate-mass black holes typically emit the gravitational
waves in the frequency range between ∼ 10−3 Hz and ∼ 1 Hz. The coalescenceof the intermediate-mass black holes is considered to be a candidate of the originof the super massive black hole that has been discovered at the galactic center.
Another target is the early Universe. The quantum fluctuation is considered tocause the gravitational waves in the wide frequency range, for example, between∼ 10−17 Hz and ∼ 104 Hz. The observations of the gravitational waves from thecoalescence of the intermediate-mass black holes and the early Universe provide
new physical and astronomical information. Therefore, some space gravitational
wave observatories are proposed to avoid the seismic motion.
In order to observe the gravitational waves in the frequency range between∼ 0.1 Hz and ∼ 10 Hz, DECi-hertz Interferometer Gravitational Wave Observatory (DECIGO) and its precursor mission, B-DECIGO, have been proposed.
The final goal of DECIGO is the observation of the gravitational waves from theearly Universe. DECIGO and B-DECIGO are Fabry–Pérot type space gravitational wave antennas. Their sensitivities are enhanced by a Fabry–Pérot cavity
while it requires sub-µm precision ranging for interferometer operation. Thisrequirement level is significantly more stringent than that for other type of thespace antennas such as Laser Interferometer Space Antenna (LISA). LISA utilizes an optical transponder configuration and the requirement for the ranging
precision is about 10 m. Therefore, in order to operate Fabry–Pérot type spacegravitational wave antennas, new advanced control scheme for precise ranging isnecessary. Specifically, control topology and a longitudinal sensing scheme haveto be considered.
The control topology of the Fabry–Pérot type space gravitational wave antennas should be addressed for the following reasons: First, the drag-free controlof satellites to suppress the eﬀect of dragging force is necessary. It was notclear whether the drag-free control can be engaged with all degrees-of-freedom
control of the cavity. The second reason is a lack of natural reference for theiii

control while, in ground-based detectors, the ground is a stable reference. Third,
a feedback system introduces additional noises through the feedback loop. Thus,
the control system has to be carefully designed, especially in the observation
frequency band.
In addition, for the gravitational wave observation, a sensing method, i.e. aninterferometer configuration, is also important. DECIGO and B-DECIGO utilize
dual-pass Fabry–Pérot cavities to form a triangular shaped interferometer toobtain the redundancy with minimum number of the test masses. In the dualpass Fabry–Pérot cavity, laser light is injected from both sides of one Fabry–Pérot cavity and the relative frequency of the two lasers cannot be controlled
independently. This is a new interferometer configuration for DECIGO and BDECIGO. In this interferometer configuration, since the Fabry–Pérot cavities
are coupled with each other, a new scheme to obtain the longitudinal signal of
all the interferometers is necessary. Therefore, the operation of the dual-pass
Fabry–Pérot cavity has to be demonstrated for DECIGO and B-DECIGO.
In this thesis, the control scheme of the Fabry–Pérot type space gravitational
wave antenna is studied. First, the control topology is considered with the numerical model of the Fabry–Pérot type antenna including a mechanical and optomechanical response, a sensing and actuation scheme, external disturbances, andsensing noises. The new model is named a full DECIGO interferometer model.
This new model reveals that the interferometer control and drag-free control canbe engaged at the same time by separating the controlled degrees of freedom
with each control, and the mirror position can be used as a reference of the control. Using the model, a solution of DECIGO parameters√ with interferometer
control achieving the target strain sensitivity of 10−23 / Hz is also found. In
addition,√ more than one-day stability is also achieved. With the sensitivity of−23
10/ Hz, for example, the model of the phase transition at the electroweak
scale in the early Universe can be tested. Second, the dual-pass diﬀerential
Fabry–Pérot interferometer is formulated and is constructed in a ground laboratory with the 55-cm-long Fabry–Pérot cavity. The operation of the dual-pass
diﬀerential Fabry–Pérot interferometer is demonstrated for the first time. Moreover, it is confirmed that cavity detuning can be reduced by the cavity length
adjustment as predicted by the formulation. The cavity detuning reduction isessential to minimize a laser intensity noise, which would be a major noise sourcein DECIGO. These results indicate that the dual-pass diﬀerential Fabry–Pérotinterferometer is correctly understood with the new formulation.
This work helps conduct the detailed design of DECIGO and B-DECIGO
and is an essential basis for opening the window of gravitational wave physics
and astronomy in the decihertz band, especially for the observation of the early
Universe.

iv

要旨
Fabry–Pérot 型宇宙重力波望遠鏡の制御手法に関する研究長野晃士

指導教員: 内山隆

重力波とは、時空の歪みが波となって伝わる現象である。重力波の存在は、一般相対性理論を考案した Albert Einstein によって指摘された。Advanced Laser
Interferometer Gravitational-Wave Observatory (aLIGO) による、ブラックホー
ル連星合体からの重力波の初検出により、重力波物理学、天文学、そして宇宙論の新しい時代が始まった。
現在の aLIGO などの地上重力波望遠鏡の観測帯域は、重力波の周波数で約 10 Hzから数 kHz に限られている。この低周波帯域は、主に地面振動に起因する雑音で決まっている。しかし、より低い周波数帯域でも、いくつかの観測目標が予言されている。一つの観測目標が、太陽質量の約 103-6 倍の質量を持ち、中間質量ブラック
ホールの合体である。中間質量ブラックホール合体は、典型的に約 10−3 Hz から 1
Hz の重力波を放出する。この中間質量ブラックホール合体は、銀河中心の超巨大ブラックホールを形成過程の候補であると考えられている。また、初期宇宙も観測目標の一つである。初期宇宙の量子ゆらぎは、約 10−17 Hz から 104 Hz の幅広い重力波を生成すると考えられている。これらの中間質量ブラックホール合体や、初期宇宙由来の重力波観測により、新しい物理的・天文学的知見を得ることができる。したがって、地面振動の影響を避け、より低い周波数の重力波を観測するための宇宙重力波望遠鏡が提案されている。
約 0.1-10 Hz の間の重力波を観測することを目指して、DECi-hertz Interferometer
Gravitational Wave Observatory （DECIGO）プロジェクトが提案されている。また、DECIGO の前哨ミッションとして、B-DECIGO プロジェクトも計画されている。DECIGO の最終的な科学目標は、宇宙誕生直後の姿の直接観測である。
DECIGO と B-DECIGO は、Fabry–Pérot 型宇宙重力波望遠鏡である。これらの望遠鏡では、Fabry–Pérot 共振器により最高感度は向上するが、干渉計の動作のためにサブ µm の精度の測距が要求される。この要求精度は、他の方式の宇宙重力波望遠鏡より遥かに厳しく、例えば、光トランスポンダ方式を採用している、Laser
Interferometer Space Antenna での干渉計動作に必要な測距の精度は 10 m 程度である。したがって、Fabry–Pérot 型宇宙重力波望遠鏡のためには、測距・測角のための新しい精密制御手法を開発する必要がある。特に、制御トポロジーと精密測距手法を検討する必要がある。
Fabry–Pérot 型宇宙重力波望遠鏡の制御トポロジーの開発は、以下の 3 つの理由のために必要である。1 つ目の理由は、レーザー干渉計制御と、外力雑音の影響を低減するための衛星のドラッグ・フリー制御が共存可能かどうかを確認する必要があ
るからである。ドラッグ・フリー制御と、Fabry–Pérot 共振器の全自由度制御が、同居できるかは確認する必要があった。2 つ目の理由は、制御のための自明な基準点がないためである。DECIGO のような宇宙重力波望遠鏡では、地上重力波望遠鏡のよ
うに地面を基準として制御を行うことはできない。3 つ目の理由は、制御による雑音の混入である。DECIGO の感度を落とさないような制御が設計可能かどうかを確認する必要がある。
また、重力波観測のためには、測定手法も、つまり干渉計方式も重要である。特に、DECIGO と B-DECIGO では、双方向 Fabry–Pérot 共振器と呼ばれる干渉計方v

式を用いて、三角形型の干渉計を構成する予定である。双方向 Fabry–Pérot 共振器
を用いると、最少数の鏡で冗長性を持つことができる。双方向 Fabry–Pérot 共振器
では、一つの Fabry–Pérot 共振器に両側からレーザー光を入射させる。そのため、2つのレーザー光の周波数を独立に制御する事ができない。この双方向 Fabry–Pérot
共振器は、DECIGO と B-DECIGO のための新しい干渉計方式である。この方式では、複数の Fabry–Pérot 共振器が互いに影響を与え合うため、すべての干渉計の信号を同時に得るための新しい手法が必要である。そのため、DECIGO と B-DECIGOのためには、双方向 Fabry–Pérot 共振器の動作を確認する必要がある。
本論文では、Fabry–Pérot 型宇宙重力波望遠鏡の制御に関する研究を行った。まず、Fabry–Pérot 型宇宙重力波望遠鏡の、機械的および光学機械的応答、センサーとアクチュエーター、外力雑音、そしてセンサー雑音を含む新しい数値的モデルを構築し、制御トポロジーを検討した。このモデル化により、干渉計制御とドラッグ・フ
リー制御は、制御自由度を分けることにより、共存可能であることが示された。また、宇宙重力波望遠鏡の制御においては、試験質量の位置を基準にすることで制御が可能であることも示された。また、構築されたモデルと制御トポロジーを用いて、DECIGOにおける現実的な外乱や測定の雑音を考慮しても、目標感度である 10−23√
1/ Hz というひずみ感度と 1 日以上の安定性を持つ、DECIGO の干渉計制御の成立解を確立した。この感度が達成されれば、初期宇宙の電弱相転移モデルの検証などが可能になる。次に、双方向差動 Fabry–Pérot 干渉計を定式化するとともに、実際に地上の実験室に 55 cm の基線長を持つセットアップを構築し、その動作を世界で初めて実証した。さらに、DECIGO の主要な雑音源となりうるレーザー強度雑音の原因となる、共振器の共振点からのずれが、共振器長さを調整することで低減されることを確認した。これらの結果は、今回新たに行った定式化により、双方向差動
Fabry–Pérot 干渉計の挙動を正しく理解できることを示している。
この研究は、DECIGO と B-DECIGO の詳細な設計に資するものであり、将来の0.1 Hz 帯での重力波観測、特に宇宙誕生直後の姿の直接観測にむけた重要な基礎となるものである。

vi

Contents
List of Figuresxi

List of Tables

xii

Glossary

xiii

Chapter 1

Introduction

1

Chapter 2
2.1
2.2
2.3
2.4
2.5

Gravitational wave
Linearized Einstein Equation . . . . . . . . . . . . . . . . .
Derivation of gravitational waves and their nature . . . . .
Generation of gravitational waves and their sources . . . .
Gravitational wave physics and astronomy . . . . . . . . .
Multi-band gravitational wave physics and astronomy . . .

5
5
8
10
13
15

Chapter 3
3.1
3.2
3.3
3.4
3.5

Interferometric gravitational wave detector
Overview of interferometric gravitational wave detectors . .
Michelson interferometer . . . . . . . . . . . . . . . . . . .
Fabry–Pérot cavity . . . . . . . . . . . . . . . . . . . . . .
Sensitivity of various detectors . . . . . . . . . . . . . . . .
Comparison of space gravitational wave antennas . . . . . .

17
17
18
22
30
31

Chapter 4
4.1
4.2
4.3

Space gravitational wave antenna: DECIGO and B-DECIGO
Scientific objectives . . . . . . . . . . . . . . . . . . . . . .
Interferometer design . . . . . . . . . . . . . . . . . . . . .
Satellite system design . . . . . . . . . . . . . . . . . . . . .

33
33
38
46

Chapter 5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9

Control scheme of DECIGO and B-DECIGO
Setup and basis definition . . . . . . . . . . . . . . . . . . .
Cavity and satellite . . . . . . . . . . . . . . . . . . . . . .
Sensing method . . . . . . . . . . . . . . . . . . . . . . . .
Actuation method . . . . . . . . . . . . . . . . . . . . . . .
Sources of the noise . . . . . . . . . . . . . . . . . . . . . .
Control topology . . . . . . . . . . . . . . . . . . . . . . . .
Control model construction . . . . . . . . . . . . . . . . . .
Investigation of the control system with the control model .
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .

51
52
52
55
57
60
64
69
70
77

Chapter 6

Demonstration of the dual-pass diﬀerential Fabry–Pérot inter-

vii

6.1
6.2
6.3
6.4
6.5
6.6

ferometer
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Formulation and behavior of the dual-pass diﬀerential
Fabry–Pérot interferometer . . . . . . . . . . . . . . . . . .
Experimental setup . . . . . . . . . . . . . . . . . . . . . .
Characterization of each component . . . . . . . . . . . . .
Results of the dual-pass diﬀerential Fabry–Pérot interferometer experiment . . . . . . . . . . . . . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .

81
81
82
85
87
92
95

Chapter 7
7.1
7.2
7.3
7.4

Conclusion
99
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
DECIGO and B-DECIGO design . . . . . . . . . . . . . . . 99
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Future prospect . . . . . . . . . . . . . . . . . . . . . . . . 102

Appendix A

Axion dark matter search with the Fabry–Pérot cavity

105

Appendix B

Power spectral density and feedback control

113

Appendix C

Noise sources of laser interferometer

119

Appendix D

Suspension system for ground-based DECIGO prototype

127

Appendix E

Electronic circuit

131

Appendix F

Hermite–Gaussian mode of the electric field

133

Appendix G

Antenna pattern

135

Appendix H

Full DECIGO interferometer model

143

Bibliography147

Acknowledgement

163

viii

List of Figures
2.1

Time variation of the free particle swarm motion caused by gravitational waves. . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

3.1
3.2
3.3
3.4
3.5
3.6

Schematic of Fabry–Pérot Michelson interferometer. . . . . . . .
Schematic of a Michelson interferometer . . . . . . . . . . . . .
Schematic of the Fabry–Pérot cavity . . . . . . . . . . . . . . .
Example of the transmitted power of the Fabry–Pérot cavity . .
Complex reflectivity of the Fabry–Pérot cavity . . . . . . . . . .
Sensitivity cases of interferometric gravitational wave detectors.

18
20
22
23
25
31

4.1
4.2

Pre-conceptual concept of DECIGO and B-DECIGO. . . . . . .
Sensitivity of DECIGO and B-DECIGO and their observationaltargets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two proposed concepts of the Fabry–Pérot type space gravitational wave antennas. . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of division of the DECIGO or B-DECIGO interferometer into three pairs of the arm cavities. . . . . . . . . . . . . .
Schematic of the frequency stabilizing system with double-passacousto-optic modulator scheme. . . . . . . . . . . . . . . . . .
Schematic of measuring the absolute length of the arm cavity .
Schematic of the heliocentric and geocentric orbit. . . . . . . . .

34

4.3
4.4
4.5
4.6
4.7
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15

35
40
42
43
44
48

Setup for control topology consideration . . . . . . . . . . . . . 53
Block diagram of length control. . . . . . . . . . . . . . . . . . . 65
Block diagram of alignment control. . . . . . . . . . . . . . . . . 67
Block diagram of the drag-free control. . . . . . . . . . . . . . . 68
Block diagram of the local control. . . . . . . . . . . . . . . . . 68
Open loop transfer function of frequency control loop. . . . . . 71
Open loop transfer function of x-axis motion of TM1b and TM2a. 71
Open loop transfer function of x-axis motion of TM1c. . . . . . 71
Open loop transfer function of alignment control. . . . . . . . . 71
Open loop transfer function of the drag-free control of Satellite 1. 72
Open loop transfer function of local control of TM2c. . . . . . . 72
Open loop transfer function of the global attitude control. . . . 72
Residual displacement between each test mass and the satellite
in DECIGO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Residual tilt between each test mass and the satellite in DECIGO. 73
Residual cavity fluctuation of DECIGO. . . . . . . . . . . . . . 73

ix

5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
5.25
5.26
5.27

6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
A.1
A.2

Residual cavity fluctuation of B-DECIGO. . . . . . . . . . . . .
Amount of the applied force for control of DECIGO. . . . . . .
Amount of the applied torque for control of DECIGO. . . . . .
Amount of the applied force for control of B-DECIGO. . . . . .
Amount of the applied torque for control of B-DECIGO. . . . .
DECIGO noise budget (interferometer 1) with the full control
implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
DECIGO noise budget (interferometer 2) with the full control
implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
DECIGO noise budget (interferometer 3) with the full control
implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
B-DECIGO noise budget (interferometer 1) with the full control
implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
B-DECIGO noise budget (interferometer 2) with the full control
implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
B-DECIGO noise budget (interferometer 3) with the full control
implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of DECIGO detection ranges with all noises andonly with quantum noise. . . . . . . . . . . . . . . . . . . . . . .

73
74
74
74
74
75
75
75
76
76
76
78

Relation between the cavity resonant frequency and laser frequency. 82
Schematic of the dual-pass diﬀerential Fabry–Pérot interferometer experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Photograph of the main dual-pass Fabry–Pérot cavities. . . . . 86
Block diagram of the dual-pass diﬀerential Fabry–Pérot interferometer experiment. . . . . . . . . . . . . . . . . . . . . . . . . . 87
Example of the result for cavity scanning to measure actuationeﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Enlarged plot of figure 6.5 . . . . . . . . . . . . . . . . . . . . . 89
Lock sequence of the dual-pass diﬀerential Fabry–Pérot interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Measured open loop gain of each control loop . . . . . . . . . . 93
Error and feedback signals when two cavity lengths are shifted.
94
Error and feedback signals when two cavity lengths are adjusted. 94
Measured oﬀsets of the Pound–Drever–Hall signal at the five different cavity lengths. . . . . . . . . . . . . . . . . . . . . . . . . 95
Measured calibrated noise spectra of the dual-pass diﬀerential
Fabry–Pérot interferometer . . . . . . . . . . . . . . . . . . . . . 96
Measured coherence of the dual-pass diﬀerential Fabry–Pérot interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Noise spectra with common noise subtraction. . . . . . . . . . . 97
Schematic of the experimental setup for axion dark matter searchwith a Fabry–Pérot cavity. . . . . . . . . . . . . . . . . . . . . . 108
Sensitivity comparison of the detectors with several parameter
sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

x

B.1
B.2
B.3
B.4

Block diagram of a single-input single-output system. . . . . . .
Block diagram of the feedback control system. . . . . . . . . . .
Example of the Bode plot of open loop transfer functions of stableand unstable feedback control. . . . . . . . . . . . . . . . . . . .
Block diagram of the multi-input multi-output feedback control
system and equivalent set of the single-input single-output feedback control systems. . . . . . . . . . . . . . . . . . . . . . . . .

114
116
117

118

C.1

Considered diﬀerential Fabry–Pérot interferometer. . . . . . . . 120

D.1

Schematic and CAD image of the suspension system of the prototype. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

E.1
E.2
E.3
E.4

Circuit diagram of the photodetectors. . . . . . . . . . . . . . .
Circuit diagram of the driver for the laser frequency (Koheras
BASIK X15). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Circuit diagram of the filter of a length loop. . . . . . . . . . . .
Circuit diagram of the filter of a DC boost. . . . . . . . . . . . .

G.1
G.2
G.3
G.4
G.5

Some definitions for antenna pattern calculation. . . . . . . . . 136
Antenna pattern functions for the L-shaped interferometer. . . . 138
Antenna pattern functions for triangular-shaped interferometer 1. 139
Antenna pattern functions for triangular-shaped interferometer 2. 140
Antenna pattern functions for triangular-shaped interferometer 3. 141

H.1
H.2
H.3

Top layer of the full DECIGO interferometer model. . . . . . . 144
Main system of the full DECIGO interferometer model. . . . . . 144
Plant system of the mechanical and opto-mechanical system of
the full DECIGO interferometer model. . . . . . . . . . . . . . . 145

xi

131
131
132
132

List of Tables
3.1

4.1

4.2

5.1

5.2

Summary of detector configuration and parameters. FPC,
Fabry–Pérot cavity; MI, Michelson interferometer. . . . . . . . .
Specification and requirement of DECIGO and B-DECIGO in
dual-pass diﬀerential Fabry–Pérot interferometer. Spectrum
items are defined at 0.1 Hz. . . . . . . . . . . . . . . . . . . . .
Comparison of the two concepts of the proposed Fabry–Pérot
type space gravitational wave antennas. . . . . . . . . . . . . . .
Mechanical property of test masses and satellites. In the parentheses, B-DECIGO parameters are shown if they are diﬀerent
from DECIGO one. . . . . . . . . . . . . . . . . . . . . . . . . .
Summary of control topology. . . . . . . . . . . . . . . . . . . .

30

39
41

54
70

7.1

Summary of the design of DECIGO and B-DECIGO consideredin this thesis. In the brackets, the achieved values in the previous
works are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . 100

A.1

Sets of parameters of the considered detectors. . . . . . . . . . . 110

D.1

Resonant frequencies for each mode of two suspension types. . . 128

xii

Glossary
Symbols
i
c
ℏ
e
λ
ν
ωl
ωGW
h
kB
dL
z
r
t
l, ξ
L
F
G
m, M
I
H0
M⊙
Einc
Pinc
Rµ νδσ
Rµν
R
Rc
T
Tµν
ΩGW (f )(for indices)µ, ν, δ, ...
i, j, k, ...

√imaginary unit i = −1
speed of light c = 299792458 m/s
reduced Planck constant ℏ = 1.054 × 10−34 J selementary charge e = 1.602 × 10−19 Cwavelength of laser lightfrequency of laser light (ν = c/λ)angular frequency of laser light (ωl = 2πν)angular frequency of gravitational wave
strain of gravitational wave
Boltzmann constant kB = 1.38 × 10−23 m2 kg/(s2 K)
luminosity distance
red shift
amplitude reflectiveity
time, or amplitude transmissivity
length of interferometer
length of cavity
finessegravitational constant G = 6.67 × 10−11 m3 /kg s2
mass
moment of inertia
Hubble constantsolar mass
incident electric field
incident laser power
Riemann tensor
Ricci tensor
scalar curvatureradius of curvature of mirror
temperatureenergy-momentum tensor
energy per unit logarithmic interval frequency of stochastic
gravitational wave
0, 1, 2, 3
1, 2, 3xiii

Abbreviations
AdVirgo Advanced VirgoaLIGO
Advanced LIGO
AOM
acousto-optic modulator
ASCalignment sensing and control
BBO
Big Bang ObserverBDbeam dump, or Brans–Dicke
BSbeam splitter
CAST
CERN Axion Solar TelescopeCE
Cosmic Explorer
CBCcompact-binary coalescenceDCdirect current
DECIGO DECi-hertz Interferometer Gravitational Wave Observatory
EOMelectro-optic modulator
ETM
end test massFI
Faraday isolator
FP (FPC) Fabry–Pérot (Fabry–Pérot cavity)
FSR
free spectral range
FWHM
full width at half maximum
GRB
gamma-ray burst
GW
gravitational wave
HWP
half-wave plate
ISCO
inner most stable circular orbit
ITM
input test mass
LIGO
Laser Interferometer Gravitational-Wave Observatory
LISA
Laser Interferometer Space AntennaLS
local sensor
MmirrorMI
Michelson interferometerNS
neutron starOC
optical circulator
PBS
polarizing beam splitterPDphoto detector
PDH
Pound–Drever–Hall
PSR
pulser
PZT
piezoelectric transducer
QPD
quadrant photo detector
SATsatellite
SNsupernova
STM
steering mirrorTM
test massUTC
coordinated universal time
UGF
unity gain frequency
WFS
wavefront sensor

xivChapter 1

Introduction
Gravitational wave is a ripple of spacetime. The existence of the gravitational
wave was predicted in 1916 by Albert Einstein [1], who pointed out that gravity is
a curvature of spacetime in general relativity. General relativity predicts several
new gravitational phenomena, such as apsidal precession [2], gravitational lensing [3–5], black holes [6, 7], and gravitational waves [1]. The gravitational wave
is one of the most important predictions. Therefore, many studies for detecting
gravitational waves have been performed after the Einstein’s predication [8–10].
Especially, astronomical objects are main targets of the study.
The observation of gravitational waves is motivated by two major scientificobjectives. The first motivation is the test of general relativity [11, 12]. Since
general relativity is the most reliable theory of gravity so far, most of the physical studies on the Universe are based on general relativity. Thus, testing generalrelativity leads to improving our knowledge about the Universe. The second motivation is astronomical one [13, 14]. To gravitational waves, almost everythingis highly transparent. Although electro-magnetic waves are used in conventional
astronomy, they can be absorbed and/or scattered by normal matter and muchinformation is lost. Accordingly, gravitational waves provide essentially new
astronomical information, such as mechanism of gamma ray bursts [15] and supernovae [16,17]. With the gravitational observation, the mass and the dynamics
of the astronomical event can be observed.
The first but indirect evidence of the gravitational wave was found by Russell
Alan Hulse and Joseph Hooton Taylor, Jr., who observed an orbital period of
the binary pulsar, PSR B1913+16 [18]. They found that the orbital period of
the binary pulsar was getting short and it was consistent with the predicationof general relativity. This result indicates that the binary pulsar loses a part ofits energy with gravitational wave emissions as predicted by general relativity
and the existence of the gravitational wave. However, the dynamics of strainof space-time cannot be observed in the binary pulsar observation since, in theobservation of the orbital period, just the energy of the binary system can bemeasured.
For the direct observation of strain of space-time, laser interferometric gravitational wave detectors are widely used. The laser interferometric gravitational
wave detecters are based on a Michelson interferometer [19]. The Michelsoninterferometer transduces the phase diﬀerence of two splitted laser lights into1

the laser intensity variation. Therefore, by using the Michelson interferometer,
the optical path length fluctuation due to the gravitational waves can be observed. In addition, a Fabry–Pérot cavity is implemented in the base-line of theMichelson interferometer to enhance the gravitational wave signal [20, 21]. For
now, several km-class ground-based laser interferometric gravitational wave detectors are in operation all over the world, such as Advanced Laser Interferometer
Gravitational-Wave Observatory (aLIGO) in US [22], Advanced Virgo (AdVirgo)
in Italy [23], GEO600 in Germany [24–26], and KAGRA in Japan [27, 28]. The
observation frequency of these observatory is from ∼ 10 Hz to a few kHz. The
lower limit of the frequency is determined by the seismic motion.
In 2015, aLIGO observed the first gravitational wave event, which is called
GW150914 [29]. GW150914 is the gravitational wave from binary black holes ofwhich masses are 36 M⊙ and 29 M⊙ (M⊙ is the solar mass) at redshift z = 0.09.
After GW150914, more than 10 black hole binaries including GW150914 weredetected [30]. By using these events (and a neutron star binary event explainedlater), general relativity was tested while no inconsistency with general relativity
was found [31, 32].
In addition to the black hole binaries, the neutron star binary has been observed with aLIGO and AdVirgo in 2017. The neutron star binary event iscalled GW170817 [33]. In GW170817, besides the gravitational wave observation, electro-magnetic follow-up observations were also performed [34]. Especially, there was coincidence between the gravitational wave event and the shortgamma-ray burst event GRB 170817A [35]. This multi-observation strongly indicates that the short gamma-ray burst is caused by the neutron star binary
coalescence. According to these black hole and neutron star events, it can besaid that a new era of the gravitational wave and multi-messenger physics andastronomy has started.
Although the observational frequency range of the ground-based laser interferometric gravitational wave observatory is from ∼ 10 Hz to ∼ 1000 Hz, someobservational targets outside the range of the frequency band are predicted. One
target is the coalescence of the intermediate-mass black holes, i.e. the black holes
with masses in the range between ∼ 103 M⊙ and ∼ 106 M⊙ , which typically emit
the gravitational waves in the frequency range between ∼ 10−3 Hz and ∼ 1 Hz.
The coalescence of the intermediate-mass black holes is considered to be a candidate of the origin of the super massive black hole that has been discovered at
the galactic center [36, 37]. Another target is the early Universe [38, 39]. The
quantum fluctuation is considered to cause the gravitational waves in the wide
frequency range, for example, between ∼ 10−17 Hz and ∼ 104 Hz [40]. The
observations of the gravitational waves from the coalescence of the intermediatemass black holes and the early Universe provide new physical and astronomical
information.
Therefore, for further expansion of gravitational wave physics and astronomy,
it is important to expand the observation frequency range. The observation oflower frequency gravitational waves (< 10 Hz) than those have been observed
allows us to study various sciences, such as distinction of the formation scenario ofsuper-massive black holes in galactic centers [36, 37, 41], study on the evolution2

of the compact binary stars, such as white dwarf binaries [42], precise test of
the modified gravity theories [43, 44] and so on. Especially in 0.1-Hz range,
there is a possibility to observe the gravitational waves generated in the early
Universe [38, 39] and to verify the inflation theory [40, 45, 46].
So far, two types of the next generation laser interferometric gravitational wave
detectors are proposed for further expansion of the gravitational wave science: 10km-class ground-based detectors and space detector with 100-km to 2.5-Gm scaleinterferometer. Examples of the former type detectors are Einstein Telescope [47]and Cosmic Explorer [48]. They are aimed to observe relatively high frequency
gravitational waves in between 5 Hz and ∼ kHz. For example, gravitational waves
in that frequency range are caused by stellar mass compact binaries. Examplesof the latter type detectors are Laser Interferometer Space Antenna (LISA) [49],
Big Bang Observer (BBO) [50], TianQin [51], Taiji [52], TianGO [53], DECihertz Interferometer Gravitational Wave Observatory (DECIGO) [54], and BDECIGO are proposed. Since the low frequency sensitivity of the ground based
detectors are limited by ground motion [55,56], space detectors are necessary for
low frequency gravitational wave observation.
Space gravitational wave detectors are classified into mainly two types: opticaltransponder type (LISA, BBO, TianQin, and Taiji) and Fabry–Pérot type detectors (DECIGO and B-DECIGO). By using the Fabry–Pérot cavity, the laser
beam is circulated in the cavity. In other words, the time for the interaction of
the laser interferometer and the gravitational waves is enhanced with the cavity.
As a result, the highest sensitivity of the laser interferometer can be improved.
Consequently, DECIGO can be so sensitive that it has a possibility to observe
the gravitational wave background generated in the inflation era of the early
Universe [38, 39].
The Fabry–Pérot type space gravitational wave detectors, however, need precise control system since the Fabry–Pérot cavity has small linear range shorter
than the laser wavelength. In gravitational wave detectors, laser with a wavelength of ∼ µm is used*1 so that sub-µm level control is necessary. Although
the precise control of ∼ 10−11 m has been achieved in the current ground-based
gravitational wave detectors, new control scheme is necessary for space detectors
since they have a number of challenges that are not in the ground-based detectors. One of the challenges is lack of natural reference for the control while, inground-based detectors, the ground is a stable reference. Another challenge isthe necessity of drag-free control that is not used in the ground-base detectors.
The drag-free control is the control of a satellite to follow a test mass. The test
mass floats in the satellite and is isolated from the dragging force, such as solar
radiation pressure, applied to the satellite [57]. It was not clear whether the
drag-free control can be engaged simultaneously with all the degrees-of-freedom
control of the Fabry–Pérot cavity. Moreover, a feedback system for precise control introduces additional noises through feedback loops. The additional noise
could disturb the gravitational wave observation. Therefore, the control system
has to be carefully designed especially in the observation frequency band [58,59].
*1 This wavelength is used for the gravitational wave observatory due to the technical reason.

There is a stable and powerful laser with a wavelength of ∼ 1 µm, for example 1.064 µm.

3In addition, for gravitational wave observation, a sensing method to measure
the length of cavity, i.e. an interferometer configuration, is also important. DECIGO and B-DECIGO (and most of the other space detectors) adopt a triangularshaped interferometer geometry for improving calibration precision, false alarm
ratio, and sensitivity to polarizations of gravitational waves [60, 61]. These features are important to improve the accuracy of the parameter estimation of thegravitational wave sources from the observation and the reliability of the gravitational wave detection. DECIGO and B-DECIGO utilize a dual-pass Fabry–Pérot
cavity to form the triangular shaped interferometer. One advantage of this configuration is that the redundancy can be obtained with minimum number of thetest masses. This is a new interferometer configuration. In the dual-pass Fabry–Pérot cavity, laser light is injected from both sides of one Fabry–Pérot cavity
and the relative frequency of the two lasers cannot be controlled independently.
In this interferometer configuration, since the Fabry–Pérot cavities are coupledwith each other, we need a new scheme to obtain the longitudinal signal of all theinterferometers while the Fabry–Pérot cavity is widely used in optics field, e.g.
spectroscopy, optical quantum computing, laser telecommunications. Therefore,
the operation of the dual-pass Fabry–Pérot cavity should be demonstrated for
DECIGO and B-DECIGO.
In this thesis, in order to realize the Fabry–Pérot type space gravitational
wave detectors, such as DECIGO and B-DECIGO, and expand the gravitational
wave physics and astronomy, the control scheme of the Fabry–Pérot type spacegravitational wave detectors is addressed. Specifically, we investigate the control
topology of the Fabry–Pérot type space gravitational wave detectors and experimentally construct the dual-pass Fabry–Pérot interferometer for demonstration.
This thesis is organized as follows: Chapter 1 is this chapter. Chapter 2 introduces features of the gravitational wave from general relativity, sources of
the gravitational wave, and scientific cases with the observation of the gravitational waves. Chapter 3 shows a review of the laser interferometric gravitational
wave detectors and comparison among the ground-based detectors and spacedetectors (LISA, the optical transponder type; DECIGO, Fabry–Pérot type).
Chapter 4 explains the scientific objectives and overall concept of DECIGO and
B-DECIGO. Chapter 5 describes the study of control scheme of DECIGO and
B-DECIGO. The result of the study includes the confirmation of the control stability and noise performance of feedback control. Chapter 6 provides the resultof the experimental demonstration of the dual-pass Fabry–Pérot cavity. Chapter 7 concludes the study of the control scheme of the Fabry–Pérot type spacegravitational wave detectors.

4

Chapter 2

Gravitational wave
Gravitational waves are ripples in space-time. They were predicted in generalrelativity by Einstein [1]. Hulse and Taylor proved the existence of the gravitational wave indirectly via a reduction in the revolution cycle of the binary pulsar,
PSR B1913+16 [18]. Their observation indicated that reduction in the energyof the binary pulsar was able to be explained with general relativity. In otherwords, the ripple in space-time was not detected in Hulse and Taylor observation.
Afterward, research toward the direct observation of the gravitational wave
was conducted. Subsequently, aLIGO detected the gravitational wave from abinary black hole on Sep. 14, 2015 (UTC) for the first time [29]. This event
is called GW150914. After GW150914, more than 10 gravitational wave events
have been detected. Among them, a neutron star binary event, GW170817, wasalso detected [33]. In GW170817, many electro-magnetic observatories detected
afterglow of the neutron binary coalescence [34]. These gravitational wave events
led to many scientific results, such as the test of general relativity [31, 32], andopening the gravitational wave physics and astronomy era.
In this chapter, we linearize Einstein equation for understanding the gravitational waves. From the linearized Einstein equation, the gravitational wave isderived. Then, we introduce a number of gravitational wave sources that canbe or is expected to be detected. Finally, expected sciences with gravitational
waves are reviewed.

2.1 Linearized Einstein Equation
In general relativity, time and space are represented in four dimensions (one time
coordinate and three space coordinates). In the four-dimensional space-time,
two very near events, xµ *1 and xµ + dxµ , have a gap, ds, which is expressed inquadratic form as
ds2 = gµν dxµ dxν ,(2.1)
where gµν is a metric tensor, which is often abbreviated simply as a metric [62].*2
Equation (2.1) shows that, in general relativity, nature of space-time is repre*1 In this thesis, x0 , and (x1 , x2 , x3 ) indicate time coordinate (ct) and space coordinate,respectively. Also, Greek indices (Roman indexes) take the value of 0, 1, 2, 3 (1, 2, 3).
*2 In this thesis, we follow Einstein summation convention.

5

sented by the metric tensor. In addition, equation (2.1) indicates that gµν is asymmetric tensor.
When matter exists, the metric tensors follow the Einstein equation,8πG
1Rµν − gµν R = 4 Tµν ,
2
c

(2.2)

where Rµν is a Ricci tensor, R is scalar curvature, G is the gravitational constant,c is the speed of light, Tµν is an energy-momentum tensor. *3 The left and righthand side of Einstein equation (2.2) correspond to space-time and matter field,
respectively. In other words, Einstein equation indicates that matter makes
curvature of space-time, i.e. gravitational field. In addition, Einstein equationshows that moving matter varies the gravitational field.
In the range well far from the source of the gravitational field, we can understand the nature of the space-time considering the perturbation from the flatspace-time. When a Minkowski metric, ηµν , is expanded with small perturbation, hµν , and we drop the second order term of |hµν |, the metric, gµν , under the
existence of matter is represented asgµν = ηµν + hµν ,−1 0 0 0
 0 1 0 0ηµν =  0 0 1 0 ,0 0 0 1(2.3)|hµν | ≪ 1.

(2.4)

This kind of treatment is known as the linearized theory.
From here, we drives Einstein equation in linearized theory. First, we defineh ≡ η µν hµν ,

(2.5)

1
h̄µν ≡ hµν − ηµν h.
2

(2.6)

When we inversely define h̄ ≡ η µν h̄µν = h − 2h = −h, the metric can be
represented as,1hµν = h̄µν − ηµν h̄.
(2.7)
2

µ
*3 In this thesis, Christoﬀel symbol, Γρµν , Riemann tensor, R νδσ , Ricci tensor, Rµν , scalar

curvature, R, are defined as follows [63]Γρµν =1 ρσ
g (∂µ gσν + ∂ν gσµ − ∂σ gµν )
2

(
∂µ ≡

∂
∂xµ

µ
µ
α
µ
α
Rµ νδσ = ∂ρ Γµνσ − ∂σ Γνρ + Γαρ Γνσ − Γασ Γνρ ,Rµν = Rα µαν ,R = g µν Rµν .

6),

Ricci tensor, Rµν , and scaler curvature, R, is denoted as,
11Rµν = − (□h̄µν − ∂ ρ ∂µ h̄ρν − ∂ ρ ∂ν h̄ρµ − ηµν □h̄),
2
2
1
ρ µ
R = − (−□h̄ − 2∂ ∂ h̄ρµ ),
2

(2.8)
(2.9)

where □ ≡ ηµν ∂ µ ∂ ν = ∂µ ∂ µ . Consequently, Einstein equation (equation (2.2))
is expressed as,)
1(8πG□h̄µν + ηµν ∂ ρ ∂ σ h̄ρσ − ∂ ρ ∂ν h̄µρ − ∂ ρ ∂µ h̄νρ = − 4 Tµν .
2c(2.10)

Finally, when we apply Lorenz gauge,∂ ν h̄µν = 0,(2.11)

equation (2.10) is denoted as□h̄µν = −

16πG
Tµν .
c4

(2.12)

This equation is the linearized Einstein equation. The linearized equation canbe used to understand gravitational field in linearized theory.
With equation (2.11), 10 degrees of freedom of symmetry tensor, hµν , is reduced to 6. In addition, with equations (2.11) and (2.12), the energy-momentum
tensor is represented as∂ ν Tµν = 0.
(2.13)
This equation corresponds to the law of conservation of the energy and momentum.
As the end of this section, we show that Lorenz gauge can be applied anytime.
First, let’s consider coordinate transformation, xµ → x′µ = xµ + ξ µ (x), where|∂µ ξν | is the same order of magnitude as |hµν |. Then, h̄µν and its derivatives aretransformed ash̄µν (x) → h̄′ µν (x′ ) = h̄µν (x) − ∂µ ξν (x) − ∂ν ξµ (x) + ηµν ∂ρ ξ ρ (x).
∂ ν h̄µν (x) → (∂ ν h̄µν )′ (x′ ) = ∂ ν h̄µν (x) − □ξµ (x)(2.14)(2.15)

Thus, if ξµ meeting the equation□ξµ (x) = ∂ ν h̄µν (x)(2.16)

exists anytime, Lorenz gauge can be applied anytime. Since we can denote∂ ν h̄µν (x) = fµ (x) using a function fµ (x), equation (2.16) is expressed as□ξµ (x) = fµ (x).

(2.17)

Then, the answer of equation (2.17) can be denoted with Green function G(x)meeting□G(x − y) = δ 4 (x − y),(2.18)7∫asξµ (x) =d4 y G(x − y)fµ (y).

(2.19)

Since ξµ (x) in the left hand side of equation (2.19) formally exists always, there
is an answer of equation (2.16) anytime. It means that Lorenz gauge can beapplied anytime.

2.2 Derivation of gravitational waves and their nature
In vacuum, that is the region where Tµν = 0, equation (2.12) is denoted as□h̄µν = 0.

