THESIS

Optical Loss Study of Molecular Layers Using a
Cryogenic Folded Cavity for Future
Gravitational-wave Detectors

Satoshi Tanioka
Department of Astronomical Science
School of Physical Sciences
SOKENDAI (The Graduate University for Advanced Studies)
January, 2021

ii

Abstract
The first direct detection of gravitational-wave (GW) opened a new window
to the Universe. A number of detections of GWs have been reported, and theexistence of intermediate mass black hole was revealed. The Einstein Telescope (ET), a 3rd-generation gravitational-wave detector (GWD) in Europe,and LIGO Voyager, a substantial upgrade of aLIGO, are planning to employ
cryogenic silicon mirrors to reduce thermal noise. It is expected that such
future cryogenic GWDs will enable us to constrain the formation process ofsuch massive black holes or discover new GW sources with improved sensitivity. The realization of cryogenically cooled mirror is an important key and big
challenge to achieve the future cryogenic GWDs.
In the future GWDs such as the ET and LIGO Voyager, a molecular layer
formed on the cryogenic mirror surface can become one of the problems due
to its optical loss. We theoretically estimated the optical loss induced by
the molecular layer and revealed that the optical absorption induced by themolecular layer can exceed the cooling capacity even if its thickness is only
1 nm, which can hinder the cryogenic operation.
We developed 10 K folded optical cavity to investigate the optical loss of not
only the coatings but also the molecular layer. In this experiment we realized
the cryogenic optical cavity with folded configuration for the first time ever,
and achieved cavity enhanced ellipsometry at both room temperature andcryogenic temperature. These measurements enabled the characterization of
the molecular layer even if its thickness is only a few nm.
Besides, we simulated the molecular layer formation rate for the case of the
ET and LIGO Voyager. The results show the molecular layer formation canbecome a problem for long term operation. Finally, based on the experimental
study here, we propose a monitoring system which probe the cryogenic mirror
surface and measure the thickness of molecular layer. Such monitoring system
will be able to identify the molecular layer formation and be useful not only
operation but also commissioning works.
The theoretical and experimental works have investigated the feasibility of
the future cryogenic GWDs. This thesis reports the details of these results.
iii

iv

Contents
Abstractiii

Contents

viii

Introduction

1

1 Background
1.1 Nature of Gravitational-waves . . . . . . . . . . . . . . . . . . .
1.1.1 GW as a Solution of Einstein’s Equation . . . . . . . . .
1.1.2 Polarization . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Sources of Gravitational-waves . . . . . . . . . . . . . . . . . . .
1.2.1 Compact Binary Coalescence . . . . . . . . . . . . . . .
1.2.2 Spin of Compact Stars . . . . . . . . . . . . . . . . . . .
1.2.3 Supernovae . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Stochastic Gravitational-wave Background . . . . . . . .
1.3 Laser Interferometric Gravitational-wave Detectors . . . . . . .
1.3.1 Working Principle . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Enhancement by the Fabry-Perot Arm Cavities . . . . .
1.3.3 Further Improvement . . . . . . . . . . . . . . . . . . . .
1.3.4 Noise Sources . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Summary of this Chapter . . . . . . . . . . . . . . . . . . . . . .

5
5
6
7
8
10
10
12
12
13
13
13
16
21
23
25

2 Cryogenic Gravitational-wave Detectors
27
2.1 CLIO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 KAGRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Configuration . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.2 Vibration Isolation System . . . . . . . . . . . . . . . . . 29
2.2.3 Cryogenic Systems of KAGRA . . . . . . . . . . . . . . . 30
2.2.4 Sensitivity of KAGRA . . . . . . . . . . . . . . . . . . . 32v

CONTENTS
2.3

2.4

2.5

2.6
2.7

CONTENTS

Future Gravitational-wave
Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 The Einstein Telescope . . . . . . . . . . . . . . . . . . .
2.3.2 LIGO Voyager . . . . . . . . . . . . . . . . . . . . . . . .
Technologies for Future
Gravitational-wave Detectors . . . . . . . . . . . . . . . . . . .
2.4.1 Filter Cavity . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Substrate Material . . . . . . . . . . . . . . . . . . . . .
2.4.3 Coatings . . . . . . . . . . . . . . . . . . . . . . . . . . .
Technical Problem— Molecular Layer Formation . . . . . . . . . . . . . . . . . . .
2.5.1 Previous Studies . . . . . . . . . . . . . . . . . . . . . .
Target of this Study . . . . . . . . . . . . . . . . . . . . . . . .
Summary of this Chapter . . . . . . . . . . . . . . . . . . . . . .

34
34
35
37
38
38
39
41
43
43
44

3 Theoretical Study of
Cryogenic Molecular Layer
47
3.1 Optical Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Optical Absorption . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Optical Loss Estimation . . . . . . . . . . . . . . . . . . . . . . 53
3.4.1 KAGRA . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.2 The Einstein Telescope . . . . . . . . . . . . . . . . . . . 56
3.4.3 LIGO Voyager . . . . . . . . . . . . . . . . . . . . . . . . 57
3.5 Performance of GWD . . . . . . . . . . . . . . . . . . . . . . . . 58
3.6 Implications to Heat Input . . . . . . . . . . . . . . . . . . . . . 60
3.7 Summary of this chapter . . . . . . . . . . . . . . . . . . . . . . 64
4 Development of Cryogenic Folded Cavity
67
4.1 Optical Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1 Static Response . . . . . . . . . . . . . . . . . . . . . . . 68
4.1.2 Characteristic Quantities . . . . . . . . . . . . . . . . . . 69
4.2 Locking Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Folded Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.1 Static Response . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.2 Input and Output Mirrors . . . . . . . . . . . . . . . . . 75
4.3.3 Folding Mirror . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.4 Geometrical Parameters . . . . . . . . . . . . . . . . . . 76
4.4 Optical Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5 Spacer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.6 Vacuum and Cryogenic System . . . . . . . . . . . . . . . . . . 79
vi

CONTENTS

CONTENTS

4.7
4.8
4.9

Laser Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Finesse Measurement Methods . . . . . . . . . . . . . . . . . . .
Cavity Enhanced Ellipsometry(P-S Splitting) . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9.1 Characteristic Matrix . . . . . . . . . . . . . . . . . . . .
4.9.2 Frequency Shift . . . . . . . . . . . . . . . . . . . . . . .
4.9.3 Errors in CML Thickness Measurement . . . . . . . . . .
4.10 Summary of this chapter . . . . . . . . . . . . . . . . . . . . . .

81
82
84
84
86
89
89

5 Experimental Characterization of Cryogenic Molecular Layers
Using a Folded Cavity
91
5.1 Optical Loss Characterization . . . . . . . . . . . . . . . . . . . 92
5.1.1 Finesse of Fused Silica Fabry-Perot Cavity . . . . . . . . 92
5.1.2 Optical Loss in the Folding Mirror . . . . . . . . . . . . 93
5.1.3 P-S Splitting . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 Optical Loss at Cryogenic Temperature . . . . . . . . . . . . . . 96
5.2.1 Cavity Length Change . . . . . . . . . . . . . . . . . . . 97
5.2.2 Reflectance of the Mirror . . . . . . . . . . . . . . . . . . 97
5.2.3 Optical Loss Measurements . . . . . . . . . . . . . . . . 98
5.3 Measurements with a Modified Spacer . . . . . . . . . . . . . . 102
5.3.1 Geometrical Parameters . . . . . . . . . . . . . . . . . . 102
5.3.2 Cooling of the Cavity . . . . . . . . . . . . . . . . . . . . 104
5.3.3 Finesse and P-S Splitting Measurements . . . . . . . . . 104
5.3.4 Optical Loss and Thickness . . . . . . . . . . . . . . . . 105
5.4 Summary of This Chapter . . . . . . . . . . . . . . . . . . . . . 107
6 Discussions and Implications
111
6.1 Heat Input due to the CML . . . . . . . . . . . . . . . . . . . . 111
6.1.1 Optical Loss and Thickness of the CML . . . . . . . . . 112
6.1.2 Heat Input to the Test Mass by the CML . . . . . . . . . 115
6.1.3 Other Heat Loads . . . . . . . . . . . . . . . . . . . . . . 116
6.2 Sensitivity of CEE . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Formation Speed . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3.1 Formation Speed Degradation . . . . . . . . . . . . . . . 118
6.3.2 Comparison with the Measurement in KAGRA . . . . . 119
6.4 Estimation of the CML Formation Rate
in Future GWDs . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.5 Possible Solutions
to Mitigate the Impacts of the CML . . . . . . . . . . . . . . . . 127
6.5.1 Longer Cryotrap . . . . . . . . . . . . . . . . . . . . . . 128
6.5.2 Cooling Order . . . . . . . . . . . . . . . . . . . . . . . . 129
vii

CONTENTS

6.6

CONTENTS

6.5.3 Monitoring and Desorption System . . . . . . . . . . . . 129
6.5.4 Laser Induced Desorption System . . . . . . . . . . . . . 133
Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7 Summary and Conclusion

135

A Direct Measurement of Thermal Noise
139
A.1 Coating Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . 139
A.2 Reduction of CTN . . . . . . . . . . . . . . . . . . . . . . . . . 140
A.2.1 Cryogenic Mirror . . . . . . . . . . . . . . . . . . . . . . 140
A.2.2 Beam Size and Shape . . . . . . . . . . . . . . . . . . . . 141
A.3 Direct CTN Measurement
by Using a Folded Cavity . . . . . . . . . . . . . . . . . . . . . . 142
A.3.1 Three Different Spatial Modes . . . . . . . . . . . . . . . 142
A.4 Related Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A.4.1 Seismic Noise . . . . . . . . . . . . . . . . . . . . . . . . 143
B Angular Response
147
B.1 Angular Response of a Triangular Cavity . . . . . . . . . . . . . 151
B.2 Beam Spot Response . . . . . . . . . . . . . . . . . . . . . . . . 153
B.2.1 Wave Front Sensing Signals of the IMC . . . . . . . . . . 156
B.3 Implication to Experiment . . . . . . . . . . . . . . . . . . . . . 157
B.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Acknowledgements

161

viii

CONTENTS

CONTENTS

Acronyms
AOM
ASC
AR
BS
CEE
CLIO
CML
CTN
DC
DoF
DRSE
EOM
ET
ETM
FI
FP
FSR
FWHM
GW
GWD
HR
HWP
IMC
ISCO
ITM
PBS
PD
PDH
PRG
PZT
QPD
QWP
WFS

acoust-optical modulatoralignment sensing and control
anti-reflective, anti-reflection
beam splitter
cavity enhanced ellipsometry
Cryogenic Laser Interferometer Observatory
cryogenic molecular layer
coating thermal noise
direct current
degrees of freedom
detuned resonant sideband extraction
electro-optic modulator
Einstein Telescope
end test mass
Faraday isolator
Fabry-Perot (cavity)
free spectral range
full width at half maximum
gravitational wave
gravitational-wave detector
high-reflection
half-wave plate
input mode cleaner
innermost stable circular orbit
input test mass
polarizing beam splitter
photodetector, photodiode
Pound-Drever-Hall (method)
power recycling gina
piezoelectric transducer
quadrant photodiodequarter-wave plate
wavefront sensingix

CONTENTS

CONTENTSxIntroduction
First of all, the purpose of this thesis is to study the feasibility of future
cryogenic gravitational-wave detectors in terms of the optical loss induced by
the molecular layer formed on the cryogenic mirror surfaces.
There have been large efforts to directly observe gravitational-wave (GW)
with the ground-based laser interferometric detectors. In order to detect GWs,
so-called 2nd generation interferometers such as advanced LIGO, advanced
Virgo, and KAGRA were built. Advanced LIGO achieved direct detection of
GW from a binary black hole system in 2015, and the new window to probe theUniverse was opened. Following observing run, advanced LIGO and advanced
Virgo detected a GW from a binary neutronstar merger. So far, a number ofdetection of GW events are reported by them.
Future gravitational-wave detectors (GWDs) are planning to employ cryogenically cooled mirrors to improve the sensitivity by reducing thermal noise.
A cryogenically cooled mirror is an important key and a challenge for therealization of future GWDs.
Difficulties in such cryogenic mirror come from several facts. One is that thedevelopment of high-end mirror substrate and coatings. Both mirror substrateand coatings need to have not only small optical loss, but also small mechanicalloss. Development of such state-of-the-art mirrors is indispensable. Another isthat the issue of frosting of mirrors, which is caused by the adsorption of residual gas molecules. An adsorption of molecules on the mirror surface induces
unwanted optical loss, resulting in degradation of the sensitivity. Therefore,
the investigation in the impacts by frosting on a cryogenic mirror is importantto realize the future GWDs.
In this thesis, a development of optical loss characterization system, anda theoretical study and experimental studies of the molecular layer formationon the mirror surface are presented. It is shown that the optical loss in the
cryogenically cooled mirrors in future GWDs can exceed the requirement due
to the adsorption of molecules. In addition, the optical loss estimation by acryogenic folded cavity is demonstrated. Moreover, a study using Monte-Carlosimulation, which can predict the molecular layer formaton rate on the mirror
1

CONTENTS

CONTENTS

surface is presented, and a monitoring system for cryogenic mirror surface arepresented based on the folded cavity experiment. Thus the study presented inthis thesis should play an important role to design and construct the future
GWDs.
The following paragraphs are dedicated to explain the contents of thisthesis.
Chapter 1 briefly describes the background of this research, In this chapter,the derivation of GW, its astrophysical sources and an introduction of the laserinterferometric GWDs are given.
Chapter 2 describes several cryogenic GWDs and their features. KAGRA
is a cryogenic GWD, and the author contributed to its first cryogenic test operation and first observation. In addition, one of the most critical problems in
KAGRA which can also happen in the future GWDs is described — molecularlayer formation on a cryogenically cooled mirror. This problem is the one of
the motivation of this thesis.
Chapter 3 describes the theoretical calculation of the optical loss induced
by cryogenic molecular layer (CML). Theoretical estimation shows remarkableoptical absorption by the CML in future cryogenic GWDs, and exceed thecooling capacity even if its thickness is a few nanometer. The results are
published as an article by the author.
Chapter 4 explains a cryogenic folded cavity to measure an optical loss at a
mirror surface which is developed by the author. The cryogenic folded cavity
has not been realized so far, and this is the first time ever. This device currently
enables the optical loss measurement at broadband temperature range. The
optical setup and measurement methods are introduced. The cavity enhanced
ellipsometry is also introduced in this section. The spacer for the cavity was
designed by the author using 3D CAD soft, and manifactured by the company.
The installation of this setup was done by the author.
Chapter 5 describes the experimental results to evaluate the optical loss inthe mirror. Furthermore, the measurement of optical loss induced by the CML
is also estimated by combining the ringdown and cavity enhanced ellipsometry.
The author built the model to analyze the results of cavity enhanced ellipsometry. The characterization of the CML by cavity enhanced ellipsometry ispresented for the first time by the author.
Chapter 6 describes the implications to the cryogenic GWDs. Here a simulation is done by the author to estimate the CML formation for the case of
future GWDs. The model for the simulation was made by the author. Possible
solutions to avoid the affects of CML is also introduced here. The monitoring
system for CML formation on mirror surface is proposed here by the author
inspired by the cryogenic folded cavity experiment.
2

CONTENTS

CONTENTS

Chapter 7 describes the summary and conclusion of this thesis.
Appendix A is dedicated for the upgrade of this setup for direct measurement of coating thermal noise. Low thermal noise coatings at cryogenic
temperature is a key to realize the future GWDs. The cryogenic folded cavity
will enable direct measurement of coating thermal noise directly which promotes the development of high-end coatings. A possible upgrade plan toward
the direct measurement of coating thermal noise, and related noise sources are
introduced.
Appendix B is dedicated for the analytical calculation of angular response
which enables the compute WFS signals of an optical cavity with less cost. The
original ides of this method was proposed by the collaborator of the author.
A further analysis and interpretation was done by the author.