(2.20)

This is a wave equation for h̄µν and shows that h̄µν , which is perturbation from
the Minkowski metric, is propagated at the speed of light. This h̄µν is a gravitational wave.
From here, we investigate the nature of the gravitational wave. In the range far
enough from the gravitational wave source, Einstein equation can be represented
in a simple form using Lorenz gauge, equation (2.11). However, Lorenz gauge
does not decide the gauge form completely. In other words, even with Lorenz
gauge, we can still perform coordinate transformation. In fact, according toequation (2.15), when we transform the coordinate, xµ , meeting Lorenz gaugeasxµ → x′µ = xµ + ξ µ ,(2.21)□ξµ = 0,(2.22)

Lorenz gauge is invariant. When □ξµ = 0, □ξµν = 0, whereξµν ≡ ∂µ ξν + ∂ν ξµ − ηµν ∂ρ ξ ρ ,(2.23)is also valid since □ and ∂µ can be commutated. Equation (2.14) indicates
that, under the coordinate transformation as equation (2.21), equation (2.20) is
invariant. Therefore, by using equation (2.22), degrees of freedom of h̄µν can be
reduced by four. As a result, according to this symmetry and Lorenz gauge, thedegree of freedoms of h̄µν is two.
To denote h̄µν with two degrees of freedom, we deform equations (2.11) and(2.22). First, ξ 0 is chosen to make trace zero, that is h̄ = 0. Notice that, whenh̄ = 0, h̄µν = hµν . Second ξ i (x) is chosen to make h0i (x) = 0. Then, sinceh̄µν = hµν , from Lorenz gauge, equation (2.11), with µ = 0,
∂ 0 h00 + ∂ i h0i = 0,(2.24)and h0i = 0, ∂ 0 h00 is denoted as∂ 0 h00 = 0.

8(2.25)

This indicates that h00 is temporally constant. From physical point of view, thetime independent term, h00 , represents static gravitational field, i.e. Newtonian
potential of the gravitational wave source. Since gravitational waves are timedependent, ∂ 0 h00 = 0 means h00 = 0 when we consider gravitational waves.
Consequently, h0i (x) = 0. Therefore, only spacial components, hij , which meet
Lorenz gauge (∂ j hij = 0) and traceless condition (hi i = 0) are left. In summary,hµν is donated as,
h0µ = 0, hi i = 0, ∂ j hij = 0,(2.26)
which are called a transverse-traceless gauge (TT gauge). Since equation (2.26)
is composed of eight formulas, 10 degrees of freedom of hµν are reduced to be
two. Hereafter, the metric in TT gauge is written as hTT
ij .
Note that TT gauge is not valid in gravitational wave sources since □h̄µν ̸= 0.
In the gravitational wave sources, once Lorenz gauge is applied, the degree of
freedom of □ξµ = 0, i.e. □ξµν = 0, remains. However, we cannot make thecomponents of h̄µν to meet □h̄µν ̸= 0, zero even by subtracting ξµν meeting□ξµν = 0.
ikx, where kx =Equation (2.20) has a plane wave solution, hTTij (x) = eij (k)eµµk xµ , k = (ωGW /c, k), ωGW /c = |k|, eij (k) is a polarization tensor, and ωGW
is angular frequency of the gravitational wave. When a monochromatic plane
wave that is denoted with a wave-number vector, k, equation (2.26) indicatesthat non-zero components of hTT
is orthogonal to the propagation directionij
vector of the gravitational wave, n̂ = k/|k|. This is because in the plane wave,∂ j hij = 0 means nj hij = 0, i.e. the transverse wave. When n̂ is chosen along z
axis, because of equation (2.26) and the symmetry analysis of hTTij , the metric iswritten as,
h+ h× 0
 h× −h+ 0  cos[ωGW (t − z/c)],hTT(2.27)ij (t, z) =000 ij
or simply,(hTT
ab (t, z) =h+ (t) h× (t)h× (t) −h+ (t)),(2.28)abwhere h+ (t) ≡ h+ cos[ωGW (t − z/c)], h× (t) ≡ h× cos[ωGW (t − z/c)], and a, b =1, 2. h+ (t) and h× (t) are called plus mode (+ mode) and cross mode (× mode),
respectively. By using them, a square of the infinitesimal world line, ds2 isdenoted as
ds2 = − c2 dt2 + dz 2 + [1 + h+ (t)]dx2 + [1 − h+ (t)]dy 2 + 2h× (t)dxdy.

(2.29)

Figure 2.1 shows time variation of the free particle swarm motion caused by plusand cross-mode gravitational waves.

9Figure 2.1: Time variation of the free particle swarm’s motion caused by plusand cross-mode gravitational waves. Here, gravitational waves are assumed topropagate along z-axis.

2.3 Generation of gravitational waves and their sources
2.3.1 Generation of gravitational waves
Using the linearized Einstein equation (equation (2.12)), generation of the gravitational wave can be understood. When the source, i.e. the energy-momentum
tensor, generating gravitational waves is locally distributed, the solution of equation (2.12) is denoted as∫ µν
4G
T (t − |x − x′ |/c, x′ ) 3 ′h̄µν (t, x) = 4
d x.
(2.30)c|x − x′ |Moreover, if the size of the gravitational wave source is much smaller than the
wave length of the gravitational wave generated by the source and an observeris in far from the source, equation (2.30) is written ash̄ij (t, x) =2G
Q̈ij (t − ro /c),c4 ro

(2.31)

where ro is a distance between the gravitational wave source and the observer,
and Qij is a quadrupole momentum of the source, which is defined as
(
)
∫
1
′
′ ′
′i ′j
Qij (t − ro /c) ≡ ρ(t − ro /c, x ) xi xj − δij x x
d3 x ′ .
(2.32)
3
Equation (2.31) indicates that the lowest degree of the gravitational wave generation is quadrupole. Using equation (2.31), the amount of the gravitational
wave generated by its source is evaluated.

10

2.3.2 Sources of the gravitational wave
Compact-binary coalescenceAll gravitational waves detected by ground-based detectors by the end of 2017
are generated by compact-binary coalescences, known as CBC, [30]: binary
black holes, and binary neutron stars. Other compact-binary-coalescence candidates that have not been detected directly are a neutron-star-black-hole binary [64, 65],*4 a binary white dwarf [66], a white-dwarf-neutron-star [67] andwhite-dwarf-black-hole binary [68], and so on. Strain equivalent amplitude,hCBC (t), of gravitational waves from equal-mass compact-binary coalescences
during inspiral, i.e. the evolution phase of the compact binary of which orbital
radius decreases gradually before the merging, is written as [63], in the detector
frame,)5/3 ()2/3
GMCBC
πfGW (τ )cos(2πfGW (τ )t + ϕ0 ),c
21/5 c3()3/8 (
)−5/8
5 1
GMCBC
1
fGW (τ ) =(τ ≡ tcoal − t),
π 256 τ21/5 c3

4hCBC (t) ≃dL((2.33)(2.34)

where MCBC is the mass of the source object, dL is the luminosity distance,
fGW is the frequency of the gravitational wave, ϕ0 is a phase factor, and tcaol
is the time at coalescence. Equation (2.34) indicates that the frequency of thegravitational wave is gradually increasing during the inspiral and is the largest
at coalescence. This characteristic signal of the gravitational wave is called achirp signal. Strain equivalent amplitude in frequency space, hCBC (f ) [1/Hz], is
written as [63], in detector frame,hCBC (f ) ≃

1
π 2/3

(

5
24

)1/2

c
dL

(

GMCBC
21/5 c3)5/6

f −7/6 .

(2.35)The inspiral is finished at the innermost stable circular orbit. The gravitational
wave frequency at the innermost stable circular orbit is given by
1
c3
fISCO = √.
6 6(2π) GMCBC(2.36)Let us consider the amplitude of the gravitational wave during inspiralfrom compact-binary coalescences, especially black-hole binaries.
From*4 Thereis a probable neutron-star-black-hole-binary event in Observational Run 3 ofaLIGO and AdVirgo, so called S190814bv, which is publicly alerted in GraceDB(https://gracedb.ligo.org) while detailed parameters have not been published yet.

11equations (2.35) and (2.36),)()−7/6
100 Gpc
f
,
dL
fISCO
( 4
)
10 M⊙
fISCO ≃ 0.2
[Hz].
M

−21

hCBC (f ) ≃ 4 × 10

(

M
4
10 M⊙

)5/6 ((2.37)(2.38)

Supernova
It is possible to detect gravitational waves from core collapse of supernovae if
there is a mass asymmetry in the core collapse [17]. By using numerical simulation, amplitude and frequency of gravitational waves from supernova is estimated. Typically the amplitude is h ∼ 10−22 − 10−21 and the frequency is ∼ 1
kHz in 10 kpc far from the supernova [69].
Isolated pulsar
A pulsar is an astronomical object emitting periodic electromagnetic pulse in
gamma ray, X-ray, radio wave, and visible light [70]. The pulsar is considered to
be spinning magnetized neutron star that is a supernova remnant [70]. If these
pulsars have a slight mountain or valley because of the deformation or the fluid
oscillation, they can generate continuous gravitational wave [71]. For example,
mili-second pulsars that have 1 ms or a few 10 ms duration emit gravitational
waves in the frequency range between 10 Hz and 1000 Hz. The amplitude of thegravitational waves from the isolated pulsar is given by [63]3/2

hpulsar ≃16π 2 Gfrot
ϵI,
c4 dL

(2.39)

where frot is the rotation frequency of the pulsar, ϵ is the ellipticity of the pulsar,
and I is the momentum inertial along the rotation axis of the pulsar. If the CrabPulsar (PSR J0534+2200), which is 2 kpc away from the Earth, has ellipticity ϵ ∼10−5 , the gravitational wave of which amplitude is estimated to be h ∼ 10−26 atthe Earth is considered to be generated [63]. Although this amplitude is relatively
small compared with the amplitude of the gravitational waves from compactbinary coalescences, the gravitational waves from the pulsars can be amplified
by integrating the signal for long time since they are continuous. Therefore,
the pulsar is one of the promising astronomical objects generating detectablegravitational waves.
Early Universe
During inflation in the early Universe, the quantum fluctuation of the spacetime, i.e. the tensor metric, can generate stochastic gravitational waves calledstochastic gravitational wave background [38, 39]. Since the quantum fluctuation is considered to be scale invariant, the generated gravitational wave isalso scale invariant. In other words, the power of the generated gravitational
wave is frequency independent. In addition to the space-time fluctuation, cosmic

12strings [72, 73], domain walls [74, 75], and preheating of the Universe after inflation [76] in the early Universe are considered to be the sources of the stochastic
gravitational wave background. The stochastic gravitational wave background ischaracterized with its energy per unit logarithmic interval of frequency, Ωgw (f ),and its power spectral density is written as
SSGWB (f ) =3H02Ωgw (f ),4π 2 f 3(2.40)

where H0 is the Hubble constant [63] (the definition of the power spectral density
is shown in Appendix B).

2.4 Gravitational wave physics and astronomy
So far, aLIGO and AdVirgo have been operated for two Observational Runs
and 10 black hole binaries and one neutron star binary have been detected [30].
Here, we review the scientific results of the two Observational Runs. Noticethat although the third Observational Run started from April, 2019, and many
gravitational waves were detected, the results are not explained here since thedetailed parameters of the events have not been published yet. In addition to thecurrent result, future prospect of the gravitational wave physics and astronomy
are shown, too.

2.4.1 Test of general relativity
One of the most important objectives of the gravitational wave physics is testing
general relativity [31,32]. The simplest way to do the test is comparing the waveform predicted from general relativity to that observed with the real detectors.
This method is called parameterized test [77, 78]. Another way is comparing
the parameters of the gravitational wave source estimated from two durations,for example, inspiral and ringdown modes [79]. Since gravity in the ringdown
modes is much stronger than that in the inspiral, the gravitational eﬀect beyond
general relativity is considered to be visible in the ringdown modes. Althoughcontradiction of general relativity has not been found yet, many modified gravitational theories are proposed [80–82]. One of the major predictions of modified
theories is gravitational wave of vector and/or scalar mode in addition to theconventional tensor mode [83]. The additional mode aﬀects the waveform of
the gravitational waves or the result of the parameter estimation. The numberof gravitational wave modes that can be tested is in principle equal to or lessthan the eﬀective number of the detectors. Thus, further observation with more
detectors is necessary [84, 85].

2.4.2 Source of the gamma-ray burst
In the neutron star binary observation, GW170817, not only gravitational
wave but also electro-magnetic signals were observed [86]. Especially, theshort gamma-ray burst, GRB 170817A, is coincident to the gravitational wave

13

event [35]. Although the gamma-ray burst is discovered in 1960s [87], its source
and mechanism are not fully concluded [88]. Joint observation of gravitational
wave and gamma-ray burst was considered to be promising method to address
the mechanism of the gamma-ray burst [89]. The observations of GW170817
and GRB 170817A indicate that the neutron star binary can be the sourceof the short gamma-ray burst. However, there is an unresolved point in thisevent: the observed gamma-ray bust is much weaker and closer than the typicalevent. The observed luminosity distance and the flux of GW170817 and GRB
170817A is 42.9 Mpc and 2 × 1047 erg, respectively, while the nearest andweakest gamma-ray burst except for GRB 170817A is about 400 Mpc awayfrom the Earth and 2 × 1049 erg [35]. Thus, it has not been concluded if GRB
170817A is the same type of gamma-ray burst as the other events or not andfurther observations are still necessary for the final conclusion.

2.4.3 Synthesis of heavy nuclei
Multi-messenger astronomy related with GW170817 provided important information about the synthesis of heavy nuclei. In the follow-up electro-magnetic
observation, a kilonova was found [86]. The spectra of the kilonova afterglow
observed by various observatories with electro-magnetic waves, such as infra red,visible, ultra violet, and X-rays, indicate that the r-process occurred in the neutron star binary merger and heavy nuclei were generated [90]. However, it has
not been qualitatively concluded that the amount of heavy nuclei is enough to
explain the current amount of that in the Universe. If more events are observed,
we can investigate the synthesis of heavy nuclei in more quantitative way.

2.4.4 Verifying inflation
If the gravitational wave from the early Universe is observed, its nature, for example, inflation mechanism [91, 92], can be directly investigated. Gravitational
waves from the early Universe is expanded by the inflation. Since spectra of thestochastic gravitational wave background is frequency invariant, there are wide
frequency range observation and measurement. In very low frequency (10−18 10−16 Hz), manifold microwave telescopes are trying to detect B mode of cosmic
microwave background that is characteristic signal generated by the stochastic
gravitational wave background [93, 94]. Observations in the frequency range between 10−9 Hz and 10−3 Hz were performed pulsar timing array [95–97] anddoppler tracking using Cassini [98]. Between 1 Hz and 103 Hz, the ground-based
interferometric gravitational wave detectors provided the upper limits [99–101].
The gravitational waves at very high frequency (100 MHz-1 GHz) were searched
for with a special laser interferometer, called synchronous recycling interferometer [102,103], and magnon detectors [104]. Although no observation has detectedstochastic gravitational wave background, cosmic microwave background observation gave the most strict upper limit to Ωgw (0.2 Hz) of ∼ 10−16 [105].

14

2.5 Multi-band gravitational wave physics and astronomy
It is worth noting that there are advantages of the multi-band gravitational wave
physics and astronomy.

2.5.1 Precise Characterization of the compact binary
As shown in equations (2.33) and (2.34), the frequency of the gravitational wave
increases with time. If the gravitational wave is observed from the low frequency, we can observe the gravitational wave event from the early phase. Thus,
the multi-frequency band observation of the compact binaries leads to the longobservation duration. For example, the 30-30 M⊙ binary black hole can be observed for ∼ 2 years if the observation band is between 0.01 and 1000 Hz. Such along observation provides the high signal-to-noise ratio observation. As a result,the compact binary can be precisely characterized. Consequently, early alerts
before the coalescence for the electromagnetic observatories and the precise test
For example, if there is anof general relativity can be conducted [106–108]. √−22
observatory with the strain sensitivity of ∼ 10
/ Hz between 0.1 Hz and 10
Hz (this corresponds to the B-DECIGO sensitivity explained later), the coalescence time can be estimated with statistic error with in 5 s in six days before
the coalescence [108].

2.5.2 Mass distribution of the black hole
One of the astrophysically important questions is the way the super-massive
black hole (∼ 108 M⊙ ) in the galactic center is formed. There are a number ofscenarios proposed to the formation mechanism. One scenario is so-called hierarchical scenario [36]. In this scenario, a lot of seed black holes of which masses are∼ 103 M⊙ merge and form relatively heavy black holes (∼ 106 M⊙ ). These black
holes having masses between ∼ 103 M⊙ and ∼ 106 M⊙ are called intermediatemass black holes. Subsequently, the coalescences of the intermediate-mass black
holes produce the super-massive black hole. This formation scenario predicts theexistence of the black holes with wide mass range. In order to distinguish theformation scenario, we need to investigate the mass distribution of the black holein the Universe. As shown in equation (2.36), the frequency of the gravitational
wave from the black hole binary varies depending on the black hole mass. Therefore, the multi-band observation is necessary to obtain the mass distribution of
the black hole.

2.5.3 Observing the structure of the stochastic gravitational wave
background
Another important target of the multi-band gravitational wave observation isthe examination of the stochastic gravitational wave background. It is a main
target of multi-band gravitational wave observation since the energy of stochastic

15

gravitational wave background from the inflation is considered to be frequency
independent as explained in Subsection 2.3.2. Actually, the multi-band observation of the stochastic gravitational wave background is important even more
as described below. In addition to the energy of stochastic gravitational wave
background, its spectrum shape is used to probe the evolution of the early Universe [109]. For example, after the inflation, the Universe was considered to bereheated [110]. The reheating makes a cut-oﬀ frequency in the spectrum of thestochastic gravitational wave background depending on the duration of reheating. This is because the expansion law of the Universe during the reheating is
the same as that during the matter-dominant phase where Ωgw (f ) is reduced.
The observation of cut-oﬀ frequency provides a direct evidence of the reheating [111, 112]. Moreover, there are other proposed models predicting diﬀerent
spectra of the stochastic gravitational wave background [113, 114]. Therefore,
by observing the stochastic gravitational wave background in the wide frequency
range, the proposed models can be tested.

16

Chapter 3

Interferometric gravitational wave
detector
In this chapter, we summarize the feature of laser interferometers: a Michelson interferometer and a Fabry–Pérot cavity. Then the designed sensitivity ofground-based and space gravitational wave detectors are shown. In the end, two
space gravitational wave detectors, LISA and DECIGO, are compared.

3.1 Overview of interferometric gravitational wave
detectors
For direct detection of gravitational waves, it is necessary to measure the strain
between two free masses, at least, of h ∼ 10−21 caused by gravitational waves.
This tiny space-time ripple was detected with aLIGO for the first time. aLIGO isa laser interferometric gravitational wave detector [22]. Most of the gravitational
wave detectors that are under operation, such as AdVirgo [23], KAGRA [27, 28],
and GEO600 [24–26] are also laser interferometers. Moreover, future space gravitational wave detectors, such as DECIGO [54], LISA [49], BBO [50], TianQin [51], Taiji [52], and TianGO [53] are interferometric detectors.
Laser interferometric gravitational wave detectors are based on a Michelsoninterferometer [19]. The Michelson interferometer converts the variation of twoarms length, i.e. phase variation, to the variation of intensity. Since a wavelengthof laser light (∼ 1 µm) is used as a reference of the arm length, length change
can be measured precisely. However, if we use a simple Michelson interferometer, optimal arm length of the response to the gravitational waves at 100 Hz (0.1
Hz) is 750 km (750 Mm). Such long interferometer cannot be constructed on
the Earth. Therefore, the Fabry–Pérot cavity is used as a base-line of the interferometer to obtain an eﬀective long arm by circulating the laser in the cavity
as shown in figure 3.1 [21]. Another way to obtain long arm interferometer isconstructing the interferometer in space. For example, LISA will have 2.5 × 109 m arms without cavities to detect 1-mHz gravitational waves [49]. DECIGO
will have 1000-km-long Fabry–Pérot cavity with finesse, i.e. the number of theroundtrips of laser in the cavity (defined in equation (3.35) later), of 10 to detect17gravitational waves in 0.1-10 Hz [54].
Laser interferometers are prevented from detecting gravitational waves by various noise sources, such as quantum noise caused by the quantum mechanics oflaser [115, 116], thermal noise due to the brownian motion of the mirror andsuspension systems [117–122], a noise caused by seismic motion [123,124], Newtonian force applied to the mirror [55,125,126], and so on. We need to reduce theeﬀect of all noise sources below that of the gravitational waves to detect them.

Figure 3.1: Schematic of the Michelson interferometer with the Fabry–Pérot
arm cavity, called a Fabry–Pérot Michelson interferometer. BS is beam splitter,
ITMX(Y) and ETMX(Y) are input and end test masses, and PD is a photodetector.

3.2 Michelson interferometer3.2.1 Electric field response of Michelson interferometer
Schematic view of a Michelson interferometer is shown in figure 3.2. Laser lightis emitted from the source and the electric field injected to the beam splitter,
Einc , is expressed as [127]Einc = E0 eiωl t ,(3.1)
where E0 is amplitude of the electric field, and ωl is angular frequency of thelaser light. When amplitude reflectivity and transmissivity of the beam splitteris √12 , and that of the two end mirrors is unity, the electric field of the laser lightreceived by the photodetector, EPD , is written as
EPD =

)
(
1
E0 eiωl t e−2iϕX − e−2iϕY ,
2

18

(3.2)

where 2ϕX and 2ϕY are phase shifts due to the round trip in arm along x- andy-axis, respectively.*1
Laser power, P (x), at x can be expressed by the following relation:
P (x) = α|E(x)|2 ,α ≡ ϵ0 Ac,(3.3)

where E(x) is the electric field at that place, ϵ0 is vacuum permittivity, and A isan eﬀective cross section of the laser light along its propagation direction. Thus,
laser power received by the photodetector, PPD , is1
Pinc (1 − cos 2ϕ− ),
(3.4)
2
)
(
where ϕ− ≡ ϕX − ϕY , and Pinc ≡ α|Einc |2 = α|E0 |2 is a laser power injectedto the beam splitter. Equation (3.4) indicates that the Michelson interferometertransforms the phase diﬀerence between the two arms to the laser power on thephotodetector.
PPD ≡ α|EPD |2 =3.2.2 Michelson interferometer response to gravitational waves
To understand the response of the Michelson interferometer shown in figure 3.2,
let’s consider a plus-mode gravitational wave comes from z-axis (z > 0), i.e.
gravitational wave expressed in equation (2.29) with h+ (t) = h(t) and h× (t) =0. In this condition, the response of photon in the Michelson interferometer isconsidered [128]. The square of the infinitesimal world line ds2 meets
ds2 = −c2 dt2 + [1 + h(t)]dx2 = 0.

(3.5)

If |h(t)| ≪ 1 and dx
dt > 0, it is denoted as[]
1
1 − h(t) c dt = dx.
2

(3.6)

When we integrate both sides of equation (3.6) with time from t − ∆tX to t,
2ξX
1
∆tX =
+
c
2

∫ t

h(t′ )dt′

(3.7)

t−∆tX

where ∆tX is the time for round trip of photon between the beamsplitter andthe end test mass, and ξX is a length of the x-arm. By recursively substituting
equation (3.7) for the ∆tX in the right-hand side of equation (3.7), we obtainthe following equation:
2ξX
1
∆tX =
+
c
2

∫ t

h(t′ )dt′ .

(3.8)

t−2ξX /c

*1 There is a phase shift occurring during traveling from the beamsplitter to the photode-

tector and the factor should be in the right-hand side of equation (3.2). However, we canset the factor to be unity without losing generality.

19

Figure 3.2: Schematic of a Michelson interferometer. Base lines along x-axis andy-axis are called x-arm and y-arm, respectively. TMX and TMY are end test
masses (mirrors). In the panel on top right, transformation of the electric field,
E, when it is reflected and transmitted by the beamsplitter, is shown. Note thatthe phase of the electric field is flipped when it is injected from the substrateside of the beamsplitter.

Therefore, the phase shift of that photon receives during round trip in the x-arm,
2ϕX (t), is expressed as
2ϕX (t) = ωl ∆tX=

2ξX ωl
ωl
+
c
2

∫ t

(3.9)
h(t′ )dt′ .

(3.10)

t−2ξX /cIn the same manner, we obtained the formula about the phase shift of whichphoton receives during round trip in the y-arm, 2ϕY (t), as2ξY ωlωl
2ϕY (t) =
−
c
2

∫ t

h(t′ )dt′ ,(3.11)

t−2ξY /cwhere ξY is a length of the y-arm. When both arms are almost the same but
have a bit diﬀerent length, i.e. ξX ≃ ξY ≃ l and l− ≡ ξX − ξY ̸= 0, the diﬀerence20

of the phase shift that occurs in both arms, 2ϕ− (t), is written as,∫ t
2l− ωl2ϕ− (t) =+ ωlh(t′ )dt′ct−2l/c2l− ωl≡+ δϕGW (t).
c(3.12)(3.13)

The first and second terms of equation (3.13) indicate the static phase shift justdue to the length diﬀerence between the beamsplitter and two end mirrors andthe dynamical phase variation due to the gravitational waves, respectively.
From here, we consider the frequency response of the Michelson interferometer.
When we perform Fourier transform of h(t),∫ ∞h(t) =
h(ω)eiωt dω,(3.14)−∞

from equation (3.13), δϕGW (t) is written as∫ t
δϕGW (t) = ωlh(t′ )dt′t−2l/c∫ ∞=−∞where we define

HMI (ω)h(ω)eiωt dω,2ωlsin
HMI (ω) ≡
ω

(

ωl
c

)

(3.15)

e−i c .

(3.16)On the other hand, Fourier transform of δϕGW (t) is∫ ∞δϕGW (t) =δϕGW (ω)eiωt dω.

(3.17)

ωl−∞

By comparing equations (3.15) and (3.17), we obtain the following relationship:
δϕGW (ω) = HMI (ω)h(ω).

(3.18)

This equation indicates that HMI (ω) is the (angular) frequency response of theMichelson interferometer to the gravitational wave at angular frequency of ω.
The gain of the frequency response of the Michelson interferometer, |HMI (ω)|,
is maximized atπcω=.
(3.19)
2l
This means that the sensitivity of the Michelson interferometer cannot be improved by making long arm length at high frequency region. This is because
the phase variation caused by gravitational waves is cancelled while the photon
travels in the arm of the Michelson interferometer. For example, the optimalarm length for the gravitational wave at 100 Hz (0.1 Hz) is 750 km (750 Mm).
Such long interferometers cannot be constructed on the Earth. In space, very
long arm interferometers are planned, such as LISA, which has 2.5 × 109 -m-longarms to detect gravitational wave below 0.01 Hz [49].

21

3.3 Fabry–Pérot cavity
As explained in the previous section, the response of the simple Michelson interferometer on the Earth to gravitational waves at 0.1 Hz or 100 Hz is not
optimized for its length limit. To solve this problem, the Fabry–Pérot cavity as
shown in figure 3.3 is used to obtain an eﬀective long arm interferometer, i.e.
large response to the gravitational waves [21]. The current ground-based observatories have a few kilometer-long cavities that make eﬀectively about 1000 km
arms.*2 Also, Fabry–Pérot type space gravitational wave antennas are planned,
such as DECIGO and B-DECIGO. The detail of these detector is explained later.
In this section, the feature of the Fabry–Pérot cavity is reviewed.

Figure 3.3: Schematic of the Fabry–Pérot cavity. OC is an optical circulator,
ITM and ETM are input and end test masses (mirrors), respectively. Whenthe light coming from the substrate side is reflected by the mirror, the phase ofelectric field is sign flipped.

3.3.1 Electric field response of the Fabry–Pérot cavity
Schematic of the Fabry–Pérot cavity is shown in figure 3.3. From here, we discusselectric fields in the cavity, Ea and Eb , incident electric field, Einc , reflected andtransmitted electric fields, Er and Et . Without losing generality, the incident
electric field is written as
Einc = E0 eiωl t .
(3.20)
The amplitude reflectivity and transmissivity of ITM (ETM) are defined as ri
and ti (re and te ), respectively. We assume that there is no optical loss in thecavity, unless otherwise stated. The intracavity electric field is expressed as
Ea = Einc ti
Eb = te Ea e∞∑(ri re )n e−2iϕn ,

n=0
−2iϕ

,

*2 GEO600 utilizes a folded Michelson configuration.

22(3.21)(3.22)

where 2ϕ is the phase shift that photon receives during one round trip. By taking
summation, these equations are denoted asti,
1 − ri re e−2iϕti re e−2iϕ
Eb = E0 eiωl t.
1 − ri re e−2iϕ
Ea = E0 eiωl t(3.23)(3.24)

Consequently, the reflected and transmitted electric fields are written as−ri + (ri 2 + ti 2 )re e−2iϕ,1 − ri re e−2iϕti te e−iϕiωl t
Et = E0 e.
1 − ri re e−2iϕ
Er = E0 eiωl t(3.25)(3.26)

Using equations (3.23), (3.25), and (3.26), the intracavity power, Pa , reflected
power, Pr , and transmitted power, Pt , are written as,ti 2
Pinc ,
1 + (ri re )2 − 2ri re cos 2ϕ
ri 2 + [(ri 2 + ti 2 )re ]2 − 2(ri 2 + ti 2 )ri re cos 2ϕ
Pr ≡ α|Er |2 =Pinc ,
1 + (ri re )2 − 2ri re cos 2ϕ(ti te )2
Pinc .
Pt ≡ α|Et |2 =
1 + (ri re )2 − 2ri re cos 2ϕ

Pa ≡ α|Ea |2 =(3.27)(3.28)(3.29)The example of the transmitted power, Pt , is illustrated in figure 3.4.

Power [a.u.]

1

Trans.

0.5

0
−2

−1

0
φ [× π rad]1

2

Figure 3.4: Example of the transmitted power of the Fabry–Pérot cavity. In thisfigure, the case of the cavity with finesse, F, of 15 is shown.
The intracavity power is maximized at 2ϕ = 2πn (n ∈ N, N is a set of natural
numbers). This condition is called resonance. The resonance condition is denotedas2Lωl= 2πn,(3.30)c23

where L is the cavity length. The resonance frequency, i.e. resonance conditionin terms of frequency, is written as
νres = nc≡ nνFSR ,2L(3.31)

where νFSR is called the free spectral range.
When the cavity is composed of high reflectivity (≳ 0.9) mirrors, the cavity has the peak of the intracavity power around resonance frequency. Using
equation (3.27), the full width at half maximum of the peak, 2ϕ1/2 , meetscos 2ϕ1/2 =
If

ϕ1/2
π

−1 + 4ri re − (ri re )2.
2ri re(3.32)≪ 1, 2ϕ1/2 is denoted as
ϕ1/2 =

1 − ri re.
√
2 ri re(3.33)The half width at half maximum is also written in terms of frequency asν1/2 =

1 − ri re c.
√
2 ri re 2πL(3.34)

ν1/2 is called the cavity pole frequency. Here, the ratio of the free spectral range(νFSR ) to the full width at half maximum of the peak (2ν1/2 ) is finesse, which
represents the sharpness of the resonance peak. Specifically, the finesse, F, iswritten as√
π ri reνFSR
F≡=.
(3.35)
2ν1/2
1 − ri reWith equation (3.25), the complex reflectivity of the Fabry–Pérot cavity, rFP ,is denoted as−ri + (ri 2 + ti 2 )re e−2iϕrFP =.
(3.36)
1 − ri re e−2iϕFigure 3.5 shows that absolute value and phase of rFP .

3.3.2 Response of the Fabry–Pérot cavity to the mirror motion
We will consider the response of the Fabry–Pérot cavity to the mirror motion [128]. The time for laser to make n-times round trips, which is ∆tn , isdenoted as,
n
2L
2 ∑
∆tn ≃
n+
δL(t − (2m − 1)L/c),(3.37)cc m=1
where δL(t) (|δL(t)| ≪ L) is the cavity length variation, i.e. mirror motion. By
performing Fourier transformation of δL(t),∫ ∞δL(t) =
δL(ω)eiωt dω,(3.38)
−∞

24

−4

−3

−2

−1

0

1

2

3

4

5

−4

−3

−2

−1
0
1
φ [×10−3 rad]

2

3

4

5

Abs.

Phase [deg]

1
0.999
0.998
0.997
0.996
0.995
0.994
−5
180
135
90
45
0
−45
−90
−135
−180
−5

Figure 3.5: Complex reflectivity of the Fabry–Pérot cavity. Here, we adopt
ri = 0.998, ti = 0.063, and re = 0.999995.

∆tn is written as
2L
2
∆tn ≃
n+
c
c

∫ ∞

1 − e−2iγ(ω)n iωtδL(ω) iγ(ω)
e dω,e− e−iγ(ω)−∞(3.39)

where γ(ω) ≡ Lωc .
In the same way as equation (3.21), when the cavity length is fluctuating, the(L)intracavity electric field of the cavity, Ea , is written as
Ea(L) = Einc ti∞∑(ri re )n−1 e−2i∆tn ωl .

(3.40)

n=1

By substituting equation (3.39) for equation (3.40) and using |δL| ≪ L, weobtain the intracavity electric field as,
(L)

∫ ∞

2ωl
e−iγ(ω)δL(ω)eiωt dω
−2i(ϕ+γ(ω))
c
1
−
r
r
e
i e
−∞
(3.41)
∫ ∞
ti
ti
(L)
−i
(3.42)HFP (ϕ, ω)δL(ω)eiωt dω,
≡
−2iϕ
−2iϕ
1 − ri re e
1 − ri re e
−∞

Ea
ti
ti
=
−
i
Einc
1 − ri re e−2iϕ1 − ri re e−2iϕwehre

e−iγ(ω).
c 1 − ri re e−2i(ϕ+γ(ω))2ωl(L)
HFP (ϕ, ω) ≡(L)

(3.43)

This HFP (ϕ, ω) is the response of the Fabry–Pérot cavity to the mirror motion. When ω/(2π) ≪ νFSR and 2ϕ = 2πn, the response can be approximately
presented as2ωl 2F
1(L)
HFP (ω) =.
(3.44)
c π 1 + iω/(2πν1/2 )25

In the same manner as the intracavity case, the formula of the reflected electric(L)field, Er , is obtained as
Er(L) = −ri Einc + ti re Ea(L) e−2iϕ .

(3.45)As a result, the response of the reflected electric field is denoted as−ri + re e−2iϕti 2 re e−2iϕ
Er
=
−
i
Einc
1 − ri re e−2iϕ1 − ri re e−2iϕ
(L)

∫ ∞
−∞

(L)

HFP (ϕ, ω)δL(ω)eiωt dω.

(3.46)

Equations (3.42) and (3.46) indicate that, in addition to the electric fieldat angular frequency of ωl , the electric field at angular frequency of ωl ± ω isgenerated by the mirror motion at angular frequency of ω. The electric field at
angular frequency of ωl and ωl ± ω are called a carrier and a side-band light,
respectively.
In the end, we consider the amplification of the phase variation with the Fabry–Pérot cavity. Here, we assume ri ≪ re to compare the response of the cavity
with that of just the end mirror (ri = 0). Note that the Fabry–Pérot cavity
is usually used on a condition that the end mirror is more reflective than the
input mirror (ri ≪ re , over-coupled cavity) or the end mirror is as reflective as
the input mirror (ri = re , critically-coupled cavity). The response of the cavity
is diﬀerent from when it is in resonance condition or anti-resonance condition.
Anti-resonance condition is given by()
1
2ϕ = 2π n +.
(3.47)
2
At anti resonance, the intracavity power is the weakest. When we consider thefrequency range meeting |γ(ω)| ≪ 1 or |δL(ω)| ≪ 1, the reflected electric fields
at resonance and anti resonance are written as
)
2ωl
2
F
δL(ω)eiωt dω (resonance),= 1 − 2i
π
c
−∞
)
∫ ∞(
(L,anti)Er
π 1 2ωl= −1 + 2i
δL(ω)eiωt dω (anti resonance).
Einc
2F
c
−∞
∫ ∞(

(L,reso)Er
Einc(3.48)(3.49)The electric filed just reflected by the end mirror without the input mirror isexpressed as(L,NC)

ErEinc= 1 − 2i∫ ∞2ωl
δL(ω)eiωt dω
−∞ c(no cavity).

(3.50)

Here, we compare the phase quadrature, i.e. imaginary part, of each case. They

26

are denoted as
]
[
)
∫ ∞(
(L,reso)
Er
2
2ωl
= −2
F
δL(ω)eiωt dω (resonance),
Im
Einc
π
c
−∞
[
]
)
∫ ∞(
(L,anti)Er
π 1 2ωlIm=2
δL(ω)eiωt dω (anti resonance),
Einc
2
F
c
−∞
]
[
∫ ∞
(L,NC)Er2ωl= −2
δL(ω)eiωt dω (no cavity).
Im
Einc
c
−∞

(3.51)(3.52)(3.53)

These equations show the phase variation of the electric field reflected by thecavity is 2F/π times larger at resonance than that of just the end mirror, while
2F/π times smaller at anti resonance. In other words, the photons in the Fabry–Pérot cavity make 2F/π-times round trips on average in the resonance condition.
Therefore, to obtain the eﬀective long interferometer, we need to keep the Fabry–Pérot cavity at the resonance. For this purpose, Pound–Drever–Hall technique
is often used [129]. Notice that the right hand side of equations (3.51) and (3.52)
for the critically-coupled cavity is obtained by modifying π4 F → π1 F.

3.3.3 Response of the Fabry–Pérot cavity to the gravitational waves
We consider the response of the Fabry–Pérot cavity to the gravitational
wave [128]. Here, gravitational wave, h(t), making space fluctuation along the(h)cavity axis. The time for laser to make n-times round trips is ∆tn , which isdenoted as,
∫ t
2L
(h)
∆tn ≃
n+
h(t′ )dt′ .
(3.54)
ct−2Ln/c
By performing Fourier transformation of h(t) (equation (3.14)),c∫ t

1 − e2iγ(ω)h(ω)dω.
2 t−2Ln/c
iω

2L
1
∆t(h)
n+
n ≃

(3.55)In the same manner as consideration of the response to the mirror motion, theintracavity and reflected electric fields are written as
(h)

Ea
ti
ti
=
−i
−2iϕ
Einc
1 − ri re e
1 − ri re e−2iϕEr−ri + re e−2iϕti 2 re e−2iϕ
=
−
i
Einc
1 − ri re e−2iϕ1 − ri re e−2iϕ
(h)

where

∫ ∞
−∞

(h)

(3.56)(h)(3.57)HFP (ϕ, ω)h(ω)eiωt dω,∫ ∞−∞HFP (ϕ, ω)h(ω)eiωt dω,sin[γ(ω)]e−iγ(ω).
ω 1 − ri re e−2i(ϕ+γ(ω))ωl(h)
HFP (ϕ, ω) ≡(h)

(3.58)

This HFP (ϕ, ω) corresponds to the frequency response of the Fabry–Pérot cavity
to the gravitational wave.

27

(h)

(L)

When HFP (ϕ, ω) is compared with HFP (ϕ, ω), we obtain the relation, in thefrequency range of ω ≪ 1, as(h)c sin[γ(ω)] (L)HFP (ϕ, ω)2ω
L (L)≃ HFP (ϕ, ω).
2

HFP (ϕ, ω) =(3.59)(3.60)

Thus, the cavity response to the gravitational wave with amplitude of h is equivalent to the response to the length fluctuation of Lh/2.

3.3.4 Pound–Drever–Hall technique
Here, Pound–Drever–Hall technique is reviewed [129]. Pound–Drever–Hall technique is used to obtain the phase fluctuation signal of the cavity by modulating
the phase of the incident laser as follows:
PDH
Einc= E0 ei(ωl t+β sin ωs t)≃ E0 eiωl t +ββ
E0 ei(ωl +ωs )t − E0 ei(ωl −ωs )t ,2
2

(3.61)(3.62)

where ωs is the modulation frequency that is typically a few tens MHz and β isthe modulation index that is typically ∼ 0.1. In the second equality of the above
equation, we assumed |β| ≪ 1. Normally, ωs is chosen for the second and thirdterms of equation (3.62) to be oﬀ the resonance, i.e. to be just reflected by the
input mirror. Consequently, from equation (3.25), the reflected electric field iswritten asββ−ri + re e−2iϕE0 eiωl t − E0 ei(ωl +ωs )t + E0 ei(ωl −ωs )t−2iϕ
1 − ri re e
2
2
)]
[
(
re − ri
1 − ri 2F
iωl t
2∆ϕ + β sin ωs t .
≃ E0 e−i
1 − ri re
1 − ri re π

ErPDH =

(3.63)(3.64)

Here, we assume 2ϕ = 2πn + 2∆ϕ and |∆ϕ| ≪ 1. The power of the reflected
electric field at angular frequency of ωs is
1 − ri 2F(2∆ϕ)β sin ωs t
1 − ri re π
1 − ri 2F= 2Pinc(2∆ϕ)β sin ωs t.
1 − ri re π

PrPDH (ωs ) = 2α E0 eiωl t

2(3.65)(3.66)

By demodulating this signal at the angular frequency of ωs and applying a low
pass filter to the demodulated signal, we obtain the linear signal to ∆ϕ as
PDH
Pr,demod∝ 2Pinc

1 − ri 2F(2∆ϕ).
1 − ri re π

(3.67)

This can be used to measure the cavity phase signal and to keep resonance ofthe cavity.

28

3.3.5 Eﬀect of the frequency and intensity fluctuation of the incident
beam to the Fabry–Pérot cavity
In the measurement with the Fabry–Pérot cavity, the frequency and intensity
fluctuation of the injected beams are noise sources.
First, how the frequency fluctuation of the incident laser aﬀects the obtained
demodulated signal of the Pound–Drever–Hall technique is considered. The
phase variation, ∆ϕ, is decomposed to the cavity length and frequency fluctuation into two terms,()
2Lωl ∆L ∆ωl
2∆ϕ =
+
.
(3.68)cLωlThis equation indicates that, with the Pound–Drever–Hall technique, the diﬀerence of the normalized cavity length from the normalized laser frequency cannot
be distinguished. In other words, when we measure the cavity length, the frequency fluctuation behaves as noise,∆L(freq) = L

∆ωl
.
ωl

(3.69)and vice versa for frequency measurement.
Second, we will consider the eﬀect of the laser power fluctuation, ∆P . Here,
we assume that there is some detuning of the cavity, i.e. ∆ϕ → ϕ0 + ∆ϕ. In thiscase, the demodulated signal of the Pound–Drever–Hall technique is denoted as
Pr,PDHdemod ∝ (Pinc + ∆P )(ϕ0 + ∆ϕ) = Pinc ϕ0 + Pinc ∆ϕ + (∆P )ϕ0 .

(3.70)In the second equality, the second order term, |∆P ∆ϕ|, is ignored. In the righthand side, the first term corresponds to the DC term that does not aﬀect fluctuation measurement, the second term is the signal term, and the third term isthe noise term due to the power fluctuation. This means that the noise due tothe power fluctuation increases depending on the detuning ϕ0 .

3.3.6 Alignment sensing with the Fabry–Pérot cavity
With the Fabry–Pérot cavity, not only the length (or frequency) fluctuation but
also alignment signal, e.g. the diﬀerence of the cavity axis from the injected axisof laser, or miscentering from the center of mass of the mirrors, can be measured. The spatial translation or tilt of laser axis is expressed with higher-order
Hermite–Gauss mode expansion as explained in Appendix F. If the translation(tilt) of the laser axis is small, it is written with 10/01-modes of Hermite–Gaussmode [130]. The cavity axis can be expressed in the same way as the laser axis
since the laser light in the cavity passes the centers of the curvature of the mirrors. Therefore, by measuring the 10/01-modes, we can perform the alignment
sensing and control of the cavity. For example, a number of methods of alignment sensing have been proposed, such as dither technique [131], transmission
beam monitoring [132], and wave front sensing method [133, 134].

29

Table 3.1: Summary of detector configuration and parameters. FPC, Fabry–Pérot cavity; MI, Michelson interferometer.
Detector

DECIGOLISA

Advanced LIGOInterferometer

FPC

FPC + MI

Location
FlyingschemeBaseline
Frequency

Space
Formation
flying1000 km10−1 -101 Hz

MI(Transponder)
Space
Constellation
flying
2.5 × 106 km
10−4 -10−1 Hz

GroundN/A
4 km
10 -103 Hz• Seismic(< 101 Hz)• Thermal(101 -102 Hz)• Quantum(> 102 Hz)
1Major noise
sources• Quatnum(10−1 -101 Hz)• Force(< 10−3 Hz)• Quantum(> 10−3 Hz)

Displacement
noise
requirement
Force noise
requirement√∼ 10−18 m/ Hz√∼ 10−11 m/ Hz√∼ 10−20 m/ Hz√∼ 10−16 N/ Hz√∼ 10−14 N/ Hz√∼ 10−13 N/ Hz

3.4 Sensitivity of various detectors
The interferometric gravitational wave detector can be understood by the combination of the Michelson interferometer and the Fabry–Pérot cavity. Here, theconfigurations of various detectors and their principal noises are summarized,
especially for DECIGO and LISA as examples of space detectors and aLIGO
as an example of the ground-based detector. Table 3.1 shows the summaryof the detectors. aLIGO (and most of the other ground-based detectors) has
a combined interferometer of Michelson interferometer and Fabry–Péort cavity [21, 135]. Similarly, DECIGO uses the Fabry–Pérot cavity to enhance theresponse to the gravitational waves. The configuration of LISA is a kind ofMichelson interferometer. The arm length of LISA is so long that diﬀraction
loss is large and just reflecting the laser by the mirror producing too much signal
loss. Thus, LISA measures the phase of the transmitted laser once and transmits
the laser with the same phase after amplifying the signal. This configuration iscalled optical transponder configuration.
The designed sensitivity of DECIGO, LISA, and aLIGO is shown in figure 3.6(the detailed explanation of noise source is presented in Appendix C). The lower
frequency limit of aLIGO (and other ground-based detectors) is ∼ 10 Hz due tothe seismic motion and Newtonian force noise. As explained in Section 2.3.2, inorder to observe heavier mass black holes, we have to go to lower frequency bandthat can be achieved with space detectors.

30

10 -16

Strain sensitivity [1/rtHz]10

DECIGOLISA
Advanced LIGO
NS binary30-30 M sun-17

10 -18
10

-19

10

-20

10

-21

10

-22

10 3 -10 3 M sun
10 5 -10 5 M

sun

10 -2310 -2410 -25 -4
10

10 -3

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

Frequency [Hz]Figure 3.6: Sensitivity cases of interferometric gravitational wave detectors (solid
curves): DECIGO [54], LISA [136], and aLIGO [137]. Dashed lines indicate
inspiral signals from neutron star binary (1.4-1.4 M⊙ ) and black hole binaries(30-30 M⊙ , 103 -103 M⊙ , and 105 -105 M⊙ ) at redshift z = 1. The upper frequencyof the inspiral signal is set at the innermost stable circular orbit frequency, fISCO .

3.5 Comparison of space gravitational wave antennas
For now, most of the interferometric space gravitational wave antenna are classified into two types: a Fabry–Pérot type and an optical transponder type*3 .
Fabry–Pérot type is represented by DECIGO and B-DECIGO, whereas opticaltransponder type is represented by LISA, BBO, Taiji, and TianQin. Here, we
compare the two type configurations by taking DECIGO and LISA as examples.
DECIGO and LISA have satellites that include two test masses. The relative
position and angle of the satellite to the test mass are controlled with thrustersof the satellite. This control is called a drag-free control [138,139]. The drag-freecontrol is an essential technique for space detectors for reducing noise introducedby the satellite motion [140] that is caused by the thruster noise [141–143] or solar
radiation pressure fluctuation [142, 144].
One of the major diﬀerences between DECIGO and LISA is the base-line*3 TianGO [53] adopts a Michelson type configuration.

31

length. DECIGO has 1000-km-long arms while LISA has 2.5×106 -km-long arms.
The arm length corresponds to the observational frequency band. Although
longer arm makes the response to the gravitational wave better, the diﬀraction
loss becomes larger and signal becomes smaller. As a result, by changing the arm
length, we can change the frequency band depending on the scientific objectives
as shown in figure 3.6. The arm length also varies the noise requirement as shownin table 3.1.
With the short arm length, i.e. the small diﬀraction loss, the Fabry–Pérot
cavity can be applied to enhance the highest sensitivity by enlarging the responseto the gravitational waves. DECIGO utilizes Fabry–Pérot type interferometer
configuration that is another diﬀerence from LISA. LISA uses optical transponder
configuration because of its large diﬀraction loss. This diﬀerence leads to thediﬀerence of the highest strain sensitivity as shown in figure 3.6.
One challenge with Fabry–Pérot type space gravitational wave antenna is necessity of the precise control. Specifically, the combined control of the laserinterferometer and the satellite including drag-free control and formation flying of the satellite is necessary. This is because the Fabry–Pérot cavity has asmall linear range that is shorter than the wavelength of the laser as explainedin Section 3.3.2. LISA does not require the formation flying and LISA style isconstellation flying, i.e. a kind of the flying with rough formation. Therefore,
for the Fabry–Pérot type space gravitational wave antenna, we need to develop
precise control scheme including sensing scheme and feedback topology.

32

Chapter 4

Space gravitational wave antenna:
DECIGO and B-DECIGODECIGO is a proposed space gravitational wave antenna using the Fabry–Pérot
arm cavities as shown in figure 4.1 [54]. B-DECIGO, which is a proposed precursor mission to DECIGO, is a relatively small detector while it has almost thesame system as DECIGO. B-DECIGO aims to observe the gravitational wave
in the same frequency band, i.e. 0.1-10 Hz, as DECIGO. Compared with theother pursued space antennas, LISA, TianQin, Taiji, and TianGO, DECIGO
and B-DECIGO have better sensitivity in relatively high frequency range thanks
to the Fabry–Pérot arm cavity. BBO has somewhat similar design sensitivity to
DECIGO by increasing laser power and telescope size without the Fabry–Pérot
cavity. The arm cavity requires precise formation flight. Therefore, we need todevelop a new control scheme as explained in Chapter 5.
In this chapter, we review the science targets of DECIGO and B-DECIGO
first. Then, their designs are described.

4.1 Scientific objectives
The target sensitivity of DECIGO and B-DECIGO is shown in figure 4.2 (thedetailed parameters to obtain the sensitivity is explained later). Once this sensitivity is achieved, fruitful scientific insights are obtained.

4.1.1 Verifying and charactering inflation and thermal history after the
inflation√
When DECIGO has a sensitivity of 10−23 / Hz at 0.1 Hz, the stochastic gravitational wave background equivalent to ΩGW ∼ 1 × 10−16 can be observed bytaking the correlation of the two clusters (see Section 4.3.4) with 3-year observation [63]. ΩGW ∼ 1 × 10−16 is comparable to the current limit of the slowroll model inflation [105]. A number of inflation models that predict this power
level of the stochastic gravitational wave background can be verified with theobservation with DECIGO [40,45,46]. Moreover, in certain parameters, the cutoﬀ frequency due to reheating of the Universe after inflation could be around33

Thruster
Laser
Drag-free satellite
PD

PD

MirrorMirrorPD

Dual-pass
Fabry-Pérot cavity

ThrusterMirror
PD
Laser

Laser
PD

ThrusterPD

Drag-free satellite

Drag-free satellite

Figure 4.1: Pre-conceptual concept of DECIGO and B-DECIGO.

0.1 Hz. If DECIGO observes the frequency structure of the gravitational wave
background spectrum, the thermal history of the Universe is characterized [145].
For the observation of the stochastic gravitational wave background, the compact binary coalescences that cannot be resolved as a single event make thestochastic gravitational wave foreground noise [146, 147]. Especially, galactic
white dwarfs [66] and extra-galactic white dwarfs [148] are considered to be
the sources of the foreground noise. Since their eﬀect is expected to be small
above 0.1 Hz [148], the lower frequency band of DECIGO is 0.1 Hz although thestochastic gravitational wave signal is larger in the lower frequency. One of theobjectives of B-DECIGO is characterization of the number of the white dwarf
to understand the feature of the foreground noise for the future DECIGO observation for the gravitational wave background. By understanding the foregroundnoise, we can develop the technique to reduce the eﬀect of the foreground noise.

4.1.2 Early alert to other electromagnetic and gravitational wave observatories
One of the most important abilities of B-DECIGO (and DECIGO) is to alert
other electromagnetic and gravitational wave observatories about the astrophysical event well in advance. For example, B-DECIGO can typically detect theneutron star binary at z = 0.1 in signal-to-noise ratio of ∼ 30 about one week
before the merger [108]. Approximately, this leads to the sky localization of34

10 -18

Strain sensitivity [1/rtHz]

10

B-DECIGO
DECIGODECIGO (3-year corr.)
NS binary30-30 M sun-19

10 -2010

10 3 -10 3 M

sun

10 4 -10 4 M sun

-21

GW

=1.5x10 -16 )10 -22
10 -2310 -2410 -25
10 -26 -2
10

10 -1

10 0

10 1

10 2

10 3

Frequency [Hz]Figure 4.2: Sensitivity of DECIGO and B-DECIGO and their observational targets. Solid lines are B-DECIGO and DECIGO sensitivities. The blue dot-dashedline is DECIGO sensitivity for stochastic gravitational wave with 3-year correlation of two clusters. The dashed lines indicate inspiral signals from the neutronstar binary (1.4-1.4 M⊙ ) and black hole binaries (30-30 M⊙ , 103 -103 M⊙ , and
104 -104 M⊙ ) at redshift z = 1. The upper end of the frequency of the inspiralsignal is set at the innermost stable circular orbit frequency, fISCO . The black
dot-dashed line is spectrum of the stochastic gravitational wave background with
ΩGW = 1.5 × 10−16 .

∼ 1 deg2 and coalescence time prediction of ∼ 10 s. Using these estimations,
B-DECIGO raises an alert. If the moment of the astrophysical transient is observed, totally new information, such as the mechanism of the gamma-ray burst,
can be provided [149].

4.1.3 Hubble constant measurement
When the sensitivity shown in figure 4.2 is achieved, DECIGO and B-DECIGO
can observe inspiral of neutron star binaries in signal-to-noise ratio of ∼ 300 and∼ 30 at redshift, z = 0.1, respectively [108]. As a result, the luminosity distanceof the source is estimated precisely. In addition, the source can be localized so
precisely that the host galaxies can be identified for many gravitational wave
events. Such a precise parameter estimation with the gravitational-wave observation provides enough information for the electro-magnetic observatories tomeasure the redshift. Since the luminosity distance can be estimated from general relativity and the redshift can be measured from the electro-magnetic wave
spectrum and its Doppler shift, the Hubble constant is expected to be mea-

35

sured with small systematic error that will be determined by the instrumental
systematic error [150].
DECIGO has an ability to measure the Hubble constant in another way onlywith the gravitational wave observation [151]. DECIGO can observe the neutron star inspiral for several years if the inspiral is observed from 0.1 Hz to 10
Hz. During the observation, the phase of the inspiral signal is gradually shiftedfrom the waveform predicted without Universe expansion. The waveform of theneutron star binary inspiral can be given by general relativity and the redshiftof the neutron star can be determined by investigating the discrepancy fromthe predicted waveform. Thus, the Hubble constant can be measured from theluminosity distance and the redshift measured with the gravitational wave observation. The precision of the waveform prediction is determined by the statisticalerror of the parameter estimation of the observed neutron star binary. The precision is expected to be around the inverse of the signal-to-noise ratio of theneutron star binary observation. The Hubble constant is expected to be measured in this precision. This measurement provides totally new information of
the Hubble constant even without conventional electro-magnetic observation.

4.1.4 Dark matter searchPrimordial black holes are candidates of the cold dark matters. The primordial
black holes with masses of MPBH are generated in the very early Universe via
MPBH
gravitational collapse if the quantum density fluctuation is larger than ∼ (cH−1 )3 ,
−1
where cH is the horizon scale [152]. Since the quantum fluctuation is considered to have a white spectrum, the primordial black hole masses can be widelydistributed between 10−38 M⊙ and 103 M⊙ . When DECIGO and B-DECIGO
have the sensitivity shown in figure 4.2, the primordial black holes with masses
in the range between 10−13 M⊙ and 10−7 M⊙ can be observed [153]. Although
this mass region has been limited by the gravitational micro lensing eﬀect [154],
the gravitational wave observation provides the independent information about
the primordial black hole dark matter.
DECIGO and B-DECIGO are sensitive instruments for another dark mattercandidate, axion dark matter. Axion is a pseudo-scalar field which is originallyproposed a half-century ago and high energy physics also predicts manifold ofaxion-like particles. From cosmological point of view, axion or axion-like particle
is considered to be a candidate of dark matter. Axion dark matter varies thevelocities of the phase of the circular-polarized light. This phase velocity variation can be measured precisely with the Fabry–Pérot cavity [155]. DECIGO and
B-DECIGO can improve the current upper limit of the axion-photon coupling
constant provided by the haloscope observation by three orders or so. This new
scheme was proposed by the researchers including the author of this thesis as aleading person and is explained in detail in Appendix A.

36

4.1.5 Distinguish formation scenario of supermassive black holes
B-DEICGO (and DECIGO, of course) observes numerous black hole binary coalescences even at very high redshift that corresponds to the age of the Universeof√the first star formation. For example, when the strain sensitivity of 10−22/ Hz is achieved with B-DECIGO, it can detect 30-30 M⊙ black hole binary
at z = 10, which corresponds to the age of the Universe of the first star formation [156]. Also the intermediate-mass black hole binaries having mass of∼ 1000 M⊙ at z = 10 can be observed with signal-to-noise ratio of 100. Thesevarious mass black hole observations allow us to distinguish formation scenarioof the supermassive black holes, which are observed in galactic centers [36, 37].

4.1.6 Testing the modified gravitational theory
Using DECIGO and B-DECIGO, the modified gravitational theories can betested precisely. Although only tensor modes of the gravitational waves areallowed in general relativity, the other modes of vector and scalar can exist in
the alternative modified gravity theories [83]. For example, Brans–Dicke theory,
which is the simplest tensor-scalar theory, predicts the existence of scalar mode in
addition to tensor mode [80]. DECIGO and B-DECIGO have an ability to probe
Brans–Dicke theory with the observation of intermediate mass ratio inspiral, e.g.
neutron star and black hole binaries with 1.4-10 M⊙ [44]. The inspiral evolutionin Brans–Dicke theory diﬀers from that predicted by general relativity because of
the scalar dipole radiation. Since the observable number of inspiral cycle of theneutron star and black hole binaries is large in DECIGO and B-DECIGO, theBrans–Dicke parameter, ωBD , which is the characteristic parameter of Brans–
Dicke theory, can be tested precisely. The current limit, ωBD > 4 × 104 , provided
by the doppler tracking with Cassini [157] can be improved by four orders with
DECIGO. In addition to the Brans–Dicke theory, other modified gravitational
theory can be tested with DECIGO and B-DECIGO [158–160].

4.1.7 Probing for Type Ia supernova progenitor
Type Ia supernovae are considered to be related with the carbon-oxygen whitedwarfs although there is no direct observational evidence. Among the whitedwarf related events, white dwarf binary merging is one promising candidate of
the Type Ia supernova progenitor. White dwarf binary merges typically between
0.05 and 0.1 Hz, which is almost in the DECIGO and B-DECIGO band, i.e.
[161]. Given the event rate of the Type Ia supernovae, DECIGO (B-DECIGO)
can observe white dwarf binaries of ∼ 20000 (∼ 20) per one year. Since thethree-dimensional localization volume of the white dwarf binary is estimated to
be about 500 Mpc3 with DECIGO, a host galaxy can be identified for many
events. Even with B-DECIGO, the nearby but rare event could be associatedwith its host galaxy. If the coincidence of the white dwarf binary and Type Ia
supernova is observed with DECIGO or B-DECIGO, the progenitor of the TypeIa supernova may be able to be probed.

37

4.2 Interferometer design
According to the scientific objectives explained above,the strain sensitivity of√−23
DECIGO is required to be better than 10/ Hz between√ 0.1 and 10 Hz.
B-DECIGO aims to achieve the strain sensitivity of 10−22 / Hz in the same
frequency range. Here, we explain the interferometer design of DECIGO and
B-DECIGO to meet their sensitivity requirement, at least, in terms of the principal noise discussed in Appendix C. The summary of this section is shown intable 4.1. Notice that the design in this chapter is mainly considered just withthe longitudinal things and simple control scheme. For more detailed design, we
need to consider the control scheme of all degrees of freedom and control scheme
as shown in Chapter 5.

4.2.1 Interferometer configuration
DECIGO and B-DECIGO are Fabry–Pérot type gravitational wave antennas.
The pre-conceptual design is shown in figure 4.1. DECIGO and B-DECIGO areconstructed as a triangular shaped interferometer. DECIGO (B-DECIGO) has
a 1000-km-long (100-km-long) arm cavity of which finesse is 10 (100). DECIGO
and B-DECIGO have shorter arms than LISA. This is because of constructing
the cavity by reducing the optical loss due to the diﬀraction, i.e. divergence ofthe laser light [127]. As a result, DECIGO and B-DECIGO have a better shotnoise level than LISA.
Compared with the L-shaped interferometer such as the current ground-based
detectors, the triangular shaped interferometer has the following advantages.
First, the calibration and false alarm ratio are improved by constructing nullstreams, which are time-series data where the gravitational wave signals are cancelled, from the linear combination of each interferometer output [61]. Since theinstruments are limited in space detectors, in-situ calibration method is important. Second advantage is the improvement of the sensitivity to the polarizationof gravitational waves [60, 61]. As discussed above, detection of non-tensor gravitational waves leads to the test of the modified gravity theories. Third, thetriangular shaped interferometer provides the redundancy which is importantfor space gravitational wave detectors.
In DECIGO and B-DECIGO, the laser lights from two satellites are injected
one arm cavity from both sides and its length can be measured with the twolasers. This type of the interferometer is called dual-pass Fabry–Pérot interferometer. The dual-pass Fabry–Pérot interferometer leads to the redundancy
with minimum number of the main mirrors. However, since the dual-pass Fabry–Pérot interferometer is new configuration for DECIGO and B-DECIGO, we needto develop it.
In order to achieve the DECIGO and B-DECIGO interferometers, two-typeconcepts as shown in figure 4.3 have been proposed up to now. Notice that
the Fabry–Pérot Michelson interferometer similar to the ground-based detector
is not eﬀective in DECIGO and B-DEICGO since it is diﬃcult to implement

38

Table 4.1: Specification and requirement of DECIGO and B-DECIGO in dualpass diﬀerential Fabry–Pérot interferometer. Spectrum items are defined at 0.1
Hz.
Items
Arm cavity
Arm length L [km]Finesse F
Wavelength λ [nm]Mirror mass m [kg]Mirror diameter [cm]Mirror thickness [cm]Radius of curvature [km]Pre stabilized laser
Output power P0 [W]
Relative intensity
noise√δP/P0 [/ Hz]
Frequency√ noiseδf [Hz/ Hz]
Interferometer sensing
and actuation
Actuator √noiseδFact [N/ Hz]
Actuator range
(max)
Fact
[µN]
Satellite
Mass mSAT [kg]Cross section [m2 ](typical)
Coupling from sat. disp.
to mirror acc. K [/s2 ]Drag-free control
Local sensor√ noiseδxLS [m/ Hz]
Control gainof satellite at 0.1 Hz
Control gainof mass at 0.1 Hz
ThrusterThruster noise√δFthruster [N/ Hz]
Thruster range(max)
Fthruster [µN]

Specification/Requirement
DECIGO
B-DECIGO

Limiting factor

103
10
515
100
166
2

102
100
515
30
70
4

607

60.7

Angular opticalanti-spring> 10> 0.02

Sensitivity< 2 × 10−9

< 10−8Sensitivity<1<1

Sensitivity< 2 × 10−17< 2 × 10−17

Sensitivity>2> 0.06(> 6)

Depend on
orbit design

1000

1000

2

2

< 5 × 10−7

< 5 × 10−7

Sensitivity< 10−12< 10−12Sensitivity

> 250

> 250

Sensitivity< 10−6< 10−6

Sensitivity< 1 × 10−7

< 1 × 10−7

Sensitivity

> 100

> 100
(> 200)Depend on
orbit design

39

Sensitivity
Diﬀraction loss
Sensitivity
Diﬀraction loss

so-called recycling techniques in space [21, 135]. The left and right concepts infigure 4.3 are called a single-source and a two-source concepts, respectively. The
advantages and disadvantages of these two concepts are summarized in table 4.2.
In the two-source concept, there is a major concern about the detection instrument, i.e. phase meter [162, 163]. To achieve the shot-noise-limited sensitivity,
a new low noise phase meter should be developed, especially for DECIGO sensitivity. On the other hand, in the single-source concept, the shot-noise-limited
measurement can be achieved with Pound–Drever–Hall technique similarly to
the ground-based gravitational wave detectors [164, 165]. Thus, we focus on the
single-source concept in this thesis and this concept is called a dual-pass diﬀerential Fabry–Pérot interferometer. Note that the two lasers confronting in thecavity have orthogonal polarization for decoupling the dual-pass interferometer.
(a)Drag-free satellite(b)Drag-free satellite

Laser

Laser
PM

PM

EOM
PD

Laser

PD

EOM
PD

EOM

PD

ThrusterThrusterMirrorThrusterCavityCavityPD

PD
Mirror

Laser
EOM

EOM

CavityThruster

EOM

PM

PD

PM
Laser

PD

Drag-free satellite

Drag-free satellite

Laser
PM

EOM

Laser

PD
Drag-free satellite

PD

EOM

MirrorCavityLaser EOM

ThrusterCavityCavityPD

PM

PD
Thruster

Laser
Drag-free satellite

Figure 4.3: Two proposed concepts of the Fabry–Pérot type space gravitational
wave antennas. PM means phase meter.
The dual-pass diﬀerential Fabry–Pérot interferometer can be divided into threearm-cavity pairs as shown in figure 4.4. In the interferometer (1), one cavity is
kept to resonate with the feedback control of the laser frequency and the resonance of the other cavity is kept by controlling the mass position. This kindof interferometer is called a diﬀerential Fabry–Pérot interferometer (or locked
Fabry–Pérot interferometer), which was used in the ground-based gravitational
wave detectors [166, 167]. The interferometer (2) can be operated as anotherdiﬀerential Fabry–Pérot interferometer same as the interferometer (1). The interferometer (3) is not simple. Since the length of Cavity b is determined by
the interferometer (1), the mass position cannot be controlled arbitrarily. In
addition, the laser frequencies are shifted for each other for decoupling the threeinterferometers. Therefore, in order to resonate Cavity b for both lasers of interferometer (1) and (3) simultaneously, we need to adjust the length of Cavity
a and b. They can be adjusted, for example, with auxiliary ranging instruments
explained in Section 4.3.5. For more detail, see Chapter 6 also.

40

Table 4.2: Comparison of the two concepts of the proposed Fabry–Pérot typespace gravitational wave antennas.
Concept type

Single source

Two sources

Advantage• Simple configuration,• Shot noise limited sensitivity can be achieved
with the state-of-the-art
interferometer technique.

• Three arms can be regarded independently,• Mass control is not necessary (in length control),i.e. mass actuator noise
does not matter.

Disadvantage• To obtain length signals of three arms, absolute length should be adjusted,• Low noise mass actuatoris necessary,• Three arms should becontrolled in phase.

• To obtain shot noise limited sensing, new low
noise sensing technique
should be developed,• Doubled space and electric power are necessary
for two laser sources.

4.2.2 Arm cavity design
DECIGO has 1000-km-long arm cavities. The finesse of the cavity is 10 andcorresponds to the amplitude reflectivity of the mirrors of 0.84. The test masses
that float in the satellites are made of fused silica and each test mass weighs
100 kg. The geometry of the test mass is as follows: the diameter, 166 cm; thethickness, 2 cm; the radius of curvature of high reflectivity surface, 607 km. The
diameter is designed to receive enough amount of light to reduce diﬀraction loss
in terms of optical power less than 6%. The radius of curvature is designed toobtain stable cavity and to weaken the optical angular anti-spring eﬀect, known
as Sidles–Sigg eﬀect [168], as explained in Section 5.2.2. B-DECIGO cavity is 100
km and its finesse is 100. The mirror diameter is 70 cm to make diﬀraction loss
less than 0.6%. Above parameters and the others are summarized in table 4.1.

4.2.3 Pre stabilized laser
The seed laser source of DECIGO and B-DECIGO is the Yb-doped fiber distributed feedback laser of which wave length is 1030 nm [169]. The seed laser is
injected to a second harmonic generator and converted to the green light, 515
nm. The laser has to be stabilized and amplified in√order to meet the√ require−9
ment of DECIGO (P0 = 10 W, δP/P0 = 2 ×√
10 / Hz, δf =√1 Hz/ Hz) and−8
B-DECIGO (P0 = 0.02 W, δP/P0 = 10 / Hz, δf = 1 Hz/ Hz), which aresummarized in table 4.1.

41

Figure 4.4: Schematic of division of the DECIGO or B-DECIGO interferometer
into three pairs of the arm cavities.

For the power amplification to meet the B-DECIGO requirement, the seedlaser is amplified with Yb doped fiber amplifier in 1030 nm using master oscillator
power amplifier technique. Although the way to amplify the laser to 10 W has notbeen decided, the coherent combing scheme, which is used in the ground-based
gravitational wave detector [170, 171], is one candidate to meet the requirementof DECIGO.
The laser intensity is stabilized with feedback control [169]. The amplified
laser is splitted into two paths. One beam goes to the main interferometer andthe other is received by the photodetector. The latter signal is fed back to thecurrent of the laser diode used as a pump laser for the above power amplification.
As a result, the intensity of the laser introduced to the main interferometer isstabilized.
The frequency stabilization is performed with the iodine cell [169]. In general,
stable frequency reference is necessary for the frequency stabilization. In thefrequency stabilization with the iodine cell, the saturated absorption line is used
as frequency
reference. The laser frequency fluctuation is stabilized better than 1√
Hz/ Hz. Since this stability is not enough to detect gravitational wave, the laser42

is more stabilized with main arm cavity. For this purpose, as shown in figure 4.5,
the laser before the iodine cell is introduced to the acousto-optic modulator,
which can shift the laser frequency proportional to the applied voltage. With
this scheme, the laser frequency can be controlled with the main interferometersignal [172]. Hereafter, we call the pre-stabilized laser just the laser source unlessotherwise noted.
Signal from main interferometer(can control frequency of the laser
injected to the main inteferometer)

To maininterferometer

Laser system
Seed laser

PBS

0th

AOM

1st

Iodine cell

PD

HWP

FBS

Beam dump
QWP

Frequency stabilization

Figure 4.5: Schematic of the frequency stabilizing system with double-passacousto-optic modulator scheme. FBS is a fiber beam splitter; QWP is a quarterwave plate.
It is important to note that the central frequencies of the three laser sources ineach satellite are shifted for each other by a few hundred MHz for decoupling thedual-pass interferometers. Without the frequency shifting, the relative motion of
the laser source, i.e. the satellite, makes large phase noise due to Doppler shift.
However, this frequency shift requires the absolute cavity length adjustment tooperate the three dual-pass interferometers as explained in Chapter 6.

4.2.4 Sensing and actuation
In DECIGO and B-DECIGO, the same sensing method is planned to be used.
The longitudinal signal of the cavity is obtained with the Pound–Drever–Hallmethod [129]. The absolute length of the cavity is measured with the beatnote
frequency of the two dual-pass laser resonating in the same cavity as shown infigure 4.6. The absolute cavity length, L, and the beatnote frequency, ∆ν, havea relationship as
Nc
L=
,
(4.1)
2∆ν
where N is the number of the beatnote frequencies in terms of the free spectral range. Since N can be determined from the independent measurement,
e.g. see [173], the absolute length of the cavity can be determined from equation (4.1) [174, 175]. The precision is wavelength level, i.e. sub-µm or so. The
alignment signals are obtained with the wave front sensing method [133, 134]
and the dithering method [131]. In addition to these global sensings, in thesatellite, the local Michelson interferometer to measure diﬀerential optical path43

fluctuation is also implemented. The diﬀerential optical path fluctuation could
be a noise source due to the diﬀerential Doppler shift. In order to suppress the
Doppler noise, the local Michelsoninterferometer should have better sensitivity√√−15 0.1 Hz
than 2.4 × 10 (/f ) m/ Hz for DECIGO and 2.4 × 10−14 (0.1 Hz/f ) m/ Hz
for B-DECIGO.

For anothersatellite

Laser

PDbeatS pol.

HWP
PBS
P pol.

+
PD

Laser

L

Satellite1

Satellite 2

Figure 4.6: Schematic of measuring the beatnote, ∆ν, of the lasers from twosatellites. PDbeat is the photodetector for the beatnote measurement. HWP is
the half wave plate, and PBS is the polarizing beam splitter. The half wave plateis used to adjust the polarization and make interference between the lasers from
Satellite 1 and 2. Note that the two laser lights in the arm cavity are shifted in
the illustration while they are overlapped in fact.
The actuator for cavity control is the frequency actuator in the laser source
and the displacement actuator on the test mass. The frequency actuator in√ thelaser is the acousto-optic modulator and has to have lower noise than 1 Hz/ Hz.
For the test mass actuator, electro-static actuator [176,177]√ is planned to be used
−17
and its requirement of noise is better than 2 × 10
N/ Hz for DECIGO and
B-DECIGO.
The actuator range is designed to have force enough to compensate relative
acceleration to the test masses. In the space detectors, the main causes of theforce generating the relative acceleration are the laser radiation pressure and thegravity from the satellite and the Earth. The amount of the force varies by thedetector configurations. In DECIGO, the laser radiation pressure is dominant tobe 0.1 µN. When B-DECIGO is in heliocentric orbit, gravity from the satellitecauses dominant noise which is given byG(∆m)mTM
,
(∆r)2

(4.2)

where ∆m is the mass asymmetry and ∆r is the distance between the test massand the mass asymmetry. Since we assume ∆m = 3 kg and ∆r = 1 m, thetest mass of B-DECIGO feels the static force of about 6 × 10−9 N. Note that44

the laser radiation pressure in B-DECIGO is about 2 nN which is subdominant
force. When B-DECIGO is in geocentric orbit, gravity from the Earth causes the
major force that is about 0.6 µN as discussed in Section 4.3.4. Consequently, therequirements for the actuators of DECIGO and B-DECIGO in heliocentric orbit
is > 1 × 10−6 N and > 6 × 10−8 N, respectively. We assume the safety factorof 10 in terms of the maximum force range of the actuator. For B-DECIGO in
geocentric orbit, actuator range should be > 6 × 10−6 N. Again, we assumed thesafety factor of 10. In order to achieve the dynamic range of the mass actuatorfor DECIGO and B-DECIGO in geocentric orbit, we need to design frequency
dependent actuation eﬃciency: low gain and low noise in observation frequency(0.1-10 Hz), high gain and large noise below observation frequency (≪ 0.1 Hz).
Actually, since actuators with these values are too weak to achieve the cavity
resonance at first moment, known as lock acquisition, stronger actuators (morenoisy) will be used during the lock acquisition stage. For alignment actuation,
steering mirrors are implemented in each optical path.
For the DECIGO and B-DECIGO operation, the control scheme, i.e. how to
use the sensed signal and actuators, is important. This will be discussed in detailin Chapter 5.

4.2.5 Input and output optics
After the laser is output from the laser source, the laser is transferred to theoptical bench via optical fiber. During the transfer, phase modulation is appliedwith an electro-optic modulator for the Pound–Drever–Hall method and thewave front sensing method. In the optical bench, the light is divided into twopaths for two arm cavities and the polarization is adjusted for discriminationof the dual-pass interferometer. Note that most of the optics are fixed on the
optical bench monolithically [178] except for a few steering mirrors, and so on.
Then, the light is expanded with telescope system before injected to the arm
cavity. The telescope system is one of the components that should be newly
developed since the very long cavity requires precise adjustment of the spacialmode of the incident laser. The reflected light is contracted in reversing way.
The reflected light and the transmitted light coming from the other satellite are
distinguished with the polarization and frequency as explained in Section 4.2.1
and Section 4.2.3.
Using the laser distinguished with the polarization and frequency, the noisecaused by the other laser can be reduced. The noise due to the other laser ismainly caused by the intensity fluctuation and is written as
hlaser ∼αλ δP (∆ν)
√
βLF
P0

1
(
1+

∆ν
ν1/2

)2 ,

(4.3)

where α is the amplitude of the non-desired polarized electric field normalized by
the amplitude of the desired polarized electric field, and β is the modulation index
of the laser for the Pound–Drever–Hall technique. When we√ design ∆ν ∼ 50
δP (∆ν)−7∼ 10/ Hz [179, 180]. As
MHz, the typical intensity fluctuation is P045

a√result, the noise caused by the other laser is estimated to be hlaser ∼ 10−28/ Hz and is 105 (106 ) times smaller than the target sensitivity of DECIGO (BDECIGO). Here, we assume α = 0.01 and β = 0.1. For λ, L, and F, the values
shown in table 4.1 are used. ν1/2 can be calculated to be 7.5 Hz from L and F.

4.3 Satellite system design
4.3.1 Overview of satellite
Each satellite is equipped with two test masses, their actuators, a pre-stabilized
laser, and local input/output optics. In addition, local sensors around masses,
thrusters, auxiliary sensors, data acquisition/processing/transfer systems, andpower suppling system are also incorporated into the satellite. The latter two
are not discussed in this thesis.
The summary of the satellite parameters is shown in table 4.1. The mass ofsatellite will be around 1000 kg and the typical cross section will be about 2
m2 . To avoid generating mechanical vibration noise, movable components, e.g.
momentum wheel, are used as little as possible. Since the mast system canweaken the structure of the satellite and enlarge a mechanical vibration noise,
the solar cell will be mounted on the body.
The test mass is stored in the satellite without any mechanical connection.
However, there is a small coupling from the satellite motion to the test massacceleration because of gravitational force or electromagnetic force. The coupling
factor, K [/s2 ], strongly depends on the geometry of the test mass and satellite
while it is estimated to be 10−6 /s2 , at most [140]. In DECIGO and B-DECIGO,
K is designed to be less than 5 × 10−7 /s2 with a geometry management.

4.3.2 Drag-free satellite
The drag-free control is the relative motion and angle control of the satellite
following the test mass [57]. The drag-free control minimizes the dragging force
eﬀect to the test mass [139,140,181,182]. In DECIGO and B-DECIGO, main objective is reduction of solar radiation pressure fluctuation eﬀect. In the drag-freecontrol, local sensor to measure the relative motion/angle between the satellite
and the mirror and low noise thrusters are used and their noise level and range
are important. The contribution from the satellite motion is written asDF
δaDF
TM = KδxSAT
)
(
GDF
1
δFthruster=K
δxLS +,
1 + GDF
1 + GDF mSAT ω 2

(4.4)
(4.5)

where δaDF
TM is the acceleration of the test mass due to the drag-free control,DF
δxSAT is the displacement of the satellite, and GDF is the open loop transfer
function of the drag-free control (about the open loop transfer function, see
Appendix B).
As the local sensor, interferometric sensors are planned to be used for DECIGO and B-DECIGO while LISA will use electro static sensors [177, 183]. The

46

ultimate specification of drag-free control is limited by the sensor noise. The requirement for drag-free control is more strict in DECIGO and B-DECIGO thanthat of LISA because of the force noise√requirement diﬀerence. Interferometricsensor has low noise level, ∼ 10−12 m/ Hz, which meets the DECIGO and BDECIGO requirement [184, 185]. About the range, orthogonal quadratures typeinterferometer has a wide range [185].
The thruster for the drag-free control should have low noise although its noise
can be suppressed by the drag-free control. Lower noise thrusters make thelower requirement for the control gain of the drag-free control. In addition,
the thruster should be continuously variable for the precise drag-free control.
Otherwise,quantization noise could be a problem. If the thruster noise is 1×10−7√
N/ Hz, the control gain should be larger than 250 at 0.1 Hz. There are various
thruster candidates which can meet the above features [186], e.g. cold-gas jet,ion thruster, and field-emission electric propulsion. However, they have not been
fully characterized and some of them have not been tested in space. Therefore,
the design will be finalized after further study.
The largest force applied to the satellite is solar radiation pressure in heliocentric orbit and gravity of the Earth in geocentric orbit. Since the typical solar
radiation power density is 1360 W/m2 [140, 144], the solar radiation pressure
force is estimated to be ∼ 10 µN. Here, we assume that the typical satellite
cross section is 2 m2 . To compensate this amount of force, the low noise thruster
should have maximum force larger than 100 µN in heliocentric orbit when we use
safety factor of 10. The gravity from the Earth in geocentric orbit is about 20µN as discussed in Section 4.3.4. This leads to the requirement for the thruster
range in geocentric to be larger than 200 µN with safety factor of 10.
Considering that the three-dimensional mechanical degree of freedom of thesatellite is six and that of the two test masses is 12 in total,*1 we cannot control all
the relative motions and angles between the satellite and the two test masses onlywith the drag-free control. Thus, one test mass should be controlled to follow thesatellite with the test mass actuators, i.e. electro static actuators.*2 However,if the control band width is as high as the observation band, the sensitivity canbe easily contaminated. To avoid it, the test mass control gain is designed to beless than 10−6 at 0.1 Hz. The detailed discussion is described in Chapter 5.

4.3.3 Test mass housing
The test mass is placed in the housing in the satellite. The housing includes
electrostatic actuators, interferometric local sensors, vacuum system, discharging
system [187], and launch locking system. Here, they are explained briefly. The
former two are mentioned above. The vacuum system is necessary in DECIGO
and B-DECIGO to reduce noise caused by the molecular gas around the test
mass. The molecules stochastically hit the test mass and make the test massfluctuate. This noise is called residual gas thermal noise. To obtain enough small*1 The satellite and the mirror are each considered to be a rigid body.
*2 There is other choice of which degrees of freedoms are used for the drag-free control. For

example, the motion of two test masses (3 + 3) can be followed by the satellite [139, 182].