3

CONTENTS

CONTENTS

4

Chapter 1
BackgroundA direct detection of gravitational-wave (GW) is a challenging because theinteraction of gravity with masses is remarkably weak. A creation of GW
with a detectable amplitude in a laboratory is impossible with the current
technologies. In order to detect GWs, one have to rely on astronomical sources,
which can be extremely massive and highly accelerated. From this point ofview, astronomy and GW are inseparable. Moreover, the detection of GWs
from astronomical sources provides us unique information of the Universe.
This is because GWs are unaffected by the matter they encounter.
The laser interferometric GW detector is in mainstream, and it is mostsensitive at around 100 Hz. In 2015, Advanced LIGO (aLIGO) constructed in
America detected the GW from binary black hole merger. This first direct
detection of GW opened a new window to the Universe — this window iscalled Gravitational-wave Astronomy. In 2017, aLIGO and Advanced Virgoin Europe detected a GW from binary neutron star merger, and the electromagnetic counterpart was also observed by telescopes. The sensitivity of thedetector has been continuously improved over the course of the years anddozens of detection are achieved by aLIGO and Virgo.
This chapter is dedicated to explaining GW astronomy and the laser interferometric detector as a background of this thesis. First, we briefly introduce
the derivation of GW based on the general theory of relativity. Second, we
describe the astrophysical sources of GW. Then, we introduce the laser interferometric detector and its noise sources.

1.1

Nature of Gravitational-waves

GW is a ripple of space-time and propagates as a wave at the speed of light.
GW was predicted by A. Einstein on the basis of the general theory of relativity
51.1 Nature of Gravitational-waves

1 Background[1, 2].
GW is derived form Einstein’s equation within the scheme of the general
theory of relativity. In the theory, gravitation is considered as distortion in
space-time, expressed by a physical quantity called the metric. Moreover, themetric follows Einstein’s equation which contains the second derivative of themetric with respect to time. Thus, the metric can evolve as a function of time.
In fact, a small perturbation in the metric can propagate as a wave throughspace-time.

1.1.1

GW as a Solution of Einstein’s Equation

In this subsection, we review the derivation of GWs which can be derived asa solution of Einstein’s equation when a small perturbation in space-time isintroduced [3].
The interval of two separated events in space-time is described with themetric tensor gµν as,
ds2 = gµν dxµ dxν ,(1.1)

where Greek indices run from 0 to 3, so that µ, ν = 0, 1, 2, 3. With thisnotation, space-time can be simply expressed by xµ = (ct, x, y, z).
Einstein’s equation, which is the most important equation, is expressed as
8πG
1Rµν − Rgµν = 4 Tµν ,
2
c

(1.2)

where the left-hand side represents the geometrical nature of space-time andthe right-hand side represents the mass and energy of the system. Here, Rµν ,R = g µν Rµν , and Tµν are the Ricci tensor, the Ricci scalar curvature, and thestress-energy-momentum tensor, respectively.
We consider the case that a small perturbation, hµν , is introduced to theflat space-time asgµν = ηµν + hµν .

(1.3)

Assuming vacuum, Tµν = 0, then applying the above equation to Eq. 1.2, andneglecting the higher order terms while leaving the first order terms of h, one
can obtaing the linearized Einstein’s eq,


∂2
1 ∂2
(1.4)
− 2 2 + 2 hµν = 0.
c ∂t∂xiThis describes the propagation of GW hµν as a three-dimensional wave which
travels at the speed of light.
6

1 Background

1.1 Nature of Gravitational-waves

When we choose a spatial coordinate such that a GW travels in the z
direction, the plane wave solution can be summarized as,hµν = Aµν exp[iωg (t − z/c)],

(1.5)

where


0 0
0
0 h+ h×Aµν = 
0 h× −h+
0 0
0


0
0
.
0
0

(1.6)

Here ωg is the angular frequency of the GW, h+ and h× are independent
constants.

1.1.2

Polarization

We consider a case where two free particles are placed at separated locations— one at the origin of the coordinate and the other at xi = (L, 0, 0). Supposethat each of them is at rest initially, and GW passes along the z-axis. The
proper distance between them, L0 , can be expressed as
Z
Z L
1
0
2 1/2
(1.7)
|g11 |1/2 dx ≈ L + h11 L.
L ≡ |ds | =20Therefore, GW changes the proper distance when it passes through. The
amount of change in the distance, ∆L ≡ L0 − L, is proportional to both theinitial distance L and amplitude h.
As shown in Fig. 1.1, there are two independent polarizations in GWs —amplitude of which are expressed as h+ and h× , and they are referred as plusand cross-mode, respectively. In order to understand the characteristic of
those modes, consider a similar thought experiment in which a plane wave
passes through along z-axis with a number of free particles arranged on thex-y plane. According to Eq. 1.7, if an initial proper distance between themwas Li = (Lx , Ly , 0), the displacement ∆L can be written as  x
 x
1 h+ h×L∆L
exp[iωg (t − z/c)],
(1.8)
=
h
h
Ly
∆Ly
2
×
+
 x
 y
1
1
L
L
= h+
exp[iωg (t − z/c)]. (1.9)
y exp[iωg (t − z/c)] + h×
L
Lx
2
2
Fig. 1.1 shows how free particles respond to GWs for both the plus- and crossmodes. The plus mode distorts space as a ”plus”-shape, while the cross mode
makes it a ”cross”-shape.
71.1 Nature of Gravitational-waves

1 BackgroundFigure 1.1: Polarization of gravitational-wave. Upper figure shows the plusmode, and lower shows the cross-mode, respectively.

1.1.3

Radiation

GWs can be radiated by masses when they accelerate similarly to the radiationof electro-magnetic waves which are generated by a charged particle when it
accelerates. A big difference, however, is that the radiation of GW requires aquadrupole moment though electro-magnetic radiation is induced by a dipole
moment.
First, we consider the Lamor formula as an example which denotes theluminosity of an electro-magnetic waveLe asLe =where d =Pi qi r i

2 2
d̈ ,
3c2

(1.10)

is the sum over the electric dipole moment of charged
8

1 Background

1.1 Nature of Gravitational-waves

particles.
For the case of the radiation of GW, the GW luminosity, Lg , becomesLg ∝X
G 2 2 X
d̈
,
d̈
=
m
r̈
=
Ṗ i ,ii
c3
ii(1.11)

where P i is the kinematic momentum of i-th particle. Instead of charged particles, the kinematic momentum is introduced as GW is generated from masses.
However, as the kinematic momentum is always conserved in an isolated system, the kinematic dipole d does not evolve as a function of time. Therefore,
GW cannot be radiated by a kinematic dipole moment.
The next lowest moment is the quadrupole moment. By introducing themass quadrupole moment Ijk , a GW version of Lamor formula [4] can be
expressed as,
G ... 5h( I jk ) i,
5c5


X

1
A A
A 2
Ijk ≡
mA xj xk − δjk x
,
3
A
Lg =

(1.12)(1.13)

where A represents the label of A-th particle. Therefore, radiation of GW
requires a quadrupole moment so as to evolve as a function of time.
Order Estimation of Radiated Energy
Here, we briefly show the energy of gravitational wave radiation from Eq. 1.12.
The third derivative of the quadrupole moment with respect to the time scales
as
...
M v3
M R2
∼
,
I ij ∼
T3
R(1.14)

where M, R, T , and v are the typical mass, spatial size, time scale, and velocityof the system, respectively. Plugging Eq. 1.14 into Eq. 1.12, one can obtain2
Rsch  v 658
Lg ∼ 1.8 × 10
[erg/s],(1.15)
Rc
where Rsch ≡ 2GM/c2 is the Schwarzschild radius. Considering the case of anastronomical system which is bounded by its self-gravity, the kinetic energy
and the potential energy are comparable, so thatM v2 ∼
9

GM 2.
R(1.16)

1.2 Sources of Gravitational-waves

1 Background

With this relation, the luminosity 1.15 can be rewritten as
5

Rsch
57
Lg ∼ 2.3 × 10
[erg/s].
R(1.17)

This result indicates that the GW luminosity depends on the size of theobject. This is the reason why the compact stars are considered as the main
GW sources.
Order Estimation of Amplitudes
We estimate the order of the amplitude of GW, h, based on the approximation
[4]
h∼

Rsch
(v/c)2 ,r(1.18)

where r is the distance between the GW source and an observer. The emission
efficiency of GW can be parameterized as

7/2
Rsch
ε∼
.
(1.19)
R
Plugging Eq. 1.19 into Eq. 1.18, one can obtain ε 2  M   r −21
2/7 Rsch∼ 1.5 × 10
/7
.
h∼ε
2r
0.1
M
17 Mpc(1.20)

Here, the mass M and the distance r are scaled by the solar mass M =1.989 × 1030 [kg] and that from the earth to the Virgo cluster of about 17 Mpc,
respectively.

1.2

Sources of Gravitational-waves

In this section, we briefly review the GW sources, which have been observedor expected for the detection. More details can be seen in the reference [5].

1.2.1Compact Binary Coalescence

A binary system formed by two compact objects such as a black hole or a neutronstar is one of the most promising sources of GW, and a number of compact
binary coalescence have been detected [6]. A wave form of the compact binary
coalescence is known as the chirp signal.
10

1 Background

1.2 Sources of Gravitational-waves

100

Amplitude h [×10−23]

75
50
25
0
−25
−50
−75
−100
−50

−40

−30
−20
−10
Time to Coalescence [msec]

0

Figure 1.2: The calculated chirp signal from a 1.4M -1.4M neutronstar binary at the Virgo cluster of about 17 Mpc.

A chirp signal, fGW , grows up as a function of the time. This is a consequence of the fact that the angular momentum of the system is depositedas the GW radiation. Therefore, the separation distance of the stars gradually becomes smaller, resulting in the frequency evolution of a signal. The
frequency evolution of a chirp signal can be approximated as5/8  −3/8

1s
1.21M
,
(1.21)
fGW ≈ 134 HzMτ(m1 m2 )3/5,
(1.22)
M≡(m1 + m2 )1/5
where τ is the time to coalescence, M is the chirp mass defined by the massesof the binary stars, m1 and m2 . Fig. 1.2 shows how chirp signal evolves overtime. The chirp signal from 1.4M binary neutronstar merger at the Virgocluster is assumed.
A GW radiation in a chirp waveform ends when the orbital radius becomes11

1.2 Sources of Gravitational-waves

1 Backgroundsmaller than the innermost stable circular orbit (ISCO), where stable circularorbits are no longer allowed. Then the two stars plunge toward each other andcoalesce. For the case of a nutronstar-neutronstar binary, when they are at
the ISCO, the expected frequency fISCO is about 800 Hz.
In addition, such binary systems can be used as a standard siren [7]. Whena chirp signal is detected, the amplitude provides the information of the luminosity distance. In addition to that, the frequency gives the information about
the red shift. Thus, the detection of GW from such binary systems allows one
to perform cosmological distance measurement.

1.2.2

Spin of Compact Stars

Spinning compact stars, such as pulsars, are expected to be sources of continuous GWs, frequency of which can be stable. In order to radiate GWs by thespin, the mass density must not be axisymmetric about the spin axis because
the axisymmetric system does not have a quadrupole moment.
The expected amplitude of GW from a spinning of compact stars can be
estimated as2  
I
10 kpcfg−25,(1.23)h ≈ 1.06 × 10
10−61038 kgm2r
1 kHz
where  is the ellipticity, I is the inertia along the spin axis. It should be notedthat the frequency of the GW is twice of the actual spinning frequency fspin ,hence fg = 2fspin .

1.2.3

Supernovae

A supernova explosion can become a GW source because it dramatically accelerates a large amount of masses within a short time scale. If non-axisymmetriccomponents are involved in the core collapse of supernovae, burst GW signals
can be emitted.
The prediction of the waveform is, however, difficult because the explosion
mechanism is uncertain. Assuming that the mass of a system is about M andthe characteristic velocity of v/c ∼ 0.1, the amplitude can be estimated as
10 kpc−20.
(1.24)h ∼ 9.6 × 10r
Therefore, a supernova in the Galaxy can be a target. Most of the explosion models suggest that the typical frequency is below 1 kHz, hence in theobservational band of GWDs.
12

1 Background

1.2.4

1.3 Laser Interferometric Gravitational-wave Detectors

Stochastic Gravitational-wave BackgroundA stochastic GW background is expected to arise from a superposition of anumber of unresolved GW sources. It can carry unique features from the early
epochs of the Universe, which cannot be accessed by standard astronomical observations. Therefore, direct measurements of the amplitude of the stochastic
background are important to understand the evolution of the Universe.
Recently a simultaneous observation of GWDs has set an upper limit on
the spectral density of the GW background, hence the cosmological energy
density of GW [8].

1.3

Laser Interferometric Gravitational-wave
Detectors

Current terrestrial gravitational-wave detectors (GWDs) adopt laser interferometric technique and are based on the Michelson interferometer. The Michelson interferometer can be considered as an optical phase detector, while it isbroadly used as a Fourier transform spectrometer in astronomical telescopes.
Comparing the optical phase difference between each arm in the Michelsoninterferometer, one can detect a GW. The working principle is described in
the following section.