47

residual gas thermal noise, the vacuum level should be better than 10−10 Pa. The
discharging system is practically important to operate the interferometer stably.
Since the test mass does not touch anywhere mechanically, it is gradually charged
by the cosmic rays. When the charge of the test mass is large, its coupling
to the electrostatic actuator gets strange. Therefore, the test mass should bedischarged occasionally with, e.g., ultraviolet light and so on. The launch locksystem protects the test mass from the large vibration during launch. The test
mass will be softly released after the satellite arrives proper position.

4.3.4 Orbit
DECIGO is planned to be in the heliocentric orbit as shown in figure 4.7. DECIGO is composed of the four clusters, i.e. 12 satellites [188]. Two of fourclusters are overlapped in opposite shape to take correlation for the stochastic
gravitational wave background. Using the four clusters, we can observe the multimode gravitational waves, i.e. not only tensor mode gravitational wavs but also
vector/scalar mode gravitational waves, for testing the modified gravitational
theories.
B-DECIGO orbit will be heliocentric orbit or geocentric orbit as shown in figure 4.7. In geocentric orbit a bit far or near orbit from geostationary orbit isconsidered for now. In contrast to DECIGO, B-DECIGO will have one cluster.
Note that even with one cluster, we can localize gravitational wave events considering the time variation of the angular response of the detector, called antenna
pattern discussed in Appendix G, during the event. For example, it is expectedthat one neutron star inspiral is typically in B-DECIGO observation band in ∼ 1
year.

Earth
Sun

Sun

Heliocentric orbit

Geocentric orbit

Figure 4.7: Schematic of the heliocentric and geocentric orbit.
In geocentric orbit, the gravity of the Earth, especially its J2 term, limits therange of actuator and thruster. J2 term is an ellipticity measure of the Earthand makes a diﬀerential acceleration of the two masses, aJ2 , given by
2
3GME rE= 8.2 × 10−6
aJ2 ≃ J24
2rD(

42000 km
rD

)4

[
]
m/s2 ,(4.6)

where, J2 (= 1.08263 × 10−3 ) is the J2 constant, G is gravity constant, ME is
the mass of the Earth, rE is the radius of the Earth, and rD is orbital radius of48

B-DECIGO. The diﬀerential acceleration between two masses in distance of L is
written as
(
)(
)5
]
L
42000 km [L
−8
∆aJ2 ≃
aJ2 = 1.9 × 10
m/s2 .
(4.7)
rD
100 kmrD

4.3.5 Auxiliary sensors and thrusters
Auxiliary sensors
In addition to the main interferometer, DECIGO and B-DECIGO have various
auxiliary sensors. For example, star trackers, beam scanning systems with an
acousto-optic deflector, sun sensors, global positioning systems, microwave ranging systems [189], and so on. They are used for rough ranging in orbit before
interferometer lock acquisition [190]. During observation, their signals are used
for the calibration of satellite position and orientation. The calibration precisionaﬀects the systematic error of the gravitational wave localization. For example,
typical star tracker has the angular resolution ∼ 10 arcsec and it corresponds
to ∼ 10−5 deg2 , which is much better than the statistic localization error while
relative calibration of the star tracker and interferometer axis is necessary.
Thrusters
In addition to the low noise thruster, two types of auxiliary strong thrusters
will be incorporated. The first one is bipropellant catalytic hydrazine thruster(N2 O4 /N2 H4 ) for apogee kick. The thruster has typically 500-N thrust force.
The second one is 1-N-class cold gas jets. They are used for orbital control
which will be performed when we need to move the satellite a lot, e.g. during
interferometer lock acquisition. These two strong thrusters are turned oﬀ duringobservation for avoiding causing the vibration noise.

49

Chapter 5

Control scheme of DECIGO and
B-DECIGO
DECIGO and B-DECIGO are currently designed to be unique space gravitational
wave antennas with Fabry–Pérot arm cavities and drag-free satellites. Eachtechnique has been independently demonstrated, e.g. the former in ground-based
detectors, and the latter in LISA path finder [139,182], while the combination ofthem has been neither demonstrated nor considered. Actually, although the twocontrols can be regarded almost independently in LISA configuration [191], they
are strongly related to DECIGO and B-DECIGO configurations. Therefore,
the control topology in which the interferometer and drag-free control should
be developed for DECIGO and B-DECIGO. Moreover, feedback control system
introduces additional noise through the feedback loop. The additional noise could
disturb the gravitational wave observation. Therefore, control system should be
carefully designed especially in observation frequency band [58, 59].
In this chapter, we study on the control topology of DECIGO type space detector and construct the numerical control model. Moreover, using the model, thedetailed requirements including control design, the stability, and the sensitivityof DECIGO and B-DECIGO are considered.
This chapter is organized as follows: In Section 5.1, we show the setup andbasis definition used in this chapter. From Section 5.2 to Section 5.4, as a preparation of the numerical model construction, we newly modeled the mechanicaland opto-mechanical response, and sensing and actuation scheme in DECIGO
and B-DECIGO. This modeling clarifies which degrees of freedom should becontrolled in the interferometer control or the drag-free control. In Section 5.5,
dominant sources of noise and instability are shown. In Section 5.6, using theresult prepared in the above sections, we design the control topology of DECIGO
and B-DECIGO. After that, in Section 5.7, the numerical control model of DECIGO and B-DECIGO is constructed with the designed control topology. The
model is the new model including full responses of the Fabry–Pérot type spaceantenna and is named a full DECIGO interferometer model. Subsequently, in
Section 5.8, using the full DECIGO interferometer model, we consider the detailed requirement including control design for DECIGO and B-DECIGO, andinvestigate the performance of the design control scheme. Especially, the sta-51

bility of the control and the interferometer operation, and noise performance of
DECIGO and B-DECIGO are investigated. Finally in Section 5.9, the obtained
results are discussed.

5.1 Setup and basis definition
The setup that we consider is shown in figure 5.1. Note that not all componentsof DECIGO and B-DECIGO are shown here. For example, the electro-optic
modulator is not shown. All components except for the test masses are fixed
on the satellite. Hereafter, we assume all test masses have the same mass mTM ,reflectivity r, and the radius of curvature Rc and the optical loss of the cavity is
negligible. The length of each cavity is La(b,c) and they are macroscopically thesame (L ≃ La,b,c ). Each laser frequency is represented as f1(2,3) .
The basis with which we consider the interferometer and satellite motion andangle is shown in figure 5.1-(b). Hereafter, we assume that the center of mass
and geometry of the satellite are located at the same position, and the satellites
and the test masses are rigid bodies when we consider the control topology. The
origin of each basis is defined at initial position of the center of satellite mass forthe satellites, and the center of mass for the test mass. The optical basis origin
is defined at the initial waist position that is located at the center of the cavity
since the cavity is composed of the mirror having the same radius of curvature.
The mechanical position and angle vector of Test Mass iα and Satellite i are represented as xiα ≡ (xiα , yiα , ziα , θrol,iα , θpit,iα , θyaw,iα )⊤ andxSATi ≡ (xSATi , ySATi , zSATi , θrol,SATi , θpit,SATi , θyaw,SATi )⊤ (i = 1, 2, 3, and α =a,b,c). The optical alignment of Cavity α and incident beam from Satellite icavα cavα cavα ⊤are expressed as xcavα≡ (xcavα , y cavα , z cavα , θrol, θpit , θyaw ) and xinciα ≡iαinciα inciα inciα ⊤(xinciα , y inciα , z inciα , θrol, θpit , θyaw ) . In terms of Satellites 1(2,3) basis, test
mass TM1b(2c,3a) is at (a cos(π/6), a sin(π/6), 0)⊤ and test mass TM1c(2a,3b) is
at (a cos(π/6), −a sin(π/6), 0)⊤ . The force and torque applied to the Test mass iα
and Satellite i are expressed as Fiα ≡ (Fxiα , Fyiα , Fziα , Tθrol,iα , Tθpit,iα , Tθyaw,iα )⊤and FSATi ≡ (FxSATi , FySATi , FzSATi , Tθrol,SATi , Tθpit,SATi , Tθyaw,SATi )⊤ , respectively.

5.2 Cavity and satellite
5.2.1 Mechanical property
Table 5.1 shows the mechanical property of the test masses and satellites. They
are regarded as rigid bodies in the consideration of control. The satellite is
assumed to be cylindrical shaped along z-axis and to have uniform density. In
addition, we assume that they are damped with the surrounding gas of whichpressure is 10−10 Pa and damping ratio is the same for all degrees of freedom as1011 Hz. As a result, the equations of motion of the test mass and the satellite

52

(b)

(a)
LAS1
SAT1
PD1bc

PD1b
BD

pitch

TM1c zCavity b
TM3b

PD2Lc
PD2La

Cavity a

TM2c

y

PD3b

a

PD3Lb

y

PD3ab
LAS2

TM2a

PD2ca
SAT2

TM3a

z

SAT3

6 6 z
y TM1bzyPD3La PD3a

PD2a

y
6

a z

roll

y

y x
x zCavity cCavity a

xx

TM2a

LAS3

x

x y
z Cavity
b
x
x
TM

6

x

y
x a

a

x

Cavity c

y

z

PD1Lc
TM1b

PD2c

yaw

Thruster

PD1Lb

TM1c

TM2c

z

SAT1

PD1c

z

y

x

z

a

6

z6 x
y a

TM3a

3b

z
y

SAT3

SAT2

Figure 5.1: (a) Setup for control topology consideration. PD is photodetector,
LAS is laser, SAT is satellite, TM is test mass, and BD is beam dump. The
photodetector is composed of the photodetector and quadrant photodetectorsthat are used for length and alignment sensing although the quadrant photodetectors are not explicitly shown in the figure. The mirrors located between the
beamsplitter just in front of the laser source and each test mass are used assteering mirrors. The black boxes around the test masses are each their housing
including actuators and local sensors. Note that not all components on DECIGO
and B-DECIGO are shown here. (b) Basis of each component. Black and blue
colors indicate the bases for satellites and test masses, respectively. Green color
indicates the basis for the optical axis for cavity and injected beam. The top
right subpanel shows the definition of six degrees of freedom of the basis.

are given by
MTM (ẍiα + γTM ẋiα ) = Fiα ,(5.1)

MSAT (ẍSATi + γSAT ẋSATi ) = FSATi ,(5.2)

where Miα ≡ diag(mTM , mTM , mTM , ITM,rol , ITM,pit , ITM,yaw ) and MSAT ≡diag(mSAT , mSAT , mSAT , ISAT,rol , ISAT,pit , ISAT,yaw ) are the mass matrices(mTM(SAT) is the mass of the test mass (satellite) and ITM[SAT],rol(pit,yaw) is
the moment of inertia along x(y, z)-axis) of the test mass and the satellite,
respectively, and γTM ≡ γTM I6 and γSAT ≡ γSAT I6 (I6 is a 6-by-6 identity
matrix) are the damping matrices of the test mass and the satellite, respectively.
γTM(SAT) is a damping ratio of the test mass (satellite). By performing Laplacetransformation (for detail, see Appendix B), equations (5.1) and (5.2) aredenoted as′
MTM (−ω 2 I6 + iωγTM )x̃iα ≡ MTM
x̃iα = F̃iα ,

(5.3)

′
MSAT (−ω I6 + iωγSAT )x̃SATi ≡ MSATx̃SATi = F̃SATi ,

(5.4)

2

where the tilde indicates the frequency domain parameters.

53

Table 5.1: Mechanical property of test masses and satellites. In the parentheses,
B-DECIGO parameters are shown if they are diﬀerent from DECIGO one.
Mass [kg]Diameter [m]Thickness [m]Moment of inertia [kg·m2 ]along x-axis (Roll)along y-axis (Pitch)
along z-axis (Yaw)
Damping ratio [Hz]Test mass
100 (30)
1.66 (0.70)
0.02 (0.04)

Satellite
1000
1.6
2.0

36 (1.8)
18 (0.90)
18 (0.90)
10−11

490
490
320
10−11

The satellite motion is coupled to the acceleration of the test mass. The
relationship is given byẍ1b(2c,3a) = KRSATto M1 xSAT1(2,3) ,(5.5)

ẍ1c(2a,3b) = KRSATto M2 xSAT1(2,3) ,

(5.6)

where
( )
( )

cos (π6 ) sin ( π6 ) 0000− sin π
cos π600( )0 ( ) a
6


π
π


0
0
1
a
sin
−a
cos
0
,
( π )6( π6)RSATto M1 ≡ sin ( 6 )000
0 cos (6 )



0
0
0 − sin π6
cos π6
0
0
0
0
0
0
1
(
)
(
)

cos (− π6 ) sin (− π6 ) 000π− sin − π
cos − 6
0
0(
0(
6

)
)
π
π

0
0
1
a
sin
−
−a
cos
−
( 6)
(
)6
RSATto M2 ≡ 000 cos (− π6 )sin (− π6 )000 − sin − π6
cos − π6
0
0
0
0
0


(5.7)


0
a

0
,
0

0
1
(5.8)
are rotation matrices from the satellite position and angle to the mass position
and angle.

5.2.2 Opto-mechanical property
The test masses for the Fabry–Pérot cavity are coupled via radiation pressure of
the laser. This coupling is called opto-mechanical coupling. The angular optomechanical coupling, known as Sidles–Sigg eﬀect [168], aﬀects the cavity stability
since one opto-mechanical mode behaves as an anti-spring. Also, translational
opto-mechanical couplings are considered [192]. These couplings are represented54

as  2P cav
2P cav− c
00Fy,1θyaw,1cRccav
2P cav   Fy,2   0

− 2Pc
0
cRc  θyaw,2  ,
=

cav
mc
2P
Tθyaw,1   0
00   y1 ccav
2PTθyaw,2
y2mc
0
0
0
c


  2P cav

2P cav
00Fz,1θpit,1ccRc
2P cav
2P cav   Fz,2   00ccRc  θpit,2  ,

=
cav
mc
2P
Tθpit,1   00− c
0   z1 cav
2PTθpit,2
z2mc
0
0
0
− c


(5.9)

(5.10)

where P cav is intracavity power,(Fy(z),1 , Fy(z),2 ) = (Fy(z)1b , Fy(z)3b ), (Fy(z)2c , Fy(z)1c ), (Fy(z)3a , Fy(z)2a ),(5.11)(Tθyaw(pit),1 , Tθyaw(pit),2 ) = (Tθyaw(pit),1b , Tθyaw(pit),3b ), (Tθyaw(pit),2c , Tθyaw(pit),1c ),(Tθyaw(pit),3a , Tθyaw(pit),2a ),(5.12)(θyaw(pit),1 , θyaw(pit),2 ) = (θyaw(pit),1b , θyaw(pit),3b ), (θyaw(pit),2c , θyaw(pit),1c ),(θyaw(pit),3a , θyaw(pit),2a ),(5.13)

mc
mc
and y(z)mc
1(2) is the mis-centering on the test mass and defined as (y(z)1 , y(z)2 ) ≡
mc
mc
mc
mc
mc
(y(z)mc
1b , y(z)3b ), (y(z)2c , y(z)1c ), (y(z)3a , y(z)2a ). The mis-centering on eachtest mass is denoted asmc
y1b(2c,3a)=−

L cavb(c,a)θyaw+ y cavb(c,a) ,2

L cavb(c,a)
mc
z1b(2c,3a)= θpit+ z cavb(c,a) ,2

L cavb(c,a)
mcy3b(1c,2a)= − θyaw− y cavb(c,a) ,2
L
cavb(c,a)mc
z3b(1c,2a)= − θpit+ z cavb(c,a) .
2(5.14)

5.3 Sensing method
Hereafter, for simplicity, we use sensing eﬃciency normalized with its DC value.
This treatment does not lose generality.

5.3.1 Interferometer longitudinal sensing
In each cavity, its longitudinal fluctuation signal is taken with the Pound–Drever–Hall technique. As discussed in Section 3.3.4, each photodetector measures thephase fluctuation of the cavity. When we measure the phase in terms of length,55

the measured signals at photodetectors are each written as)((L)
HFP (ω)δf1
sPD1b(1c) = (L),−δx3b(2c) − δx1b(1c) + L
f1
HFP (0)()(L)
HFP (ω)δf2−δx1c(3a) − δx2c(2a) + L,
sPD2c(2a) = (L)f2
HFP (0)()(L)
HFP (ω)δf3−δx2a(1c) − δx3a(3c) + L.
sPD3a(3c) = (L)f3
HFP (0)(5.15)

Moreover, the absolute length of each cavity can be measured as
sPD1Lb (3Lb ) = Lb ,sPD2Lc (1Lc ) = Lc ,sPD3La (2La ) = La .

(5.16)

5.3.2 Alignment sensing
In the alignment sensing of DECIGO and B-DECIGO, the wavefront sensing andthe dithering method are used. Using the wavefront sensing method [133,134], we
can measure the diﬀerence between the alignment of the incident beam (y(z)incinc
cav
and θyaw(pit)) and the cavity axis (y(z)cav and θyaw(pit)). The signal is expressedassQPD1b ,y(z ) = y(z)cavb − y(z)inc1b ,

cavb
inc1b
sQPD1b ,yaw(pit) = θyaw(pit)− θyaw(pit),sQPD1c ,y(z ) = y(z)cavc − y(z)inc1c ,

cavc
inc1c
sQPD1c ,yaw(pit) = θyaw(pit)− θyaw(pit),sQPD2c ,y(z ) = y(z)cavc − y(z)inc2c ,

cavc
inc2c
sQPD2c ,yaw(pit) = θyaw(pit)− θyaw(pit),sQPD2a ,y(z ) = y(z)cava − y(z)inc2a ,

cava
inc2a
sQPD2a ,yaw(pit) = θyaw(pit)− θyaw(pit),sQPD3a ,y(z ) = y(z)cava − y(z)inc3a ,

cava
inc3a
sQPD3a ,yaw(pit) = θyaw(pit)− θyaw(pit),sQPD3b ,y(z ) = y(z)cavb − y(z)inc3b ,

cavb
inc3b
sQPD3b ,yaw(pit) = θyaw(pit)− θyaw(pit),(5.17)

where the cavity axis can be written with the position and angle of the test massasy1b(2c,3a) − y3b(1c,2a)θyaw,1b(2c,3a) − θyaw,3b(1c,2a)
+ Rc
,
2
2
y1b(2c,3a) + y3b(1c,2a)
Rc θyaw,1b(2c,3a) + θyaw,3b(1c,2a)1cavb(c,a)
+
,
θyaw
=
L
2
2
Rc − 2Rc − L2y cavb(c,a) =z1b(2c,3a) + z3b(1c,2a)θpit,1b(2c,3a) + θpit,3b(1c,2a)
− Rc
,
2
2
z1b(2c,3a) − z3b(1c,2a)
1
Rc θpit,1b(2c,3a) − θpit,3b(1c,2a)cavb(c,a)
θpit
=−
+
.
L
2
2
Rc − 2Rc − L2(5.18)z cavb(c,a) =56

Here, we assume L2 < Rc < L. Note that the quadrant photodetectors arelocated in the photodetector that has the same index.
With the dither sensing method [131], we can measure the mis-centering of
the beam spot on each test mass in the longitudinal signal. Note that the beam
mis-centering is denoted with the same basis as each test mass. The beam miscentering on each test mass can be distinguished with the dithering frequency.
For example, when we dither TM1b and TM3b at 1.03 Hz and 1.39 Hz, the miscentering signals on TM1b and TM3b appear at 1.03 Hz and 1.39 Hz, respectively.
The signals that we can measure are expressed as
mc
mc
mc
mc
sPD1b[3b] ,y1b(3b)= y1b(3b)= z1b(3b)− y1b(3b) , sPD1b[3b] ,z1b(3b)− z1b(3b) ,mc
mc
sPD2c[1c] ,y2c(1c)= y2c(1c)− y2c(1c) ,mc
mc
sPD2c[1c] ,z2c(1c)= z2c(1c)− z2c(1c) ,mc
mc
sPD3a[2a] ,y3a(2a)= y3a(2a)− y3a(2a) ,mc
mc
sPD3a[2a] ,z3a(2a)= z3a(2a)− z3a(2a) . (5.19)

In addition to the above single cavity alignment signal, the diﬀerence of thetwo cavity axes can be measured with QPD1bc(2ca, 3ab) similarly to the wave
front sensing as
sQPD1bc(2ca,3ab) ,y = y cavb(c,a) + y cavc(a,b) ,cavb(c,a)cavc(a,b)sQPD1bc(2ca,3ab) ,yaw = θyaw
− θyaw,sQPD1bc(2ca,3ab) ,z = z cavb(c,a) − z cavc(a,b) ,cavb(c,a)sQPD1bc(2ca,3ab) ,pit = θpit

cavc(a,b)+ θpit.

(5.20)

5.3.3 Local sensing
Using local sensors, i.e. interferometric sensors, the test mass positions from thesatellite can be measured. The obtained signal is written as
sLS,1b(2c,3a) = x1b(2c,3a) − RSATto M1 xSAT1(2,3) ,sLS,1c(2a,3b) = x1c(2a,3b) − RSATto M2 xSAT1(2,3) ,(5.21)

where sLS,iα ≡ (sLS,xiα , sLS,yiα , sLS,θrol,iα , sLS,θpit,iα , sLS,θyaw,iα )⊤ is the obtained
signal.

5.4 Actuation method
Hereafter, for simplicity, we use actuation eﬃciency normalized with its DC
value. This treatment does not lose generality. Note that we assume that the test
masses and the satellites float in free space though they rotate around the center
object (Sun or Earth) in reality. In other words, we neglect the centrifugal force
applied to the test mass and the satellite due to the actuation. In heliocentric
orbit, the additional centrifugal force applied to the test mass (satellite) due
to the actuation is ∼ 0.1 (1) nN and it is much smaller than the other forces.
In geocentric orbit, the additional centrifugal force applied to the test mass

57

(satellite) due to the actuation is ∼ 30 (1000) µN. Although these values arecomparable with the other forces that should be compensated, this is just a
problem of the actuator diagonalization.

5.4.1 Frequency actuation
The laser frequency can be actuated by applying the voltage to the laser source.
When the voltage, vf1(2,3) , is applied to the laser source, the frequency is variedasδf1(2,3) = vf1(2,3) .
(5.22)

5.4.2 Test mass actuation
The test mass position and angle are actuated with the test mass actuator, i.e.
electro-static drive. By applying the voltage, viα , we can apply the force andtorque δFiα to the test mass asδFiα = viα ,(5.23)

viα ≡ (vxiα , vyiα , vziα , vθrol,iα , vθpit,iα , vθyaw,iα )⊤ ,(5.24)

whereδFiα ≡ (δFxiα , δFyiα , δFziα , δTθrol,iα , δTθpit,iα , δTθyaw,iα )⊤ .

(5.25)

Here, δFx(y,z)iα and δTθrol(pit,yaw),iα are the force along x(y, z)-axis and the torquealong roll (pitch, yaw) angle applied to the test mass.
Notice that when the force and torque are applied to the test mass, its backbrreaction, δFSAT1(2,3)
is applied to the satellite as

where

br
δFSAT1(2,3)= −RM1to SAT δF1b(2c,3a) ,(5.26)br
δFSAT1(2,3)= −RM2to SAT δF1c(2a,3b) ,(5.27)( )
cos ( π6 ) sin π6
 0
RM1to SAT ≡ 
 0

 0
0

( )
− sin( π6)cos π6
0
0
0
a

(
)
(
)

cos ( − π6 ) − sin( − π6) sin − πcos − π6600RM2to SAT ≡ 

0
0


0
0
0
a

0
0
1( )
a sin (π6 )−a cos π6
0
0
0
1(

0
0
0( )
cos ( π6 )sin π6000)(0 π )
πa sin (− 6 ) cos ( − 6 )−a cos − π6
sin − π6
0
0

58


0
0

0
,
0

0
1
(5.28)
0
0
0
0

0(
0
.
)
− sin( − π6) 0cos − π6001(5.29)000( )− sin( π6)cos π60

are rotation matrices from the satellite position and angle to the test mass position and angle.

5.4.3 Satellite actuation
The satellite is actuated with low noise thrusters. With the applying voltage,
vSATi , the force and torque applied to the satellite are expressed as
δFSATi = vSATi ,(5.30)

where
vSATi ≡ (vxSATi , vySATi , vzSATi , vθrol,SATi , vθpit,SATi , vθyaw,SATi )⊤ ,(5.31)⊤δFSATi ≡ (δFxSATi , δFySATi , δFzSATi , δTθrol,SATi , δTθpit,SATi , δTθyaw,SATi ) . (5.32)

5.4.4 Incident beam alignment
The incident beam to the cavity is aligned with the satellite and steering mirrors
on the satellite. The incident beam is given byxinc1b(2c,3a) = RSATto Inc1 xSAT1(2,3) + xSTM1b(2c,3a) ,(5.33)

xinc1c(2a,3b) = RSATto Inc2 xSAT1(2,3) + xSTM1c(2a,3b) ,(5.34)STMiα STMiα STMiα ⊤where xSTMiα ≡ (xSTMiα , y STMiα , z STMiα , θrol, θpit, θyaw ) denotes thesteering mirror position and angle, which are defined in the basis of the incident
beam, and
( )
( )


cos (π6 ) sin ( π6 ) 0000− sin π
cos π600( )0 ( ) L2 
6


L
L
π
π


0
0
1 2 sin( 6) − 2 cos
0 , (5.35)( π )6
RSATto Inc1 ≡ 
π

sin
0
0
0
0
cos


(6π )
( 6π )
000 − sin 6
cos 6
0
0
0
0
0
0
1
( )
( 5π )

cos (5π
sin
0
0
0
6 )
6 ) 0
(
− sin 5π
cos 5π00( )0( ) − L2 
6
6


5π
L

0
0
1 − L2 sincos( 5π
0 
,

6
2
6
)
)
(
RSATto Inc2 ≡ 
5π
5π

0
0
0
cos
sin
0


6 )
(65π )( 5π
00
0 − sin 6
cos 6
0 
0
0
0
0
0
1
(5.36)
are rotation matrices from the satellite position and angle to the incident beam
position and angle.

595.5 Sources of the noise
In order to estimate the noise level, the sources of noise in DECIGO and BDECIGO are listed here. The listed noise sources are also used to estimate theactuator loading. Noises not shown here are discussed in Appendix C and they
are negligible for the investigation of the control.

5.5.1 Interferometer noise
Quantum noise
As discussed in Appendix C, the most fundamental limit of the laser interferometer is determined by quantum noise due to the quantumness of photon. The
quantum noise is divided into two components: shot noise and radiation pressure
noise. They are given by [115]
v
[
u
)2 ]
(
1 u
πℏcλ
f
t
xshot (f ) =1+,
4F 2Pincν1/2v
u 2πℏP
1
16F
incu
xrad (f ) =)2 ,(tπ(2πf )2 mTM
cλf
1 + ν1/2(5.37)(5.38)

where m is the mass of the test mass, ℏ is reduced Planck constant, and λ is
the wavelength of the laser. Note that this is equivalent to the diﬀerential cavity
length fluctuation, i.e. not to one test mass fluctuation.
Frequency noise
Equation (3.69) indicates that the laser frequency fluctuation is a noise sourceof the laser interferometer. The frequency noise of the laser source in DECIGO
and B-DECIGO is given by√δf1(2,3) = 1 Hz/ Hz(5.39)
Intensity noise
According to equation (3.70), there is a kind of sensing noise due to the intensity
fluctuation coupled with the cavity detuning, which is written as
xint = ∆x

δP,P0

where ∆x is the cavity detuning in terms of length.

60(5.40)

Alignment sensing noise
The noises of the wave front sensing for shift and tilt of the optical axis are givenby [193]{√
δyWFS = δzWFS ≃ 2 × 10−10 m/ Hz,√(DECIGO)(5.41)
δθpit,WFS = δθyaw,WFS ≃ 8 × 10−16 rad/ Hz,{√δyWFS = δzWFS ≃ 1 × 10−9 m/ Hz,√δθpit,WFS = δθyaw,WFS ≃ 5 × 10−14 rad/ Hz.

(B-DECIGO)(5.42)For the global attitude sensing, we assume the same noise level as the wave front
sensing.
The noises for the dithering method for beam spot shift measurement are givenby [194]{√
3 × 10−8 m/ Hz, (DECIGO)√
δydith = δzdith ≃(5.43)
5 × 10−9 m/ Hz. (B-DECIGO)
Here, we assume that dithering frequency is 1 Hz and applied force is 0.06 µN.
This leads to the root-mean-square of the beam spot fluctuation of about 0.1
mm.
Test mass tilt coupling
The tilt motion of the test mass causes length noise through the beam spot
mis-centering on the test mass. This tilt coupling noise is approximately givenby [195]
δxtilt = δθyaw(pit) ∆y(z)RMS ,(5.44)
where δθyaw(pit) is yaw (pitch) motion of the test mass and ∆y(z)RMS is the
root-mean-square of the beam mis-centering along y(z)-axis.
Absolute length sensing noise
The absolute length sensing is more noisy than sensing with the Pound–Drever–Hall technique. We assume the absolute length sensing noise is to be√δxAbsL = 1 × 10−13 m/ Hz.
(5.45)

5.5.2 Force noise on test masses
The major force noise on the test mass is caused by electrostatic actuator, thermal radiation pressure, and the gravity. To investigate the force noise eﬀect
on the control scheme, it is enough to consider these noises since they are thedominant noise sources.

61

Actuator noise
The actuator noise is important above 0.001 Hz. The actuator of DECIGO and
B-DECIGO is planned to be electrostatic actuator. The force noise is assumed
to be√δFact ≃ 2 × 10−17 N/ Hz,(5.46)√
δTact ≃ 2 × 10−17 Nm/ Hz.
(5.47)
Note that the force noise level depends on the required actuator range [176].
Thermal radiation pressure
When temperature of the housing around test mass fluctuates, the thermal radiation pressure applied to the test mass causes force noise. The thermal radiationpressure force fluctuation, δFTrad , is given by [140]δFTrad = ϵTrad

4σSB STM T 3 δT,c

(5.48)

where σSB is Stefan–Boltzmann constant, STM is the cross section of the test
mass, δT is temperature fluctuation of housing, and ϵTrad is common mode rejection ratio. When the temperature fluctuations in front and back of the test
mass are correlated, the thermal radiation pressure is reduced. When we√ assume
ϵTrad = 0.1, STM = 2 m2 , T = 300 K, δT = 10−8 (0.1 Hz/f )1/3 K/ Hz, andtypical scale of temperature fluctuation is 0.5 m, the force and torque noises dueto the thermal radiation pressure are written as
δFTrad ≃ 4 × 10

−17

δTTrad ≃ 2 × 10

−17

(
(

0.1 Hzf
0.1 Hz
f

)1/3
)1/3

√
N/ Hz,(5.49)√
Nm/ Hz.

(5.50)

Gravity
The gravity force is significant in the low frequency. In heliocentric orbit, thegravity field caused by the satellite applies static force of 20 nN according toequation (4.2). In geocentric orbit, the gravity of the Earth J2 term is ∼ 0.6 µN
at ∼ 10−5 Hz (∼ /1 day). We need to confirm that this force is successfully
suppressed with the feedback control. This point is discussed in the end of
Section 5.8.2

5.5.3 Force noise on satellites
About the force noise on the satellite, solar radiation force and thruster noise
should be considered. In the geocentric orbit, we also need to consider gravity
from the Earth.

62

Solar radiation pressure
The solar radiation pressure and its fluctuation are given by [140, 142, 144]
WSun SSAT= 9 × 10−6 N,c()1√
δWSun SSAT
0.1 Hz 3−9δFSrad == 3 × 10
N/ Hz,
cf
FSrad =(

(5.51)(5.52)) 13√
where WSun = 1360 W/m and δWSun = 0.2
W/m2 / Hz are the solar
radiation power density and its fluctuation, respectively. The solar radiationpressure torque depends on the satellite shape. When we assume the typicalscale of the satellite of 1 m and 10% coupling from the force to torque, the solar
radiation torque is denoted as
2

0.1 Hzf

1
FSrad = 9 × 10−7 Nm,
10
(
)1
√
1
0.1 Hz 3−8δTSrad =Nm/ Hz.
δFSrad = 3 × 1010f
TSrad =

(5.53)(5.54)

Thruster noise
In DECIGO and B-DECIGO, 100-µN class thrusters are planned to be used.
Their force and torque noises are approximately given by [177]√δFthr = 1 × 10−7 N/ Hz,√δTthr = 1 × 10−7 Nm/ Hz.

(5.55)(5.56)

Here, we assumed that the two thrusters are placed at a distance of 1 m.
Gravity of the EarthIn B-DECIGO with the geocentric orbit, the gravity of the Earth makes someamount of force and torque at ∼ 10−5 Hz in the same manner as the gravity on
the test masses. They are written as
FGrE ∼ 2 × 10−5 N,(5.57)

TGrE ∼ 3 × 10−7 Nm.

(5.58)

5.5.4 Sensing noise of local sensors
As a local sensor, interferometric sensors are planned to be used. The sensitivity
is given by [185]√δxLS = 1 × 10−12 m/ Hz.
(5.59)

63

By putting six local sensors to measure the six mechanical degrees of freedomof the test mass, we measure the tilt of the test mass. Approximately, the tilt
sensing sensitivity is written as√δθLS = 3 × 10−12 rad/ Hz.
(5.60)
Here, the distance of the local sensors are assumed to be ∼ 0.5 m.
The noise of the star tracker is assumed to be√
δθSTT = 2 × 10−6 rad/ Hz.

(5.61)

5.6 Control topology
Here, we show the basic control topology of length, alignment, drag-free, andlocal control of DECIGO and B-DECIGO. In this section, the coupling between
the satellite and the test mass, back reaction, and opto-mechanical coupling are
not considered for simplicity. The eﬀects of these couplings can be suppressed
when the control gain is enough. Note that they considered in the constructed
model shown in Section 5.7 to quantitatively evaluate the eﬀects of the couplings.
For more description about the feedback control, such as Laplace transformation,
the block diagram, and stability analysis, see Appendix B. Note also that there
is no inter-satellite communication in terms of the data processing in the following design since the precise control with the inter-satellite communication ischallenging.

5.6.1 Length control
The block diagram of the length control is shown in figure 5.2. The signals
obtained from Cavity c with PD1c and PD2c, i.e. sPD1c and sPD2c are fed
back to Laser 1 and Laser 2, respectively. The output of PD3a, i.e. sPD3a is
fed back to Laser 3. Signals from PD1b and PD2a are fed back to TM1b andTM2a (longitudinal motion), respectively. The longitudinal motion of TM1c iscontrolled with the absolute length measurement of Cavity c, i.e. sPD1Lc . The
gravitational wave signal is taken from PD1b, PD2a, and PD3b.
It is worth noting that PD3b can measure the signal of Cavity b even without
the feedback of any signal. This is achieved by adjusting the length of Cavity aand Cavity b in the dual-pass diﬀerential Fabry–Pérot interferometer as shownin Chapter 6.

5.6.2 Alignment control
Since the alignment control having the same topology is implemented to all thecavities, we focus on one cavity, Cavity b, first. In the alignment control, theposition of the test mass is set as a reference of the alignment control. In otherwords, these two parameters are regarded as parameters external to the control

64

L

f3

HFP

f3

sPD3a

Gf3

x3a
x2a

HFP

sPD2a

G2a
L

f2

HFP

f2

Gf2

x2c
x1c

HFP
G1La

L

sPD2c

f1
f1

sPD3b

HFP
sPD1c

sPD1La
Gf1
G1b

x1b
x3b

sPD1b

HFP
(L)

(L)

Figure 5.2: Block diagram of length control. We define HFP ≡ HFP (ω)/HFP (0).
Gf1(2,3) and G1b(2a) are the control gains of each control loop.

loop. From this point of view, the alignment sensing vector is denoted asθyaw,1bsQPD1b(3b) ,y( )()
sQPD1b(3b),yaw  θyaw,3b 
y
x
1b
SAT1
′
′′



sASC1b(3b),y ≡  sPD1b(3b) ,y mc  = Ry y STM1b(3b)  + Ry y3b + Ry xSAT3 ,
1b
STM
mc
sPD1b(3b) ,y3b
θyaw 1b(3b)(5.62)

θpit,1bsQPD1b(3b) ,z( )()
sQPD1b(3b),pit  θpit,3b 
z1b′′′ xSAT1sASC1b(3b),z ≡ = Rz z STM1b(3b)  + Rz
+ Rz
,
mc 
sPD1b(3b) ,z1b
z3b
xSAT3
STM
mc
sPD1b(3b) ,z3b
θpit 1b(3b)(5.63)′′′
where Ry(z) , Ry(z)and Ry(z)are the transfer matrices from each component
to the measured signals. Among these matrices, Ry and Rz are important to65

design the control topology and written as


Rc
2
Rc

 2(Rc − L2 )

Ry = − Rc (L−Rc ) 2(Rc − L2 )

Rc2
−2 R −( c L2 )



− R2c

Rc

 2(Rc − L2 )

Rz =  Rc (L−Rc ) 2(Rc − L2 )

Rc2
−2 R −( c L2 )


0
Rc
−1

2(Rc − L2)Rc2,−2 R −L00( c 2)
(L−Rc )
− R2 cR00( c − L2 )

− R2c
−1 0
c
− 2 RR−0 −1
( c L2 )
Rc2.
−2 R −L00( c 2)
Rc (L−Rc )002(Rc − L
2)
− R2c

−1
0

(5.64)(5.65)The processed signals are fed back to the test mass angles and incident beamalignments. The relation between them and the feedback voltages is written asθyaw,1b)( θyaw,3b 
11
 STM1b(3b)  = diag,
, 1, 1y
ITM,yaw (−ω 2 + iγω) ITM,yaw (−ω 2 + iγω)STMθyaw 1b(3b)()⊤· vθyaw,1b , vθyaw,3b , vySTM1b(3b) , v STM1b(3b)
θyaw





≡ (MASCy )−1 vASCy,1b(3b) ,(5.66)

θpit,1b()
 θpit,3b 
11
 STM1b(3b)  = diag,
, 1, 1zITM,pit (−ω 2 + iγω) ITM,pit (−ω 2 + iγω)STMθpit 1b(3b))⊤(· vθpit,1b , vθpit,3b , vzSTM1b(3b) , v STM1b(3b)θpit≡ (MASCz )−1 vASCz ,1b(3b) ,(5.67)

where vy(z)inc1b(3b) and vθinc1b(3b) are the voltages fed back to the steering mirror
yaw(pit)1b(3b).
Figure 5.3 shows the block diagram of alignment control of one cavity. According to the discussion in Appendix B, the control matrix needs to meet thefollowing condition:(MASCy )−1 Gy Ry = (diagonal matrix),(MASCz )−1

Gz Rz = (diagonal matrix).

66(5.68)(5.69)

The simple choice of the control gain, Gy(z) , is [57])(Gy = MASCy diag Gθyaw,1b , Gθyaw,3b , Gyinc1b(3b) , Gθinc1b(3b) Ry−1 ,yaw)(Gz = MASCz diag Gθpit,1b , Gθpit,3b , Gzinc1b(3b) , Gθinc1b(3b) Rz−1 ,(5.70)(5.71)

pit

where Gθyaw(pit),1b(3b) , Gy(z)inc1b(3b) , and Gθinc1b(3b) are the control open loop gains
yaw(pit)of each loop. The above discussion in this section including equations (5.62)(5.71) is valid for Cavity b and Cavity a by replacing (1b, 3b, SAT1, SAT3) to(2c, 1c, SAT2, SAT1) and (3a, 2a, SAT3, SAT2) in equations (5.62)-(5.71).
ySAT
yTM
yASC

R'
y'

sASC1b(3b),y

R'
y

zSAT

R'
z'

zTM

R'
z

zASC

Ry
My-1

Gy

sASC1b(3b),z

Rz
Mz-1

Gz

Figure 5.3:
Block diagram of alignement control.
We deSTM1b(3b) STM1b(3b) ⊤fine yASC≡(θyaw,1b , θyaw,3b , y, θyawzASC≡) ,STM1b(3b) STM1b(3b) ⊤⊤⊤(θpit,1b , θpit,3b , z, θpit) , yTM ≡ (y1b , y3b ) , zTM ≡ (z1b , z3b ) ,⊤and y(z)SAT ≡ (xSAT1 , xSAT3 ) .

5.6.3 Drag-free and local control
In the drag-free control, TM1c, TM2a, and TM3a are used as references for
SAT1, SAT2, and SAT3, respectively. The obtained signals are fed back to
the thruster of the satellite. The block diagram for these controls is shown infigure 5.4. With the similar discussion in the previous section, we choose thegain matrix, GDF1(2,3) , as
GDF1 = (RSATto M2 )−1′· diag(GxSAT1 , GySAT1 , GzSAT1 , Gθrol,SAT1 , Gθpit,SAT1 , Gθyaw,SAT1 )MSAT,(5.72)

GDF2 = (RSATto M2 )−1′· diag(GxSAT2 , GySAT2 , GzSAT2 , Gθrol,SAT2 , Gθpit,SAT2 , Gθyaw,SAT2 )MSAT,(5.73)

GDF3 = (RSATto M1 )−1′· diag(GxSAT3 , GySAT3 , GzSAT3 , Gθrol,SAT3 , Gθpit,SAT3 , Gθyaw,SAT3 )MSAT,(5.74)

where GxSAT1(2,3) , GySAT1(2,3) , GzSAT1(2,3) , Gθrol,SAT1(2,3) , Gθpit,SAT(2,3)1 , andGθyaw,SAT1(2,3) are the open loop gains of each degree of freedom.
The test masses other than the reference of the drag-free control are locallycontrolled with local sensors partially. Specifically, the positions and rolls of67

Satellite 1
x1c
RSATtoM2

xSAT1

sLS,1c

-1

(M'
SAT)

Satellite 2
xSAT2

x2a

Satellite 3

sLS,2a

RSATtoM2
-1

(M'
SAT)

GDF1
x3a
RSATtoM1

xSAT3

sLS,3a

-1

(M'
SAT)

GDF2

GDF3

Figure 5.4: Block diagram of the drag-free control.

TM2c and TM3b are controlled and the positions only along y- and z-axes androll of TM1b are controlled. The block diagram of this control is shown infigure 5.5. The control gain is designed as
GLS,1b = diag(0, GLSy,1b , GLSz ,1b , GLSθroll ,1b , 0, 0),(5.75)

GLS,2c = diag(GLSx ,2c , GLSy,2c , GLSz ,2c , GLSθroll ,2c , 0, 0),(5.76)

GLS,3b = diag(GLSx ,3b , GLSy,3b , GLSz ,3b , GLSθroll ,3b , 0, 0),(5.77)

where GLSx ,2c(3b) , GLSy,1b(2c,3b) , and GLSz ,1b(2c,3b) are the open loop gains ofeach degree of freedom.
Satellite 1
GLS,1bxSAT1

x1b
RSATtoM1

sLS,1bSatellite 3

Satellite 2

GLS,3b

GLS,2cxSAT2x2c
RSATtoM1

xSAT3

sLS,2cx3b
RSATtoM2

sLS,3bFigure 5.5: Block diagram of the local control.

5.6.4 Global attitude control of satellite formation
In addition to the above control, global attitude of satellite formation needs to becontrolled. For this purpose, the cavity axis diﬀerence in yaw motion of Cavity
b and c is fed back to y-axis motion of TM1c. The cavity axis diﬀerence in pitch
motion of Cavity c and a (a and b) is fed back to the z-axis motion of TM2a68

(TM3a). The obtained signal, sQPD1bc ,y(z ) , is denoted as1) y1c + f (y1b , y3b , y2c , θyaw,1b , θyaw,3b , θyaw,2c , θyaw,1c ),sQPD1bc ,y = − (2 Rc − L2(5.78)
1) z2a + f ′ (z2c , z1c , z3a , θpit,2c , θpit,1c , θpit,2a , θpit,3a ),sQPD2ca ,z = − (2 Rc − L2(5.79)
1) z3a + f ′′ (z3b , z2a , z1b , θpit,3a , θpit,2a , θpit,3b , θpit,1b ).
sQPD3ab ,z = (L
2 Rc − 2(5.80)In addition, the roll motion of the drag-free control reference masses, i.e.
TM1c, TM2a, and TM3a, is controlled with the star trackers.

5.6.5 Summary of control
In the end of this section, the summary of control topology is shown in table 5.2.
Note that x and the roll of incident beam alignment do not need to be controlled
and x- and y-axis motion of TM3a, y-axis motion of TM2a, and z-axis motionof TM1c define the reference frame of the control. In other words, the attitudeof the triangular interferometer can be changed by actuating these test masses.
This topology is implemented in the model constructed in Section 5.7.

5.7 Control model construction
In order to analyze the control of DECIGO and B-DECIGO, we construct a
control model with Matlab and Simulink. In this model, mechanical and optomechanical properties (Section 5.2), sensing and actuation method (Section 5.3
and 5.4), and control topology (Section 5.6) are implemented with graphicallycoding using block diagrams of Simulink. The constructed model is shown inAppendix H. We named this model a full DECIGO interferometer model. We
used the parameters that we considered in this chapter and shown in table 4.1.
With the model, we can calculate the transfer function from each excitation port
to each sensing port and noise spectrum based on the calculated transfer function(for more details of the noise estimation method, see Appendix B). Therefore,
whether the controls of all degrees of freedom shown in table 5.2 are stable andthe requirements of noises and their root-mean-square for the sensitivity andstability are met can be investigated.

69

Table 5.2: Summary of control topology. Used signal name to control each degree
freedom is shown. Align, alignment control; Global, global attitude control; DF,drag free control; STT, star tracker.
x
y
z
Roll
Pitch
Yaw

TM1b
Length
Local
Local
Local
Align
Align

y
z
Pitch
Yaw

x
y
z
Roll
Pitch
Yaw

SAT1
DF
DF
DF
DF
DF
DF

TM1c
LengthGlobal
Ref
STT
Align
Align

Inc1b
Align
Align
Align
Align
SAT2
DF
DF
DF
DF
DF
DF

TM2c
Local
Local
Local
Local
Align
Align

Inc1b
Align
Align
Align
Align

TM2a
Length (Abs.)
Ref
Global
STT
Align
Align

Inc1b
Align
Align
Align
Align

SAT3
DF
DF
DF
DF
DF
DF

f

Inc1b
Align
Align
Align
Align

LAS1
Length

TM3a
Ref
Ref
Global
STT
Align
Align

Inc1b
Align
Align
Align
Align

TM3b
Local
Local
Local
Local
Align
Align

Inc1b
Align
Align
Align
Align

LAS2
LengthLAS3
Length

5.8 Investigation of the control system with the control
model
5.8.1 Open loop transfer function
Using the full DECIGO interferometer model, we designed the open loop gainsof each loop described in Section 5.6. Figures 5.6-5.12 show the designed open
loop gains for DECIGO. The figures indicate that all degrees of freedom of eachcomponent are stably controlled (the stability can be checked with the phase at
the unity gain frequency as discussed in Appendix B). The peaks around 10−4
Hz, which appear in figures 5.7, 5.10, and 5.11, are caused by the coupling, K,
between the satellite and the test masses. Below the peak frequency, we cannot
increase the gain of the control loop. For B-DECIGO, the control gains aredesigned in the same way and all controls are made to be stable.

70

Gain

Laser 1
Laser 2
Laser 3

10-4

10-4

10-3

10-3

10-2

10-2

10-1

10-1

100

100

101

101

102

102

103

103

104

Phase [deg]GainPhase [deg]

109
8
10
7
10
6
10
105
104
3
10
102
101
100
10-1
10-5
180
90
0
-90
-180
10-5

104

Frequency [Hz]

1012
11
1010
10 9
108
10
7
106
105
104
103
102
101
100
10
10-1
10-2
10-3
10-5
180
90
0
-90
-180
10-5

TM1b xTM2a x

10-4

10-3

10-2

10-4

10-3

10-2

10-1

100

101

102

103

104

10-1

100

101

102

103

104

Frequency [Hz]Phase [deg]GainFigure 5.6: Open loop transfer function of frequency control loop. The
open loop gain for Laser 1 is overlapped by that for Laser 2.

105
104
103
102
101
100
10-1
10-2
10-3
-4
10
10-5
-6
10
10-7
10-5
180
90
0
-90
-180
10-5

Figure 5.7: Open loop transfer function of x-axis motion of TM1b andTM2a.

TM1c x

10-4

10-3

10-2

10-4

10-3

10-2

10-1

100

101

102

103

104

10-1

100

101

102

103

104

Frequency [Hz]Figure 5.8: Open loop transfer function of x-axis motion of TM1c.

Figure 5.9: Open loop transfer function of alignment control. The open
loop gain for TM1b Yaw is overlapped by that for TM3b Yaw. Also
the open loop gain for STM1b (yand Yaw) and STM3b (y) is overlapped by that for STM3b Yaw.

71

Gain
Phase [deg]

105
104
3
10
2
10
1
10
0
10
10-1
-2
10
10-3
10-4
10-5
10-6
10-7
10-5
180
90
0
-90
-180
10-5

TM2c xTM2c yTM2c zTM2c Roll

10-4

10-3

10-2

10-4

10-3

10-2

10-1

100

101

102

103

104

10-1

100

101

102

103

104

Frequency [Hz]Phase [deg]GainFigure 5.10: Open loop transfer
function of the drag-free control ofSatellite 1.

105
104
103
102
101
100
10-1
10-2
10-3
10-4
10-5
10-5
180
90
0
-90
-180
10-5

Figure 5.11: Open loop transfer
function of local control of TM2c.
The open loop gain for y-axis motion
is overlapped by that for z-axis.

TM1c yTM2a zTM3a z

10-4

10-3

10-2

10-4

10-3

10-2

10-1

100

101

102

103

104

10-1

100

101

102

103

104

Frequency [Hz]Figure 5.12: Open loop transfer function of the global attitude control.

5.8.2 Noise suppression performance
We investigate the noise suppression performance of the designed control system.
Specifically, the residual motion between each mass and satellite and the residual cavity fluctuation are estimated. If the former is large, the actuator response
could be non-linear. If the latter is large, the intensity noise coupling could be
so large that the sensitivity can be degraded. Figures 5.13 and 5.14 show theresidual displacement and tilt between each test mass and the satellite for DECIGO case, respectively. They meet the requirement for obtaining the linear
response of the electro-static actuator (displacement, 0.1 mm; tilt, 10−4 rad). In
B-DECIGO case, it is confirmed that the requirement is met. Figures 5.15 and5.16 show the residual cavity fluctuation of DECIGO and B-DECIGO, respectively. In DECIGO and B-DECIGO, the root-mean-square values of the residualfluctuation, i.e. cavity detuning, are 5 × 10−10 m and 1 × 10−10 m, respectively.
The intensity noise with this cavity detuning is estimated in the next subsection.

72

Figure 5.13: Residual displacement
between each test mass and thesatellite in DECIGO.

Figure 5.14: Residual tilt between
each test mass and the satellite in
DECIGO.

Figure 5.15: Residual cavity fluctuation of DECIGO.

Figure 5.16: Residual cavity fluctuation of B-DECIGO.

For B-DECIGO in geocentric orbit, the largest fluctuation of the cavity iscaused by the gravity of the Earth. As mentioned in Section 5.5.2, we needto confirm the fluctuation caused by the gravity of the Earth is suppressed by
the control system. In order to keep the longitudinal cavity fluctuation to beless than 10−10 m, i.e. not to make the gravity of the Earth aﬀect the cavity
detuning, the transfer function from the force applied to the test mass to thelongitudinal cavity fluctuation should be less than 10−5 m/N at around 10−5 Hz(≃ /1 day). According to the estimation with the constructed full mode, thetransfer function is less than 10−5 m/N in each cavity requirement.

5.8.3 Actuator range
In Chapter 4, we estimate the required actuator range considering the external
force that should be compensated. However, the sensing noise also takes a partof the actuator range through the feedback system. Thus, we investigate theamount of the applied force and torque for control as shown in figures 5.17 and5.18 for DECIGO and figures 5.19 and 5.20 for B-DECIGO. The range required73to each actuator of the test mass is 2 × 10−7 N in DECIGO and 8 × 10−7 N in
DECIGO.

Figure 5.17: Amount of the applied
force for control of DECIGO.

Figure 5.18: Amount of the applied
torque for control of DECIGO.

Figure 5.19: Amount of the applied
force for control of B-DECIGO.

Figure 5.20: Amount of the applied
torque for control of B-DECIGO.

5.8.4 Noise budget of DECIGO and B-DECIGO
Using the full DECIGO interferometer model with the designed control loop, weestimate the expected noise budget of DECIGO and B-DECIGO. Figures 5.215.23 show DECIGO cases and figures 5.24-5.26 show B-DECIGO cases. Note
that, in figures 5.21-5.26, L-shaped interferometer equivalent sensitivities areshown. Here, we consider the noise sources described in Section 5.5. We assumed
the cavity detuning of 5 × 10−10 m for DECIGO and 1 × 10−10 m for B-DECIGO
as shown in figures 5.15 and 5.16 and the root-mean-square of the beam spot
mis-centering of 0.1 mm. The latter is mainly caused by the dithering. In thesefigures, the curves labeled ‘ASC’ and ‘Global ASC’ mean the noises equivalent tothe strain that issues from the sensing noise of the alignment control and global
attitude control, respectively. For the frequency noise of DECIGO, we assume
the noise cancellation by a factor of 10 with post processing.

74

Figure 5.21: DECIGO noise budget (interferometer 1) with the full control implementation.

Figure 5.22: DECIGO noise budget(interferometer 2) with the full control implementation.

Figure 5.23: DECIGO noise budget(interferometer 3) with the full control implementation.

75

Figure 5.24: B-DECIGO noise budget (interferometer 1) with the full control
implementation.

Figure 5.25: B-DECIGO noise budget (interferometer 2) with the fullcontrol implementation.

Figure 5.26: B-DECIGO noise budget (interferometer 3) with the fullcontrol implementation.

765.9 Discussion
5.9.1 Control filter in the real detector
In order to design the control filter, we should know the interferometer responses
while, in the constructed model, they are known in advance as described in
Section 5.2-5.4. In the real detector, the interferometer response should be diagonalized in orbit after launch. Similar investigation was performed in the
ground-based detectors, especially, for suspension systems after their construction. This kind of work is known as commissioning. By using the auxiliary
sensors including the local interferometric sensors, the commissioning can also
be performed in space.
As shown in figure 5.6, the unity gain frequency of the control loop for thefrequency control loop in DECIGO is about 2 kHz that is higher than the free
spectral range of the arm cavity, 150 Hz. Such a high unity gain frequency
is necessary to reduce the residual cavity fluctuation shown in figure 5.15. The
source of the residual motion is the laser frequency fluctuation. At the frequencies
of multiple of the free spectral range, the gain of the cavity response goes zeroand the phase is delayed by 90 deg. Therefore, compensation of the phase delay
is necessary to obtain the unity gain frequency above the free spectral range.

5.9.2 Actuator range
The required actuator range for the developed control system is 2 × 10−7 N
in DECIGO. This is within the range considered in terms of the external force
in Chapter 4. To obtain the dynamic range of the actuator in DECIGO, newtechniques, such as frequency-dependent actuators, are necessary. The required
actuator range for the control system in B-DECIGO is 8 × 10−7 N, which ismainly for the noise of the star tracker for the global attitude control. The
actuator range is larger than that of the heliocentric-orbit case mentioned in
Chapter 4. This result indicates that the frequency-dependent actuators or otherimproved actuator are necessary for B-DECIGO even in the heliocentric orbit.
Another solution to save the actuator range is the development of the low noisestar tracker or compatible sensors.

5.9.3 Dominant technical noise source
Figures 5.21-5.26 indicate that the noise performance of the feedback control
meets the sensitivity requirement of DECIGO and B-DECIGO. As shown infigures 5.21-5.26, the dominant noise sources are coupling noise from the sensing
noise of the local sensor in 0.1-0.2 Hz and the alignment sensing noise below 0.1
Hz. Local sensing noise is mainly coupled from the x-axis local control shown inequations (5.76) and (5.77). As shown in figure 5.11, although the control gain
is rolled oﬀ at 0.1 Hz, we can improve the shape of control filters by considering
the measured open loop gain in actual operation. Notice that above 0.1 Hz,77the sensitivity is limited by the residual motion of the satellites due to the local
sensing noise. Alignment sensing coupling is caused by the global attitude control
in equation (5.78). Cavity c can be mis-aligned by the motion of TM1c and they-axis motion of TM1c can couple with the x-axis motion through the motion ofSatellite 1. Thus, the global attitude control can make a spurious large couplingfrom the alignment sensing noise to the length of Cavity b which is included inInterferometer 1 and 3. This coupling can be reduced by improving the shapeof the control filter shown in figure 5.12, if necessary. These discussions also
support the qualitative reliability of the full DECIGO interferometer model.

5.9.4 Eﬀect of the technical noise source on the gravitational-wave
observation.
As shown in figures 5.21-5.26, the sensitivity of DECIGO and B-DECIGO islimited not only by the fundamental noise, i.e. quantum noise, but also by thetechnical noise. We investigate the eﬀect of the sensitivity degradation fromthat only with quantum noise that is considered in the pre-conceptual designof DECIGO [54]. Here the detection range (sometimes called inspiral range) of
the compact binary coalescence is investigated. The detection range is a kindof measure of the detectability of the compact binary coalescence and is definedas the luminosity distance for the binary detected at a certain signal-to-noise

10 5

Detection range [Gpc]

Full
Only quantum noise

10 4

10 3

10 2

10 1
10 0

10 1

10 2

10 3

10 4

10 5

Source mass [M sun]

Figure 5.27: Comparison of DECIGO detection ranges with all noises and onlywith quantum noise. Note that the source mass is represented in the detector
frame.

78

ratio, typically eight.*1 Using the equations (2.35) and (B.4), we calculate thedetection range of the sensitivity with the full noise sources and that only withquantum noise shown in figure 5.21. The result is shown in figure 5.27. Notethat the integration band is between 0.01 Hz and the frequency of the innermost
stable circular orbit (equation (2.36)). Figure 5.27 indicates that the detection
range is degraded by about 30% below 104 M⊙ and is decreased steeply above
104 M⊙ . The degradation below 104 M⊙ is caused by the intensity noise. The
detection range above 104 M⊙ is decreased since the heavy binary merges below
the sensitivity band, which is 0.06 Hz or so. In spite of the degradation of
the detection range due to the technical noises, most of the scientific targets
discussed in Chapter 4 are still within the reach of DECIGO and B-DECIGO.
Especially, the detection range longer than 200 Gpc is enough to observe thecompact binaries since the first star of the Universe is thought to be formed at aluminosity distance of 200 Gpc [196]. Actually, the probability of the detection of
the light white-dwarf binaries, which are estimated to merge at around 0.05 Hz,
is relatively low. For investigating the wide mass range distribution of the black
holes (1-106 M⊙ ), multi-band observation with the detectors that are sensitive
below 0.01 Hz, such as LISA, is important.

*1 The signal-to-noise ratio of eight is a kind of gravitational wave detection threshold in

the gravitational wave research field. The signal-to-noise ratio of eight corresponds tothe false detection probability of ∼ 10−15 [63].

79Chapter 6

Demonstration of the dual-pass
diﬀerential Fabry–PérotinterferometerIn this chapter, we show the result of the demonstration of the dual-pass diﬀerential Fabry–Pérot interferometer in the ground laboratory.

6.1 Objective
The dual-pass diﬀerential Fabry–Pérot interferometer is a new interferometer
configuration for DECIGO and B-DECIGO. In the dual-pass diﬀerential Fabry–Pérot interferometer, its longitudinal signals cannot be obtained with the same
way as the conventional diﬀerential Fabry–Pérot interferometer since, in general,
all the cavities cannot meet the resonance condition simultaneously as explainedin Section 6.2.1. In order to obtain the signal of the dual-pass diﬀerential Fabry–Pérot interferometer, the length of each cavity has to be adjusted to be similarly(quantitative discussion is in the next section). By adjusting the cavity lengths,
the oﬀset of the laser frequency from the cavity resonant frequency, known ascavity detuning, can be reduced. This is an essential feature of the dual-pass
diﬀerential Fabry–Pérot interferometer not only for signal extraction but also
for noise reduction to the gravitational wave observation. When there is cavity
detuning, intensity fluctuation of the laser couples with the detuning and can bea dominant noise source of DECIGO and B-DECIGO. Therefore, we develop thedual-pass diﬀerential Fabry–Pérot interferometer to demonstrate its operationand to investigate its behavior. In particular, the dependence of the cavity
detuning on the cavity length is investigated. This time, in order to focus on
the operation of one dual-pass diﬀerential Fabry–Pérot interferometer, the setup
constructed in the ground loboratory has two arm cavities with two laser sources.

816.2 Formulation and behavior of the dual-pass diﬀerential
Fabry–Pérot interferometer
6.2.1 Dependence of the cavity detuning on the cavity length diﬀerence
Here, we explain the behavior of the dual-pass diﬀerential Fabry–Pérot interferometer in DECIGO and B-DECIGO. In the dual-pass diﬀerential Fabry–Pérotinterferometer, one cavity is composed of two lasers, which have diﬀerent frequency in order to reduce interference eﬀect of them. Even in this case, two of
the three arm cavities in DECIGO and B-DECIGO can be resonant for bothlasers with feedback control as shown in figure 6.1. Laser 1 and Laser 2 arecontrolled to follow the resonant frequency of Cavity c. The length of Cavity
a and b are controlled to follow Laser 2 and 1, respectively. The frequency ofLaser 3 is controlled to use Cavity a as a reference. Thanks to these feedback
control, Cavity a (c) is resonant for Laser 2 and 3 (1 and 2) and Cavity b isresonant for Laser 1. However, Cavity b cannot be resonant for Laser 3 using
feedback control in general. This is because, if there is the additional feedback
control loop, the feedback control is cancelled out due to the other control loops.
Therefore, without a new technique, we cannot obtain the longitudinal signal of
Cavity b with Laser 3.
(a)(b)
Satellite 1

PD
ThrusterMirrorCavity b

Thrusterfeedback
PD

PD

MirrorMirror

10

-4

(c)

Cavity a

10

c

2La

10

f3

Cavity b

10 0
10 -2

c

2Lb

-4

10 0
10 -2

Cavity a0Cavity c

f1

c

2Lc

-1

f3 Cavity b

10 0

10 -1
10

-3

Cavity c543Frequency [Hz]Frequency [Hz]EOM
Laser 3

Laser2 EOM

Satellite 2

-2

10 -4

Cavity aPD
feedback

10

f2

Intracavity Power [a.u.]

feedback

PD

Cavity c

10 0

10

EOM

Intracavity Power [a.u.]

fee
db
ac
k

Laser 1PD
Thrusterfeedback
Satellite 3

Figure 6.1: (a) Schematic of the dual-pass diﬀerential Fabry–Pérot interferometer
length control topology. (b) Relation between the cavity resonant frequency andlaser frequency. fi (i = 1, 2, 3) is the frequency of each laser. (c) Enlarged figureof (b) around f3 .
In order to make the cavity resonant for both lasers without feedback control,
it is found that we need to adjust the absolute cavity length. The frequency
diﬀerence of the frequency of Laser 3, f3 , from the resonant frequency of Cavity
b, fb , is written as∆f3 ≡ f3 − fb = f3 − N

82c. (N ∈ N)
2Lb

(6.1)

Since Laser 3 is controlled to resonate with Cavity a, the frequency of Laser 3 isdenoted as f3 = N ′ 2Lc a (N ′ ∈ N). Consequently, ∆f3 is written as∆f3 = N ′

c
c
−N
.
2La
2Lb

(6.2)

We can choose N ′ in 0.995N ≲ N ′ ≲ 1.005N . This range corresponds to theactuation range of the laser source. Obviously, when N ′ = N and La = Lb , thefrequency diﬀerence ∆f3 is zero, i.e. Cavity a is resonant with Laser 3. WhenN ′ = N + ∆N (∆N ∈ N) and La = Lb + ∆L (|∆L|/Lb ≪ 1), ∆f3 is denoted as()∆Lc∆N + N
.
(6.3)
∆f3 ≃
2Lb
Lb
When |∆L|/Lb ≲ 0.005 (this level can be achieved with auxiliary sensors suchas microwave ranging system), we can choose ∆N to be the nearest integer to−N ∆L
Lb . As a result, the frequency diﬀerence ∆f3 is within,|∆f3 | ≤

c|∆L|
.
2L2b

(6.4)

The frequency diﬀerence corresponds to the cavity detuning. When the cavity
detuning is smaller than the resonant width of the cavity, i.e. 2ν1/2 , the longitudinal and alignment signal of the cavity can be obtained in the same manneras the conventional cavity. The stringent requirement for the cavity detuning isdetermined by the intensity noise coupling (equation (3.70)) to be |∆f3 | ≤ 0.6 Hz
for B-DECIGO (|∆f3 | ≤ 0.3 Hz for DECIGO). This leads to the length adjustment requirement of |∆L| ≤ 40 m for B-DECIGO (|∆L| ≤ 2 km for DECIGO).
This precision can be achieved with the auxiliary ranging instruments such as
micro-wave ranging system [189]. The dependence of the cavity detuning on thecavity length is a representative feature of the dual-pass diﬀerential Fabry–Pérotinterferometer and should be confirmed in the experiment.

6.2.2 Optical spring eﬀect
Here, we discuss the eﬀect of the optical spring due to the cavity detuning. When
there is detuning in a cavity, the circulating power of the cavity varies dependingon the mirror motion. As a result, the radiation pressure of the circulating laser
behaves as a spring. This eﬀect is called the optical spring [197]. The spring
constant of the optical spring is given by [198]K opt ≡ k opt + iωγ opt ,k opt ≡

∆f
4Pcirc ωl2 ],Lc 2π[(∆f )2 + ν1/2

γ opt ≡ −

ν1/2 (∆f )8Pcirc ωl2 ]2 ,
Lc 4π 2 [(∆f )2 + ν1/2

83

(6.5)
(6.6)
(6.7)

where Pcirc is the circulating power in the cavity and ∆f is the cavity detuning
in terms of frequency. Equations (6.5)-(6.7) indicate that the optical springprovides the positive and anti-damping spring (or the negative and damping
spring) depending on the detuning. In the dual-path diﬀerential Fabry–Pérotinterferometer shown in Chapter 5, the optical spring exists in Cavity c since
there is cavity detuning due to the diﬀerent length of the cavity as discussed in
Section 6.2. In other words, TM1b and TM3b are considered to be connectedwith the optical spring.
One way to deal with the optical spring eﬀect due to the diﬀerent length ofthe cavity is detuning the resonance of Cavity c from f1 . When the resonanceof Cavity c is detuned from f1 , the laser beam from Satellite 1 behaves as the
optical spring, too. When the amount of the detuning of the resonance of Cavity
c from f1 is the same as that from f3 but the sign is reversed, the optical springeﬀect is cancelled. Notice that the optical spring by detuning the cavity has
been demonstrated [199, 200] and the amount of the detuning can be adjusted.
Therefore, the results in Chapter 5 are valid when the optical spring eﬀect iscancelled by adjusting the detuning of the resonance of Cavity c from f1 .
Actually, even without detuning the resonance of Cavity c from f1 , the opticalspring eﬀect due to the diﬀerent length of the cavity can be dealt with using thecontrol discussed in Chapter 5. If we assume ∆f = ∆f3 = 0.3 Hz for DECIGO
and ∆f = ∆f3 = 0.6 Hz for B-DECIGO, the spring constants of the opticalspring between TM1b and TM3b for DECIGO and B-DECIGO are given by,according to equations (6.5)-(6.7),{
k opt = 0.63 N/m,
(DECIGO)(6.8)
γ opt = −2.7 × 10−2 N/(m/s),{
k opt = 0.25 N/m,γ opt = −1.1 × 10−2 N/(m/s).

(B-DECIGO)(6.9)

Here, we also use the parameters shown in table 4.1. Note that, in the dualpath diﬀerential Fabry–Pérot interferometer, we can choose the positive andanti-damping optical spring by adjusting f3 . In the condition with the positive
and anti-damping optical spring, we discuss the optical spring eﬀect. The resonant frequency of the optical spring in Cavity c is 0.017 Hz for DECIGO and0.020 Hz for B-DECIGO. These resonant frequencies are smaller than the unity
gain frequency of the control of the x-motion of TM1b according to figure 5.7.
Since the optical spring aﬀects the control system below the resonant frequency,
the optical spring eﬀect can be suppressed with the control system considered in
Chapter 5. We also confirmed the above discussion using the full DECIGO interferometer model including the optical spring eﬀect, and could obtain the stable
control system and the sensitivity better than the target sensitivity between 0.1
Hz and 10 Hz.

84

6.3 Experimental setup
Figure 6.2 shows the schematic of the dual-pass diﬀerential Fabry–Pérot interferometer experiment. A photograph of the main dual-pass Fabry–Pérot cavities
is shown in figure 6.3. We use single-frequency distributed-feedback fiber lasers
with a wavelength of 1550 nm (Laser1, Koheras AdjustiK C15; Laser 2, Koheras
BASIK X15). The powers of output of Laser 1 and Laser 2 are about 10 mW
and 30 mW, respectively. Each of their outputs is a polarization-maintaining
single-mode optical fiber. The single-mode optical fiber is the optical fiber thatis designed to transmit only the fundamental spacial mode of the laser beam [127](about the spacial mode of the laser beam, see Appdendix F). The emitted lasers
go to the fiber coupled Faraday isolators (Thorlabs IO-G-1550-APC for bothlasers, not shown in figure 6.2) and electro-optic modulators (EO Space PM0K5-00-PFA-PFA-UL-S for Laser 1 and PM-0S5-00-PFA-PFA-NT for Laser 2).
The phase modulation frequencies for Laser 1 and Laser 2 are 14.7 MHz and 15.3
MHz, respectively. We use the diﬀerent modulation frequencies for signal decoupling of the two lasers. After the electro-optic modulator, the laser is splittedwith the fiber coupled beam splitter: one goes to the main dual-pass interferometer (main beam) and the other goes to the monitoring system for absolute cavity
length measurement as explained later. The main beams are introduced to freespace through collimators (Newport F-COL-9-15-FCAPC) and splitted with thebeam splitter for the two arm cavities. After adjusting the mode matching andalignment, the beam is incident to the cavity from both sides. This is the main
dual-pass Fabry–Pérot cavity. The cavity is 55-cm-long and its finesse is designed
to be 210. The radius of curvature of the mirrors is 2 m and their amplitudereflectivity is 0.992 (Lattice Electro Optics, Inc. BS-1550-R98.5-UF-MPC-05252000). These cavity mirrors are placed on the two optical benches isolated withrubber stacks. The reflected light and the transmitted light from the other side
are received by each photodetector. Note that the two beams have orthogonallinear polarization to avoid interference eﬀect. The cavity longitudinal signals
are obtained with the Pound–Drever–Hall technique.
The block diagram of the control in this experiment is shown in figure 6.4.
Signals from PD1a and PD2a are fed back to the frequencies of Laser 1 and Laser
2, respectively. With the signal from PD2b, the position of M2b is controlledusing a piezoelectric transducer. We defined open loop transfer functions of each
feedback loop as
G1a ≡ S1a E1a A1a ,(6.10)

G2a ≡ S2a E2a A2a ,(6.11)

G2b ≡flas
S2b E2b A2b .
L

85(6.12)

Isolated optical bench
PD1b

Laser 1yb

M1bfilter

PD2a

Cavit

Laser 2

M2a

ya

M1a

f1mod

Cavit

M2b

f2mod
PD2b

PD1a

filterfilterPDbeatoptical fiber

Lasercollimator

mirrors
mirror with PZTfilter filter for control

fiber
beamsplittersbeamsplitter
photodetector

fiber EOM

movable stage
with a micrometersignal generator

beam dump

mixer

oscilloscope/data logger

lowpass filter

frequency counter

Figure 6.2: Schematic of the dual-pass diﬀerential Fabry–Pérot interferometer
experiment.

Figure 6.3: Photograph of the main dual-pass Fabry–Pérot cavities.

Each error signal (about the notation, see Appendix B) is denoted as
S1a flas
S1a(x1a − x2a ) +f1 ,1 + G1a L
1 + G1aS2a flasS2ae2a =(x1a − x2a ) +f2 ,1 + G2a L
1 + G2ae1a =86(6.13)(6.14)

(

)
G2b
G1a
flas
G2a
e1b =
−
S1b (x1a − x2a )1 + G2a 1 + G2b
1 + G1a
L
1
flas
1
1
G2b
+
S1b f1 −S1b f2 ,S1b (x1b − x2b ) +1 + G2b L
1 + G1a
1 + G2a 1 + G2b(6.15)
G2a
1
flas
1
flas
S2b (x1a − x2a ) +S2b (x1b − x2b )1 + G2a 1 + G2b L
1 + G2b L
1
1+
S2b f2 .
(6.16)
1 + G2a 1 + G2b

e2b = −

When |G1a | ≫ 1 and |G2a | ≫ 1, we can write e1b and e2b as
1
flas
1
flas
S1b (x1a − x2a ) +S1b (x1b − x2b ) ,1 + G2b L
1 + G2b L
flas
flas
1
1
e2b = −
S1b (x1a − x2a ) +S1b (x1b − x2b ) ,1 + G2b L
1 + G2b L
e1b = −

(6.17)(6.18)

so that the same signal should be measured at the two ports.
S1a

f1

E1a
A1a

x1a
x2a

flas

L

S2a

f1a
e2a

S1b

e1b

E2a

f2
x1b
x2b

e1a

A2a
flas

L

S2b

f2a
e2b
E2b

A2b

f2b

Figure 6.4: Block diagram of the dual-pass diﬀerential Fabry–Pérot interferometer experiment. eiα and fiα are the error signal and feedback signal of each port,
respectively, Siα is the sensing eﬃciency, Eiα is the control electric filter, Aiα isthe actuator eﬃciency (i = 1, 2 and α = a, b), flas = 5.8 × 1014 Hz is the laser
frequency, and L is the cavity length.

6.4 Characterization of each component
After the setup was constructed, the finesse of the cavities, the actuation eﬃciencies, the sensing eﬃciencies, the control filters, and the cavity lengths were
evaluated.

87

6.4.1 Finesse of the cavity
The finesse of the cavity was measured by taking the ratio of the free spectral
range to the resonance peak full width at half maximum by sweeping the frequency of Laser 2. This method is called cavity scanning. Since Laser 2 haslarger power than Laser 1, the measurement with better signal-to-noise ratio canbe performed with Laser 2. We scanned Cavity a and Cavity b and saw several
pairs of the resonant peaks next to each other for each cavity. One pair of theresonant peaks is shown in figures 6.5 and 6.6 as an example. The largest peak infigures 6.5 is the fundamental Hermite–Gaussian mode (about Hermite–Gaussianmode, see Appendix F). The finesse is measured with the following equation:
F=

∆tFSR
,
∆tFWHM(6.19)

where ∆tFSR is the temporal diﬀerence of the peaks and ∆tFWHM is the full
width at half maximum of the peak in terms of time. As a result, the finesses ofCavity a and b were measured to be,
Cavity a: 290 ± 70,(6.20)

Cavity b: 270 ± 80,(6.21)

where statistical errors are shown. The measured finesses of the two cavities are
the same within error ranges. Although the measured finesse of Cavity a does
not agree with the designed value within the error range, it does not significantly
aﬀect the whole result in this chapter.

88

10-4
8

8

Trans. power
Applied voltage

7
6

4
5
2
4
0
3
-2

Applied voltage [V]Transmietted power [W]

6

2
-4
1
-6

0
0

0.05

0.1

0.15

0.2

0.25

Time [s]Figure 6.5: Example of the result for cavity scanning to measure actuation eﬃciency. In this plot, the applied voltage to Laser 2 for scanning its frequency andtransmitted power measured with PD1b is shown. During this measurement, thebeam from Laser 1 was blocked.

Transmitted light power [W]

8

10 -4
MeasuredFitted

7
6
5
4
3
2
1
0
-4

-3

-2

-1

0

Time [s]

1

2

3

4
10

-4

Figure 6.6: Enlarged plot of figure 6.5 around 0.08 s. We can see the peak of thefundamental mode. The origin of the horizontal axis (i.e. 0 s) is set to be thepeak point.

896.4.2 Actuator eﬃciency
Actuator eﬃciencies, i.e. laser frequency change of Laser 1 (Laser 2) when 1
V is applied to Laser 1 (Laser 2) and the eﬃciency of piezoelectric transduceron M2b, were measured with two methods. The first one is cavity scanningagain for overall eﬃciency measurement and the second one is frequency response
measurement. The cavity has a peak at every half of the wavelength, which is a
precise length reference. Thus, by comparing the voltage diﬀerence between thetwo cavity peaks during cavity scanning with each actuator to the half of the
wavelength, we can measure the overall eﬃciency of the actuator. If we scan thecavity with the piezoelectric transducer on M2b, the eﬃciency of the cavity ismeasured. The frequency response can be measured with the transfer function
measurement from the input of the actuator to the sensor output when the cavity
is locked with each actuator. As a result, the actuator eﬃciencies were measured
to beLaser 1 (freqency): A1a (f ) = A1 exp (−if /f1 ) ,(6.22)

A1 = (1.31 ± 0.04) × 10 [Hz/V], f1 = 5500 ± 20 [Hz],8

Laser 2 (freqency): A2a (f ) = A2

1
1
exp (−if /f2′′ ) ,1 + if /f2 1 + if /f2′

(6.23)

A2 = (3.89 ± 0.08) × 107 [Hz/V], f2 = 328 ± 3 [Hz],f2′ = 5250 ± 40, [Hz], f2′′ = 17.0 ± 0.2 [kHz],

Piezoelectric transducer: A2b (f ) = APZT

1′
exp (−if /fPZT) , (6.24)
1 + if /fPZTAPZT = (4.7 ± 0.4) × 10−7 [m/V],′
fPZT = 2500 ± 700 [Hz], fPZT
= 5500 ± 20 [Hz].

6.4.3 Sensing eﬃciency
In the cavity experiment, the laser frequency diﬀerence from the resonant frequency of the cavity can be measured with the Pound–Drever–Hall technique
as explained in Chapter 3. The eﬃciency of the measurement of the frequency
diﬀerence between the laser frequency and the resonant frequency of the cavity
can be represented by sensing eﬃciency, which is the transfer function from thediﬀerence of the resonant frequency to the output voltage. The sensing eﬃciency
can be measured from the measurement of the transfer function from the inputof the actuator to the sensor output if the actuator eﬃciency is known. In thisdual-pass diﬀerential Fabry–Pérot interferometer experiment, sensing eﬃciency
is assumed to have a flat response since ν1/2 is ∼ 500 kHz, which is much larger

90

than the frequency that is considered in this experiment, e.g. at most 10 kHz(see also Section 3.3.2). The measured sensing eﬃciencies are
PD1a : S1a = (2.64 ± 0.01) × 10−8 [V/Hz],(6.25)

PD1b : S1b = (1.655 ± 0.006) × 10−8 [V/Hz],(6.26)

PD2a : S2a = (8.76 ± 0.03) × 10−8 [V/Hz],(6.27)

PD2b : S2b = (1.40 ± 0.03) × 10−7 [V/Hz].

(6.28)

Note that these results are expressed in terms of the cavity resonant frequency.

6.4.4 Control electric filter
The control electric filter is designed to obtain stable control. The measured
transfer functions are′′
1 + if /f1a
1 + if ′′′ /f1a
Laser 1 loop: E1a (f ) = E1a′
1 + if /f1a 1 + if /f1a′
E1a = 3598 ± 2, f1a = 2.80 ± 0.01 [Hz], f1a= 32.6 ± 0.2 [Hz],(6.29)′′′′′
f1a= 310 ± 2 [Hz], f1a= 1.1 ± 0.4 [kHz],

1 + if ′′ /f2a 1 + if ′′′ /f2a1
(6.30)′′′′′ ,
1 + if /f2a 1 + if /f2a 1 + if /f2a′
E2a = 94830 ± 60, f2a = 0.390 ± 0.008 [Hz], f2a= 34.1 ± 0.3 [Hz],Laser 2 loop: E2a (f ) = E2a

′′
′′′
′′′′
f2a
= 286 ± 3 [Hz], f2a= 1.67 ± 0.03 [kHz], f2a= 9.7 ± 0.2 [kHz],M2b loop: E2b (f ) = E2b′′
1 + if ′′′ /f2b
1 + if /f2b
′ ,1 + if /f2b 1 + if /f2b(6.31)′
E2b = 55.32 ± 0.01, f2a = 2.76 ± 0.02 [Hz], f2b= 31.9 ± 0.4 [Hz],
′′
′′′
f2b
= 353 ± 6 [Hz], f2b= 1.434 ± 0.002 [kHz].

6.4.5 Absolute cavity length
The absolute cavity length can be determined if the cavity free spectral range
is measured according to equation (3.31). In this experiment, the free spectral
range can be measured by observing the beatnote frequency with PDbeat and afrequency counter when the two lasers are made to resonate at the next resonanceof the cavity. The measured lengths of Cavity a and b are
Cavity a: La = 0.55340 ± 0.00001 [m],(6.32)

Cavity b: Lb = 0.55340 ± 0.00001 [m].

(6.33)

Note that the length of Cavity a can be changed with the stage on which M2a
is located and the value of La shown in equation (6.32) is the length of Cavity aafter the adjustment to be the same as the length of Cavity b.

91

6.5 Results of the dual-pass diﬀerential Fabry–Pérotinterferometer experiment6.5.1 Lock sequence
Figure 6.7 shows the lock sequence of the dual-pass diﬀerential Fabry–Pérotinterferometer. At 11 s, the frequency of Laser 1 is controlled to be the resonant
frequency of Cavity a. Then, at 19 s, the frequency of Laser 2 is controlled tobe the resonant frequency of Cavity a. Finally, at 32 s, the length of the cavity
b is controlled with the signal obtained with PD2b. The error signal from thePD1b can be observed although it is not fed back to any actuator. Note that at
7 s, Cavity a is held in the non-intended higher order mode.