1.3.1

Working Principle

As shown in Fig. 1.3, the concept of current ground-based GWDs are based on
the Michelson interferometer with mirrors suspended by wires. The suspended
mirror not only provides but also makes the mirrors effectively free masses
longitudinally with respect to the propagation direction of the laser beam in
the observational frequency range.
When GW passes through the interferometer, it modulates the opticallength of each arm by ∆L at the same frequency. The lengths change isdifferential in each arm, resulting the change in the interferometric conditionof the output laser field.
Therefore, GWs can be detected as an intensity variation at a photo detector (PD) placed at the output side of the Michelson interferometer. When
the wavelength of the GW is much longer than thearm lengths, the amount of
the change is proportional to the macroscopic arm lengths L, so that
∆L ∼ hL,13(1.25)

1.3 Laser Interferometric Gravitational-wave Detectors

1 BackgroundFigure 1.3: Schematic drawing of the Michelson interferometer.

where h is the amplitude of the GW. Thus, it is reasonable to construct a large
size interferometer to increase the sensitivity.
In the following subsection, we see the principle of the detection of GWs
by the interferometer.
Detection with Michelson Interferometer
The Michelson interferometer (MI) is an interferometric device which provides
us the optical phase of laser field. It consists of a beam splitter (BS) andtwo end mirrors as shown in Fig. 1.3. The BS splits the laser beam into twopaths — one is transmitted through it and the other is reflected. Each beamis reflected back to the BS by the end mirror and is re-combined which results
in interference. At the detection port, called anti-symmetric (AC) port, a PD
observes the power of the electric field.
We assume that a GW, polarizing in the plus-mode, passes along the z-axis
through the MI placed in the x-y plane. The interval of two separated objects
14

1 Background

1.3 Laser Interferometric Gravitational-wave Detectors

can be expressed as
ds2 = −(cdt)2 + {1 + h(t)}dx2 + {1 − h(t)}dy 2 + dz 2 .

(1.26)As the light travels along the world line of ds2 = 0, the light traveling on thex-axis satisfiesp
cdt = 1 + h(t)dx.
(1.27)
By integrating dx over the round trip path from the BS to the end mirror, one
can obtain

Z t
Z t 
1 0
2lx
t0
0
p
1 − h(t ) dt0 =dt ≈,(1.28)
2c
1 + h(t0 )
t−τx
t−τx
where τx represents the duration for the light to round-trip in the x arm andlx denotes the length of x arm. Therefore, τx can be calculated asZ
2lx 1 tτx =+h(t0 )dt0c2 t−τxZ
2lx 1 t+h(t0 )dt0 .
(1.29)≈
c2 t−2lx /c
As h  1, τx can be approximated by 2lx /c. Therefore, the net change in the
optical phase isZ
2ω0 lx ω0 t
φx = ω0 τx =+h(t0 )dt0 ,(1.30)
c2 t−2lx /cwhere ω0 is the angular frequency of the laser field. In the right-hand-side ofthe equation, the first term denotes the static phase rotation due to the lighttraveling a length of 2lx while the second term denotes the phase derivationinduced by the GW. The net phase change of the light traveling along the yarm is acquired in the same manner and expressed asZ
2ω0 ly ω0 t
φy = ω0 τy =−h(t0 )dt0 ,(1.31)
c2 t−2ly /cwhere the ly is the length of the y arm.
Assuming that the arm lengths are almost identical, lx ≈ ly = l, the phasedifference between that of x and y arm is
φx − φy =2ω0 (lx − ly )+ ∆φg ,c15

(1.32)

1.3 Laser Interferometric Gravitational-wave Detectors

1 Background

where
Z t

h(t0 )dt0 .

∆φg ≡ ω0(1.33)

t−2l/cThe GW changes the interference condition, and it can be observed as powerfluctuation at the PD. Assuming that a GW has a long wavelength such thatthe induced phase shift does not vary as fast as the round trip time of 2l/c, it
is approximated as,
∆φg ≈

2ω0 l
h.
c

(1.34)

Frequency Response of the Michelson Interferometer
Assuming that a GW is monochromatic and has an angular frequency of ωg ,it can be expressed as,h(t) = heiωg t .

(1.35)

Plugging this into Eq. 1.33, one can obtain the frequency response of a Michelson interferometer to a GW as [9]ωg l −i ωg l
2ω0
∆φg
sin
(1.36)e c .
HMI (ωg , l) ≡ iωg t =heωgc
The response curve is shown in Fig. 1.4 when the length of both arms is set tobe 3 km. At low frequencies the response can be approximated as
HMI ≈

2ω0 l
.
c

(1.37)

When the frequency of GW becomes higher than the inverse of the roundtrip time c/2l, the above approximation is not valid any more because thephase of the GW signal starts cancellation during the light trips in the arms.
This effect appears as a cut off in the response curve in figure 1.4.

1.3.2

Enhancement by the Fabry-Perot Arm Cavities

The optical cavity or equivalently Fabry-Perot (FP) cavity is a device formedby two or more partial reflectors to store photons (or to reject them) in thecavity. In particular, those formed by two mirrors facing each other is known
as the FP cavity, as depicted in Fig. 1.6. In order to increase the sensitivity
with a given arm length, the Fabry-Perot (FP) cavity is embedded in each arm
16

1 Background

1.3 Laser Interferometric Gravitational-wave Detectors

1014

Michelson
FPMI

Responses [rad/strain]

1013
1012
1011
1010
109
108 0
10

101

102

103

Frequency [Hz]

104

105

Figure 1.4: Frequency responses of the Michelson and Fabry-Perot Michelsoninterferometers. In both cases, the length of the arms are assumed to bel = 3 km. For the Fabry-Perot Michelson configuration, t21 = 0.014, and t22 =50 ppm are assumed.

of the Michelson interferometer. The optical configuration is shown in Fig. 1.5.
Employing the FP arm cavity enables the laser field to bounce multiple times
in the arm, and the interaction time of the light with GWs is increased. In
other words, the effective length of the arm is elongated.
Here we describe the response of the FP Michelson interferometer. First,
we will focus on a single FP cavity as shown in Fig. 1.6. Then we introduce a
perturbation which is due to GW. Successively, we derive the response of thesingle cavity. Finally, we compute the response of the FP Michelson interferometer.
171.3 Laser Interferometric Gravitational-wave Detectors

1 BackgroundFigure 1.5: Schematic drawing of FPMI.

Response of a Single Fabry-Perot Cavity to a Perturbation
We consider a static case of a single FP cavity. A set of the field equations
can be written as
Ecav (t) = t1 Ein + r1 r2 Ecav (t − 2T )e−i∆φ(t) ,Er (t) = −r1 Ein + t1 r2 Ecav (t − 2T )e−i∆φ(t)(1.38),(1.39)

where Ein (t), Ecav (t), and Er (t) are the incident, intra-cavity, and reflected
fields, respectively. rj and tj are the amplitude reflectivity and transmissivityof j-th mirror. L represents the distance between two mirrors, T ≡ L/c isthe one way trip time, and ∆φ denotes the phase rotaion. From Eq. 1.30, thephase shift can be redefined as
ω0∆φ(t) =
2Z t
t−2L/c18h(t0 )dt0 .

(1.40)

1 BackgroundEin

1.3 Laser Interferometric Gravitational-wave Detectors

r1, t1

r2, t2

Ecav

Et

Er
L
Figure 1.6: The schematic drawing of the Fabry-Perot cavity with two mirrors.

Here we assumed that the FP cavity is kept on resonance, i.e., the static phaserotation satisfies 2ω0 L/c = 2πn with n and integer. Therefore, ∆φ is induced
purely by GW in this case.
Suppose that ∆φ introduces a small perturbation in the intra-cavity field.
In this case, the intra-cavity field can be redifined as
E(t) = Ē + δE(t),(1.41)

where Ē is the static solution and δE(t) represents the deviation from thestatic solution. We assume that the phase resolution is so small that the phaseterm can be approximated bye−i∆φ ≈ 1 − i∆φ.

(1.42)

Plugging Eqs. 1.41 and 1.42 into Eq. 1.38, we obtainĒcav + δE(t) ≈ −r1 Ein + r1 r2 Ē + r1 r2 δE(t − 2T ) − ir1 r2 Ē∆φ.

(1.43)

Here we ignored higher order terms which contain the product of ∆φ andδEcav . Using the static solution, Ē = t1 Ein + r1 r2 Ē, we can obtain
δE(t) = r1 r2 δE(t − 2T ) − ir1 r2 Ē∆φ.
19(1.44)

1.3 Laser Interferometric Gravitational-wave Detectors

1 BackgroundBy applying the Laplace transform, the fluctuation in the intra-cavity fields
can be expressed by
δEcav (s) = −ir1 r2 Ē∆φ(s).
1 − r1 r2 e−2sT(1.45)

This is the deviation introduced by the GW in the intra-cavity field.
Combining Eqs. 1.38 and 1.39, one can obtain
Er (t) =t2 + r12
t1
E(t) − 1
Ein .
r1
r1

(1.46)

A deviation in the intra-cavity field δEcav leads to a small deviation in thereflected field via
δEr (s) =t1
δEcav (s).
r1

(1.47)

From Eqs. 1.45 and 1.47, one can derive the phase rotaion in the reflected fielsas∆φ(s)t21 rrδEr (s).
= 2∆φ(s) ≡
(t1 + r12 )r2 − r1 1 − r1 r2 e−2sTĒr(1.48)

This equation shows that the induced phase shift ∆φ is amplified by the frequency dependent factor when it is observed at the reflected field.
Response of the Fabry-Perot Michelson Interferometer
Usually, the arm lengths of the short Michelson interferometer is much shorter
than the baseline of the arm FP cavity. Thus, it is reasonable to neglect thecontribution from the Michelson interferometer. In this case, the Michelsoninterferometer works as a device which subtracts the optical phase difference
between the x and y arms.
When a plus-mode GW passes along the z-axis through the FP Michelsoninterferometer placed in the x-y plane, the phase shift due to GW in the xarm has a different sign compared to that of y arm. Therefore, the output
field from the Michelson has a phase shift of 2∆φr if both arms have the samelength. As a result, the frequency response of the FP Michelson interferometeris expressed as
2∆φr (ωg ),h(ωg )
t21 r2
HMI (ωg , L)
= 2.
2
(t1 + r1 )r2 − r1 1 − r1 r2 exp(−2iωg L/c)

HFPMI ≡

20

(1.49)(1.50)

1 Background

1.3 Laser Interferometric Gravitational-wave Detectors

Figure 1.7: Schematic of GWD’s configurations. (A) Power recycling configuration. (B) Signal recycling or resonant sideband extraction configuration.
(C) Dual-recycling configuration.

Here we used the fact that ∆φ = HMI h/2. Thus, the addition of the FP arm
cavities changes the frequency response and improve the sensitivity as shownin Fig. 1.4.

1.3.3

Further Improvement

In addition to the FP arm cavities, current GWDs employ the scheme called
the dual-recycling. This scheme improves the sensitivity and optimizes theobservational frequency band by adding two more mirrors to the FPMI. The
FPMI enhanced by the dual-recycling system is referred to as dual-recycled
Fabry-Perot Michelson interferometer (DRFPMI). The configuration is shown21

1.3 Laser Interferometric Gravitational-wave Detectors

1 Backgroundin Fig. 1.7.
The dual-recycling is a conjunction of two independent techniques — power
recycling and signal recycling techniques. The signal recycling technique canbe further divided into two modes — the signal recycling and resonant sideband
extraction modes.
Power Recycling Technique
The power recycling technique uses an additional mirror, called power recycling
mirror (PRM), at the input side of the Michelson interferometer as shown in
Fig. 1.7. By employing the additional mirror, one can reflect the light back to
the interferometer again. Thus, one can increase the laser power stored inside
the arm cavities to reduce shot noise.
Signal Recycling Technique
The signal recycling technique utilizes an additional mirror, called signal recycling mirror (SRM), at the output side of the Michelson interferometer asshown in Fig. 1.7. This scheme enables us to recycle GW signals and propagate
them back to the interferometer [10]. The SRM couples with the differential
motion of two FP arm cavities to increase the effective finesse for GW signals.
Therefore, the storage time increases, resulting in a higher sensitivity at low
frequencies. However, it leads to a narrower observational bandwidth.
Resonant Sideband Extraction
The resonant sideband extraction is the scheme which employs the same optical configuration as that for the signal recycling. In this scheme, the effectivefinesse is reduced in order to expand the observational bandwidth at the expense of the sensitivity at low frequencies.
The signal recycling and the resonant sideband extraction can be switchedto each other by changing the interference condition without modifying theoptical configuration.
Dual-Recycling
The power recycling, and the signal recycling (or resonant sideband extraction)
schemes can be independently employed together. This configuration is called
the dual-recycling configuration. The dual-recycling configuration is shown in
Fig. 1.7.
22

1 Background

1.3.4

1.3 Laser Interferometric Gravitational-wave Detectors

Noise Sources

The sensitivity of GWDs is determined by the summation of lots of noise
sources. Here we introduce several noises which limit the sensitivity.
Quantum Noise
Quantum noise is a fundamental noise which limits the sensitivity at broadband frequency. At the lower frequency band, radiation pressure noise is oneof the dominant noise sources. The radiation pressure noise√ fluctuates themomentum of the suspended mirrors and is proportional to P . Therefore,
the smaller beam power and heavier test mass mirror can reduce the radiationpressure noise. The shot noise, which limits the sensitivity at higher frequency,can√be reduced by increasing the laser power because it is in proportional to
1/ P . Therefore, these two quantum noise are in trade-off relationship and
one needs to choose appropriate laser power to optimize the sensitivity.
Employing a filter cavity is a promising approach to reduce the quantum
noise. This technique reduces both radiation pressure and shot noise withoutchanging the parameters such as laser power. In order to realize this technology
in GWDs, several studies are ongoing [11, 12]
Seismic Noise
Seismic noise is one of the biggest and inevitable noise sources which dominants at the lower frequency band. It introduces displacement to the positionof the mirrors and results in variation in the optical lengths. Besides, themisalignment in the cavity axis can be induced by the seismic noise. Such a
misalignment can deteriorate detector sensitivity and stability by increasinga few noise couplings including those to length fluctuation and beam jitter.
Moreover, angular misalignment causes light in the fundamental mode to becoupled into higher-order spatial modes, reducing the amount of power stored
in the main part of gravitational wave detector. This directly decreases thedetector sensitivity. Although the mirrors are suspended as pendulums to reduce the seismic noise, there still remains non-negligible amount of vibration
which hinders one to keep resonant condition. Therefore, a feedback control is
indispensable to reduce the amount of vibration.
Laser Noises
A free running laser shows fluctuation in both the amplitude and frequency
of its field. As any photon detection measurements involves an intensity measurement of the laser, the amplitude noise can couple with the resultant photo23

1.3 Laser Interferometric Gravitational-wave Detectors

1 Backgroundcurrent in various ways. Therefore, the amplitude noise can introduce errors
via photon detection. The frequency noise directly couples with the interferometric condition of the detector, resulting in an issue of the length sensing
control. Thus, the frequency noise must be suppressed.
Thermal Noise
The mirrors and their suspensions are thermally fluctuated because they have afinite temperature. These thermal fluctuations induce variations in the optical
path lengths of the interferometer, called thermal noise. When the suspension
is at a finite temperature, it excites the mechanical mode of suspension, resulting in the displacement of the suspended mirror [13]. The thermal noise causedby the suspension wires is called suspension thermal noise. The mirror itself
also exhibits the thermally induced fluctuation. They are categorized into two
kinds: Browninan noise and thermo-optic noise [14]. Cryogenic operation of
a GWD is a promising approach to reduce the thermal noise, resulting in the
improved sensitivity. In order to achieve higher sensitivity, several GWDs areoperated or planned to be operated with cryogenically cooled test masses.

24

1 Background

1.4

1.4 Summary of this Chapter

Summary of this Chapter• Gravitational-wave is predicted on the basis of the general theory of
relativity.
• The first direct detection of GW by aLIGO opened a new window to the
Universe, called gravitational-wave astronomy. A number of GW eventsare detected by terrestrial gravitational-wave detectors.
• The current gravitational-wave detectors are based on the Michelson interferometer enhanced by the additional mirrors for recycling technique.
• The sensitivity of laser interferometric gravitational-wave detector is determined by the various noises. Reduction of these noises are indispensable to improve the sensitivity of the detector.

25

1.4 Summary of this Chapter

1 Background26

Chapter 2
Cryogenic Gravitational-wave
Detectors
A laser interferometric GWD achieved first detection of GW and opened a
new window to the Universe. A GWD with improved sensitivity enables wider
range of detection of GW sources and enrich the GW astronomy. Thermal
noise induced by coatings deposited on the test mass surface is a dominantnoise source around 100 Hz where the current GWDs have the highest sensitivity. As the thermal noise is proportional to the temperature of the system,
the use of cryogenically cooled test masses and suspensions is a promising way
to reduce the thermal noise. Therefore, cryogenic operation of GWDs efficiently improves the sensitivity and enables to achieve further discoveries in
the Universe.
This chapter is dedicated to introduce the cryogenic GWDs which is constructed or planned to be constructed. First, we introduce CLIO as a prototype
for KAGRA. Second, we explain KAGRA which employs cryogenically cooled
sapphire test masses. Then, we introduce the future cryogenic GWDs, theEinstein Telescope and LIGO Voyager. After that, the technologies which arebeing developed to realize the future GWDs are described. Finally, the issue
which originated from the cryogenic system in KAGRA is described.

2.1

CLIO

CLIO (Cryogenic Laser Interferometer Observatory) is a prototype interferometric GWD toward KAGRA [15]. CLIO was constructed underground in
Kamioka mine, which can reduce the seismic noise. CLIO has two 100 m
length arm, and a Fabry-Perot cavity is embedded in each arm. One of thenotable features of CLIO is the employment of cryogenically cooled sapphire27

2.2 KAGRA

2 Cryogenic Gravitational-wave Detectors

test masses, aiming at the demonstration of thermal noise reduction. In order to achieve the cryogenic operation of CLIO, a number of studies wereconducted such as measurement of cryogenic contamination [16], and development of a low-vibration cryocooler [17]. Through these works, CLIO achieved acryogenic operation, and demonstrated the thermal noise reduction in a GWD[18]. Thus, CLIO paved a path to large-scale cryogenic interferometric GWDs.

2.2

KAGRA

KAGRA, a gravitational-wave detector constructed in Japan, is operated withcryogenically cooled sapphire mirrors, aiming to reduce the thermal noises
and to improve the sensitivity [19, 20]. In this section, we briefly describe thecharacteristics of KAGRA. More details can be found in Refs [21, 22]. The
parameters are listed in Tab. 2.2.

2.2.1

Configuration

KAGRA employs DRFPMI configuration with 3 km arm cavities as shown in
Fig. 2.1. The optical configuration of KAGRA can be divided into five systems— the input optics, FPMI, power recycling optics, signal recycling optics andoutput optics.
The input optics consist of three parts. An input mode cleaner (IMC),
which is a triangular cavity in the form of isosceles triangle with the length of
one side much shorter than the other two, is employed in the input optics. The
IMC provides three critical functionalities — spatial mode cleaning, polarization selection and frequency stabilization [23, 24] The input mode matchingtelescopes (IMMTs) are used to adjust the spatial mode of input beam. The
FPMI part consists of the BS, which splits the beam into two, and two FP
cavities embedded in each 3 km length arm. The test mass mirrors which compose arm cavities are sapphire mirrors with temperature of 22 K. The power
recycling and signal recycling systems are adopted to improve the sensitivity
by reducing the quantum noise. The output optics remove the unwanted output beam such as higher-order modes which are generated by misalignments,
impurities of mirrors, and so on.
KAGRA is also constructed underground to reduce the seismic noise which
limits the sensitivity at lower frequency range as shown in Fig. 2.7. These
two features, the cryogenically cooled mirrors and underground location, areunique for KAGRA which are not found in current other GWDs.
28

2 Cryogenic Gravitational-wave Detectors

2.2 KAGRA

Figure 2.1: Configuration of KAGRA [21]. Each arm has 3-km length which iscomposed of the sapphire mirrors. Several types of suspensions are installed in
KAGRA. Test masses which compose arm cavities are suspended by Type-A
suspension in order to reduce the impacts of seismic motion. The types ofsuspensions are shown in Fig. 2.2.

2.2.2

Vibration Isolation System

Seismic motion is an inevitable noise source for ground-based precision measurements. The vibration isolation system which reduces the seismic motion of
the mirrors are indispensable for GWDs. Fig. 2.2 shows the vibration isolation
systems employed in KAGRA. A km-scale GWD requires large seismic attenuation factor of 10−8 - 10−10 around 10-100 Hz. In order to achieve such highvibration isolation ratio, 13.5 m height mutlistage pendulum suspension, called
Type-A, is used for test mass mirrors. At the bottom of Type-A tower, a cryogenically cooled payload, called cryopayload, is hanged. A cryopayload plays
an important role to achieve a cooling and vibration isolation simultaneously.
Other main optics are also suspended for seismic attenuation. Type-B
29

2.2 KAGRA

2 Cryogenic Gravitational-wave Detectors

Figure 2.2: The vibration isolation systems used in KAGRA [25]. Type-A suspensions are used for test masses. Type-B is a suspension for signal recycling
cavity mirrors and beam splitter. Type-Bp suspension is for power recycling
mirrors. Type-C suspensions are employed for other important optics such as
input mode cleaner (IMC) or input mode matching telescope (IMMT).

is a suspension for signal recycling mirrors and the beam splitter. Type-B
suspension has approximately 3.1 m height. Type-Bp suspension, simplified
compared to Type-B suspension, is used for power recycling mirrors. TypeC suspensions are employed for other important optics such as input modecleaner (IMC) or input mode matching telescope (IMMT). Thanks to thesesuspensions, displacements induced by the seismic motion are mitigated, andthe detector is operated with higher sensitivity.

2.2.3

Cryogenic Systems of KAGRA

The use of cryogenically cooled sapphire test mass mirrors is one of the notablefeatures of KAGRA. Its diameter, thickness, and weight are 22 cm, 15 cm,
and 23 kg, respectively. On the sapphire mirror surface, dielectric multilayer
coatings are deposited in order to reflect or anti-reflect the laser beam. The
multilayer coatings are composed of SiO2 and Ta2 O5 stacked alternately in30

2 Cryogenic Gravitational-wave Detectors

2.2 KAGRA

Table 2.1: Cooling methods and temperature of GWDs.
Name
KAGRA
LIGO Voyager
The Einstein Telescope

Temperature
22 K
123 K
10 K

Cooling method
conduction
radiation
conduction

order to achieve high reflection to the laser beam.
The sapphire test masses are suspended by sapphire fibers which have large
thermal conductivity at cryogenic temperature. In addition, sapphire fiber has
low mechanical loss, resulting in in low thermal noise. On the other hand, thediameter of the fiber is thicker than that of fused silica fiber used in aLIGO toachieve enough heat extraction from sapphire test mass to maintain cryogenic
temperature. This leads to larger suspension thermal noise and lower resonant
frequencies of fibers which is called violin modes as shown in Fig. 2.7.
Tab. 2.1 describes the cooling methods of various GWDs. The way of cooling down the mirror depends on its target temperature. As radiative coolingis dominant above ∼ 100 K, LIGO Voyager plans to employ radiative cooling.
On the other hand, radiative cooling is not dominant below 100 K. Therefore, in order to cool down the mirror below 100 K, one have to employ heat
conduction for cooling down the mirror. For the case of KAGRA and the ET,heat links is or will be used which can introduce vibration contamination. Low
vibration cryocoolers are used for cooling down the mirrors in order to reduce
the amount of vibration which can be transferred to the mirror [17]. Besides,
aluminum heat links are employed, aiming effective cooling and mitigate vibration contamination [26]. However, the vibration induced by heat links can stilldegrade the sensitivity, and the heat link vibration isolation systems (HLVISs)
are installed to reduce the vibration noise in KAGRA.
Fig. 2.3 shows the cryogenic part of KAGRA. The sapphire mirror is suspended by cryopayload to reduce the seismic noise. Heat links are connectedto the cryopayload though the HLVISs for conductive cooling. Fig. 2.4 shows
how the test mass is cooled by conductive cooling.
The cryopayload is hanged by 13.5 m length suspension, called Type-A
suspension. Cryogenically cooled duct shields are installed to reduce the radiation from room temperature environment. The length of duct shield isapproximately 5350 mm, and its diameter is 370 mm as shown in Fig. 2.5.
Baffles are installed inside the duct shield in order to mitigate the impacts ofscattered light which can appear in output signal of GWD by recombining to31

2.2 KAGRA

2 Cryogenic Gravitational-wave Detectors

Figure 2.3: Schematic wiwe of the cryogenic payload (Left) and tower for test
mass (right) in KAGRA [21]. The sapphire mirror is suspended by cryopayload. Heat links are connected to the cryopayload for conductive cooling. The
beam from the arm cavity comes though the duct shield which reduces theradiation from room temperature environment.

the main beam. This shield also reduces the thermal input by radiation fromthe room temperature environment to 0.1 W. The extractable heat is designed
as 0.72 W. This value is considered enough for maintain the temperature ofsapphire mirror at 22 K. Thanks to the cryogenic system, KAGRA can operatewith cryogenically cooled sapphire mirrors.
Fig. 2.6 shows the cooling curve of sapphire test mass. In order to reach thecryogenically cooled sapphire test mass, it takes about one month. Therefore,
the time which can be spent for commissioning before an observing run becomes shorter than other room temperature GWDs due to the cooling period.
Studies for shortening the cooling time are important for cryogenic GWDs.

2.2.4

Sensitivity of KAGRA

The designed sensitivity of KAGRA is shown in Fig. 2.7. Quantum noise
dominates broad band frequency range — radiation pressure noise and shotnoise limit the sensitivity at lower and higher frequency range, respectively.
Thermal noise induced by sapphire mirrors is still one of the dominant noise32

2 Cryogenic Gravitational-wave Detectors

2.2 KAGRA

Figure 2.4: Schematic of the thermal connection of cryogenic payload in KAGRA. The sapphire test mass is suspended by sapphire fibers which extractheat, and other stages are connected by aluminum heat links.

sources around 100 Hz where KAGRA hase the highest sensitivity. Especially,
thermal noise from coatings on the sapphire mirrors are dominant noise source.
Not only quantum noise, but also thermal noise, especially coating thermal
noise, are needed to be reduced for further improvement in the sensitivity.
One can see several peaks in suspension thermal noise. Two peaks at lower
frequency is due to the resonance of the suspension. The peak around 180 Hz
is so-called violin mode which originates from the sapphire fibers hanging test
mass mirrors.
Historically, the binary neutron star inspiral range has been commonly used
as an index of the sensitivity of GWD. The binary neutronstar inspiral range
is the merit of the GWD which denotes the detectable range of 1.4 M binary
neutronstar merger with S/N = 8. The binary neutron star inspiral range canbe calculated as [28]
5/6 Z fmax −7/3 1/2
 1/2cGMc
f
0.442 5
df
,
(2.1)
R=
2/3
3
ρth
6
π
c
fmin Sn (f )where ρth = 8 is the threshold, G the gravitational constant, and Mc the chirp
mass of the binary system. The factor 0.442 is the sky average constant [29].
The lower frequency limit fmin is set to be fmin = 10. The upper end frequency
is set to be the gravitational-wave frequency at the ISCO of the Schwarzshild33

2.3 Future Gravitational-wave
Detectors

2 Cryogenic Gravitational-wave Detectors

Figure 2.5: Schematic picture of cross section of the duct shield in KAGRA.
The length of duct shield is 5350 mm and the diameter is 370 mm. The baffles
are installed inside the duct at 1 m intervals to reduce the scattering noise.

metric which is described as
fmax =

c3,
63/2 πGMtot(2.2)

where Mtot = m1 + m2 is the total mass. The designed binary neutron star
range of KAGRA is 153 Mpc with Detuned Resonant Sideband Extraction(DRSE) configuration.

2.3

Future Gravitational-wave
Detectors

2.3.1

The Einstein Telescope

The Einstein Telescope (ET) is planned to be constructed in Europe with 10 km
arm length xylophone. The ET will be constructed underground and consists of
two part, the ET-LF and the ET-HF. The ET-HF will adopt room temperature
test masses which explores high frequency GWs. On the other hand, the ETLF, which is specialized for low frequency GWs, will employ cryogenicallycooled test masses at 10 K with 1550 nm wavelength laser. In order to achieve
10 K test mass mirrors, conductive cooling systems are planed to be installedin the ET. As conductive cooling can induce vibration contamination via heatlinks, vibration isolation systems for heat links also will be employed [30]. In
order to realize 10 K cryogenically cooled mirrors, a number of studies areon-going [31, 32].
As shown in Fig. 2.8, the ET-LF has remarkable sensitive below 10 Hz where
one cannot access with current GWDs. The design sensitivity is approximately
34

2 Cryogenic Gravitational-wave Detectors

2.3 Future Gravitational-wave
Detectors

Figure 2.6: The cooling time of KAGRA test mass [21]. Due to the restart ofcryocoolers, one can see the bump around day 30. It takes about one monthto obtain cryogenically cooled sapphire test mass.