0.5

1a PDH

1
0
-1

0

20

40

1b DC

0.5
0

5

40

0

20

40

5

1b PDH

5
0
-5

20

40

0

20

40

10 5

1

Freq. [Hz]

10 -3
2a DC

0.5
0

10 8

Freq. [Hz]02a PDH

1
0

2

2a FB

0
-2

-1
0

20

40

0

40

2b DC

1
0

2b PDH5
020Time [s]

40

20

40

10 7

5

-5
0

0

10 5

Freq. [Hz]

10 -3

2

20

Freq. [Hz]Power [W]

20
10

10

1

1a FB

6

4
0

-3

Freq. [Hz]10Power [W]10 8

Freq. [Hz]

1a DC

0

Power [W]10 5

2Freq. [Hz]Power [W]

10 -3
1

2b FB

4.5
4

0

20

Time [s]

40

0

20

40

Time [s]Figure 6.7: Lock sequence of the dual-pass diﬀerential Fabry–Pérot interferometer. Left panels show the laser power received by each photodetector, center
panels show the error signals of each cavity, and right panels show the feedback
signals in each control loop. The error and feedback signals are written in termsof the resonant frequency, which is the origin of these signals. At the timings
indicated by the dashed lines, each control servo was engaged successively.

92

105
10

1a (Measured)
1a (Fitted)
2a (Measrued)
2a (Fitted)
2b (Measured)
2b (Fitted)

4

Gain

103
102
101

Phase [deg]

100
10-1
100
180
90
0
-90
-180
100

101

102

103

104

101

102

103

104

Frequency [Hz]Figure 6.8: Measured open loop gain of each control loop. The upper and lower
panels show the gain and phase, respectively. The solid lines are the open loop
gains predicted by the characterization of each component in Section 6.5.

6.5.2 Open loop transfer function
We measured open loop transfer functions of the two frequency servos and piezoelectric transducer servo as shown in figure 6.8. The open loop transfer functions
predicted by the characterization of each component described in Section 6.5 arealso shown. Here, we used the mean values. The measured gain and predicted
gain agree with each other around their unity gain frequency.

6.5.3 Detuning dependence on the cavity length
We measured the dependence of the error signal oﬀset from the resonant frequency on the cavity length with the Pound–Drever–Hall technique. The error
signals correspond to the cavity detuning. The cavity length was adjusted bymoving the stage with a micro meter where M2a is placed. Two examples of
the time series data of the error signals when the two cavity lengths are oﬀ andadjusted are shown in figures 6.9 and 6.10. We measured the error signal oﬀsets
with five diﬀerent positions of M2a as shown in figure 6.11. During this measurement, the cavity alignment is kept to be almost the same. We determinedthe error signal oﬀset dependence of the mirror position by fitting the result infigure 6.11 to be (−5.1 ± 0.5) × 108 Hz/m.

93

4

5

-5
0

Freq. [Hz]

20
10

5

40

60

80

10

3
2
1
0
-1

100

1a FB

0

20

40

60

80

100

5

0

0

20

40

60

80

100

10 4

4

10 7
2a PDH

2
0
-2
-4

2

2a FB

0
-2

0

20

40

60

80

100

0

10 5
2

2b PDH

0
-2
0

20

40

60

80

100

10 7

Freq. [Hz]Freq. [Hz]7

1b PDHmean-5

Freq. [Hz]Freq. [Hz]1a PDH0

Freq. [Hz]Freq. [Hz]

10

20

40

60

80

2

2b FB

1
0
-1

100

0

20

Time [s]

40

60

80

100

Time [s]Figure 6.9: Error and feedback signals when two cavity lengths are shifted.

20

40

60

80

1a FB

1
0
-1
-2

100

0

40

60

80

100

1b PDH
mean

0
-5

0

20
10

4

40

60

80

100

4

10
2a PDH

2
0
-2
-4

0

20

40

60

80

2a FB

0
-1
-2

100

0

20

40

60

80

100

10 7

2

Freq. [Hz]2b PDH

0
-2
0

7

1

10 5

Freq. [Hz]

20

10 5

5

Freq. [Hz]Freq. [Hz]0-50Freq. [Hz]10 7
1a PDH

Freq. [Hz]Freq. [Hz]

10 4
5

20

40

60

80

100

4

2b FB

2
0
-2
0

Time [s]

20

40

60

80

100

Time [s]Figure 6.10: Error and feedback signals when two cavity lengths are adjusted bymoving the position of M2a as described in Section 6.4.5.

94

3

105
MeasuredFittedPDH offset [Hz]

2

1

0

-1

-2

-3
-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Mirror position [mm]Figure 6.11: Measured oﬀsets of the Pound–Drever–Hall signal, i.e. error signals,
at the five diﬀerent cavity lengths. The oﬀset of the Pound–Drever–Hall signal
corresponds to the cavity detuning.

6.5.4 Noise spectrum measurement
We measured noise spectra of the dual-pass diﬀerential Fabry–Pérot interferometer. The result is shown in figure 6.12. The ‘1a’ and ‘2a’ curves are calibrated
from the feedback signal and the ‘2b’ curve is calibrated from the error signal.
The ‘1b’ and ‘2b’ curves represent the diﬀerential length fluctuations of Cavity a and b. The spectra of ‘1a’ and ‘1b’ indicate that the frequency noise is
successfully stabilized by the frequency servo and they work as a diﬀerential
Fabry–Pérot interferometer eﬀectively without feeding back the signal of PD1b.

6.6 Discussion
When the dual-pass diﬀerential Fabry–Pérot interferometer works well, the same
signal, i.e. the diﬀerential length fluctuation of the two cavities, is obtained in
the two interferometer. To show that, we calculated the coherence of e1b ande2b as shown in figure 6.13. Since the average number is 100, the two signals are
regarded as coherent when the coherence is larger than 0.06 [201]. Figure 6.13
indicates that the two signals are coherent between 1 and 1500 Hz. The coherence
below 10 Hz decreases since the noise is large. The main noise source is considered
to be the diﬀerential component of the seismic motion. The coherence above
1500 Hz is small since the gain of the frequency feedback loop is small. In
this case, the independent frequency noise aﬀects the measurement according toequations (6.15) and (6.16). The discussion in this paragraph indicates that the95

Displacement noise [m/rtHz]

10 -8
10

1a
1b
2a
2b

-9

10 -10
10 -11
10 -12
10 -1310 -14 0
10

10 1

10 2

10 3

10 4

Frequency [Hz]Figure 6.12: Calibrated noise spectra of the dual-pass diﬀerential Fabry–Pérotinterferometer measured with PD1a, PD1b, PD2a, and PD2b.

dual-pass diﬀerential Fabry–Pérot interferometer is considered to be successfully
operated.
In order to see the behavior of the dual-pass diﬀerential Fabry–Pérot interferometer with another way, we also check if the common motion can be subtracted
by taking the diﬀerence between the output signals of PD1b and PD2b. Since
there could be relative calibration error between PD1b and PD2b, we need toapply a factor to the output signal of PD2b for the subtraction of the output ofPD2b from the output of PD1b. We survey the optimal factor of the subtraction
with 1D-grid searching. The best subtracted signal is shown in figure 6.14. This
figure suggests that the two interferometers measure the same signal, i.e. thediﬀerential cavity length fluctuation. The optimal factor for the subtraction of
the output of PD2b from the output of PD1b was 0.89. This indicates that there
is relative calibration error of 11% between the two interferometers. The 11%
calibration error can be explained by the sensing eﬃciency error estimated in
Section 6.5.
It is worth noting that the procedure mentioned above for the subtraction
from the signal of PD1b with the signal of PD2b is a kind of way to make anull-stream and can also be used for the relative calibration of each sensing port
even in DECIGO and B-DECIGO. Since the mass and volume of the instrument
are limited in the space mission, this kind of in-situ scheme is beneficial.

96

1

Coherence

0.8
0.6
0.4
0.2

Phase [deg]b/w 1b and 2b
0
10 0
180
90
0
-90
-180
10 0

10 1

10 2

10 3

10 4

10 3

10 4

Frequency [Hz]

10 1

10 2

Frequency [Hz]Figure 6.13: Measured magnitude-squared amplitude (upper panel) and phase(lower panel) of coherence of the error signal from PD1b and PD2b.

Frequency fluctuation [Hz/rtHz]

10 4

10 3

10 2

10

1

10 0
10 0

"1b" - "2b"x0.89"1b""2b"x0.89
lower limit from coherence
Laser 1
Laser 2

10 1

10 2

10 3

10 4

Frequency [Hz]Figure 6.14: Noise spectra with common noise subtraction (blue curve). ‘1b’ and‘2b’ are measured error signals with PD1b and PD2b, respectively. ‘Laser 1’ and‘Laser 2’ are estimated frequency noises of Laser 1 and Laser 2, respectively. The
purple line indicates the lower limit of the noise subtraction estimated from thecoherence measurement as shown in figure 6.13. The curves of the subtracted
signal and ‘lower limit from coherence’ are overlapped.

97

Here, the result in Section 6.5.3 is discussed to see the dependence of thecavity detuning on the cavity length. The fitting result of the relation between
mirror position and cavity detuning from figure 6.11 is (−5.1 ± 0.5) × 108 Hz/m.
Our formulation and the cavity length measurement in Section 6.4.5 predictthat the dependence of the detuning of Cavity b to the position of TM2a is to be(−4.8976±0.0002)×108 Hz/m. They are consistent within the error ranges. This
discussion indicates that, by adjusting the cavity length, the longitudinal signal ofthe dual-pass diﬀerential Fabry–Pérot interferometer can be obtained. Moreover,there is the position of TM2a where the cavity detuning is zero as shown infigure 6.11. Therefore, intensity noise coupling explained in Section 3.3.5 can be
reduced in the dual-pass diﬀerential Fabry–Pérot interferometer.

98

Chapter 7

Conclusion
7.1 Summary
In this thesis, the control scheme of the Fabry–Pérot type space gravitational
wave antenna is studied. First, the control topology is considered with the numerical model of the Fabry–Pérot type antenna including a mechanical and optomechanical response, a sensing and actuation scheme, external disturbances, andsensing noises. The new model is named a full DECIGO interferometer model.
This new model reveals that the interferometer control and drag-free control canbe engaged at the same time by separating the degrees of freedom controlled
by the control system: one mirror in the satellite is used as the reference ofthe drag-free control and the other mirror is used for the interferometer control
and the local control. The model also reveals that the mirror position can beused as a reference of the control to obtain the stable control system. Using themodel, a solution of DECIGO parameters√ with interferometer control achieving−23
the target strain sensitivity of 10/ Hz is also found as shown in figure 5.21.
In addition, more than one-day stability was also achieved. Second, the dualpass diﬀerential Fabry–Pérot interferometer is formulated and is constructed ina ground laboratory as shown in figure 6.2. The operation of the dual-pass diﬀerential Fabry–Pérot interferometer is demonstrated for the first time. Moreover,
it is confirmed that cavity detuning can be reduced by the cavity length adjustment as predicted by the formulation as shown in figure 6.11. The cavity
detuning reduction is essential to minimize a noise coupled with the laser intensity fluctuation, which would be a major noise source in DECIGO. These results
indicate that the dual-pass diﬀerential Fabry–Pérot interferometer is correctly
understood with the new formulation.

7.2 DECIGO and B-DECIGO design
In this thesis, control topology and the dual-pass diﬀerential Fabry–Pérot interferometer are developed. These two components are essential for DECIGO
and B-DECIGO. Here, including the obtained results, we summarize the designof DECIGO and B-DECIGO as shown in table 7.1. The dual-pass diﬀerential
Fabry–Pérot interferometer is utilized. The sensing method for the longitudinalsignal is the Pound–Drever–Hall method. The methods for the alignment signal

99

are the wavefront sensing method and the dithering method. With these sensing
methods, the interferometer is controlled stably with almost shot-noise-limited
sensitivity between 0.1 and 10 Hz. We can implement the drag-free control tocontrol the satellite motion at the same time as the interferometer control with
control scheme developed in this thesis. The degrees of freedom of the test massthat is not controlled with the interferometer control is controlled with the local
sensors and the star trackers. In this design, the dominant technical noise iscaused by the noise of the local sensor and the significant part of the actuator
range is taken by the star tracker noise. The√ local sensors and the star trackers−12should be more sensitive than 1√
× 10
m/ Hz above 0.1 Hz (for interferometer
sensitivity), and 2 × 10−6 rad/ Hz below 0.01 Hz (for interferometer stability),
respectively. The test mass actuators shouldhave the maximum range of 2×10−6√−17
N and the noise level of 2 ×
N/ Hz above 0.1 Hz. The dynamic range of√ 10−11/ Hz, which is greater than that of the conventional
the actuator is ∼ 10
simple actuators, so that we need to develop a new actuator. One candidate is
the frequency-dependent actuator that has the large range (larger than 2 × 10−6
N with the safety factor of√ 10) outside the observation band, and is low noise−17(better than 2 × 10
N/ Hz ) in the observation band. The coupling from thesatellite motion to the test mass acceleration should be less than 5 × 10−7 /s2
not only for the sensitivity but also for the stability below 10−4 Hz. Using thesecomponents and control scheme developedin this thesis, we achieved√√ the target−23−22
sensitivity of DECIGO (10/ Hz) and B-DECIGO (10/ Hz) between
0.1 and 10 Hz and one-day-long stability in the simulation.
Table 7.1: Summary of the design of DECIGO and B-DECIGO considered inthis thesis. In the brackets, the achieved values in the previous works are shown.
Interferometer
configuration
Sensing methodlongitudinalalignment
Control topology
Local sensor
sensitivity
Star tracker
sensitivity
Actuator
range
noise
Coupling between
satellite and test mass

Dual-pass diﬀerential
Fabry–Pérot interferometer
Pound–Drever–Hall method
Wavefront sensing and dithering
Combined control of the interferometer control
with the drag-free control√
1 × 10−12 m/ √Hz (> 0.1 Hz)[ 2 × 10−9 m/√ Hz (in space) [182],
1 × 10−12 m/ Hz√ (ground) [185] ]−6
2 × 10 rad/ √Hz (< 0.01 Hz)−6[ 2 × 10 rad/ Hz [182] ]−6
2 × 10N√−17
2 × 10N/ Hz (> 0.1√
Hz)
−6
−15
[ ∼ 5 × 10 N, ∼ 5 × 10
N/ Hz [176] ]5 × 10−7 /s2 [ 4 × 10−7 /s2 [140] ]100

7.3 Conclusion
The observation frequency of the ground-based gravitational wave observatory,
such as aLIGO, is limited between ∼ 10 Hz and a few kHz. The lower frequency
limit is mainly determined by the seismic motion. Some observational targets
below 10 Hz are predicted. One target is the coalescence of the intermediatemass black holes with masses in the range between ∼ 103 M⊙ and ∼ 106 M⊙ .
The intermediate-mass black holes typically emit the gravitational waves in
the frequency range between ∼ 10−3 Hz and ∼ 1 Hz. The coalescence of theintermediate-mass black holes is considered to be a candidate of the origin of thesuper massive black hole that has been discovered at the galactic center. Anothertarget is the early Universe. The quantum fluctuation is considered to cause thegravitational waves in the wide frequency range, for example, between ∼ 10−17
Hz and ∼ 104 Hz. Therefore, some space gravitational wave observatories areproposed to avoid the seismic motion.
DECIGO and B-DECIGO are the planned Fabry–Pérot type interferometricspace gravitational wave antennas to observe the gravitational waves between√
0.1 Hz and 10 Hz. The design sensitivity of DECIGO is better than 10−23 / Hz
so that the stochastic gravitational wave background generated by the quantum
fluctuation in the early Universe could be observed. These space antennas have
two unique features: the precise interferometer control system including the
Fabry–Pérot cavity and drag-free satellite, and the dual-pass diﬀerential Fabry–Pérot interferometer.
The Fabry–Pérot cavity is necessary to achieve high shot-noise-limited sensitivity. The drag-free satellite is also necessary to isolate the test masses from theextra-satellite noise sources. Although these two techniques have been demonstrated independently, there was a concern about the simultaneous operationof them, especially in terms of control method. Thus, we numerically modeled
the full control system of DECIGO and B-DECIGO. The model is named thefull DECIGO interferometer model. This numerical model reveals that the interferometer control and drag-free control can be engaged at the same time byseparating the degrees of freedom controlled by the control system; one mirror
in the satellite is used as the reference of the drag-free control and the othermirror is used for the interferometer control and the local control. Using themodel, we also considered the detailed requirement including control design for
DECIGO and B-DECIGO. As a result, we found a solution of interferometer√
control with target strain sensitivity of DECIGO, which is 10−23 √/ Hz, andmore than one-day stability. If we achieve the sensitivity of 10−23 / Hz at 0.1
Hz, some features at the early Universe can be tested, for example the phasetransition at the electroweak scale [113] and the cosmic strings [114].
The dual-pass diﬀerential Fabry–Pérot interferometer is a totally new interferometer configuration. We formulated the dual-pass diﬀerential Fabry–Pérotinterferometer and found that absolute length adjustment is needed to obtain
all interferometer signals. For the demonstration, the ground-based dual-pass
diﬀerential Fabry–Pérot interferometer was constructed. In the actual experi-101

ment performed in the ground-base laboratory with the 0.55-m-long dual-pass
Fabry–Pérot cavities, we succeeded in operating the dual-pass diﬀerential Fabry–Pérot interferometer and confirmed that adjustment of the absolute arm length
reduced the cavity detuning as expected with our formulation. This leads to
the reduction of the laser intensity noise coupling. These results indicate thatthe behavior of the dual-pass diﬀerential Fabry–Pérot interferometer is correctly
understood. In future, the constructed interferometer can be expanded for theexperimental demonstration of the full control scheme, which we developed withthe full DECIGO interferometer model.
We conclude that the sensing and control scheme of the interferometer inFabry–Pérot type interferometric space gravitational wave antenna, especially
DECIGO and B-DECIGO, has been demonstrated theoretically and experimentally. We hope that this work provides an essential basis for opening the windowof gravitational wave physics and astronomy in the decihertz band.

7.4 Future prospect
In this section, we describe the future prospect towards the actual DECIGO
and B-DECIGO observation and further expansion of gravitational wave physics
and astronomy. Specifically, further investigation of interferometer simulation,
construction of ground-based DECIGO and B-DECIGO prototype, and spacedemonstration are shown.

7.4.1 Further investigation of interferometer simulation
In the simulation expressed in this thesis, the orbital motion is incorporated
as a typical external force noise. However, the actual behavior of the orbital
motion varies depending on the orbital parameters, such as semi-major orbital
axis, orbital inclination, longitude of the ascending node, launch date, and soon. For more realistic simulation, orbital simulation should be combined with
the interferometer simulation.
Moreover, though we showed one solution to achieve shot noise limited sensitivity of DECIGO and B-DECIGO, there are other possibilities to achieve thesimilar sensitivity. For example, interferometer configuration could be changed
to the interferometer with two laser sources in one satellite while the configuration requires new research and development. If it is confirmed in future that the
new interferometer configuration can achieve the shot noise limited sensitivity,
we need to perform a trade-oﬀ study between the scheme studied in this thesis
and the newly developed configuration.

7.4.2 Construction of ground-based DECIGO and B-DECIGO prototype
The dual-pass diﬀerential Fabry–Pérot interferometer experiment constructed inthis work can be expanded into the future small-scale ground-base prototype of
DECIGO and B-DECIGO. The prototype will have the following two features:

102

the same control topology including interferometer of sensors and actuators andsoft suspension systems. The latter is necessary to mimic the situation of space.
About the control topology, the alignment sensing system will be implemented
in addition to the main dual-pass diﬀerential Fabry–Pérot interferometer demonstrated in this thesis. Since the alignment sensing method was demonstrated in
the previous work [133], just an implementation is necessary. About the soft
suspension system, we will use torsion pendulums. The conceptual design has
been done as shown in Appendix D. By using the prototype, the control scheme
discussed in this thesis can be experimentally demonstrated. After the principal demonstration, the integrated test with the actual components such as test
mass actuators, local sensors, thrusters and so on, should be performed. This
integration test can also be performed with the modification of the prototype.

7.4.3 Space demonstration
Towards DECIGO and B-DECIGO, even small scale demonstration in space,
if any, is very beneficial. The implementation of the control scheme discussed
in this thesis including the interferometer characterization leads to an essential
milestone for DECIGO and B-DECIGO. Another thing which would be tested inspace is noise characterization. Using the measured data in space , the accuracyof the performance estimation in this thesis can be improved. In addition, it isimportant to demonstrate the end-to-end test of the gravitational wave observation, i.e. initial alignment, detector operation, data transfer from the satellite to
the Earth, and data analysis.

103

Appendix A

Axion dark matter search with the
Fabry–Pérot cavity
The Fabry–Pérot cavity is a very precise measurement instrument which is usedeven to detect gravitational waves. In addition to the gravitational wave observation, there are many applications of precision measurements, for example,
axion dark matter search. Here, we explain the new approach for the axion dark
matter search with interferometric gravitational wave detectors. This work isreported in K. Nagano et al. [155]. In this chapter, the natural unit ℏ = c = 1 is
used.

Axion dark matter and its probe technique
The axion is a pseudo-scalar field which is originally proposed a half-century
ago by Roberto Daniele Peccei and Helen Rhoda Quinn to solve the strong
Charge-conjugation Parity problem in quantum chromodynamics physics, known
as “quantum chromodynamics axion” [202]. Recently, high energy physics, e.g.
string theory, also predicts manifold of axion-like particles from the compactification of extra dimensions [203–206]. Typically, the axion is considered to havea light mass m ≪ 1 eV. Also, it behaves like non-relativistic fluid in the current
universe because of its oscillatory feature. Therefore, from cosmological pointof view, the axion or axion-like particle is considered to be a candidate of dark
matter. Hereafter, they are jointly called “axion”.
In general prediction of high energy physics, the axion has a weak but finite
coupling to gauge bosons, such as photons. This feature allows us to search
for axion dark matter with optical techniques. Conventionally, many experiments and observations have tried to detect axions under the magnetic fieldbackground, where axions are converted to photons (and vise versa) [207–213].
For example, in the solar axion observation such as CERN Axion Solar Telescope(CAST) [214, 215], the axion that is generated in the Sun is converted to X-ray
with magnets and then the axion could be observed by the X-ray detector. Another example is the observation of the axion generated in the supernova such as
SN1987A [216]. The axion generated in SN1987A is considered to be converted
to gamma ray by the galactic magnetic field. Thus the axion could be observed105with the gamma-ray observation. Note that the result of the axion observation
using SN1987A depends on the model of the galactic magnetic field. Although
the various observations explained above were performed, no characteristic signalof axion dark matter has been detected and a new method to search for axion
dark matter is needed.
Lately, a number of new approaches to search for axion dark matter using
optical cavity have been proposed [217–220]. These proposals do not need astrong magnet which could be a noise source (or make an experiment expensive),
while they utilize phase velocity variation between two circular-polarized lights
caused by axion dark matter [221–225]. Inspired by these experimental proposals,
we propose a new scheme to search for axion dark matter by using a Fabry–Pérot cavity, especially the arm cavity of the interferometric gravitational wave
detectors.

Phase velocity modulation caused by axion dark matterIn this section, how the dispersion relations of two circular-polarized lights are
modified in the axion background field. The axion-photon coupling constant, gaγ ,which is the characteristic parameter of axion’s coupling to photons is denoted
as Chern-Simons interaction
gaγ
a(t)Fµν F̃ µν = gaγ ȧ(t)ϵijk Ai ∂j Ak + (total derivative),(A.1)
4
where a(t) is the value of the axion field, the dot is the time derivative, ϵijk is
the three-dimentional Levi-Civita tensor, and Aµ is the vector potential obtained
from a curl of the electromagnetic field strength tensor. The electromagnetic
field strength tensor is defined as Fµν ≡ ∂µ Aν − ∂ν Aµ . The Hodge dual of Fµν
is defined as F̃ µν ≡ ϵµνρσ Fρσ /2, where ϵµνρσ is the four-dimentional Levi-Civita
tensor. Here, cosmic expansion is ignored and the Coulomb gauge, ∂i Ai = 0 andthe temporal gauge, A0 = 0 are adopted. By the Fourier transformation of Ai ,they are decomposed to the two circular-polarized photons,∑ ∫ d3 k
Ai (t, x) =Aλ (t, k) eλi (k̂) eik·x ,
(A.2)
3
(2π)
λ=L,R

where k is the vector of the wave number (k ≡ |k|), and the circular-polarization
vectors meet
′

′

λ
λ∗
λλ
eλi (k̂) = eλ∗, iϵijm kj eL/Ri (−k̂), ei (k̂)ei (k̂) = δm (k̂) = ±kei

L/R

(k̂).

(A.3)

Here the indices of L and R mean the left- and right-handed circular lights,
respectively. The upper and lower signs of the double sign in equation (A.3)
correspond to the left- and right-handed circular lights, respectively. Hereafter,
the same notation for the double sign is used as equation (A.3). The equation of2
motion for electromagnetic field, ÄL/R + ωL/R
AL/R = 0, needs the modificationof the dispersion relation because of the axion field,2
ωL/R= k 2 (1 ∓ gaγ ȧ/k) ,

106

(A.4)

where ωL/R is the angular frequency of the left- and right-handed polarized light.
Equation (A.4) also modifies the phase velocity of the two circular polarized
lights,gaγ ȧ
c2L/R = 1 ∓
.
(A.5)
k
Notice that, since axion dark matter is non-relativistic, its momentum is ignored.
The field of the axion dark matter is written asa(t) = a0 cos[mt + δτ (t)],(A.6)

where m is the axion mass which corresponds to the oscillation frequency, f =m/(2π) ≃ 2.4 Hz (m/10−14 eV), and δτ (t) is a phase factor of the axion oscillation. The phase factor, δτ (t), can be considered to be constant within the timescale of the coherent axion oscillation, τ = 2π/(mva2 ), where va is the velocityof the axion dark matter. Plugging the local dark matter velocity, ∼ 10−3 , weobtain
( −16
)
10
eV
τ ∼1
years.
(A.7)
m
According to equations (A.6) and (A.5), the phase velocity of the left- and righthanded light is written as
cL/R (t) ≃ 1 ± δc(t) ≡ 1 ± δc0 sin[mt + δτ (t)],(A.8)

where δc0 ≪ 1 is assumed andgaγ a0 m
δc0 ≡≃ 1.3 × 10−242k(λ1550 nm)(gaγ
10−12 GeV−1

)
.

(A.9)

Here the present energy density value of axion dark matter around the Earth,3ρa = a20 m2 /2 ≃ 0.3 GeV/cm is used. By measuring the phase velocity modulation, i.e. δc0 , we can search for axion dark matter.

Axion search with a Fabry–Pérot cavity
Figure A.1 shows the schematic of the experiment with a Fabry–Pérot cavity
that we propose. The cavity is kept on resonance with Pound–Drever–Hall technique [129] and increases the phase velocity diﬀerence, δc, as explained later. The
linearly-polarized laser with the wavelength of λ is injected. Here, p-polarized incident laser can be considered without loosing generality. The cavity is composedof the input and output mirrors with amplitude reflectivities and transmissivitiesof (r1 , t1 ) and (r2 , t2 ). Hereafter, the axion is regarded as oscillating coherently
while the laser circulating in the cavity interacts with axion dark matter unlessotherwise mentioned. In other words, δτ (t) is treated as a constant. This is valid√
for the light axion mass range where τ > 4πL r1 r2 /(1 − r1 r2 ).
The signal of axion dark matter is detected as polarization modulation since
it is generated by the phase velocity modulation of the circular polarized lights.
The modulation is detected in the detection ports (a) and (b) using polarizing

107

optics. Specifically, interference between p-polarized light (incident light) ands-polarized light (generated by axion dark matter) which are mixed by the halfwave plate (in port (a)) or non-ideal birefringence of the input mirror (in port(b)) is detected. Note that detection ports (a) and (b) can be installed withoutchanging the rest part of the interferometer.
Laser

PBS

CavityEcavp-pol.

HWP PBSTransmission
monitorFIp-pol.
PDH signal

s-pol.

r 2, t 2

r 1, t 1

PDtrans

PDreflDetection port (a)Detection port (b)Figure A.1: Schematic of the experimental setup for axion dark matter searchwith a Fabry–Pérot cavity [155]. FI is the Faraday isolator, PBS is the polarizing
beam splitter, HWP is the half wave plate, and PD is the photodetector. The
detection instrument for Pound–Drever–Hall method is not shown.
Here, the way to enhance, δc, is explained.
p-polarization is given byp

Ein (t) = E (t) = E0 e

ikt

( L
e

The incident light with

) 1
e √
2
R

( )
1
,
1

(A.10)

where E p (t) is the vector of the electric field of the p-polarized laser, eL andeR are basis vectors of left- and right-handed lights, respectively. In the axion
background field, the vector of the electric field in the cavity in front of the front
mirror is written as
[∞
]
( )
∑
(
)
1 1
ikt
L
R
e
e
Ecav (t) = t1 E0 e
An (t) √,(A.11)
2 1
n=1{
An+1 (t) ≡ An (t)R1 T [t − 2L(n − 1)]R2 T [t − 2L(n − 1/2)](n ≥ 1)
, (A.12)
A1 = 1
where L is the length of the cavity, T (t) is transfer matrix for one-way translation,()
Le−iϕ (t)0
T (t) ≡,
(A.13)
R0e−iϕ (t)∫ t
L/R(A.14)ϕ(t) ≡ kL ∓ kδc(t′ )dt′ ,t−L

and Ri is the matrix for the reflection of the circularly-polarized lights,
(
)
0
−ri
Ri ≡
(i = 1, 2).
(A.15)−ri0108

By calculating the summation of equation (A.11), we obtain(( ))) 1 + iδϕ(t)t1 E0 eikt ( L
1 1
0
R
√
e
e
Ecav (t) =01 − iδϕ(t)1 − r1 r2
2 1
t1=[E p (t) − δϕ(t)E s (t)] ,1 − r1 r2
where E s is the vector of the electric field of the s-polarized laser
(
)
∫ ∞
∫ ∞
imt dm−imt˜˜δc(m)Ha (m)eδc(t)edt ,δϕ(t) ≡
δc(m) =
2π−∞−∞

(A.16)(A.17)(A.18)and Ha (m) is the cavity response function to the phaser velocity modulation,( )( −imL )k 4r1 r2 sin2 mL
2Ha (m) ≡ i−e.
(A.19)−i2mLm 1 − r1 r2 eHere, we assumed that the cavity is on resonance, i.e. 2kL = 2πn (n ∈ N).
Equation (A.19) shows that the signal of the axion dark matter is enhanced√
to be proportional to the finesse, i.e. π r1 r2 /(1 − r1 r2 ), and resonates at odd
multiple of the free spectral range of the cavity, i.e. m = π(2N − 1)/L (N ∈ N).

Sensitivity of gravitational wave detectors to theaxion-photon coupling
Here, we estimate the sensitivity of the gravitational wave detectors to the axion
dark matter by considering just a quantum noise. Actually, the proposed method
is not aﬀected by the displacement noise such as the radiation pressure noise, or
the gravitational wave. Thus, it is enough just to consider a shot noise. In eachdetection port, the electric field received by the photodetector is written as{}√
Evac (t)EPD (t) =Tj [θp − δϕ(t)] +(A.20)
E s (t) (j = 1, 2),
E0
√
t1 tj
Tj ≡
(A.21),
1 − r1 r2
where Evac (t) is the electric field of vacuum fluctuation, and θp (|θp | ≪ 1, |θp | ≫|δϕ(t)|, |θp | ≫ |Evac (t)|) is the mixing angle of the polarization. Here, the second(or higher) order of |θp | is ignored. The power received by the photodetector iswritten as
PPD (t) ∝ |EPD (t)|2
[
]
√
√
√
Evac (t)2≃ θp Tj E0 θp Tj − 2 Tj δϕ(t) + 2,
E0(A.22)

where we neglect the second order terms of δϕ(t) and Evac (t) and their crossterm since |δϕ(t)| ≪ 1 and |Evac (t)| ≪ E0 . We can evaluate the sensitivity by109

Table A.1: Sets of parameters of the considered detectors. The incident power,
P0 , is enhanced by the power recycling technique for CE-, KAGRA-, and aLIGOlike detectors [21].
Similar detector
DECIGO [54]
CE [48]KAGRA [27, 28]
aLIGO [22]

L [m]
106
4 × 104
3 × 103
4 × 103

P0 [W]
5
600
335
2600

λ [×10−9 m]
515
1550
1064
1064

(r12 , r22 )(0.69, 0.69)(0.9988, 0.999995)(0.996, 0.999995)(0.986, 0.999995)

comparing the signal term (second term) and noise term (third term) of equation (A.22). The one-sided amplitude spectrum density of shot noise equivalent˜to δc(m)is denoted as,√√
Sshot (m) = √k
2P0

Tj |Ha (m)|,(A.23)

where P0 = kE02 /2 is the incident power. Here, the one-sided spectrum of Evac (t)is unity [115]. Equation (A.23) indicates that, if r1 < r2 , the detection port(b) has better sensitivity than the detection port (a). On the other hand, thedetection port (a) is better when r1 = r2 . Equation (A.23) can be translated
into the sensitivity to the axion dark matter, i.e. the axion-photon coupling gaγ ,as√()  Sshot (m)(Tobs ≲ τ )1550 nmgaγ (m) ≃ 1.5 × 1012 GeV−1, (A.24)× √ S Tobs(m) √shot
λ(Tobs ≳ τ )Tobs τ

where Tobs is the measurement time. The sensitivity improvement depends on if
Tobs is larger than the axion coherent oscillation time, τ , or not [226].
Figure A.2 shows the sensitivities of the gravitational wave detectors to gaγ
with the method that we propose. Here, we set the experimental parameters
similar to the gravitational wave detectors in operation or planned (specifically,
DECIGO [54], Cosmic Explorer (CE) [48], KAGRA [27, 28] and aLIGO [22]) as
shown in table A.1. We assume Tobs = 1 year and a lossless cavity, i.e. ri2 +t2i = 1(i = 1, 2). For DECIGO-like detector, the detection port (a) is adopted, and for
CE-, KAGRA-, and aLIGO-like detectors the detection port (b) is adopted. All
detectors overcome the upper limit provided by CAST [214, 215].
We can apply the proposed method to the gravitational wave detectors byinstalling polarizing optics in detection ports without replacing optics in themain interferometer, e.g. main test masses, and so on. Although it is not sosimple to realize the detection port (b) due to the existing apparatuses, such asa signal recycling mirror [135, 227], the axion dark matter signal, in principle,
can be obtained from the detection port behind the signal recycling mirror aswith the gravitational wave signal readout [228].

110

Coupling constant g[GeV-1 ]10 -9
10 -10
10 -11
10 -12
10 -1310 -1410 -15
10 -16
10 -17
10 -16

DECIGO
CE
aLIGO
KAGRA
CAST
SN1987A

10 -15

10 -1410 -1310 -12

10 -11

10 -1010 -9

Axion mass [eV]

Figure A.2: Sensitivity comparison of the detectors of which parameters areshown in table A.1. The sensitivity in higher mass range is peaky at the axion
mass of m = π(2N −1)/L (N ∈ N). The red and gray regions express the current
limit set by CAST [214, 215] and the cosmic ray observations of SN1987A [216],
respectevely.

Summary
Axion dark matter modulates the phase velocities of the circular polarized lights.
The eﬀect in the Fabry-Pérot cavity is modeled and its response is formulated.
With this model, we propose a new scheme to search for axion dark matterwith the Fabry–Pérot cavity in the arm of gravitational wave detectors. Theirpotential sensitivity to the axion-photon coupling constant, gaγ , is evaluated tobe so sensitive that the current limit of CAST [214, 215] can be improved with
1 year observation. Remarkably, these sensitivities can be realized by installingthe additional polarizing optics without changing the main gravitational wave
interferometer.

111

Appendix B

Power spectral density andfeedback control
Here, we review power spectral density and feedback control. In this appendix,
only linear system is considered.

Power spectral density
Power spectral density is a measure of the noise/sensitivity of the system infrequency domain. The (single-sided) power spectral density for a random fluctuation, x(t), is defined as⟩1⟨ ††Sx (f ) ≡ lim
xT (f )xT (f ) + xT (f )xT (f ) ,T →∞ T∫ T /2xT (f ) ≡x(t)e−2iπf t dt,

(B.1)
(B.2)

−T /2

where ⟨· · · ⟩ means the ensemble average. Physically, the power spectral density,
Sx (f ), means the contribution of x(t) to its variance in a frequency band aroundf√. Sometimes, linear spectral density (or amplitude spectral density) defined as
Sx (f ) is useful to consider noise eﬀect.
Here, how to calculate signal-to-noise ratio is shown. We consider the detector
that has noise level of the power spectral density S(f ) and signal h(f ), which is
Fourier space representation of h(t). In this setup, the squared signal-to-noise
ratio, ρ2 , is estimated by [63],2∫ ∞|h(f )|2
df
S(f )
0
∫ ∞ √
| f h(f )|2d(log f ).
=4
S(f )
0

ρ =4

(B.3)
(B.4)

The cross spectral density and the magnitude-squared coherence are also introduced here. The cross spectral density and the magnitude-squared coherence aremeasures of the correlation of the two systems. The (single-sided) cross spectral113density for two random fluctuation, x(t) and y(t), is defined as⟩1⟨ †Sxy (f ) ≡ lim
xT (f )yT (f ) + xT (f )yT† (f ) ,T →∞ T∫ T /2∫ T /2−2iπf txT (f ) ≡x(t)edt, yT (f ) ≡y(t)e−2iπf t dt.
−T /2

(B.5)
(B.6)

−T /2

The magnitude-squared coherence is defined as2
Cxy(f ) ≡

2
Sxy
(f )
.
Sx (f )Sy (f )(B.7)

Feedback control
Laplace transformation
Feedback control is often considered in frequency domain. The frequency response of the linear system can be easily understood with Laplace transformation. Laplace transformation of time-domain function f (t) is defined as∫ ∞˜
f (t)e−st dt,(B.8)
f (s) ≡
0where s ≡ iω. In this thesis, tilde of the function in frequency domain is sometimes omitted if there is no confusion.

Transfer function
The frequency response of the linear system is characterized with the transfer
function. When the input and output of the system are X̃(s) and Ỹ (s), respectively, the transfer function, G(s), is defined as
G(s) ≡Ỹ (s)
.
X̃(s)

(B.9)

The absolute value and angle of the transfer function, i.e. |G(s)| and ∠G(s), arecalled the gain and phase, respectively. This system is illustrated as shown infigure B.1 which is called a block diagram.

X
~

G

~Y

Figure B.1: Block diagram of a single-input single-output system.
For example, let’s consider the equation of motion of a mass. The equation of
motion in time-domain can be written asmẍ(t) = F (t),114

(B.10)

where, m is the mass, x is the position of the mass, and F is the force applied
to the mass. By performing Laplace transformation, the equation of motion is
written as− mω 2 x̃(ω) = F̃ (ω).
(B.11)
Here, we use s = iω. As a result, the transfer function from the force to theposition is written as
x̃(ω)
1
=
.
(B.12)−mω 2
F̃ (ω)
This equation indicates that the frequency response of the equation of motion
from the force to the position is proportional to ω −2 with phase delay of π rad(or 180 deg).

Open loop transfer function
Feedback control is composed of a sensor, filter, and actuator. The block diagramof the feedback controlled system is shown in figure B.2. H(s), F (s), and A(s)are transfer functions of the sensor, filter, and actuator, respectively. In thissystem, the output of each port is written as
1x̃ext (s),1 + H(s)F (s)A(s)H(s)ỹ(s) =
x̃ext (s),1 + H(s)F (s)A(s)H(s)F (s)z̃(s) =
x̃ext (s).
1 + H(s)F (s)A(s)∆x̃(s) =(B.13)(B.14)(B.15)

Here, we assumed zinj = 0. Sometimes, ỹ(s) and z̃(s) are called error signal andfeedback signal, respectively. The feedback control is characterized with the open
loop transfer function defined as Go (s) ≡ H(s)F (s)A(s) which we can measure
by injecting a large signal from the injection port and taking the ratio of z̃ ′ (s)to z̃(s) asz̃ ′ (s)= H(s)F (s)A(s) = Go (s),(B.16)z̃(s)()1
H(s)F (s)A(s)′z̃inj (s), z̃ (s) = −z̃inj (s) ,z̃(s) =
1 + H(s)F (s)A(s)1 + H(s)F (s)A(s)−where z̃inj is the injection signal from the injection port shown in figure B.2.
One important feature of the feedback control is the suppression of the external
fluctuation. Equation (B.13) shows that, when |Go (s)| ≫ 1, the residual motion
is suppressed to be x̃0 (s) ≃ x̃ext (s)/Go (s). On the other hand, when |Go (s)| ≪ 1,
the external motion is not suppressed, i.e. x̃0 (s) ≃ x̃ext (s). The above discussion
about the fluctuation suppression is also valid when we change x̃ext (s) and x̃0 (s)into the linear spectral density.