100 times better at 10 Hz than current GWDs such as KAGRA. Therefore, therealization of the ET should provide further discoveries which cannot been done
by current GWDs. One example is a detection of GW from intermediate mass
black holes with masses on the order of 102 − 103 M . The existence of suchintermediate mass black hole has been proved by direct detection [33]. In order
to constrain formation scenarios such massive black holes, multiple detections
are necessary [34]. ET-LF, which employs cryogenically cooled mirror, is suitedfor detections of massive black holes and should provide further insights into
the Universe.

2.3.2

LIGO Voyager

LIGO Voyager is a substantial upgrade of aLIGO, and will employ cryogenically cooled mirrors with DRFPMI configuration [35]. The LIGO Voyager test
mass temperature is planed to be 123 K where silicon has a zero cross of its
thermal expansion coefficient [36]. Therefore, the substrate thermal noise canbe minimized.
The cryogenic system in LIGO Voyager employs radiative cooling to maintain the test mass temperature at 123 K. This approach can avoid the complicating the suspensions with vibration contamination via conductive heat35

2.3 Future Gravitational-wave
Detectors

2 Cryogenic Gravitational-wave Detectors

seismic+gravity
suspension
mirror thermal
quantum
total

Strain [/rHz]

10-21

10-22

10-23

10-24
101

102

103

Frequency [Hz]Figure 2.7: The design sensitivity and noise sources of KAGRA [27]. Blacksolid line represents the sensitivity of KAGRA. Other curves denote the noises
which contribute to the sensitivity. The first order of violin mode can be seenaround 180 Hz, and its higher-order also can be seen at higher frequency range.
At most sensitive frequency range around 100 Hz, the mirror thermal noise,
especially coating thermal noise, is still dominant though the cryogenicallycooled mirrors are employed.

paths. In order to achieve 123 K temperature, cryogenic shields with about
10 m lengths are used which isolate the radiation from the room temperatureenvironment.
LIGO Voyager requires approximately 150 W laser power at the input tothe interferometer. As the arm cavity power in LIGO Voyager is designed
as 3 MW, the absorption in the coatings is one of the problems which can
prevent cryogenic operation. Choosing a longer wavelength, around 2000 nm,
can mitigate the absorption in the coatings and realize the designed arm cavity power. However, the improvements in substrate material, coatings, andcryogenic systems are still needed to realize LIGO Voyager.
Fig. 2.9 shows the sensitivity and noise sources of LIGO Voyager. The
inspiral range of LIGO Voyager for 1.4M· neutronstar binary system is about36

2 Cryogenic Gravitational-wave Detectors

2.4 Technologies for Future
Gravitational-wave Detectors

10−21

Strain [/rtHz]

10−22

10−23

10−24

10−25100

ET-LF
ET-HF
ET sum101

102
Frequency [Hz]

103

104

Figure 2.8: The design sensitivity of the Einstein Telescope [30]. The ET
consists of two parts — specialized for low frequency or high frequency GWs.
The blue and green solid line shows the sensitivity of the ET-LF and ET-HF,
respectively. The red line represents the total sensitivity of the ET which isthe summation of the ET-LF and ET-HF.

740 Mpc. This is about 4 times larger inspiral range, resulting 43 = 64 times
larger number of detection than aLIGO. Moreover, LIGO Voyager has a goodsensitivity such that it can detect 30 − 30M binary black hole mergers evenat z ∼ 10 [35]. Therefore, the realization of LIGO Voyager will enrich thegravitational-wave astronomy.

2.4

Technologies for Future
Gravitational-wave Detectors

In order to realize the future GWDs with improved sensitivity, a number ofstudies are being conducted intensively. In this section, we briefly introduce
several technologies to be developed for future GWDs.
37

2.4 Technologies for Future
Gravitational-wave Detectors

Strain [/rtHz]

10−23

2 Cryogenic Gravitational-wave Detectors

Total
Quantum Vacuum
Seismic
Newtonian Gravity
Suspension Thermal
Coating Brownian

Coating Thermo-Optic
ITM Thermo-Refractive
Substrate Brownian
Substrate Thermo-Elastic
Excess Gas

10−24

10−25
101

102
Frequency [H0]103

Figure 2.9: The design sensitivity and noise sources of LIGO Voyager [35].
Black solid line represents the sensitivity of LIGO Voyager. Other curves
denote the noises which contribute to the sensitivity.

2.4.1

Filter Cavity

Quantum noise which limits the sensitivity at broad band frequency range is
one of the important noise sources in GWDs. This noise originates from thequantum nature of light. The use of frequency-dependent squeezing technique
is a possible solution to reduce the quantum noise in the broad band frequency.
Filter cavity experiment in TAMA 300 interferometer demonstrated frequencydependent squeezed vacuum which can reduce the quantum noise [11]. In
future, the employment of filter cavity in GWDs will enable improvement inan overall sensitivity.

2.4.2

Substrate Material

The choice of a substrate material for a test mass mirror is one of the mostimportant parts to construct a GWD. There has been a number of studies
to achieve the high-end, low-loss mirror for GWDs [37, 38, 39]. There aresome candidates as test mass mirror substrates for future GWDs — fused38

2 Cryogenic Gravitational-wave Detectors

2.4 Technologies for Future
Gravitational-wave Detectors

silica, sapphire, and silicon. Fused silica, the material used in the current
room temperature GWDs, has a large mechanical loss at cryogenic temperature
which leads to large thermal noise and is not suitable for a cryogenic GWD [40].
Silicon is one of the most likely candidates for a cryogenically operated
GWD. Silicon has a small mechanical loss and consequently low Brownian
noise under cryogenic temperature [41]. In addition, silicon has the uniqueadvantage that its substrate thermo-elastic noise vanishes at 18 K and 123 K
where its coefficient of thermal expansion crosses zero [36]. However, as siliconis opaque for wavelengths shorter than 1100 nm and has small absorption only
for wavelengths longer than 1400 nm, the wavelength of main laser should bewithin the range of 1400 nm to 2100 nm [42]. Therefore, the wavelength ofmain laser is chosen to be 1550 nm and 2000 nm for ET and LIGO Voyager,
respectively [30, 35].
The size of test mass is important for radiation pressure noise and coating
thermal noise. Heavier and larger diameter test masses are necessary to reduce
the radiation pressure noise and coating thermal noise, respectively. Current
magnetically stabilized Czochralski (MCZ) technique realized 45 cm diametersilicon crystal with high purity. Therefore, 200 kg silicon test mass without
serious impurities is achievable in the future GWDs.
Sapphire, which is used in KAGRA, is also one of the candidates. However, the available size of high-quality sapphire mirror is not so large and a
breakthrough in technology is indispensable to fabricate larger size of sapphire
mirror. In addition, sapphire is a birefringent crystal, and has a potential toinduce polarization conversion from S-pol. to P-pol., which can make harder tooperate the interferometer stably. There are rooms to be improved to achieve
higher sensitivity of GWDs with sapphire mirrors.

2.4.3

Coatings

Thermal noise arises from the coatings on the test mass becomes one of thenoise sources which can limit the sensitivity. For the case of the ET, coatings
which have 25 times smaller thermal noise than that of current ones with about
1 ppm absorption are required. Current typical SiO2 /Ti : Ta2 O5 coating havelarge mechanical loss around 20 K, which leads to large thermal noise. Therefore, in order to achieve future cryogenic GWDs, the development of coating
materials and methods such that thermal noise satisfies the requirement are
indispensable.
392.4 Technologies for Future
Gravitational-wave Detectors

2 Cryogenic Gravitational-wave Detectors

Low Noise Coating
One of the most promised way to reduce the coating thermal noise (CTN) is
to employ low noise coatings. New materials or methods have been studied
in order to reduce the CTN [32, 43]. One notable method is to use crystalline coatings which have shown large reduction in Brownian thermal noise[44]. These coatings are deposited GaAs and AlGaAs multilayers on GaAs
substrate and can be transferred onto other substrate material. With thesecoatings, notable optical properties have been demonstrated [44, 45]. However,the size of these coatings is limited by the GaAs substrates currently available.
In addition, it requires further tests to evaluate the effect of coating-substratebonding on any possible thermal noise. An alternative technique to utilize
crystalline coatings is to grow a lattice-mathced coating directly onto a substrate. This method removes the need for a complex transfer process. In thecase of silica substrate, a possible multilayer crystalline coatings is GaP/AlGaP[46]. Especially, GaP/AlGaP coatings have shown remarkable low mechanicalloss at cryogenic temperature. Nevertheless, the GaP/AlGaP coatings needmore studies as a mirror coating and further development, particularly of the
optical properties.
Amorphous silicon is an interesting material which has large refractive index and shows low mechanical loss at cryogenic temperature [31]. Amorphoussilicon, however, has large optical absorption and needs to be tackled withthis problem [47]. One possible solution is to adopt multi-material coatings
which can achieve the requirement on absorption while amorphous silicon isdeposited on near surface of the substrate [31]. By using amorphous siliconwith appropriate composition, one can significantly reduce the coating thermal
noise combining with cryogenic test masses.
Post Deposition Treatment
One possible approach to reduce the CTN is a post deposition heat treatmenti.e., annealing of coatings. Annealing at appropriate temperature improves
the optical properties of coatings such as absorption and mechanical loss [48].
Therefore, the annealing has an important role to achieve state-of-the-art mirrors.
However, annealing causes the crystallization of amorphous coatings above
a certain temperature and the crystallization temperature varies with coating
materials. For the case of typical SiO2 /Ta2 O5 coatings, tantala layers start tocrystallize around 600 ◦ C while the silica layers can be baked up to 950 ◦ C. One
possible solution to increase the annealing temperature is to employ some optimal alloy of the materials such as TiO2 or ZrO2 [49]. Doping tantala with tita40

2 Cryogenic Gravitational-wave Detectors

2.5 Technical Problem— Molecular Layer Formation

Figure 2.10: Left picture shows the sapphire mirror at room temperature [54].
Right picture shows the frosted sapphire mirror at cryogenic temperature [55].

nia is employed in aLIGO test mass so as to reduce the coating thermal noise.
Optimally doped coating shows not only lower mechanical loss but also higher
crystallization temperature and can lead to smaller thermal noise. Another is
to use very thin nano-layers in which small thickness of the layer suppresses
crystallization of coating materials [50]. In addition, nano-layer thickness thinfilm can reduce the mechanical loss at cryogenic temperature [51].
Coating-Free Mirrors
High reflective mirrors without deposited thin films can be a solution to reducethe coating thermal noise due to the lack of large mechanical loss thin films.
Recent studies have reported some amount of achievement in developing highfinesse cavities [52, 53]. However, coating-free mirrors still require significant
development in order to be employed in the precision measurements.

2.5

Technical Problem— Molecular Layer Formation

In this section, we describe a technical problem, which can happen in future
cryogenic GWDs. One of the critical problems for cryogenic GWDs is theformation of molecular layer on cryogenic mirror surfaces. The molecular layer41

2.5 Technical Problem— Molecular Layer Formation

2 Cryogenic Gravitational-wave Detectors

Figure 2.11: Schematic of how molecular layer formed on the cryogenic mirror
surface by a molecular flow from beam duct. As the pressure inside the cryostat
is quite low compared to that of beam duct because of the cryopumping. The
residual gas molecules attached on the room temperature beam duct can easily
desorbed, and transported to the cryostat. A part of transported gas molecules
hits and be trapped on the test mass surface.

formation is induced by the so-called cryopumping effect, which is usuallyemployed in a vacuum pump [56]. When the vacuum level inside the cryostat
is insufficient, the residual molecules are adsorbed by the cryogenic mirror,
resulting the frosting of the mirror. KAGRA experienced such phenomenon
when the mirrors were cooled down to cryogenic temperature, i.e., ∼ 20 K as
shown in Fig. 2.10 [55].
The molecular layer can be formed though the vacuum level inside thecryostat is enough high to avoid the adsorption of the residual molecules inside
the chamber. As the cryostat works as a type of vacuum pump, the pressure
inside the cryostat is smaller than that of the room-temperature beam duct.
This pressure difference leads to continuous molecular transportation onto thecryogenic mirror surface from the room-temperature beam duct where watermolecules can easily detached from the surface. When gas molecules hit acryogenic mirror surface, they lose their kinetic energy and adhere onto thesurface due to the cryopumping. Therefore, the collisions of gas molecules
create a molecular layer which grows over time on top of the cryogenic mirror
surface. We call such a molecular layer a cryogenic molecular layer (CML) inthis thesis. Fig. 2.11 shows the schematic picture of how a molecular layer isformed on the cryogenic test mass.
Formation of the CML prevents the observation of cryogenic GWDs with
their best performance due to its optical loss. The possible impacts of the CML
42

2 Cryogenic Gravitational-wave Detectors

2.6 Target of this Studyon the optical loss are calculated in the next chapter. The optical loss inducedby the CML also degrades the quantum noise. The CML also can contributeto the thermal noise which can limit the sensitivity [57]. In order to remove the
CML, one have to warm up the mirror, resulting in less observation time. As
shown in Fig. 2.6, recooling of the test mass mirror in KAGRA requires about
one month, and the observation will be suspended during this period. Hence,
the CML formation is a critical problem in cryogenic GWDs as it decreases
the number of detection and observation time.

2.5.1

Previous Studies

In order to achieve cryogenic operation of KAGRA, several studies were performed to investigate the impacts of CMLs. In 2001, the optical loss of CML
was measured by a table top experiment [16]. In this table top experiment,
the finesse degradation induced by a cryogenic contamination was monitored
for about 50 days with 10 K cryogenic Fabry-Perot cavity. It revealed thatthe power recycling gain (PRG) of KAGRA, which is an important factor
to maintain the performance of KAGRA, can be maintained within tolerable
degradation for 130 days.
A further study in KAGRA was performed to estimate the CML formation
speed by using a Fabry-Perot cavity [58]. The formation speed was obtainedby oscillations in finesse of the cavity induced by the reflectivity change due tothe CML formation. The estimated CML formation speed was 27 nm/day with
5 × 10−6 Pa vacuum level. This study also revealed that the inspiral range of
KAGRA can decrease by about 30% after one year of operation. In addition,
they mentioned that a CML can increase thermal noises through large heat
input to test masses.
Both of these previous studies are performed with 1064 nm wavelength laser
sources. Future cryogenic GWDs, however, will employ longer wavelength laser
such as 1550 nm or 2000 nm [30, 35], and the optical impacts of CMLs at these
longer wavelength had not been investigated. Therefore, optical loss study
and characterization of CMLs at longer wavelength becomes very importantto estimate the impacts on future cryogenic GWDs.

2.6

Target of this Study

A cryogenincally cooled mirror is a key technology to achieve the future GWDs.
Such cryogenic mirrors can adsorb the residual gas molecules, and reduce theperformance of GWD. Previous studies investigated possible impacts of adsorbed CML on KAGRA which employs 1064 nm laser. However, there is no43

2.7 Summary of this Chapter

2 Cryogenic Gravitational-wave Detectors

study which investigated the optical loss of CMLs for future GWDs, which
will employ longer wavelength than 1064 nm, though CMLs can affect the performance even at very thin thickness. The goal of this thesis is to study the
impact of the molecular formation on optical loss of future cryogenic GWDs
by theoretical calculations, andestablish a method to characterize with a ppm
level resolution in optical loss measurement and a nanometer level resolution
in thickness determination.

2.7

Summary of this Chapter• Operation with cryogenically cooled test mass mirrors is a promising
approach to enhance the number of GW detection by improving thesensitivity.
• KAGRA operates with 22 K cooled sapphire test masses to reduce thermal noises.
• The future cryogenic GWDs have improved sensitivity and should provide new discoveries.
• However, cryogenically cooled mirrors can suffer from the formation ofmolecular layers on their surfaces.
• The molecular layer formed on the cryogenic mirror surface affects the
GW observation, and can reduce the number of detection.
• Previous studies investigated the impacts of the molecular layer on the
optical loss of KAGRA. However, the impacts on the optical loss of future
GWDs have not been studied.
• The goal of this thesis is to study the impact of the molecular formation
on optical loss of future cryogenic GWDs, and give a novel approach tocharacterize the optical loss of very thin molecular layer.

44

2 Cryogenic Gravitational-wave Detectors

2.7 Summary of this Chapter

Table 2.2: The parameters of KAGRA [21].
Parameters
SymbolArm length
Larm
Mirror mass
M
Beam radius
w0
Temperature of the test massT
ITM transmittance
TITM
ETM transmittance
TETM
PRM transmittance
TPRM
Loss inside the arm cavity
Tloss,armLaser wavelengthλLaser power
Pin
Mirror thicknessdmirAbsorption in test mass
αmir
Density of sapphireρsap
Specific heat of sapphire
Csap
Thermal expansion rate of sapphire
αsap
Thermal conductivity of sapphireκsap
Young’s modulus of sapphire
Ysap
Poisson ratio of sapphire
σsap
Loss angle of sapphire mirror
φsap
Refractive index of sapphire (λ = 1064 nm)
nsap
Diameter of sapphire fiber
df
Length of sapphire fiberlfLoss angle of sapphire fiber
φfExtractable heat
Q

45

Value
3000 m22 kg
3.5 cm
22 K
0.4%
7 ppm
10%
93 ppm1064 nm
67 W
15 cm
50 ppm/cm
3980 kg/m3
0.69 J/K/kg
5 × 10−6 1/K
1.57 × 104 W/m/K
73.2 GPa
0.25
1 × 10−8 rad
1.7545
1.6 mm
350 mm2 × 10−6 rad
0.72 W

2.7 Summary of this Chapter

2 Cryogenic Gravitational-wave Detectors

46

Chapter 3
Theoretical Study of
Cryogenic Molecular Layer
The use of cryogenic mirrors is a promising approach to improve the sensitivityof GWDs. The cryogenic mirrors in KAGRA, however, suffered from the CML
on the mirror surfaces as shown in Fig. 2.10 [55]. The properties of the mirror,i.e., the reflectance and transmittance, change due to the growth of CML.
The impact of the CML on scattering has been experimentally estimatedon a small-scale cryogenic system [16]. Recent study revealed that the CML isformed on the high-reflection (HR) side of the cryogenic mirror in KAGRA, andthe formation rate and possible impacts on KAGRA were estimated [58, 59].
Both of these studies are conducted with the laser wavelength of 1064 nm. On
the other hand, future GWDs will employ longer wavelength laser. The ET isplanning to use 1550 nm wavelength laser for the law-frequency part (ET-LF),
and LIGO Voyager will use 2 µm wavelength. At such a longer wavelength
region, the water molecules, which is considered as one of the most dominantmolecules of the CML, have much larger optical absorption compared to thatat 1064 nm. Fig. 3.1 shows the absorption of ice around 1550 nm, and 2000 nm.
The amorphous ice, which is considered as the main component of the CML,
has large absorption within this range. Hence, the optical loss due to the CML
can be more critical for future cryogenic GWDs.
In this chapter, first, we describe one of the possible impacts of the CML
on cryogenic GWDs i.e., optical losses. Then, we derive a theoretical model ofthe optical loss induced by the CML in a cryogenic GWD. Finally, we show
how the optical loss of the CML affects a cryogenic interferometric GWD. The
contents described in this chapter is published as an article by the author [61].
473 Theoretical Study of
Cryogenic Molecular Layer

3.1 Optical Loss

Figure 3.1: Absorption of ice. This figure is cited from the reference [60].
Horizontal axis is the wavenumber, which is the inverse of wavelength, 1/λ.

3.1

Optical Loss

Low optical loss mirrors are vital for precision laser interferometry [38, 62]. As
shown in Fig. 3.2, optical loss in the mirror is introduced mainly by two paths— optical scattering and absorption. The test mass mirrors used in GWDs
are manufactured with state-of-the-art technologies to meet the requirements
on optical losses [63]. A molecular layer, however, generates additional opticalloss which can hinder us from operating the detector at a cryogenic temperature and achieving the design sensitivity. Especially, the optical absorption
generates an additional heat load to a cryogenic mirror and its temperature
can be increased by this effect. In this section, we review the derivation of the
optical loss by CML on a test mass mirror. More details can be seen in thereference [59].

3.2Scattering

A beam reflected by a mirror is scattered by imperfections of the mirror surface
which causes an optical loss and decreases the arm cavity power in GWDs.
Moreover, the scattered light can become a technical noise source in GWDs byrecombining to the main beam [64, 65]. Therefore, the scattering by the CML
can reduce the sensitivity of GWDs. In this section, we theoretically derive
the amount of optical loss induced by scattering in the CML.
48

3 Theoretical Study of
Cryogenic Molecular Layer

3.2 Scattering

Scattering
Absorption

Main Beam
MirrorFigure 3.2: Schematic drawing of the optical loss in the test mass mirror.
The scattering and absorption lead to less arm cavity power which decreases
the sensitivity of GWDs. Furthermore, the optical absorption introduces anadditional heat load to a cryogenic mirror.

The ratio between the total reflected beam power and that of scattered
light, called total integrated scattering (TIS), is defined as [66]
TIS ∼

Psca
,
P0 R

(3.1)

where P0 is the incident beam power, Psca is the power of the scattered lightand R is the reflectance of the CML surface. For the case of the angle ofincidence equals to zero such as the arm cavities in GWDs, the TIS can becalculated as [67]
( 
2 )
4πσ
TIS = 1 − exp −
,
(3.2)
λ
where
2

Z 1

λ

σ = 2πPSD(f )f df.

(3.3)

0

Here λ is the laser wavelength. PSD(f ) [m4 ] represents the 2−D surface power
spectral density and f [1/m] is the spatial frequency.
Assuming uniform molecular adsorption on a cryogenic mirror surface, theincident molecular flux to the unit area follows the Poisson distribution.
Therepfore, the standard deviation of the number of molecules is given by hN i where49

3 Theoretical Study of
Cryogenic Molecular Layer

3.3 Optical Absorption

N is the average number of molecules. The relationship between the thicknessof CML, t, and N can be expressed as

1/3
M
t=N
,
(3.4)
ρNA
where ρ = 940 [kg/m3 ] represents the density of the CML, NA [1/mol] is the
Avogadro number and M = 18 × 10−3 [kg/mol] is the molecular mass. From
Eq. (3.4), the relation between the mean thickness of the CML, hti and its
standard deviation, σhti , can be written as
1/3

1/6

p
p
M
M
hN i =
hti.
(3.5)
σhti =ρNA
ρNA
PSD which follows this distribution can be calculated as [68]2 2
PSD(f ) = πσhti
ξ exp[−(πf ξ)2 ],(3.6)

where ξ is the correlation length which characterizes the periodic length of theroughness along the surface. Thus, the scattered power by the CML can be
expressed as)#"(
2 2 Z 132π 4 σhtiξλ2e−(πf ξ) f df.
(3.7)
Psca ∼ P0 R 1 − exp −2
λ0Assuming the correlation length of the CML, ξ, Eq. 3.7 allows us to estimate
the optical loss generated by the scattering.

3.3

Optical Absorption

Another way leading to an optical loss is the optical absorption. Especially,
the optical absorption in a cryogenic mirror plays a crucial role in selecting
the parameters of GWD because of an additional heat load on the test mass
mirror. For the case of a GWD, a Fabry-Pérot cavity is embedded in the arm
to improve the sensitivity [20, 69, 35]. Hence, the laser power inside the arm
cavity becomes extraordinarily large and the optical absorption introducedby the CML can become a critical heat source in the cryogenic test mass.
Absorption in the anti-reflection (AR) surface can be negligible because ofmuch lower laser power on the surface than the HR side.
The laser power density inside a medium I(z) follows the Lambert-Beerlaw asI(z) = Is exp(−αz),
50

(3.8)

3 Theoretical Study of
Cryogenic Molecular Layer

3.3 Optical Absorption

where z represents the depth of the medium from its surface, Is is the laser
power density at the surface, z = 0, and α is the absorption coefficient. Assuming Gaussian beam with the beam radius of w0 , the profile of the laser
power intensity can be written as2r2
Ir = I0 exp − 2w
 0 2
2P0
2r
(3.9)
=
exp − 2 .
2
πw0
w0
p
where r = x2 + y 2 and P0 represent the radius of the beam in cross section
and the laser power, respectively. Therefore, the intensity in the medium canbe expressed as


2r2
2P0
exp − 2 − αz .
(3.10)
I(r, z) =πw02
w0
The power absorbed by the volume element rdrdθdz of the medium can becomputed as
A(r, z) = {I(r, z) − I(r, z + dz)} rdrdθ
∂I(r, z)rdrdθdz
=−
∂z 

2P0
2r2
=
α exp − 2 − αz rdrdθdz.
πw02w0(3.11)

Thus, the total laser power absorbed by a CML of thickness dCML becomesZ ∞ Z 2π Z dCML
∂I(r, z)dz
ACML = −
dr
dθ
r
∂z
0
0
0
= P0 {1 − exp(−αdCML )} .
(3.12)
Assuming the amplitude of the laser intensity inside the medium follows theLambert-Beer law, the total amount of the absorption by the molecular layer,
ACML , with the thickness of dCML is expressed as
ACML = PCML [1 − exp(−αCML dCML )],(3.13)

where PCML is the laser power incoming to the molecular layer and αCML =4πIm(NCML )/λ is the absorption coefficient of the molecular layer where NCML
is the complex refractive index of CML.
For the case of the GWDs, the laser power inside the arm cavity becomes extraordinary large. Therefore, the optical absorption introduced by the molecular layer can become a critical heat source in the cryogenic test mass. As
51

3 Theoretical Study of
Cryogenic Molecular Layer

3.3 Optical Absorption

the laser power inside the arm cavity is much larger than that of the power
recycling cavity (PRC), the absorption on the AR side of the test mass can be
negligible.
A power recycling technique is applied to GWDs in order to reduce theshot noise by enhancing the laser power inside interferometers [70]. The intracavity power is determined by the finesse of the arm cavity and the power
recycle gain. The laser power inside the PRC can be written as
ETM
PPRC = G(dITM
CML , dCML )Pin ,(3.14)ITM,ETM
ETM
where G(dITM
the thickness of the
CML , dCML ) is the power recycle gain, dCML
molecular layer on the ITM or ETM, and Pin the incident laser power into the
ETM
PRC. The power recycle gain G(dITM
CML , dCML ) can be expressed as [58](ETM
G(dITM
CML , dCML ) =t
P PRMITM
1
1 − 2 rprm i ri (dCML,i , dETM
CML,i )

)2.

(3.15)

ETM
where ri (dITM
CML,i , dCML,i ) (i = x, y) represent the reflectivity of the x or y arm
cavity i.e., inline or perpendicular to the incoming beam to the BS, which is
given by
ETM
ITM
ri dITM
CML,i , dCML,i = −rITM,i (dCML )+ETM
t2ITM (dITM
CML )rETM (dCML )rloss,arm,
ETM
1 − rITM (dITM
CML )rETM (dCML )rloss,arm(3.16)

where rloss,arm is the loss equivalent reflectivity in the arm cavity. The reflectivity of the mirror with the molecular layer can be calculated by using thecharacteristic matrix as shown in the reference [58]. Defining the amount ofloss inside the arm cavity as Tloss,arm , the loss equivalent reflectivity can be
expressed asp(3.17)
rloss,arm = 1 − Tloss,arm .
ETM
The laser power inside the arm cavity is enhanced by 2Farm (dITM
CML , dCML )/π
ITM
ETM
where Farm (dCML , dCML ) is the finesse of the arm cavity which can be writtenas
ETM
Farm (dITM
CML ,dCML )p
ETM
π rITM (dITM
CML )rETM (dCML )rloss,arm=.
ETM
1 − rITM (dITM
CML )rETM (dCML )rloss,arm

52(3.18)

3 Theoretical Study of
Cryogenic Molecular Layer

3.4 Optical Loss Estimation

As the beam is split by a beam splitter, the intra-cavity power, Parm , is reduced
by a factor of 2 and becomes
1
ETM
Farm (dITM
CML , dCML )PPRC
π
1
ETM
ITM
ETM
= Farm (dITM
CML , dCML )G(dCML , dCML )Pin .
πParm =(3.19)The part of the laser power is reflected at the interface of the vacuum andmolecular layer surface and the incoming laser power is reduced. The powerreflectance of molecular layer, RCML , is given byn0 − nCML
RCML =
n0 + nCML

2.

(3.20)The incoming laser power into the molecular layer can be written as
PCML = Parm (1 − RCML ).