115

x
~
x
~ext

y
~

H
F

z'
~
z~inj

A
z~

Inj. port

Figure B.2: Block diagram of the feedback control system.

Signal equivalent noise estimation
In the feedback control system, we can estimate the eﬀect of noise equivalent tothe signal. For example, if x̃(s) is the signal, the eﬀect of the measured noise,ñ(s), injected from the injection port in figure B.2 is estimated as follows: first,
we calculate the transfer function from the signal and the noise to the sensingy(s)port. If we use error signal, the transfer functions are written as xext(s) ≡ Gs (s)y(s)
and zinj(s) ≡ Gn (s). Then, using these transfer functions, the signal equivalentnoise is estimated to be
Gn (s)x̃n (s) =ñ(s).
(B.17)Gs (s)The discussion in this subsection is also valid for the linear spectral density.

Stability of feedback control
For the stable operation of the system, the feedback control should be stable.
The stability of the feedback control can be checked with the open loop transfer
function. Although there are several ways to check the stability [229], we focus
on the scheme with a Bode plot here. The Bode plot is a plot set of the gain andphase of the open loop transfer functions as shown in figure B.3. The frequency
where the gain is unity is called unity gain frequency. Equation (B.13) indicates
that, if the open loop transfer function at the unity gain frequency is −1 (gain is
1 and phase is −π), the residual motion, x̃0 (s), goes to infinity, i.e. the feedback
control is unstable. Thus, the open loop transfer function should be designed tohave phase delay less than π at the unity gain frequency. In practice, the phasedelay at the unity gain frequency is designed to be less than 5π/6 to obtain therobust system. The phase delay can be checked with the Bode plot intuitively
as shown in figure B.3.

116

105
104

Gain

103
102
101
100

Phase [deg]

10-1
10-2
10-2
180
90
0
-90
-180
10-2

StableUnstable

10-1

100

101

102

103

104

10-1

100

101

102

103

104

Frequency [Hz]Figure B.3: Example of the Bode plot of open loop transfer functions of stableand unstable feedback control. Phase plots for both cases are overlapped. The
unity gain frequencies in the stable and unstable cases are 10 Hz and 1000 Hz,
respectively.

Multivariable control system
Up to here, we considered a single-input single-output system. Hereafter, we will
expand the above discussion to a multi-input multi-output system. When the
inputs and outputs of the system are n- and m-dimensional, respectively, thetransfer function can be denotes with m × n matrix asỸ (s) = G(s)X̃(s),(B.18)

where, X̃(s) and Ỹ (s) are the n-dimensional input column vector and the mdimensional output column vector, respectively.
The block diagram of the feedback control system with multi variables isrepresented in figure B.4. In order to control the n-dimensional system, weneed the n-dimensional sensors, at least. Here, we consider the n-dimensional
system.
If we choose the filter matrix as F (s) = A−1 (s)G(s)H −1 (s)(G(s) ≡ diag(G1 (s), ..., Gn (s))), the open loop transfer function is writtenas A(s)F (s)H(s) = G(s). In this condition, the n-dimensional system isdecomposed into the n feedback loops of the single-input single-output systemand the discussion up to the previous section can be applied independently toeach loop (for more detailed discussion, see [229, 230]).

117

x
~1
G1

~xext

H
A

F=A-1GH-1
x~nGn

Figure B.4: Block diagram of the multi-input multi-output feedback control system and equivalent set of the single-input single-output feedback control systems.
x̃ext ≡ (x̃1 , ..., x̃n )⊤ is an input vector, and H, F , and A are sensing, filter, andactuation matrices (n × n), respectively.

118

Appendix C

Noise sources of laserinterferometer
In order to detect gravitational waves, we need to reduce many noises. For example, quantum noise of the laser light, Brownian motion of the mirror, seismic
motion, and Newtonian force caused by the density fluctuation of the groundare considered to be major noise sources for the ground based detectors. The
latter two can be mitigated in the space detectors. Here, major noise sources
are reviewed. In this appendix, we considered one diﬀerential Fabry–Pérot interferometer as shown in figure C.1. For simplicity, we assume |Gfreq | ≫ 1 and|Gx | ≪ 1 in the observation frequency range. In this condition, the gravitational
wave signal is obtained at the error signal of the cavity length control loop. Notethat the amplitude spectral
density of the noise equivalent to the length fluc√tuation, xnoise (f ) [m/ Hz], is shown and it can be converted
to the amplitude√
spectral density of the stain equivalent noise, hnoise (f ) [/ Hz], with the relation
hnoise (f ) =1
xnoise (f ),
L

(C.1)

where f is the frequency and we assume that the mirrors are free.

General noises
Quantum noise
Quantum noise is caused by the quantumness of the laser light. Since the laser isintrinsic to the laser interferometers, quantum noise is the most principal noisefor the interferometers. Quantum noise can be divided into two parts: shotnoise and radiation pressure noise. Shot noise is the sensing noise and radiationpressure noise is the force noise. With the classical measurement, shot noise and

119

ETMY

ITMY

Gfreq

ITMX

PD

Laser

ETMX

PD

Gx
Figure C.1: Considered diﬀerential Fabry–Pérot interferometer. The open looptransfer functions of the frequency and the cavity length control loop are Gfreq
and Gx , respectively.

radiation pressure noise can be written as [115]
v
[
u
)2 ]
(
1 u
πℏcλ
f
t
xshot (f ) =,1+
4F 2Pincν1/2v
u 2πℏP
1
16F
incu
xrad (f ) =
(
)2 ,
t
2
π(2πf ) mTM
cλf
1 + ν1/2

(C.2)

(C.3)

where mTM is the mass of the mirror, ℏ is the reduced Planck constant, and λ is
the wavelength of the laser. According to equaitons√(C.2) and (C.3), shot noiseand radiation pressure noise inversely depend on Pinc . This means that, bychanging the input power, the root mean square of the shot noise and radiationpressure noise, i.e. the power spectral density of the quantum noise, cannot beless than,√4ℏ
xSQL (f ) =.
(C.4)
mTM (2πf )2xSQL (f ) is called standard quantum limit and means the lower limit of the quantum noise in the classical measurement. To beat the standard quantum limit,
quantum measurement, so called quantum-non-demolition measurement, is necessary [115,116,231]. The quantum-non-demolition measurement is the measurement scheme using the quantum correlation of the system to reduce the quantum
noise.

120

Thermal noise
Since mirrors (its supporting system, if any) have finite temperature, they fluctuate due to their Brownian motion. This motion is the source of the thermal
noise. The main thermal noise source is the mirror substrate and coating (andits suspension system).
The thermal noise is estimated with the fluctuation-dissipation theorem. According to the fluctuation-dissipation theorem, the fluctuation force is givenby [117]√Fth (f ) = 4kB T R(f ),(C.5)
where kB is the Boltzmann constant, T is the temperature of the mirror, andR(f ) is the mechanical resistance. By applying the mechanical transfer function
from the force at the dissipation point to the longitudinal motion of the mirror,
the thermal noise is obtained.
The thermal noises of the mirror substrate and coating are classified into two
types according to the dissipation mechanism: Brownian noise and thermoelastic
noise. The substrate brownian noise, xSB (f ), and thermoelastic noise, xSTE (f )are represented as [118–120]√
xSB (f ) =√xSTE (f ) =)(16kB T 1 − σs 2
2 1 − 2σs ϕc dc√,
ϕs 1 + √(2πf ) πEs w0
π 1 − σ s ϕs w 0

(C.6)

2

1
64 α 2 (1 + σs ) kB T 2 ks
√ s
,
2
3
w0 (2πf )2
πρs 2 C s(C.7)

where σs is the Poisson’s ratio of the substrate, Es is Young’s modulus of thesubstrate, w0 is the laser radius on the mirror, ϕs(c) is the loss angle of thesubstrate (coating), and dc is the depth of the coating, αs is the coeﬃcient ofthermal expansion of the substrate, ρs is the mass density of the substrate, Cs is
the heat capacity of the substrate, and ks is the thermal conductivity.
Coating brownian noise, xCB (f ), and thermoelastic noise, xCTE (f ), are givenby [122]√xCB (f ) =√xCTE (f ) =

2

2

2

16kB T dc
Ec (1 + σs ) (1 − 2σs ) + Es 2 (1 + σc ) (1 − 2σc )ϕ,c(2πf ) πw0 2
Es 2 Ec (1 − σc 2 )(C.8)
264kB T (1 + σs ) αs 2 κs√,πρs Cs w0 3 (2πf )2(C.9)

where Ec is Young’s modulus of the coating, σc is the Poisson’s ratio of thecoating, and κs (≡ ks /ρs /Cs ) is the thermal diﬀusion coeﬃcient of the substrate.
In addition to the mirror thermal noise, the residual gas could introduce nonnegligible thermal noise, so called residual gas thermal noise, even if the mirror

121

is placed in the vacuum system. The residual gas thermal noise is given byv√uu Smir Pgas T 3mgas kB
1
t16,(C.10)
xgas (f ) =mTM (2πf )2
Cgas
Tgas
where Smir is the cross section of the mirror along the longitudinal axis, Pgas is
the pressure around the mirror, Cgas is the shape factor which is almost unity
for the cylinder-shaped mirror, mgas is the mass of the residual gas molecular,
and Tgas is the temperature of the residual gas.

Seismic noise
The ground always fluctuates even when the earthquake does not occur. The
surface ground motion in quiet site above 10 Hz is approximately written as [232]x′seis (f ) = 1 × 10−10

(

10 Hz
f

)2 [

√ ]
m/ Hz .

(C.11)

Note that the actual seismic motion is diﬀerent from place to place. This seismic
motion is much larger than the expected fluctuation caused by the gravitational
wave. Thus, we need to isolate the test mass from the ground motion. One
way to do this is to float the mirror in space. Another way which is done in
the ground based detectors is suspending the mirror by pendulums. When the
mirror is suspended by the n-fold pendulum which has the resonant frequency offpend and the damping value γpend , the transfer function from the seismic motion
to the mirror motion is approximately denoted as
[
]n
2
fpend
Tpend (f ) ≃.
(C.12)
2−f 2 + fpend+ if γpend /(2π)
As a result, the cavity length fluctuation with seismic motion is written as
xseis (f ) = x′seis |Tpend (f )|.

(C.13)For example, when we use 9-fold pendulums with fpend = 2.5 Hz and γpend = 2π
rad·Hz, the seismic noise above 10 Hz is approximately written as
xseis (f ) = 1.5 × 10

−21

(

10 Hz
f

)20 [

√ ]
m/ Hz .

(C.14)

Newtonian noise
When the environmental Newtonian gravity field around the test mass fluctuates,
the gravitational force jiggles the test mass even without gravitational wave. This
force fluctuation is a source of noise called Newtonian noise. The environmental
Newtonian gravity can vary due to the mass density fluctuation of the ground,122

atmosphere, water, space craft, planets, and so on. For example, the seismic
Newtonian noise is modeled as [55]()2 10 Hz(3 Hz ≲ f ≲ 10 Hz)
f)4
xSNN (f ) = 2 × 10−20 ( 10fHz(10 Hz ≲ f ≲ 30 Hz)[ √ ]m/ Hz , (C.15)

while modeling most types of the Newtonian noises is still on going.

Interferometer noises
Frequency noise
As explained in Section 3.3.5, the laser frequency fluctuation is a noise source.
Considering the frequency stabilization with y-arm, the frequency noise is denoted as
1
L
xfreq =
δf,
(C.16)
1 + Gfreq ν
where δf is the laser frequency fluctuation of the laser source before the stabilization.
It is worth noting the frequency noise can be subtracted with the frequency
fluctuation measured in the frequency control loop. In the frequency control
loop, the dominant signal is frequency fluctuation. Thus, if we know the transfer
function from the laser frequency fluctuation to the gravitational wave signal,
the frequency noise can be subtracted as follows:′
h(t) = hgw (t) + nfreq (t) − Tfreq
δf (t) + nothers (t),(C.17)

nfreq (t) = Tfreq δf (t)(C.18)

where h(t) is the measured signal equivalent to the gravitational wave strain,
hgw (t) is the gravitational wave signal, nfreq (t) is the noise due to the laser frequency fluctuation, T is the transfer function from the laser frequency fluctuation
to the measured signal equivalent to the gravitational wave strain, δf (t) is thelaser frequency fluctuation, and nothers (t) is the noise other than the frequency′′, the frequency noise cannoise. If we can estimate Tfreq
ideally, i.e. Tfreq = Tfreq
be totally subtracted. However, when there is an error, ∆Tfreq , in the estimation′′of Tfreq, i.e. Tfreq= Tfreq + ∆Tfreq , the frequency noise after the subtraction
becomes ∆Tfreq δf (t). For example, when the precision of the transfer function
measurement is 10%, i.e. ∆Tfreq = 0.1Tfreq , the frequency noise is subtracted bya factor of 10, i.e. the frequency noise after the subtraction becomes 0.1nfreq (t).

Intensity coupled noise
As explained in Section 3.3.5, the laser intensity fluctuation is a noise source
through the cavity detuning. According to equation (3.70), the intensity coupled123

noise is written as
xint =√
δP(∆xrms )2 + (∆yrms )2,
Pinc(C.19)

where ∆x(y)rms is the root-mean-square of the cavity detuning of x(y)-arm, δP is
the intensity fluctuation, and Pinc is the incident power to the first beamsplitter.

Classical radiation pressure noise
In addition to the intensity coupled noise, the intensity fluctuation of the laser
light causes the classical radiation pressure noise due to the radiation pressure
fluctuation applied to the test masses. The classical radiation pressure noise isdenoted as
8FδP
xCRP =,(C.20)
πcmTM ω 2
where m is the mass of each test mass, and F is the finesse of the cavity.
It is worth noting that the classical radiation pressure noise can totally besubtracted with the measured intensity fluctuation. For example, if the cavity is
critically coupled, the intensity fluctuation can be measured at the transmission
port of the cavity.

Phase noise after the beam splitterIn the diﬀerential Fabry–Pérot interferometer, the phase noise between the firstbeamsplitter and the input test mass aﬀect the interferometer sensitivity to thegravitational waves since the phase noise corresponds to the diﬀerential frequency
fluctuation. The phase noise is expressed asxphase =2xdiﬀ ωL,c

(C.21)

where xdiﬀ is the displacement fluctuation between the first beam splitter andeach test mass.

Test mass actuator noise
The actuator on the test mass is essential part if we actuate the test mass to keep
the cavity resonance.*1 The force noises of the actuators are not distinguishedfrom the gravitational wave. The force noise is written as
xact =

2δfact,mTM ω 2

(C.22)

where δfact is the force fluctuation of the actuator.

*1 In some interferometer configuration where the cavity resonance is kept with the fre-quency control, the test mass actuator is not necessary.

124

Noise sources in space detectors
Here, we show the specific noise sources in space detectors, especially in DECIGO
and B-DECIGO. Specifically, force noises applied to the test masses and satellites
are explained. Although the other noises, such as local sensor noises and so on,
are noise sources as discussed in Chapter 5, their eﬀects vary depending on thedetector configuration. Thus they are not discussed here.

Force noise applied on test masses
Gravitational force
The gravitational force is a noise source especially in the geocentric orbit. The
main source is the higher order mode of the eccentricity of the Earth gravity.
Currently, the eﬀect of the higher order modes up to the 2190th order is considered [233]. However, in high orbital radius, such as the geostationary orbit, thegravitational force noise can be negligible.
Electro-magnetic force
One of the electro-magnetic force noise is caused by the magnetic fluctuation.
Since there is magnetic field in the satellite, the test mass is magnetized. Thus,
the magnetic field fluctuation couples the test mass motion. The magnetic fluctuation noise is written as [140]xmag =

2χTM VTM ′B δB,
mTM µ0 ω 2(C.23)

where χTM is magnetic susceptibility of the test mass, VTM is the volume of thetest mass, µ0 is magnetic permeability of vacuum, B ′ is a local gradient of themagnetic field, and δB is the fluctuation of the local magnetic field. In DECIGO
and B-DECIGO case, the largest magnetic√field is caused by the satellite, e.g.
B ′ = 3 × 10−8 T/m and δB = 1 × 10−9 T/ Hz [140]. Here we assume shielding
by a factor of 100 around the test mass.
The Lorentz force also makes a force noise applied to the test mass [140]. The
largest noise of Lorentz force is caused by the velocity fluctuation coupled withthe static magnetic field of the Earth [234] (in the geocentric orbit) or in theinter-planet [140] (in the heliocentric orbit). However, the Lorentz force eﬀect
can be negligible.
Thermal radiation
When there is a temperature fluctuation, the thermal radiation makes a force
noise on the test mass. As explained in Section 5.5, the thermal radiation noise
is denoted as [140]8σSB STM T 3 δT
xTrad = ϵTrad.
(C.24)
mTM cω 2125

Cosmic ray collision
Cosmic rays occasionally collides with the test mass. The momentum transferfrom a cosmic ray makes a force noise on the test mass. However, the eﬀect canbe negligible [140].

Force noise of applied on satellites
Solar radiation pressure
The major force noise applied on the satellite is caused by the solar radiationpressure fluctuation. As explained in Section 5.5, the solar radiation pressure
fluctuation is estimated to be [140, 142]
δWSun SSAT
δFSrad == 3 × 10−9c(

0.1 Hzf) 13√
N/ Hz.

(C.25)

Gravitational force
The gravity force is also applied to the satellite in the same way as the test mass.
However, the gravity force can be negligible in the observation frequency band.

Other practical noises
In addition to the principal noises explained above, there are many practicalnoise sources. Since these practical noise sources vary from detector to detector,
they should be considered in each case. The possible noise sources are scattered
light, electrical circuit noise and so on.

126Appendix D

Suspension system forground-based DECIGO prototype
Before the launch of DECIGO and B-DECIGO, the ground-based prototype
should be constructed. However, it is impossible to make a free mass on the
Earth. One way to mimic the situation in space on the Earth is using a soft
suspension system. Here, we show the suspension design study for the future
prototype construction.

Suspension design
The schematic of the suspension system is shown in figure D.1. The ground baseprototype consists of the cavity mirror suspension and the optical bench suspension. In the design, we aimed to obtain almost all the considered eigenmodes ofthe suspensions to be ≲ 1 Hz.
The cavity mirror is suspended from the frame with a double pendulum. Theintermediate beam stage is made of copper and its dimensions are 10 × 2 × 2 cm3 .
The mass of the beam is ∼ 360 g. As the suspension wire, the 10-cm-long and120-µm-diameter tungsten wire is utilized. From one edge of the intermediatebeam, the mirror holder made of aluminum is suspended with two tungsten wires.
The cavity mirror is attached to the mirror holder. The mirror holder weighs∼ 220 g and the two tungsten suspension wires are 40 µm in diameter. The
mirror holder is designed to be a plus shape to obtain a large moment of inertia
for pitch and roll rotations. On the other edge of the intermediate beam, thecounter weight is attached to compensate the mass of the mirror holder.
The optical bench that mimics the satellite is suspended with a single piano
wire. The piano wire has the diameter of 150 µm and the length of 25 cm. In
order to decrease the resonant frequency of the vertical motion, a coil spring ofa spring constant of ∼ 103 N/m is inserted between the frame of the suspension
system of the optical bench and the suspension wire. The optical bench is madeof aluminum plate of which thickness is 1.5 cm and diameter is 30 cm. On the
optical bench, some optics are fixed and the total mass of the optical bench andthe optics is estimated to be ∼ 15 kg.
We estimate the resonant frequencies of the cavity mirror suspension and the

127

(a)

Frame
Tungstenwire

Vertical
Yaw

10 cm

10 cm
2 cm

2 cm

Beam
(Copper)

(b)

Pitch

Coil spring

Roll

TranslateLongitudinal

Piano wire
25 cm

Tungsten
wireCounter weight(Copper)

30 cm

15 cm
9 cm
9 cm

Frame

Mirror holder 1.5 cm(Aluminum)
Cavity mirror

Aluminum plate

(d)

(c)

Figure D.1: Schematic (upper panel) and CAD image (lower panel) of the suspension system of the prototype. The left panel shows the cavity mirror systemand the right panel shows the optical bench system. Notice that the coil springis not shown in the CAD image of the optical bench.

optical bench suspension in terms of the center of the mirror and aluminum
plate, respectively. We use a toolkit of three-dimensional rigid-body modelingcoded in Mathematica® [235, 236]. The results are shown in table D.1. As aresult, almost all the resonant frequencies are designed to be lower than ∼ 1 Hz.
Since the roll mode of the mirror is less important due to the symmetry of themirror, its resonant frequency is designed to be relatively high. The roll resonant
frequency can be reduced using a spring system.
Table D.1: Resonant frequencies for each mode of two suspension types. Mode
name: (L, T, V, R, P, Y) indicates (Longitudinal, Translation, Vertical, Roll,
Pitch, Yaw). Since the roll mode of the mirror is less important due to thesymmetry of the mirror, its resonant frequency is designed relatively higher.
Suspension type
Cavity Mirror
Optical Bench

L
0.02 Hz
1 Hz

T
1 Hz
1 Hz

V
0.5 Hz1.4 Hz128

R(30 Hz)0.09 HzP
0.05 Hz0.09 HzY
0.1 Hz0.01 HzLocal sensors and actuators
For the ground-based prototype of DECIGO and B-DECIGO, not only the mechanical part, i.e. the suspension, but also the sensor/actuator for control isnecessary. As the sensor and actuator of the first prototype, the reflective photosensor and the coil-magnet actuator are planned to be used, respectively. The
photosensor imitates the local sensor and satellite attitude sensor in DECIGO
and B-DECIGO. The coil-magnet actuator imitates the test mass actuator andthe thruster of the satellite. Although the sensor and the actuator are not thesame as those of DECIGO and B-DECIGO, we can demonstrate the control
scheme that is developed in this thesis.

129

Appendix E

Electronic circuit
Here, we show the electronic circuit diagram used in the experiment in Chapter 6.

+5V
560
G12180-010A

50
AD811

RF

15k
DC
1n

Figure E.1: Circuit diagram of the photodetectors.

8.2k

Vin

1.366k

-2.5V
2.7k 2.7k

8.2k
1.366k

2.7k
OP27

OP27
2.7k

Vout+
-2.5V
2.7k 2.7k

2.7k
Vout-

2.7k
OP27

OP27

Figure E.2: Circuit diagram of the driver for the laser frequency (Koheras BASIK
X15).

131

82n
0-1k
Vin

680k

1k

4.7k

68k
OP27

4.7k

S/W
Vout

OP27

OP27

Figure E.3: Circuit diagram of the filter of a length loop.

S/W
5.6k 82n
56k
Vin

56k

5.6k
Vout
OP27

Figure E.4: Circuit diagram of the filter of a DC boost.

132

Appendix F

Hermite–Gaussian mode of theelectric field
Here, we show the brief review of the spacial mode of the laser beam and the laseralignment sensing. For comprehensive explanation, see [127]. In this appendix,
we consider electric and magnetic field in vacuum.

Hermite–Gaussian mode
From Maxwell equations, the electric field propagating along z-axis, E, meetsthe following equation,∇ × (∇ × E) + µ0 ϵ0∂2
E = 0.
∂t2

(F.1)

)
1
E · ∇ϵ0 .
ϵ0

(F.2)

This equation is denoted as(∇ E − µ0 ϵ0 E = −∇2

Since ∇ϵ0 = 0 and the electric field can be denoted as E = E0 ei(ωl t−2πz/λ) , we
can rewrite the equation as∇2 E + ωl µ0 ϵ0 E = 0.

(F.3)

When we set the origin of the spacial coordinates at the beam waist, i.e. theposition where the beam radius becomes smallest, the general solution of equation (F.3) is given by
(√ )
(√ )
w0
2x
2y
Hl
Hm
eiωl tEl,m (x, y, z, t) =E0
w(z)
w(z)
w(z)
[ 2
]
x + y2π(x2 + y 2 )2πz
× exp − 2−i−i
+ i(l + m + 1)η ,
w (z)
R(z)
λ
(F.4)

133

where Hl (· · · ) is the l-th Hermite polynomial, w0 and w(z) = w0
the spot sizes of (the laser) beam at waist (z = 0) and z, z0 =z2
1 + z02√2
1 + zz2 are

πw02
λ

0

is Rayleighlength, R(z) = z
is the radius of curvature of the electric field at z, and( )η = tan−1 zz0 is Gouy phase at z. Here, we assume rotationally symmetricbeam along z-axis. The electric field modes represented in equation (F.4) arecalled Hermite–Gausssian modes. Hermite–Gaussian mode with l = m = 0
is called the fundamental mode. In addition, Hermite–Gaussian modes with(l = 1, m = 0) and (l = 0, m = 1) are sometimes called 10 and 01 modes,
respectively.

Beam translation and tilt
In laser interferometric gravitational wave detectors, the fundamental mode laser
is mainly used. The small translation and tilt of the beam can be expressed by
the combination of the fundamental mode and 10/01 modes [130]. The translated(x → x+δx) and tilted ((x, z) → (x′ = x cos δθ −z sin δθ, z ′ = x sin δθ +z cos δθ))
beams are given by
Translation (δx): E0,0 (x + δx, y, z, t) ≃ E0,0 (x, y, z, t) +Tilt (δθ): E0,0 (x′ , y, z ′ , t) ≃ E0,0 (x, y, z, t) + i

δxE1,0 (x, y, z, t),
w0
(F.5)

δθ
E1,0 (x, y, z, t), (F.6)θD

λwhere θD = πw(w0 ≫ λ) is the beam divergence angle. Here, we assume0|δx| ≪ w0 and |δθ| ≪ θD .
We can measure the translation and tilt of the laser beam by measuring theamplitude of the 10/01 modes. There are some schemes to measure the 10/01
modes. One example for Fabry–Pérot cavity alignment is the wavefront sensing
method [133,134] where the 10/01 modes generated in the cavity due to the cavity
axis translation/tilt are measured by taking the beatnote with the fundamental
mode reflected by the input mirror. Note that the optical axis of the Fabry–Pérot
cavity is determined by the centers of curvature of the cavity mirrors.

134Appendix G

Antenna patternHere, we show the brief review of the antenna pattern of interferometric gravitational wave detectors, i.e. angular response. Especially, we compared the patternof the L-shaped detector and the pattern of the triangular-shaped detector.
The output signal of the gravitational wave detectors are given by [63]h(n̂, t) =∑FA (n̂, t)hA (t),

(G.1)

A

where n̂ is the unit vector along the traveling gravitational wave direction,
F A (n̂, t) is the antenna pattern function of the gravitational wave detector, andhA (t) is the gravitational wave form. Here, A indicates the existing mode of thegravitational wave in the theory. As discussed in Chapter 2, in general relativity
the allowed modes are two tensor modes, i.e. plus and cross modes (A = +, ×).
In some modified gravity theories, two vector modes and two scalar modes areallowed in addition to the tensor modes. The two additional vector modes arecalled vector-x and vector-y modes (A = x, y). Also, the two additional scalarmodes are called breathing and longitudinal modes (A = b, l). The gravitational
waves of each mode are given by [84]
+
′
′
′
′
Plus mode: h+ (t)e+ij (n̂), eij (n̂) ≡ êx ⊗ êx − êy ⊗ êy ,

(G.2)

×
′
′
′
′
Cross mode: h× (t)e×ij (n̂), eij (n̂) ≡ êx ⊗ êy + êy ⊗ êx ,(G.3)

Vector-x mode: hx (t)exij (n̂), exij (n̂) ≡ ê′x ⊗ ê′z + ê′z ⊗ ê′x ,(G.4)

eyij (n̂) ≡ ê′y ⊗ ê′z + ê′z ⊗ ê′y ,ebij (n̂) ≡ ê′x ⊗ ê′x + ê′y ⊗ ê′y ,(G.5)

2ê′z ⊗ ê′z ,(G.7)

Vector-y mode: hy (t)eyij (n̂),
Breathing mode: hb (t)ebij (n̂),

Longitudinal mode: hl (t)elij (n̂), elij (n̂) ≡

√

(G.6)

where the set of the three unit vector, {ê′x , ê′y , ê′z } (ê′x = (cos θ cos ϕ, cos θ sin ϕ, − sin θ),ê′y = (− sin ϕ, cos ϕ, 0), ê′z = (sin θ cos ϕ, sin θ sin ϕ, cos θ) = −n̂), and the coordinates of the traveling gravitational wave are defined in figure G.1-(a). Here, ⊗ is
the Kronecker product. The detection of the non-tensor mode is an importantmilestone for the test of the modified gravity theory.

135

e'

(a)

(b)

e'

GW sourcee'(c)L-shaped
detector

ITF 1

ITF 2
ITF 3

Figure G.1: Some definitions for antenna pattern calculation: (a) the coordinates(b) L-shaped detector orientation, and (c) triangular-shaped detector orientation.

The antenna pattern function is calculated as
FA (n̂) = Dij eA
ij (n̂),1
Dij ≡ (û ⊗ û − v̂ ⊗ v̂),
2

(G.8)
(G.9)

where Dij is the detector tensor, and û and v̂ are the unit vectors along the arm
direction of the detector. Here and hereafter, we ignore the time dependence of
the antenna pattern function. When we set the coordinate as shown in figure G.1,
û and v̂ are given by
L-shaped: û = (1, 0, 0), v̂ = (0, 1, 0),
(
(π)
(π) )
Triangular-shaped (1): û = − cos, − sin
,0 ,
3
3
(π) )
(
(π)
v̂ = cos, − sin,0 ,3( 3( π )(π) )Triangular-shaped (2): û = (1, 0, 0), v̂ = cos
, sin
,0 ,
3
( π )3 )
(
(π)
, sin
, 0 , v̂ = (−1, 0, 0).
Triangular-shaped (3): û = − cos
3
3(G.10)(G.11)(G.12)(G.13)

Consequently, the antenna pattern functions for the L-shaped detector are
calculated as
1(L)(1 + cos2 θ) cos 2ϕ, F× = − cos θ sin 2ϕ,
2(L)
Fx = sin θ cos θ cos 2ϕ, Fy(L) = − sin θ sin 2ϕ,
1
1
(L)
(L)
Fb = − sin2 θ cos 2ϕ, Fl = √ sin2 θ cos 2ϕ.
2
2
(L)

F+ =

(G.14)

The antenna pattern functions for the triangular shape detectors are calculated

136

as

√
√
(T1)
(T1)
3
3
2

F
=
(1
+
cos
θ)
sin
2ϕ,
F
=
−

×
2 cos θ cos 2ϕ,
 +
√4
√
(T1)
(T1)
= 23 sin θ cos θ sin 2ϕ, Fy= 23 sin θ cos 2ϕ,
ITF1: Fx√√
F (T1) = − 3 sin2 θ sin 2ϕ, F (T1) = 1 3 sin2 θ cos 2ϕ.
b

4

l

2

(G.15)

2


√
(T2)
1
2

(1
+
cos
θ)(3
cos
2ϕ
−
3 sin 2ϕ),
F
=

+
8

√

(T2)


F× = − 14 cos θ( 3 cos 2ϕ + 3 sin 2ϕ),
F (T2) = 1 sin θ cos θ(3 cos 2ϕ − √3 sin 2ϕ),
x
4
√
ITF2:
(T2)

Fy
= − 14 sin θ( 3 cos 2ϕ + 3 sin 2ϕ),


√

(T2)


Fb
= − 18 sin2 θ(3 cos 2ϕ − 3 sin 2ϕ),


√

2
1
F (T2) = √
sin
θ(3
cos
2ϕ
−
3 sin 2ϕ).
l
4 2

(G.16)


√
(T3)
1
2

=
−
F
(1
+
cos
θ)(3
cos
2ϕ
+
3 sin 2ϕ),

+
8

√

(T3)


F× = − 14 cos θ( 3 cos 2ϕ − 3 sin 2ϕ),
F (T3) = − 1 sin θ cos θ(3 cos 2ϕ + √3 sin 2ϕ),
x
4
√
ITF3:
(T3)
1

Fy
= − 4 sin θ( 3 cos 2ϕ − 3 sin 2ϕ),


√

(T3)
2
1


sin
θ(3
cos
2ϕ
+
3 sin 2ϕ),
F
=

b
8

√

2
1
F (T3) = − √
sin
θ(3
cos
2ϕ
+
3 sin 2ϕ).
l
4 2

(G.17)The results are illustrated in figures G.2-G.5.
For non-tensor mode detection, we need the interferometers which have independent antenna patterns. If we utilize a triangular-shaped geometry, there is
one pair of the interferometer. Therefore, when the number of satellites is three,
the triangular geometry is better than the L-shaped geometry for the test of themodified gravity theory even though the single L-shaped interferometer has agood response to each mode.
It is worth noting two things. First, although there are three interferometersin the triangular geometry, we can obtain only two independent signals. Thisis because the third interferometer signal can be constructed ideally from thesignals of the other two interferometers. However, the third interferometer isstill very useful for constructing null-streams, which are time-series data where
the gravitational wave signals are cancelled. The null-stream is useful for thetest of general relativity [61]. The second is the time dependence of the antenna
pattern. Although it is not ignored in the above discussion, we can obtain
more information from the observation data considering the time dependenceof the antenna pattern. This is because, when the antenna pattern varies during
the observation, the detector can be regarded as multiple detectors eﬀectively.
Especially in DECIGO and B-DECIGO, the observation duration of a single
event lasts more than one month. Thus, time dependence of the antenna pattern
is useful.

137

Figure G.2: Antenna pattern functions for the L-shaped interferometer.

138

Figure G.3: Antenna pattern functions for triangular-shaped interferometer 1.

139

Figure G.4: Antenna pattern functions for triangular-shaped interferometer 2.

140

Figure G.5: Antenna pattern functions for triangular-shaped interferometer 3.

141

Appendix H

Full DECIGO interferometer model
Here, we show the Simulink system of the full DECIGO interferometer model.
The model is organized as follows: Figure H.1 shows the top layer of the full
DECIGO interferometer model, which is mainly for defining the name of theports. From these ports, the noise explained in Section 5.5 or the excitation signal to measure the open loop transfer functions can be injected and the sensor
signals can be obtained. Note that virtual sensors to measure parameters thatcannot be measured in the real detector (e.g. a global position of the mirror)
can also be used. Figure H.2 is the main system of the full DECIGO interferometer model included in the top system (represented as ‘DECIGO’). This main
system represents the behavior of the whole DECIGO. In the orange blocks, thelongitudinal and alignment sensing methods explained in Section 5.3 are coded.
In the magenta blocks, control topology explained in Section 5.6 is coded. The
gravitational wave can be injected in the cavity length to estimate the noise
performance. Figure H.3 shows the plant system of the mechanical and optomechanical system of the full DECIGO interferometer model, which includes thetest masses and the satellites. In the subsystems of the plant model, the mechanical property (in Section 5.2), the actuation system (except for the frequency
actuation in Section 5.4) and the local sensors (in Subsection 5.3.3) are coded.
The opto-mechanical coupling is also coded in this plant block.