(3.21)

Thus, assuming the Lambert-Beer law holds for the molecular layer, theamount of the optical absorption by the molecular layer can be estimated as
ACML =

1 − RCML
ETM
ITM
ETM
Farm (dITM
CML , dCML )G(dCML , dCML )π× Pin [1 − exp(−2αCML dCML )].

(3.22)It should be noted that the index of the exponential is doubled by taking into
account of the reflection of the coating.

3.4

Optical Loss Estimation

Table 3.1: The parameters to calculate the refractive indices. C1 and C2 aregiven by the previous report [71]. D1 and D2 are obtained by fitting using
Eq. 3.23.
Symbol
Value

√

C1
71

√

C2
134

D1
0.00996

D2
0.0319Some of future GWDs are planed to use cryogenic mirrors to achieve better
sensitivity. Low optical absorption is critical for cryogenic GWDs both to53

3 Theoretical Study of
Cryogenic Molecular Layer

3.4 Optical Loss Estimation

Refractive index of amorphous ice
1.375

Refractive index

1.350
1.325

10K
30K
50K
70K
90K
110K
130K
150K

1.300
1.275
1.250
1.225
1.200
250

500

750

1000

1250

1500

Wavelength [nm]

1750

2000

Figure 3.3: Refractive index of the amorphous ice for various temperaturesderived by the Lorentz-Lorenz equation.

maintain the cryogenic temperature of the test mass and achieve the design
sensitivity. As the arm cavity power is designed to be 18 kW and 3 MW for ET
and LIGO Voyager, respectively, an additional heat absorption introduced by
the CML can become a critical problem in maintaining cryogenic temperatureof the test masses. We show the impacts of CML on the future GWDs which
adopt cryogenic mirrors.
In order to evaluate the impact of the CML, we need several assumptions.
First, we assume that the CML is composed of water as the previous works havedone [58, 57]. In addition, the water molecular layer is formed as amorphousice [72]. The refractive index of amorphous ice at the cryogenic temperature
has been studied for the wavelength of 210 − 757 nm [71]. We extrapolate therefractive index at 1064 nm, 1550 nm and 2000 nm by extrapolation using theLorentz-Lorenz equation which gives the relation between the density ρ andthe refractive index n(λ) as

R(λ) =

1 n(λ)2 − 1,ρ n(λ)2 + 2
54(3.23)

3 Theoretical Study of
Cryogenic Molecular Layer

3.4 Optical Loss Estimation

where R(λ) is the specific refraction which can be expressed assR(λ) =

D1 λ2
D2 λ2
+
.
λ2 − C 1 λ2 − C 2(3.24)

C1 , C2 , D1 and D2 are the parameters to explain the experimental result andC1 and C2 are given by the previous report [71]. D1 and D2 are obtained byfitting the data shown in the reference [71] using Eq. 3.23. The results areshown in Fig. 3.3 and the value of parameters are listed in Table 3.1. As there
is no characteristic structure of H2 O molecule within the wavelength we discuss
here, its refractive index monotonically decreases gently until 2 µm as shownin the reference [73]. Therefore, this extrapolation is a reasonable assumption
to evaluate the effect of the CML. Furthermore, we use the literature value forthe estimation of optical absorption. The absorption coefficient of amorphous
ice has been reported at the temperature of 40 K and 140 K [60]. As the temperature dependence of the absorption coefficient is so weak between 40 K and
140 K, we adopt these literature values to calculate the absorption at 123 K.
In addition, we assume that the absorption coefficient at 10 K is almost thesame as the value at 40 K. The parameters which are used for the estimation
are shown in Table 3.2.

3.4.1

KAGRA

First, let us revisit the case of KAGRA as the a reference for future GWDs.
KAGRA is operated with cryogenically cooled sapphire mirrors at the temperature of 22 K with the wavelength of 1064 nm laser source [19, 20]. For thecase of KAGRA, test masses are cooled by heat conduction through sapphirefibers which suspend them. The estimated heat extraction capacity is about
0.72 W [27], and the absorption in the coatings is less than 0.5 ppm [74, 39].
More details of the impacts of the CML formation on KAGRA are reported in
the references [58, 59].
Fig. 3.4 shows estimated optical loss by scattering and absorption for KAGRA. Here we assumed the absorption coefficient as α = 2.2 × 10−2 ppm/nm
and other related parameters are listed in Tab. 2.2 [73]. Optical loss inducedby scattering is less than 1 ppm as long as the thickness of the CML is below
1 µm even if the correlation length ξ equals to σhti . Assuming the uniform
molecular injection onto the mirror surface, the correlation length, ξ, becomes
small because it forms the tidy and smooth surface. Therefore, the impact of
the scattering can be negligible when the thickness of the CML is below 1 µm.
55

3 Theoretical Study of
Cryogenic Molecular Layer

3.4 Optical Loss Estimation

Table 3.2: The assumed parameters of future cryogenic GWDs [30, 35]. Here
we assume that the CML is composed by amorphous ice. The refractive indicesof amorphous ice are derived by the fitting using Lorentz-Lorenz equation. Itshould be noted that the refractive indices at 22 K and 123 K are the averaged value of that at 10 K and 30 K, and 110 K and 130 K, respectively. The
absorption coefficient of the CML is assumed the literature value [73, 60].

Parameters
Temperature of the test mass
ITM transmittanceETM transmittancePRM transmittance
Loss inside the arm cavity
Laser wavelength
Laser power
Mirror thickness
Refractive index of CML
Absorption coefficient of CML
Absorption in test mass
Extractable heat

3.4.2

Symbol
T
TITM
TETM
TPRM
Tloss,arm
λ
Pin
dmir
nCML
αCML
αmir
Q

ET
10 K
0.7%
6 ppm
4.6%
75 ppm1550 nm
3W
50 cm
1.19
2.0 ppm/nm
3.2 × 10−2 ppm/cm
100 mW

Voyager
123 K
0.2%
5 ppm
4.92%
10 ppm2000 nm
152 W
55 cm
1.28
8.0 ppm/nm
10 ppm/cm
10 W

The Einstein Telescope

Einstein Telescope is a planned European GWD which will use cryogenic silicon mirrors at the temperature of 10 K with the wavelength of 1550 nm laser
source for the low frequency part [69]. Extracting the heat generated by theabsorption at the mirror surfaces has to be done by the thermal conductionof the suspension fibers because of the significantly low thermal radiation at
cryogenic temperature of 10 K. The absorption in the coatings is designed tobe 1 ppm, and the capacity of heat extraction by the suspension fibers is only
100 mW [30]. Therefore, the optical absorption should be kept as small as
possible.
Fig. 3.5 shows the result of the optical loss by the scattering and absorption
for ET. The impact of scattering can be negligible when the thickness of the
CML is below 1 µm in the same manner as KAGRA. On the other hand, theabsorption is remarkably large because of the use of a longer wavelength laserand has a potential to prevent a cryogenic operation due to the additional heatload to the test mass.
56

3 Theoretical Study of
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3.4 Optical Loss Estimation

KAGRA

10−4

Optical loss

10−6
10−8

10−10
10−12
10−14

abso ption
scatte ing ξ = σ⟨t⟩⟨1
scatte ing ξ = σ⟨t⟩⟨10
scatte ing ξ = σ⟨t⟩⟨100

10−16
10−181

10100

CML thickness [nm]1000

Figure 3.4: Scattering and absorption loss for the case of KAGRA. The assumed wavelength is λ = 1064 nm. Red solid line, yellow dashed line, green
dashdot line and blue dotted line represent the absorption and scattering withthe correlation length of ξ = σhti , σhti /10 and σhti /100, respectively.

3.4.3

LIGO Voyager

LIGO Voyager is a substantial upgrade of aLIGO, aiming to improve the inspiral range by a factor of 4 to 5 [35]. Silicon is also a candidate material for LIGO
Voyager test masses and will be cooled down to 123 K. The laser wavelength
is chosen to be 2 µm for LIGO Voyager in order to take advantage of lower
absorption of amorphous silicon coating. The requirement on the absorption
in the coatings is at most 1 ppm. The absorption coefficient of water moleculeis, however, larger than that of at the wavelength of 1.5 µm. Furthermore, thearm cavity power is much larger than the case of ET, 3 MW. Therefore, theheat absorption when the CML is formed on the silicon mirror can become aserious heat source which prevents the operation at cryogenic temperature of
123 K.
Fig. 3.6 shows estimated optical loss by scattering and absorption for LIGO
Voyager. Assuming the uniform molecular injection onto the mirror surface,
the impact of scattering can be negligible as long as the thickness of the CML
is below 1 µm in the same manner as the case of KAGRA and ET. On theother hand, the loss due to the absorption is much larger than the scatteringand can be harmful for cryogenic operation.
57

3 Theoretical Study of
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3.5 Performance of GWD

10−2

Einstein Telescope

10−4

Optical loss

10−6
10−8

10−10
10−12
10−14

absorption
scattering ξ = σ⟨t ⟨1
scattering ξ = σ⟨t ⟨10
scattering ξ = σ⟨t ⟨100

10−16
10−181

10

100

Thickness [nm]1000

Figure 3.5: Scattering and absorption loss for the case of ET. The assumed
wavelength is λ = 1550 nm. Red solid line, yellow dashed line, green dashdot
line and blue dotted line represent the absorption and scattering with thecorrelation length of ξ = σhti , σhti /10 and σhti /100, respectively.

3.5

Performance of GWD

Optical loss induced by CMLs can reduce the optical performance of GWDs
such as PRG. In the same manner as the previous study [16], here we estimate
the impact on of CML PRG for future GWDs using Eq 3.15. More detailed
calculations can be found in the references [58, 59]. We will derive the tolerable
CML thickness to keep the degradation of PRG up less than 10%.
First, we revisit the PRG of KAGRA which is estimated in the previous
study [58]. In the same manner as the previous study, we assume that thestart time and the formation speed of CMLs are the same for both ITM and
ETM. Also, we take changes in the refectivities of the test masses due tothe CML formation into account as the previous study did [58]. Fig. 3.7
shows how the PRG of KAGRA changes with the CML thickness. Here weassumed that the refractive index of CML as nCML = 1.26 and the absorption
coefficient as αCML = 2.2 × 10−2 ppm/nm. The oscillation is induced by thechange in reflectivities of test masses due to the CML formation. Especially,
the reflctivities of arm cavities oscillate, which leads to the oscillation in the
PRG of KAGRA. As the CML becomes thicker, the PRG decreases due to theoptical loss of CML. The tolerable CML thickness can be as thick as 750 nm.
58

3 Theoretical Study of
Cryogenic Molecular Layer

3.5 Performance of GWD

LIGO Voyager

10−2
10−4

Op ical loss

10−6
10−8

10−10
10−12
10−14

absorp ion
sca ering ξ = σ⟨ ⟩⟨1
sca ering ξ = σ⟨ ⟩⟨10
sca ering ξ = σ⟨ ⟩⟨100
100
1000
t

10−16
10−181tt

10Thickness [nm]Figure 3.6: Scattering and absorption loss for the case of LIGO Voyager. The
assumed wavelength is λ = 2000 nm. In the same manner as the case of ET, redsolid line, yellow dashed line, green dashdot line and blue dotted line represent
the absorption and scattering with the correlation length of ξ = σhti , σhti /10
and σhti /100, respectively.

Fig. 3.8 shows how the PRGs of the ET and LIGO Voyager change due tothe CML formation, respectively. We assumed the refractive indices of CMLsof the ET and LIGO Voyager as nCML,ET = 1.19 and nCML,Voyager = 1.28, respectively. Also the assumed absorption coefficients are listed in Table 3.2. For
the case of the ET and LIGO Voyager, PRGs decrease monotonically without
oscillation. This is because the optical loss of CML is much larger than that of
KAGRA and PRGs deteriorate before starting oscillation. The tolerable CML
thickness is much smaller than that of KAGRA. For the case of the ET, thetolerable CML thickness is about 60 nm. Moreover, the tolerance of LIGO Voyager is thinner than the ET and it is about 25 nm. These values are about oneorder of magnitude smaller compared to 750 nm which is the tolerable CML
thickness of KAGRA. Therefore, the performance of future cryogenic GWDs
can be easily deteriorated due to the CML formation compared to KAGRA.
59

3 Theoretical Study of
Cryogenic Molecular Layer

3.6 Implications to Heat Input

Figure 3.7: Power recycling gain of KAGRA against the thickness of CML.

3.6

Implications to Heat Input

Not only the degradation of PRG, but also the heat absorption by CMLs canprevent a cryogenic operation due to the additional heat load to test masses.
For the case of KAGRA, as the intra-cavity power becomes about 370 kW,
absorption at ppm level exceeds the capacity of heat extraction and leads
to increase in the test mass temperature. In other words, a few tens of nm
thickness CML can increase the temperature of test mass. Here we focus ona thin thickness range to estimate the heat absorption in which the powerreflectance of the CML, finesse of arm cavity and power recycling gain canbe considered to be constant. Therefore, the absorbed power by the CML is
proportional to the thickness of CML, ACML ∝ αCML dCML .
Fig. 3.9 shows the heat absorption of the input test mass (ITM) and end test
mass (ETM) of KAGRA. In this calculation, the absorption in the substrate
and radiation from the beam ducts are taken into account. Once the thickness
becomes larger than about 10 nm, the heat input to ITM exceeds the tolerable
value, 0.72 W, and the temperature of test mass cannot reach the target value.
Therefore, the thickness of CML should be kept less than about 10 nm in orderto maintain the temperature of test mass. For ETM, the heat input is smaller
as long as the thickness of CML is smaller than about 30 nm.
60

3 Theoretical Study of
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3.6 Implications to Heat Input

Figure 3.8: Power recycling gain of the ET against the thickness of CML.