143mon_disp_SAT1[disp_SAT1]

mon_disp_SAT2[disp_SAT2]

mon_disp_SAT3[disp_SAT3]

ES_M1b[ES_M1b]3

21

act_SAT1L
4

disp_LAS1cR
62

ES_M1cP
39

disp_LAS1bP
57

disp_LAS1cP
63[act_SAT3]

act_SAT3

ES_M3a[ES_M3a]act_SAT1Y

act_M1bY

act_M1cY

act_STM1bY

act_STM1cY

ES_M1bY

ES_M1cY

disp_LAS1bY

disp_LAS1cY

ES_M3b

9

27

33

111

117

act_cav1c_thetaz
161

act_ES_M1b[ES_M3b]act_cav1b_thetaz
153

disp_SAT1Y[act_M1b]

22

46

40

64

70

[act_M1c]

act_ES_M1c

act_M2cL
28

act_M2aL
34

act_STM2cL
112

act_M2aT35

act_cav1b_az
154

act_STM2aL
118

act_STM2cT
113

act_cav1c_az
162

act_cav1b_bz

act_STM2aT
119

disp_SAT2L
23

act_cav1c_bz

ES_M2cL47

disp_SAT2T
24

ES_M2aL41

ES_M2cT
48

act_ES_M2c
act_ES_M2a

CavC_L[act_M3a]

act_ES_M3a

disp_LAS1b[act_M3b]

act_ES_M3b
act_STM1b

disp_LAS2c[disp_LAS2c]

act_STM1c

disp_LAS2a[disp_LAS2a][act_STM2c]

act_STM2c

disp_LAS3a[disp_LAS3a]

act_SAT3L
227

act_M3aL
233

act_M3bL
239

act_STM3aL
245

act_STM3bL
251

act_cav2c_dz
168

act_cav2a_dz
176

disp_SAT3L
273

ES_M3aL297

ES_M3bL291[act_STM2a]act_STM2adisp_LAS3b[disp_LAS3b]

act_M3aT234

act_M3bT
240

act_STM3aT
246

act_STM3bT
252

act_cav2c_thetaz
169

act_cav2a_thetaz
177[act_STM3a]act_SAT3T
228

disp_SAT3T
274

ES_M3aT
298

ES_M3bT
292

act_STM3a

CavB_L[act_STM3b]

act_STM3b[extF_SAT1][disp_LAS1c][CavA_A]cav1b_L

extF_SAT3
extF_M1b

cav2c_L[extF_M1c]

extF_M1c

cav2a_L[CavC_A]4
cav1b_L5

cav1c_L[extF_M2c]

extF_M2c

cav3a_L

extF_M2a

cav3b_L[extF_M3a]extF_M3acavA_AbsL

extF_M3b

cavB_AbsL[extDisp_STM1b]

extDisp_STM1b

cavC_AbsL

extDisp_STM1c

cav1b_A

extDisp_STM2c8

act_cav2c_ay
166

act_STM2aY
250

act_cav2c_by
167[act_SAT3]

act_cav3a_dy
299

act_STM3bY

act_cav3a_thetay300

act_cav2a_az
178[disp_SAT3]

39

51

45

123

135

9

extF_SAT1L
40

extF_M1bL
52

extF_M1cL
46

extDisp_STM1bL
124

extDisp_STM1cL
136

act_cav3a_dz
303

act_cav3b_dz
311

act_cav3a_thetaz
304

act_cav3b_thetaz
312

act_cav3a_az
305

act_cav3b_az
313

extF_SAT1T
41

14

Fb_cavB_AbsL

15

Fb_cavC_AbsL[Cav1b_A][extF_SAT1]

extF_SAT1V
42[Cav1c_A]

extDisp_STM1bR
127

extDisp_STM1cR
139

extF_M1cP
50

extDisp_STM1bP
128

extDisp_STM1cP
140

extF_M1bY

extF_M1cY

extDisp_STM1bY

extDisp_STM1cY

act_cavBC_dy
315

141

act_cavBC_thetay316

nES_M1ccavB-cavC[CavBC_A]

extF_SAT2R
61
extF_SAT2P
62

cavC-cavA
cavA-cavB[CavAB_A][nES_M3a]nES_M3amon_SAT1[mon_SAT1][nES_M3b]

nES_M3bmon_SAT2[mon_SAT2]

nFreq1

mon_SAT3[mon_SAT3]

nFreq2

mon_M1b[mon_M1b]69

extF_SAT2L
58
extF_SAT2T
59[extF_SAT2]

extF_SAT2V
60

extF_SAT2Y

129

extF_M2aL
64

extDisp_STM2cL
130

extDisp_STM2aL
142

extF_M2aT
65

extDisp_STM2cT
131

extDisp_STM2aT
143[extF_M2c]

extF_M2cV
72

extF_M2aV
66[extF_M2a]extDisp_STM2cV
132[extDisp_STM2c]

extDisp_STM2aV
144

extF_M2cR
73

extF_M2aR
67

extDisp_STM2cR
133

extDisp_STM2aR
145

extF_M2cP
74

extF_M2aP
68

extDisp_STM2cP
134

extDisp_STM2aP
146

extF_M2aY

268

extDisp_STM2cY

262

extF_M3aL
269

extF_M3bV
265

extDisp_STM3aV
277

act_cav3b_bz

extDisp_STM3bV
283

318

179

act_cavCA_dy319

act_cavAB_dy
180[act_cavCA_A]

act_cavCA_thetay320

act_cavAB_thetay181

act_cavCA_dz
321

act_cavAB_dz
182

act_cavCA_thetaz

act_cavAB_thetaz

extF_SAT3R
260

extF_M3aR
272

extF_M3bR
266

extDisp_STM3aR
278

extDisp_STM3bR
284

nASC1b_dy191

nASC1c_dy
211[mon_M3a]extF_SAT3P
261

extF_M3aP
273

extF_M3bP
267

extDisp_STM3aP
279

extDisp_STM3bP
285

nASC1b_thetay
192

nASC1c_thetay
212[act_cav2a_A]

extF_M3aY

extF_M3bY

extDisp_STM3aY

extDisp_STM3bY

mon_M3b[mon_M3b]

extF_SAT3Y

act_cav2a_A[act_cav3a_A]

act_cav3a_A
act_cav3b_A
act_cavBC_A
act_cavCA_A
act_cavAB_A

183

352

[mon_STM1b]

mon_STM1b[mon_STM1c]

mon_STM1c[mon_STM3a]mon_STM3b[mon_STM3b]nES_M1bV
84[mon_act_DF_SAT1]

nES_M1bR
85

nES_M1cR
79[mon_act_DF_SAT2]

nES_M1bP
86

nES_M1cP
80nES_M1bY

nES_M1cY

hGW2

mon_act_DF_SAT3[mon_act_DF_SAT3]

hGW3

mon_act_freq1

nLSC1bmon_act_freq2

nLSC1cmon_act_freq3

nLSC2cmon_act_M1b10

nLSC2a349

188

mon_act_M1c

nES_M2cV
96
nES_M2cR
97

nES_M2aV
90nES_M2cP
98

nES_M2aP
92[mon_act_M3a]nES_M2cY

nES_M2aY

nLSC_AbsLa
nLSC_AbsLb

mon_act_M3b[mon_act_M3b]nLSC_AbsLc

mon_act_STM1b[mon_act_STM1b]nASC1bmon_act_STM1c292

nASC1cmon_act_STM2c[mon_act_STM2c]nES_M3aV295

nASC2cmon_act_STM2a[mon_act_STM2a]nES_M3aR296

nASC2amon_act_STM3a[mon_act_STM3a]nASC3amon_act_STM3b[mon_act_STM3b][nASC3b]nASC3bmon_disp_M1b[mon_disp_M1b][nASC_BC]

nASC_BC

mon_disp_M1c[mon_disp_M1c][nASC_CA]nASC_CA

mon_disp_M2c[mon_disp_M2c]nASC_AB

mon_disp_M2a[nSTT1]nSTT1[nSTT2]nSTT2mon_disp_M3b[nSTT3]nSTT3cav1_L_subtracted

mon_disp_M3a[mon_disp_M2a]

mon_STM1cT
172[mon_M1b]

mon_M1bV
143

mon_M1bP
145

mon_M2aV
155

mon_STM2aR
180
mon_STM2aP
181

mon_SAT2Y

mon_M2cY

mon_M2aY

mon_STM2cY

mon_STM2aY

278

308

302

356

350

mon_M3bL
303

Cav3b_dy391

Cav3a_thetay384

Cav3b_thetay
392

mon_STM3bY

Cav3a_ay
385

188

200

206

248

254

mon_act_M1bL
201

mon_act_DF_SAT1T
190

mon_act_M1cL
207

mon_act_M1bT
202[mon_act_M1b]

mon_act_M1bV
203

mon_act_STM1bL
249

mon_act_M1cT
208[mon_act_M1c]

mon_act_M1cV
209

mon_act_STM1bV
251

mon_act_STM1cR
258

mon_act_STM1bP
253

mon_act_STM1cP
259

mon_act_M1cY

mon_act_STM1bY

mon_act_STM1cY

218

212

266

260
mon_act_STM2aL
261
mon_act_STM2aT
262

mon_act_DF_SAT2V
197[mon_act_M2c]

mon_act_M2cV
221[mon_act_M2a]mon_act_M2aV
215[mon_act_STM2c]mon_act_STM2cV
269[mon_act_STM2a]nASC2a_dz
223

mon_act_DF_SAT2R
198

mon_act_M2cR
222

mon_act_M2aR216

mon_act_STM2cR
270

mon_act_STM2aR
264

nASC2a_thetaz
224

mon_act_DF_SAT2P
199

mon_act_M2cP223

mon_act_M2aP217

mon_act_STM2cP
271

mon_act_STM2aP
265

nASC2c_az
209

nASC2a_az
225

mon_act_DF_SAT2Y

mon_act_M2cY

mon_act_M2aY

mon_act_STM2cY

mon_act_STM2aY

284

320

314

368

330

mon_act_DF_SAT3L
285

mon_act_M3aL321

mon_act_M3bL315

mon_act_STM3aL
369

mon_act_STM3bL
363

nASC3b_dy
331

mon_act_DF_SAT3T
286

mon_act_M3aT322

mon_act_M3bT316

mon_act_STM3aT
370

mon_act_STM3bT364

nASC3b_thetay
332[mon_act_DF_SAT3]

mon_act_DF_SAT3V
287

nASC3b_ay
333[nASC3a][nASC3b]nASC3b_by334[mon_act_M3a]mon_act_M3aV323[mon_act_M3b]

mon_act_M3bV317

mon_act_STM3bR
366

mon_act_STM3aP
373

mon_act_STM3bP
367

mon_act_M3bY

mon_act_STM3aY

mon_act_STM3bY

224

230

nASC3b_bz

mon_disp_M1bL
225

mon_disp_M1cL
231

mon_disp_M1bT
226

342

[mon_disp_M1b]nASC_BC_dy
343

mon_disp_M1bV
227

nASC_CA_dz
341

nASC_AB_dz
201

nASC_CA_thetaz

nASC_AB_thetaz[mon_disp_M2c]

mon_disp_M2cV245

236
mon_disp_M2aL237
mon_disp_M2aT
238[mon_disp_M2a]mon_disp_M2cR
246

mon_disp_M2aV239
mon_disp_M2aR
240

nSTT2R364

mon_disp_M2cP247

mon_disp_M2aP
241

nSTT2P365

mon_disp_M2cY

mon_disp_M2aY332

nSTT2Y326

mon_disp_M3aL
333

366
nSTT3L367

mon_disp_M3bL327

mon_disp_M3aT
334[mon_disp_M3a]nSTT3T368

mon_disp_M3aV
335[nSTT3]

CavAB_dz
127
CavAB_thetaz

mon_disp_M1cY

mon_disp_M2cT244[nASC_AB]

CavAB_thetay
126

CavCA_dz
405
CavCA_thetaz

mon_disp_M1cV233242
mon_disp_M2cL
243

nASC_AB_thetay
200

124
CavAB_dy
125[CavAB_A]

CavCA_thetay404

mon_disp_M1cP
235

mon_disp_M1bY

198
nASC_AB_dy
199[nASC_CA]

CavBC_thetaz
402
CavCA_dy
403[CavCA_A]

mon_disp_M1cR234

mon_disp_M1bP
229

nASC_BC_dz
345
nASC_BC_thetaz

CavBC_dz
401

mon_disp_M1cT
232[mon_disp_M1c]

mon_disp_M1bR
228[nASC_BC]

nASC_BC_thetay
344

338

CavBC_thetay
400

mon_act_STM3bV
365

mon_act_STM3aR
372

mon_act_M3bP319

mon_act_M3aY

nASC3a_bz

nASC_CA_dy339[mon_act_STM3b]

mon_act_M3bR318

mon_act_M3aP
325

mon_act_DF_SAT3Y

nASC3b_az
337

nASC_CA_thetay340

mon_act_STM3aV
371

mon_act_M3aR324

mon_act_DF_SAT3P
289

nASC3b_dz
335
nASC3b_thetaz
336

nASC3a_az
329[nSTT1][mon_act_STM3a]mon_act_DF_SAT3R
288

nASC3a_dz
327

Cav3b_az397
Cav3b_bz

398
CavBC_dy
399

362

322
nASC3a_dy
323

Cav3b_thetaz
396

Cav3a_az389
Cav3a_bz[CavBC_A]

mon_act_STM2aV
263

nASC2c_dz
207
nASC2c_thetaz
208

Cav3b_dz
395

Cav3a_thetaz388

mon_act_STM1cV
257

mon_act_STM1bR
252

mon_act_M1cP
211

mon_act_M1bY

Cav3b_by394

Cav3a_dz387

mon_act_STM1cT
256[mon_act_STM1c]mon_act_M1cR
210

mon_act_M1bP
205

Cav3b_ay393[Cav3b_A]

Cav3a_by386

mon_act_STM1cL
255

mon_act_STM1bT
250[mon_act_STM1b]

mon_act_M1bR
204

194

[mon_act_DF_SAT2]

390

Cav3a_dy383

mon_STM3bP
355

mon_STM3aY

mon_act_DF_SAT1Y[nASC2a]

Cav2a_bz

382

mon_STM3bR
354

mon_STM3aP
361

mon_M3bY

mon_act_DF_SAT1V
191

nASC2a_bz

Cav2a_az
123

Cav2c_bz

mon_STM3aR
360

mon_M3bP
307

mon_M3aY

mon_act_STM2cL
267

nASC3a_ay
325

Cav2a_thetaz
122

Cav2c_az115

mon_M3bR
306

mon_M3aP313

mon_act_STM2cT
268

nASC3a_by326

Cav2c_thetaz
114

mon_STM3bV
353

mon_M3aR
312

mon_SAT3Y

mon_act_M2aL
213

nASC3a_thetay324

Cav2a_dz
121

mon_SAT3P
283

mon_act_DF_SAT1L
189[mon_act_DF_SAT1]

Cav2a_by120

Cav2c_dz
113

mon_STM3bT
352[mon_STM3b]

mon_STM3aV
359

Cav2a_ay
119[Cav2a_A]

Cav2c_by112

mon_STM3bL
351

mon_STM3aT
358[mon_STM3a]

Cav2c_ay
111[Cav2c_A]

mon_SAT3R
282

nASC3a_thetaz
328

nSTT3V369

mon_STM3aL
357

mon_M3bT
304

116
Cav2a_dy
117
Cav2a_thetay
118

mon_STM2cP
187

mon_M3bV
305

Cav1c_bz

108
Cav2c_dy109

mon_STM2cR
186[mon_M3b]

Cav1c_az
107

Cav1b_bz

Cav2c_thetay
110

mon_M2aP
157

mon_M3aL
309

Cav1c_thetaz
106

Cav1b_az
99

176
mon_STM2aL
177

mon_M2aR
156

mon_M3aT
310

Cav1c_dz
105

mon_STM2aT
178

mon_M2cP
163

mon_M3aV
311

Cav1c_by104

mon_STM2aV
179

mon_act_M2aT214[nES_M3b][nSTT2]

Cav1b_thetaz
98

mon_STM1cY[mon_STM2a]

Cav1c_ay103[Cav1c_A]

Cav1b_by96

mon_STM1cP
175

182
mon_STM2cL
183
mon_STM2cT
184
mon_STM2cV
185

mon_M2cR
162[mon_M3a][Cav1b_A]

Cav1b_dz
97

mon_SAT2P
139

nES_M3bP
291

nSTT2V363

mon_STM1bY[mon_STM2c]

100
Cav1c_dy101
Cav1c_thetay
102

Cav1b_ay95

mon_STM1cR
174

mon_SAT2R
138

nES_M3bY

360

mon_STM1cV
173

mon_STM1bP
169

152
mon_M2aL
153
mon_M2aT
154[mon_M2a][mon_STM1c]

mon_STM1bV
167
mon_STM1bR
168

mon_M1cY

158
mon_M2cL
159
mon_M2cT
160
mon_M2cV
161[mon_STM1b]

mon_M1cP
151

mon_M1bY[mon_M2c]

nES_M3bR
290

nSTT2L361

411

mon_M1cV
149
mon_M1cR
150

nES_M3aP
297

nSTT2T362
mon_freq3[mon_M1c]

mon_M1bR
144

nSTT1Y408

cav3_L_subtracted
410
mon_freq2

mon_freq3

mon_STM1bT
166

mon_act_M2cL
219

nSTT1P359
407

cav2_L_subtracted
409

DECIGO

mon_M1cT
148

nES_M3aY

nSTT1V357

406

mon_freq1
mon_freq2

mon_M1bT
142

mon_act_M2cT
220nSTT1R358[mon_disp_M3b]

mon_freq1

mon_SAT1T
130

mon_act_DF_SAT1P
193

nASC2a_by
222

CavA_bz

Cav1b_dy93
Cav1b_thetay94

mon_act_DF_SAT2L
195[nASC2c]

CavA_az
83

92

170
mon_STM1cL
171

mon_act_DF_SAT2T
196

nSTT1T356

cav3_L_subtracted

disp_LAS3bY

164
mon_STM1bL
165

nSTT1L355[mon_disp_M3a]cav1_L_subtracted
cav2_L_subtracted

disp_LAS3bP
349

disp_LAS3aY

146

mon_act_DF_SAT1R
192

218

CavA_thetaz82

disp_LAS3bV
347
disp_LAS3bR
348

disp_LAS3aP
343

ES_M3bY

nASC2a_ay
221354[nASC_AB][disp_LAS3b]

disp_LAS3aV
341
disp_LAS3aR
342

ES_M3bP295

mon_M1cL
147

nASC1c_bz

nASC2a_dy219

nASC2c_bz

nES_M3bL
287

nES_M3bV
289[disp_LAS3a]

ES_M3bR
294

140

nASC2a_thetay
220

nES_M3bT288[nES_M3a]

ES_M3bV
293

ES_M3aY

mon_M1bL
141

nASC1c_az
217

202
nASC2c_dy203

nASC2c_by
206[nES_M2a][ES_M3b]

ES_M3aP
301

128

nASC2c_ay
205

286

nES_M3aL
293
nES_M3aT294[mon_act_STM1c][nASC1c][nASC2c][nASC2a][nASC3a]ES_M3aV299
ES_M3aR
300

CavA_ay79

mon_SAT1L
129

nASC2c_thetay
204

nES_M2aR
91[mon_act_M2a]mon_act_M3a[nASC1b]nASC1b_bz

nES_M2aT89[nES_M2c]

mon_act_M2anLSC_AbsLa

nLSC_AbsLb

nASC1c_dz
215
nASC1c_thetaz
216

nASC1b_az
197

nES_M2aL
88

nES_M2cT95[mon_act_M1c][mon_act_M2c]

mon_act_M2c

nLSC3b[nASC1c]nASC1c_by
214

nASC1b_dz
195[nES_M1c]87

nES_M2cL94
12

mon_act_freq3[mon_act_M1b]

nES_M1cV
78

nLSC3anLSC3a350

11

mon_act_freq2

186
nLSC2c187

nLSC3bnES_M1cL
76
nES_M1cT77[nES_M1b]

93

mon_act_freq1

184
nLSC1b185
nLSC1cnLSC2anES_M1bL
82
nES_M1bT
83

mon_act_DF_SAT2[ES_M3a]CavA_thetay
78[CavA_A][Cav3a_A]

nASC1c_ay
213[nASC1b]nASC1b_thetaz
19675[mon_STM2c][mon_STM2a]mon_STM3amon_act_DF_SAT1

hGW1

hGW2

hGW3

81

mon_STM2c
mon_STM2ahGWcross

hGW1

353

nASC1b_ay
193
nASC1b_by
194

hGWplushGWplus

CavA_dz
81

mon_SAT3L
279[mon_M2a]mon_M3a[act_cav3b_A]

disp_LAS3bL
345
disp_LAS3bT
346

mon_SAT3V
281

210

mon_M2aact_cav2c_A[act_cavAB_A]

disp_LAS3aL
339
disp_LAS3aT
340

mon_SAT3T
280[mon_SAT3]

190

mon_M2c

act_cav1c_A[act_cavCA_A][act_cavAB_A][extDisp_STM3b]

act_cav1b_A[act_cavBC_A]

CavA_by80

134[act_cav1c_A]

347

344

mon_SAT2L
135[act_cav2c_A]

hGWcross

disp_LAS2aY

338

mon_SAT2T
136

extDisp_STM3bT
282[extDisp_STM3a]disp_LAS2aP
75

disp_LAS2cY

290

mon_SAT2V
137[extDisp_STM2a]

CavC_bz
76
CavA_dy77

disp_LAS2aR
74

disp_LAS2cP
69

ES_M2aY296

mon_SAT1Y[mon_SAT2]

CavC_az
381

CavB_bz

disp_LAS2aV
73

disp_LAS2cR
68

ES_M2aP45

ES_M2cY

mon_SAT1P
133[act_cavBC_A]

CavB_az91

disp_LAS2aT
72[disp_LAS2a]disp_LAS2cV
67

ES_M2aR
44

ES_M2cP
51

mon_SAT1V
131

314

CavC_dz
379

ES_M2cR
50

mon_SAT1R
132

act_cavBC_dz
317

extDisp_STM3bL
281

extDisp_STM3aT
276[extF_M3b][mon_SAT1]

act_cavBC_thetaz

280

extDisp_STM3aL
275

extF_M3bT
264[extF_M3a]extF_M3aV
271

act_cav3a_bz

extDisp_STM2aY

274

extF_M3bL
263

extF_M3aT
270[extF_SAT3]

extF_SAT3V
259[mon_M2c]

63

extF_M2cL
70
extF_M2cT
71

extF_M2cY

256
extF_SAT3L
257
extF_SAT3T
258[mon_M1c]

extDisp_STM1cV
138

extF_M1cR
49

extF_M1bP
56[nES_M1c]

nES_M2aextDisp_STM1bV
126[extDisp_STM1c]

extF_M1bR
55[Cav3b_A]

nES_M2c

extF_M1cV
48

extDisp_STM1cT
137[extDisp_STM1b]

extF_SAT1P
44[Cav3a_A][nES_M2a]extF_M1bV
54

extDisp_STM1bT
125[extF_M1c]

extF_SAT1Y[Cav2c_A][Cav2a_A]

cav3b_A[CavCA_A]

extF_M1cT
47[extF_M1b]

extF_SAT1R
43

cav2c_Acav3a_A[nES_M2c]

extF_M1bT
53

disp_LAS2aL
71

disp_LAS2cT
66[disp_LAS2c]

CavC_by378

CavC_thetaz
380

272

disp_SAT3Y[act_cav3b_A]

ES_M2aV43

CavC_ay
377[CavC_A]

CavB_dz89

disp_SAT2Y

disp_SAT3P
277

act_cav3b_ay309
act_cav3b_by
310[ES_M2a]

CavB_by88

CavB_thetaz
90

disp_SAT2P
27

disp_SAT3R
276

306
act_cav3b_dy
307
act_cav3b_thetay308[act_cav3a_A]

ES_M2cV49

CavC_thetay
376

CavB_ay87[CavB_A]

disp_SAT2R
26

disp_SAT3V
275

act_cav2a_bz

298

cav2a_L

cav2a_A

189[act_cav2a_A]

act_cav2a_by
175[ES_M2c]disp_SAT2V
25

act_cav2a_ay
174

act_cav2c_bz

act_STM3bP
255[disp_SAT2]

act_cav2a_dy
172
act_cav2a_thetay
173[act_cav2c_A]

act_cav2c_az
170[act_STM3b]

act_STM3bV
253
act_STM3bR
254

act_STM3aY

nES_M1b348[act_STM3a]act_STM3aP
249

act_M3bY

extDisp_STM3b

mon_M1c

act_STM3aV
247
act_STM3aR
248

act_M3bP
243

act_M3aY

extDisp_STM3anFreq3[act_M3b]

act_M3bR
242

act_M3aP
237

act_SAT3Y

extDisp_STM2a1

act_M3bV
241

act_M3aR236

act_SAT3P
231[nES_M1b]nFreq1[act_M3a]

act_M3aV
235

act_SAT3R
230[extDisp_STM3b][act_cav1b_A]

171

act_cav2c_dy
164
act_cav2c_thetay
165

act_STM2aP
122

244

[extDisp_STM2a]2163[act_STM2a]act_STM2aV
120
act_STM2aR
121

act_STM2cY

238[extDisp_STM3a]nFreq2[act_STM2c]

act_STM2cP
116

act_M2aY

232

act_cav3a_ay
301

cav3b_L13

cav1c_A

act_STM2cV
114
act_STM2cR
115

act_M2aP
38

act_M2cY

226

act_cav3a_by
302

FB_cavA_AbsL[extDisp_STM2c][act_M2a]

act_M2aR
37

act_M2cP
32

act_SAT2Y

7

cav3a_L[extDisp_STM1c]

act_M2aV
36

act_M2cR
31

act_SAT2P
14

cav1c_L6
cav2c_L[extF_M2a][act_M2c]

act_M2cV
30

act_SAT2R
13

act_SAT3V
229[CavB_A]

CavB_A
CavC_AextF_SAT2[extF_M1b][act_SAT2]

act_SAT2V
12

3
CavC_L[disp_LAS1b]

CavA_AextF_SAT1[extF_SAT2][extF_SAT3][extF_M3b]2

CavB_L

disp_LAS1c

CavC_dy375

CavB_thetay86

disp_LAS1cV
61

disp_LAS2cL
65

ES_M2aT
42[act_M2c][act_M2a][act_STM1b]

disp_LAS1bV
55

374

CavB_dy85

disp_LAS1cT
60[disp_LAS1c]

disp_LAS1bR
56

ES_M1bP
33[act_STM1c]84

disp_LAS1cL
59

disp_LAS1bT
54[disp_LAS1b]

ES_M1cR38

disp_SAT1P
21[act_cav1c_A]

ES_M1cV
3758

disp_LAS1bL
53

ES_M1cT36

ES_M1bR
32

act_cav1c_dz
160

act_cav1c_by
159

ES_M1bV
31[ES_M1c]disp_SAT1R20

act_cav1b_dz
152[act_cav1b_A]

disp_SAT1V
1952

ES_M1cL35

ES_M1bT30[ES_M1b]

act_cav1b_by
151

act_STM1cP
104

act_M2cT29

act_cav1c_ay
15834

ES_M1bL29

disp_SAT1T
18[disp_SAT1]

act_STM1cR
103

act_STM1bP
110

act_SAT2L
10

act_cav1b_ay
150

28

disp_SAT1L
17

act_cav1c_thetay
157

act_STM1bR
109

act_M1cP
20

act_SAT2T
11

act_STM1cV
102

16

act_cav1c_dy
156

act_cav1b_thetay
149[act_STM1c]

act_M1cR
19

act_M1bP
26

nFreq3351

155

act_cav1b_dy
148

act_STM1cT
101[act_STM1b]

act_M1bR
25

act_SAT1P
8

1

act_STM1bV
108

147

act_STM1cL
100

act_STM1bT
107

act_SAT1R
7[ES_M2a]CavA_L

act_M1cV
18[act_M1c][ES_M2c]

ES_M2aCavA_L

act_M1bV
24

99

act_STM1bL
106

act_M1cT
17[act_M1b]

ES_M2c

act_SAT2

57

nLSC_AbsLc

105

act_M1cL
16

act_M1bT
23[act_SAT1]

act_SAT1V
6

act_SAT1[act_SAT2]

ES_M1c346

15

act_M1bL
22

act_SAT1T
5[ES_M1c][act_SAT1]

mon_disp_M3bT
328[mon_disp_M3b]

mon_disp_M3aR336

mon_disp_M3bV329
mon_disp_M3bR
330

nSTT3R370

mon_disp_M3aP337

mon_disp_M3bP
331

nSTT3P371

mon_disp_M3aY

mon_disp_M3bY

nSTT3Y

Figure H.1: Top layer of the full DECIGO interferometer model.

act_SAT1

1

disp_SAT1[FB_DF_SAT]

140

act_SAT1

mon_SAT1

2

41

act_SAT2

mon_SAT2

mon_disp_SAT1
act_SAT2

MasterSW_SAT[FB_M1b]

FB_M1b

Act_M1b[FB_M1c]

FB_M1c

Act_M1c[FB_M2c]

FB_M2c

Act_M2c342

act_SAT3

mon_SAT3

mon_disp_SAT2

4

43
mon_M1b

Act_M2aFB_M3aAct_M3a[FB_M3b]

FB_M3b

Act_M3b

MassActSum

847

act_ES_M3a

mon_M3a48

act_ES_M3b

mon_M3b

act_STM1b

ES_M2aES_M2aFB_DF_M1b[FB_ES_M1b]

ES_M3aFB_DF_M2c[FB_ES_M2c]

56

53

49

50

hGW3

hGWplushGWcross

81

ES_M3b

ES_M3bservo_sub1

51

58

CavA_L
Sig_cav3a_L
CavB_L

ES_M3b

act_STM2c

1251
mon_STM2c

1352

act_STM2amon_STM2a

FB_F

37

Freq1

15

54
mon_STM3b1654

nLSC1bFreq3

nLSC1b55

nLSC1cnLSC1c56

nLSC2cnLSC2c57

nLSC2amon_freq1

nLSC2a58

nLSC3a83

nLSC3a59

nLSC3b

Fn_2
Fn_3

82

CavC_L13

mon_freq2

extF_SAT1
17

disp_LAS1b

extF_SAT2

18

nLSC3b6084

disp_LAS1b

extF_SAT2nLSC_AbsLa
61

mon_freq3
extF_SAT3

extF_SAT3

14nLSC_AbsLb
62

disp_LAS1cnLSC_AbsLc

disp_LAS1c

19

extF_M1b20

disp_LAS2c
extF_M1c

CavC_A

disp_LAS3a

M1c

L

Length servo

A

31

40

cav1b_A

act_cav1b_A
41
act_cav1c_A

disp_LAS3b63

nASC1bnASC1b64

nASC1cnASC1c65

nASC2cnASC2c66

nASC2aCavA_A20
CavB_A
CavB_A33

42

cav2c_A

act_cav2c_A
43
act_cav2a_A

extDisp_STM2c

21

extDisp_STM2c

CavC_AnASC3anASC3a68

nASC3bnASC3b69

nASC_BCnASC_BC70

nASC_CA29

nASC_CA71

extDisp_STM2a

CavC_A

extDisp_STM3amon_disp_M1b73

nASC_AB44
act_cav3a_A
45
act_cav3b_A

L
FB_STM1b

A

Mirror67

STM1b
FB_STM1c
A

Mirror

STM2c

37

46

cavB-cavC

act_cavBC_A
47
act_cavCA_A

MirrorSTM3aFB_STM3b

cavC-cavA

38
cavC-cavA
Sig_cavA-cavB_A

A

Mirror

STM1c
L
A

L
FB_STM3a

L
A

L
FB_STM2c

cavB-cavC

Sig_cavC-cavA_A

Mirror

STM2a
L

mon_act_STM1b
68
mon_act_STM1c
69

mon_act_STM2a71

Mirrormon_act_STM3a
72

STM3b

mon_act_STM3b

A[FB_STM]

mon_act_STM2c70

FB_M1c_A2
FB_M2a_A2

cavA-cavB

nASC_ABFB_M3a_A2

Alignment sensing

mon_disp_M1b

39

48

cavA-cavB

act_cavAB_A

Alignment servo

extDisp_STM3b

extDisp_STM3b
mon_disp_M1c74
mon_disp_M1c

nES_M1bnES_M1b32

nES_M1cmon_disp_M2c

M1c75
mon_disp_M2c

nES_M1c

FB_STT_M1c[FB_STT_M1c]

FB_STT_M2a[FB_STT_M2a]

FB_STT_M3a[FB_STT_M3a]

M2a

33

nES_M2cmon_disp_M2anES_M2c76
mon_disp_M2a34

mon_disp_M3a3577
mon_disp_M3anES_M3a3672

nES_M3bmon_disp_M3b

Plant

78
mon_disp_M3b

nSTT1

nSTT1
73

nES_M3anES_M3bM3anES_M2anES_M2a

nSTT2

nSTT2
74

nSTT3

nSTT3
STT servo

Figure H.2: Main system of the full DECIGO interferometer model.

144[FB_M3a]66
mon_act_M3b

M3b

FB_STM2a36

mon_act_M3a[FB_ES_M3b]

MirrorA

FB_M3b_A

cav3b_Acav3b_A[FB_M2a]65

Mirror

M3a
L

cav3a_A35
cav3a_A

Sig_cavB-cavC_A

extDisp_STM2a

L[FB_STT_M3a]

A
FB_M2c_A
FB_M2a_A

cav2a_A34
cav2a_A

Sig_cav3b_A

nASC2a67

M2a[FB_M2c]64
mon_act_M2aA

cav2c_A

Sig_cav3a_A

extDisp_STM1c

FB_M1c_A
cav1c_A32
cav1c_A

disp_LAS3a

19

mon_act_M2c

L[FB_M1c]63[FB_STT_M2a]cav1b_A

Sig_cav2a_A

18

CavA_A[FB_ES_M2c]

MirrorcavC_AbsLFB_M3a_A

disp_LAS3b

extDisp_STM1b

extDisp_STM1c

31

Length sensing[FB_M1b]62
mon_act_M1c

FB_M3b30

nLSC_AbsLc[FB_STT_M1c]
LA

cavC_AbsL

disp_LAS2c

disp_LAS3b

extF_M3b

extDisp_STM1b

30

M1b

Mirror

Sig_cavC_AbsL

Sig_cav2c_A

extF_M3aextF_M3b

extDisp_STM3a

MirrorA

cav1c_LnLSC_AbsLb

61
mon_act_M1b

L

23

Sig_cav1c_A

17

extF_M3a28

nLSC_AbsLa

disp_LAS1b

disp_LAS2a

extF_M2adisp_LAS2a27

FB_M2a2

Mirrordisp_LAS3a26

FB_M2acav1c_L

CavB_A

disp_LAS2a

extF_M2c

extF_M2a25[FB_ES_M1b]
cav2c_L24
cav2c_L
Sig_cav1c_L

FB_M1b_A

disp_LAS1c24

FB_M1c29
cavB_AbsL
Sig_cav2c_L

Sig_cav1b_A

1623

FB_M1b2

cavB_AbsL

CavA_A

disp_LAS2c

extF_M2c
22

FB_M1bcav3b_L27
cav3b_L

M2c

extF_M1c21

cav1b_L22
cav1b_L
Sig_cav3b_L

Sig_cavB_AbsL

15

extF_M1b[FB_freq]60

Sig_cav1b_L
Freq2

FreqSumnFreq2nFreq3
12

CavC_LextF_SAT1

59

mon_act_freq3

cavA_AbsL

Freq3

act_STM3b

act_STM3b

FB_freq3

25

Fn_1

nFreq1

38
39

CavB_LcavA_AbsL

mon_act_freq1

28
MasterSW_Freq

CavB_L53
mon_STM3aFB_freq2cav2a_L

11

act_STM3a14
act_STM3aAct_STM3b

cav2a_L

mon_act_freq2

Sig_cavA_AbsL

Freq2

CavA_L

Act_STM3aFB_freq1

26

Freq1

hGW2
[FB_freq]

CavA_L

act_STM2a

cav3a_L

cav3a_L
Sig_cav2a_L

52

10

Act_STM2a79
cav1_L_subtracted

servo_sub1

CavC_L

act_STM2c

servo_sub3[FB_ES_M3b]

FB_DF_M3b

ES servo

act_STM1c50
mon_STM1c

servo_sub2
cav3_L_subtracted

mon_STM1b1180
cav2_L_subtracted

servo_sub2

ES_M3aact_STM1c[FB_DF_SAT]57
mon_act_DF_SAT3

9

hGW1

49

Act_STM2c

STM_ActSum

FB_DF_Sat3

servo_sub3
act_ES_M3b9

ES_M2cmon_act_DF_SAT2

8
ES_M3aES_M2c

act_ES_M3a

Act_STM1c

MasterSW_STM

ES_M2aact_ES_M2a46
mon_M2aFB_DF_Sat2

7
ES_M1c

act_STM1b10

FB_STM6

mon_M2c7
act_ES_M2a

FB_DF_Sat1

ES_M1c

ES_M2c45

Act_STM1b[FB_STM]

ES_M1b

act_ES_M2c

act_ES_M2c
FB_M2amon_act_DF_SAT1
ES_M1b

ES_M1cmon_M1c6
MasterSW_TM

5

44

5

[FB_M3a]4
ES_M1b

act_ES_M1c

act_ES_M1c[FB_M2a]55
3mon_disp_SAT3

disp_SAT3

act_ES_M1b

act_ES_M1b2

disp_SAT2
act_SAT3[FB_M3b]

Disp_SAT

1

actSAT

F_SAT

actM1

F_M1

actM2

F_M2

Disp_M2

4

F_M1[F_SAT12M1b]

F_M2[F_SAT12M1c]

Disp_M1

act_SAT1[F_SAT12M1b]

Couple_SAT12M1bc

act_ES_M1b[F_SAT12M1c]

actSAT
disp_SAT

5

actM1[DispSAT1]

act_ES_M1c
actM2

M_ActSAT1
16

20[DispM1b]22

disp_M2

dispM2

extF_M2[DispM1c]

ES_M1bmon_disp_M1c

nES_M131

disp_LAS1b

act_STM1b
1225

CavC_LnES_M1c13

LAS1

act_STM1

ES_M1c

M_Sens_SAT1

32

dispSAT5

ES_M2
nES_M2

nES_M1b[DispSAT1]

4

ES_M1

23

SAT1

10

disp_SAT1

dispM1

mon_disp_M1b

extF_M1[F_OptM1c]1disp_SAT

extF_SAT

extF_SAT1

19

extF_M1c

dispSAT
disp_M1[F_OptM1b]

extF_M1b

11

extDisp_STM1

extDisp_STM1b
11

CavB_L
act_STM2

14

LAS2

act_STM1c26

disp_LAS1c

extDisp_STM2

extDisp_STM1c

SAT1LAS

Kopt_y8

Kopt_y12[DispM2c][DispM1b]

1/RoC2

1/RoC6

Kopt_y6[DispM1c]

Kopt_y10

Kopt_y9
Kopt_y7[DispM3b]

1/RoC

Kopt_y13
Kopt_y11

21
CavC_A

TM1

1/RoC420
CavB_A

1/RoC1[F_OptM2c]

1/RoC3

1/RoC5

TM1

Cav_A[F_OptM1b]

1/RoC7
Cav_A

TM2

TM2

Kopt_y4
Kopt_y5

CavC_Alignment[F_OptM1c]

Kopt_z4(d_y, theta_y, a_y, b_y, d_z, theta_z, a_z, b_z)'

Kopt_y2
Kopt_y3

CavB_Alignment[F_OptM3b]

Kopt_z2(d_y, theta_y, a_y, b_y, d_z, theta_z, a_z, b_z)'

Kopt_z5

Kopt_z3
10
CavA_L

Disp_SAT

2

actSAT

F_SAT

actM1

F_M1

actM2

F_M2

F_M1

[F_SAT22M2c]

F_M2[F_SAT22M2a]

Disp_SAT

3

act_SAT2[F_SAT22M2c]

Disp_M1

6

Disp_M2

act_ES_M2c[F_SAT22M2a]

1/RoC10

Couple_SAT22M2ca

actSAT
actM1

extF_SAT

disp_SAT[DispM2c]24

disp_M2
extF_M2

extF_M2a

dispM2[DispM2a][DispSAT2][F_OptM3a]

dispSAT
act_STM1

34
LAS1

15

nES_M2a

1/RoC11

TM1[F_OptM3a]7
ES_M2aact_STM2c27

LAS2

extDisp_STM2a

extDisp_STM2

16

disp_SAT[DispM3a]26

extF_M3b

M_Sens_SAT2

dispM2[DispM3b]

nES_M3a[DispSAT3]
14

dispSAT
act_STM1

36
LAS1

29

[F_OptM2a]

ES_M2
nES_M2

9
ES_M3b

M_Sens_SAT3

nES_M3b

extDisp_STM1

extDisp_STM3a15

act_STM2
LAS2

act_STM3b30
extDisp_STM3b

SAT2LAS8
ES_M3anES_M1

disp_LAS3a

act_STM3aKopt_zKopt_z1

disp_LAS2a

17

ES_M1

27
mon_disp_M3b

SAT3

Kopt_y1

CavA_Alignment3
disp_SAT3

dispM1

mon_disp_M3a
disp_M2

extF_M2

35(d_y, theta_y, a_y, b_y, d_z, theta_z, a_z, b_z)'
act_STM2

28

dispSAT
disp_M1

extF_SAT
extF_M1

Kopt_yextDisp_STM1

extDisp_STM2c
13

actM2

18

extF_M3a

Cav_A
TM2

disp_LAS2c

act_STM2a24[F_OptM3b]
ES_M2

nES_M2[F_SAT32M3b][DispSAT3]

extF_SAT3

23
1/RoC9ES_M2c[F_SAT32M3a]

F_M2

Couple_SAT32M3ab

actM1

M_ActSAT3

1/RoC8

Kopt_y15
19
CavA_A

F_M1

disp_SAT

Kopt_y17[DispM2a]6

nES_M1

33
nES_M2c

12

ES_M1

25
mon_disp_M2a

SAT2[F_OptM2a]2
disp_SAT2

dispM1

mon_disp_M2c

extF_M1

F_M2

act_ES_M3b
dispSAT

disp_M1

17
extF_SAT2

21
extF_M2c

actM2

Disp_M2
actSAT

9

actM2

M_ActSAT2[F_OptM2c]

22

F_M1

act_ES_M3a[F_SAT32M3b][DispSAT2]

act_ES_M2a

F_SAT

actM1

Disp_M1

8

Kopt_y14disp_SAT

7

actSAT

act_SAT3[F_SAT32M3a]

Kopt_y16[DispM3a]

extDisp_STM2

18
disp_LAS3b

SAT3LAS

Figure H.3: Plant system of the mechanical and opto-mechanical system of thefull DECIGO interferometer model.

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Acknowledgement
This work is supported by a large number of people. Here, I want to express myappreciation for their supports.
Takashi Uchiyama, who is my supervisor, told me that enjoying research isimportant. Also he let me do what I wanted to do. Thanks to him, I was able toconcentrate on and enjoy my research. Actually, one of my most favorite events
during my Ph.D. course is Extended Uchiyama Group Meeting in a ‘proper’place.
Masaki Ando gave me many insightful advices for my research based on his
wide knowledge and experience. From him, I learned various research related
things, such as planning of an experiment, interpretation of the obtained results,
how to present the experimental output, and so on. They will help me in myfuture research life.
Yuta Michimura is the most familiar professional staﬀ to me. I can consult
him on everything without any hesitation. When I asked something, he mostly
gave me appropriate answers.
Ayaka Shoda gave me important suggestions related to the simulation. She
kindly told me how to make a model in Matlab, Simulink. Especially, I reallyappreciate her comment on the way to calculate open loop transfer function withthe constructed model. The stating point of the full interferometer model in thisthesis is her prototype model.
Kiwamu Izumi gave me important suggestions on the interferometer control
when I considered the control topology of the dual-pass diﬀerential Fabry–Pérotinterferometer. He also encouraged me in my experiment occasionally.
Tomofumi Shimoda helped me with the design of the suspension system basedon his deep knowledge on the suspension system with torsion pendulum. In
addition, I discussed many things with him in our group oﬃce. He also gave me
insightful comments on the manuscript of this thesis.
Hiroki Takeda helped me with the construction of the dual-pass diﬀerential
Fabry–Pérot interferometer. Moreover, from him, I learned not only experimental thing but also theoretical thing.
Kunihiko Hasegawa and Takaharu Shishido kindly washed the mechanicalparts.
Shuichi Sato and Mitsuru Musha gave me valuable comments about the formation flying and laser at occasional DECIGO-related meetings, respectively.
Isao Kawano, Takahiro Ito, Satoshi Ikari, Shuhei Matsushita, and Sho Ishiwata163

gave me various knowledge from engineering aspect.
Tomohiro Fujita, and Ippei Obata told me many theoretical things, especiallyaxion dark matter and related topics. Without them, I could not perform a
research on the axion dark matter search.
From Kazuhiro Hayama, Atsushi Nishizawa, and Sachiko Kuroyanagi, I
learned theoretical point of view of the gravitational wave physics, astronomy,
and cosmology in occasional meetings.
From Tomotada Akutsu, I learned scientific attitude for research.
Kentaro Komori gave me wide knowledge on gravitational wave physics andprecision measurement.
Yutaro Enomoto gave me various knowledge on laser interferometric gravitational wave detectors. I enjoyed the discussion with him during my master anddoctoral courses. He also gave me valuable comments on the manuscript of thisthesis.
Ching Pin Ooi proofread this thesis. With him, I also discussed some topicsof thermal noise. Also, he taught me that tempura is a steamed dish.
Discussion with Naoki Aritomi, Takuya Kawasaki, Satoru Takano, Naoki Kita,
and Yuki Miyazaki made me improve my understanding on the gravitational wave
and quantum mechanics.
Shigemi Otsuka, and Togo Shimozawa made most mechanical parts of myexperiment in the machine shop. Also, I learned how to design mechanical partsand drawing from them.
Rie Kikuchi, Mayuko Niwata, and Naomi Kudo helped me with oﬃce works,
such as application of the business trip. Thanks to them, I can concentrate on
my research.
Hiroyuki Sagawa, Junji Yumoto, Katsuaki Asano, Shinji Miyoki, and Takuro
Ideguchi were referees of this thesis. I would like to thank their insightful comments.
Some of the illustrations in this thesis are drawn using the component library
prepared by Alexander Franzen.
This work is financially supported by JSPS Grant-in-Aid for Scientific Research(A) No. 15H02087 and JSPS KAKENHI Grant No. JP17J01176.
In the end, I am greatly obliged to Seiji Kawamura. He is a researcher who I
respect the most. If I had not met him, my research life would not be so exciting
as it is now. Without what he taught me, this work could not be completed.
最後の最後に、ここまで私を支えてくれた両親に感謝します。

164