Fig. 3.10 shows the heat absorption of the ITM and ETM of ET. Oncethe thickness becomes larger than a few nm, the heat input to the test masses
exceeds the tolerable value, 100 mW, and the temperature of test mass cannot
reach the target value. Therefore, the thickness of CML should be kept lessthan 1 nm in order to maintain the temperature of the test masses.
Fig. 3.11 shows the heat absorption of the ITM and ETM of LIGO Voyager.
The heat load generated by the optical absorption is extremely large as theinjected laser power is much larger than that of ET. Even if the thickness
is less than 1 nm, the heat load is still more than 10 W for both the ITM
and ETM cases. This indicates that the test mass cannot be cooled down to
123 K. Thus, not only the optical performance degradation, but also the heatabsorption due to the CML may become a critical problem.
It should be noted that the heat input to a test mass becomes larger even ata thickness which can be tolerable from the view point of optical performance.
For the case of the ET and LIGO Voyager, about 4 W and 1.7 kW heat should
be extracted from test masses to keep the degradation of PRG less than 10%,respectively as shown in Fig. 3.12. However, it is practically almost impossible
to achieve such large cooling capacity by each cooling method. In order to extract 4 W heat from the 10 K mirror by conductive cooling, the mirror needs tobe connected close to cold head of cryocoolers with much thicker fibers, which
can increase the suspension thermal noise and vibration contamination. Also,
maximum extractable heat from the 123 K mirror by radiation is estimated as
Prad = σT 4 S ≈ 14 W,(3.25)

where σ = 5.67×10−8 W/m2 /K4 is the Stefan-Boltzmann constant, T = 123 K
61

3 Theoretical Study of
Cryogenic Molecular Layer

3.6 Implications to Heat Input

1.4
1.2

Heat input [W]

1.0

KAGRA
ITM
ETM
CML
ITM substrate
Radiation

0.8
0.6
0.4
0.2
0.00

10

20

30

CML thickness [nm]4050Figure 3.9: Heat input for the case of KAGRA. Red solid line and black solidline represent the heat input of the input test mass (ITM) and the end test
mass (ETM), respectively. Yellow dashed line shows the contribution by the
CML. Blue dotted line and green dashdot line are absorption by the ITM
substrate and radiation from the beam duct, respectively.

62

3 Theoretical Study of
Cryogenic Molecular Layer

3.6 Implications to Heat Input

Einstein Telescope

0.5

ITM
ETM
ITM substrate

Heat input [W]

0.4
0.3
0.2
0.1
0.00

1

2

3

CML thickness [nm]45Figure 3.10: Heat input to each test mass mirror in the ET induced by the
optical absorption of CML. As a result of strong absorption of amorphous ice,
the heat load to test mass exceeds 100 mW even when the CML thickness isonly a few nm. It should be noted that the radiation from the beam ducts is
not taken into account for the case of ET.

63

3 Theoretical Study of
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3.7 Summary of this chapter

80
70

ITM
ETM
ITM substrate

Heat input [W]60

LIGO Voyager

50
40
30
20
10
0
0.0

0.2

0.4

0.6

CML thickness [nm]

0.8

1.0

Figure 3.11: Heat input to each test mass mirror in LIGO Voyager induced by
the optical absorption of CML. It should be noted that the radiation from thebeam ducts is not taken into account for the case of LIGO Voyager.

is the temperature of the mirror, and S is the surface area of the mirror. Here
we assumed the radius of the mirror to be 22.5 cm and the thickness of themirror to be 55 cm [35].

3.7

Summary of this chapter• Cryogenic mirrors in GWDs can introduce the CML on their surfaces
due to the adsorption of residual gas molecules.
• The CML can become a problem not only for KAGRA, but also for the
future GWDs such as the ET or LIGO Voyager. However, its impacts
on the future GWDs have not been studied nor taken into account.
• The CML induces additional optical losses which can degrade the detector’s performance and hence less number of detection of GW events.
• As the ET and LIGO Voyager are planning to employ longer wavelength
lasers, the optical absorption by CML can become crucial due to thelarge absorption coefficient of amorphous ice.
64

3 Theoretical Study of
Cryogenic Molecular Layer

3.7 Summary of this chapter

Figure 3.12: Heat input to test masses for the case of the ET and LIGO
Voyager.

• We have estimated the tolerable CML thickness from the view point of
optical performance of GWDs. The tolerable thickness of future GWD
is 2 order of magnitudes smaller than that of KAGRA.
• The heat input induced by CMLs can exceed the cooling capacity of
the ET and LIGO Voyager due to the large absorption coefficient ofamorphous ice.

65

3 Theoretical Study of
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3.7 Summary of this chapter

66

Chapter 4
Development of Cryogenic
Folded Cavity
The optical loss induced by the CML can spoil the sensitivity or even operationof cryoegenic GWDs, hence, the detection of GWs can be hindered. In orderto mitigate or solve this problem, study about the CML is necessary. Sofar, there has not been any studies about the impact of CML on GWDs atlonger wavelength than 1064 nm though the future GWDs will presumably
employ longer wavelength laser. For the case of the ET, 1550 nm wavelength
laser will be used for lower-frequency part with 10 K test mass mirrors. In
order to explore the feasibility of future GWDs, i.e., longer wavelength laserinterferometric GWDs, the experiment conducted at the same wavelength isindispensable.
In this chapter, we explain about the setup of the measurement system
which we have developed. First, we revisit the features of an optical cavity.
Then, we describe the experimental setup including the folded cavity. In addition, the cryogenic system is a unique feature in this experiment and it takes
important roles to develop future cryogenic gravitational-wave detectors. The
cryogenic system is described briefly in the following subsection.

4.1

Optical Cavity

Optical cavities are widely used in precision measurements such as gravitationalwave detection or optical lattice clock [20, 75]. The optical cavity is a device
formed by two or more partial reflectors to store or reject photons in the cavity. In particular, those formed by two mirrors facing each other is known
as the Fabry-Perot (FP) cavity. In this section, we describe the fundamental
properties of the optical cavity [76].
674.1 Optical Cavity

Ein

4 Development of Cryogenic Folded Cavity

r1, t1

r2, t2

Ecav

Et

Er
L
Figure 4.1: Schematic picture of FP cavity.

4.1.1

Static Response

Assuming that the cavity is under a static condition where there are no rapid
variations neither in the fields nor the cavity length, a set of the fields associated with the cavity can be written as
Ecav = t1 Ein + r1 r2 Ecav e−2ikL ,t1
Er = −r1 Ein + (Ecav − t1 Ein ) ,r1
Et = t2 Ecav ,

(4.1)
(4.2)
(4.3)

where Ein , Ecav , Er , and Et are incident, intra-cavity, reflected, and transmitted fields, respectively. rj and tj are the amplitude refrectivity and transmissivity of j-th mirrors where j = 1, 2, and k and L are the wave number
and cavity length, respectively. From Eq. 4.1, the intra-cavity field can be
expressed as
Ecav =t1Ein .
1 − r1 r2 e−2ikL(4.4)

Thus the intra-cavity power Pcav can be expressed as
Pcav ≡ |Ecav |2 =t21
2
2 |Ein | .
−2ikL|1 − r1 r2 e
|
68

(4.5)

4 Development of Cryogenic Folded Cavity

4.1 Optical Cavity

This represents the amount of the laser power stored inside the cavity. Fig. 4.2
shows the intra-cavity power as a function of the round trip phase which isdefined as
φrtp ≡ 2kL =

4πL
,
λ

(4.6)

where λ is the laser wavelength. The FP cavity is on resonance when the roundtrip phase satisfies the following condition,
φrtp = 2πn,(4.7)

where n is an arbitrary integer. When the cavity is on resonance, the intracavity power becomes the maximum. The resonant condition can also be
expressed as,
2πn =

4πνL
,
c

(4.8)

where n is an arbitrary integer, and ν, L and c are the laser frequency, thecavity length and the speed of light, respectively. Taking the derivation ofboth sides in the equation above, one can find
∆L
∆ν
=−
.
ν
L

(4.9)

Thus, the length variations, ∆L, are linearly imprinted onto the laser frequency
fluctuations, ∆ν, as long as the cavity is on resonance through the frequency
locked loop.

4.1.2

Characteristic Quantities

In this subsection, we briefly introduce a number of important quantities whichcharacterize an optical cavity.
Free Spectral Range
Free Spectral Range (FSR) is a measure of the separation between the neighboring resonances in terms of the laser frequency which is defined by
fFSR =
69c.
2L(4.10)

4.1 Optical Cavity

4 Development of Cryogenic Folded Cavity

FSR

FWHM

Figure 4.2: The intra-cavity power.

Finesse and Linewidth
The finesse F denotes the sharpness of the resonances which can be a meritof optical cavities. It can be approximated as√
π r1 r2
fFSR=.
(4.11)F ≡fFWHM
1 − r1 r2Once the fincesse is determined, one can compute the corresponding linewidthin terms of the full width at half maximum (FWHM) by
fFWHM =

fFSR.
F

(4.12)

Here fFWHM represents the linewidth with respect to the laser frequency.
Storage Time and Cavity PoleThe storage time τs denotes the mean duration for a photon to stay in thecavity. It is expressed as
τs =

2L
1
2L
≈
F.
c ln(r1 r2 )πc70(4.13)

4 Development of Cryogenic Folded Cavity

Laser

4.2 Locking Scheme

Pick-off
MirrorEOM

FP cavityωm
Function
Generator
Carrier
Sidebands

Error Signal
Mixer

Low-pass
FilterFigure 4.3: Schematic picture of the PDH locking. The beat note between thepromptly reflected sidebands and the leakage of carrier is detected by RFPD.
The obtained signal is demodulated by using a mixer and then the error signal
can be extracted by the low-pass filter.

The last approximation is valid as long as r1 r2 ∼ 1.
The cavity pole fc represents the frequency above of which an induced
variation in the intra-cavity field cancels. The cavity pole essentially originatesfrom the fact that a cavity has a finite storage time. It is related with thestorage time as the following form,
fc =

4.2

1
.
2πτs

(4.14)

Locking Scheme

In order to maintain the resonant condition, the active feedback control is
indispensable to stabilize the lase frequency of cavity length. Here we stabilize
the laser frequency to keep resonant condition by using the Pound-Drever-Hall(PDH) locking scheme [77]. In this subsection, the principle of the (PDH)
scheme is described.
For the PDH locking, an electro-optic modulator (EOM) is usually employed to modulate the phase. The phase modulated input laser field can be
expressed as,
Ein = E0 exp [i(ω0 t + Γ sin ωm t)] ,71(4.15)

4.3 Folded Cavity

4 Development of Cryogenic Folded Cavity

where Γ is the modulation depth which represents the amount of the phasemodulation. By using the Jacobi-Angar expansion, one can obtain
Ein = E0 eiω0 t∞
X

Jk (Γ)eikωm t ,(4.16)k=−∞where Jk (Γ) is the Bessel function of the first kind. One can see that amonochromatic phase modulation split the injected field into multiple components. The field with k = 0 is called a carrier, and the other fields k 6= 0 are
called sidebands.
Assuming the modulation depth is small such that one can ignore the higher
order of sidebands, the incident field is expressed as a sum of three fields,
Ein = E0 J0 (Γ) eiω0 t + J1 (Γ) ei(ω0 +ωm )t − J1 (Γ) ei(ω0 −ωm )t .

(4.17)

Here we use the fact that −J1 = J−1 . When the incident field is coupled tothe optical cavity, the reflected field can be expressed as
Er = E0 eiω0 t J0 ((Γ)r(0) + J1 (Γ)r(ωm )eiωm t − J1 (Γ)r(−ωm )e−iωm t ,(4.18)

where r(ωm ) represents the amplitude reflectivity of the cavity at angular frequency of ω0 + ω. Then the reflected beam power is calculated as2
Pr (t) = r (δω) J0 (Γ) + J1 (Γ) r+ (δω) eiωm t − r− (δω) e−iωm t Pin ,(4.19)

where Pin is the input beam power. Here we define a reflectivity r± (δω) asr± (δω) ≡ r(δω ± ωm ),(4.20)

where δω is an angular frequency difference between the carrier and the resonant frequency. An error signal can be extracted by demodulating the obtained
signal as
4J0 (Γ) J1 (Γ) F

=
δω.
Pin
πνFWHM(4.21)

Thus one can derive the error signal which is proportional to δω by using the
PDH scheme. Fig. 4.3 shows the schematic picture of the PDH locking.
Fig. 4.4 shows the PDH signal. The linear range of a typical PDH signal
is narrow as indicated in the figure. In order to keep the resonance condition,
the feedback control is necessary.
72

4 Development of Cryogenic Folded Cavity

1.00

4.3 Folded Cavity

PDH signal

PDH siganl [a.u.]

0.75
0.50
0.25
0.00

−0.25
−0.50
−0.75
−1.00

−2π

π

0

Phase [rad]π2πFigure 4.4: The typical PDH signal.

4.3

Folded Cavity

A folded cavity, consisting of input and output mirrors and one folding mirror,
is employed in this experiment. Fig. 4.5 shows the schematic of the folding cavity. Though a typical FP cavity involved measurement sets the upper
limit on the reflectivity of mirror, a folded setup enables us to measure theperformance of high reflective coatings, resulting in tiny transmitted beam.
Furthermore, one can conduct measurement to various coating materials justby changing only one folding mirror, which reduces the cost for the development of coatings. Therefore, the folded cavity is a notable method to measure
and compare coating performance, and develop high-end coatings.
In our measurement system, the folded cavity is installed inside a cryostat
chamber in order to cool down the cavity. Details of the set up is describe in
the following sections.
73

4.3 Folded Cavity

4 Development of Cryogenic Folded Cavity

Figure 4.5: Schematic picture of the folded cavity. The folded cavity consistsof three mirrors — input and output mirrors, and one apex mirror which foldthe beam axis.

4.3.1

Static ResponseFirst, we describe the static response of the folded cavity.
Ecav = t1 Ein + r1 r2 r2 Ecav e−2ik(L1 +L2 ) ,t1
Er = −r1 Ein + (Ecav − t1 Ein ) ,r1
Et = t2 Ecav ,(4.22)(4.23)(4.24)

where ri2 = Ri (t2i = Ti ) (i = 1, 2) are the reflectivity (or transmittivity)of the input and output mirrors, respectively, and r2 = R and t2 = T are
the reflectivity and transmittivity of the folding mirror. From Eq. 4.22, the74

4 Development of Cryogenic Folded Cavity

4.3 Folded Cavity

intra-cavity field can be expressed as
Ecav =t1Ein .
1 − r1 r2 e−2ikL(4.25)

Hence, the intra-cavity power of the folded cavity is expressed as
Pcav =

t21

2 Pin .

(4.26)

2 Pin(4.27)|1 − r1 r2 r2 e−2ik(L1 +L2 ) |Then, transmitted power can be written as
Ptrans =t21|1 − r1 r2 r2 e−2ik(L1 +L2 ) |With the active control loops, they satisfy the resonance condition,
2πn =4πν(L1 + L2 ),c

(4.28)

where n is an arbitrary integer, and ν, L and c are the laser frequency, thecavity length and the speed of light, respectively.
In the same manner as the FP cavity, one can define the free spectral range(FSR) and finesse asc,2(L1 + L2 )√π r1 r2F =.
1 − r1 r2 r2

fFSR =

4.3.2(4.29)(4.30)

Input and Output Mirrors

Fused silica mirrors are used for the input and output mirrors. Both of them arehalf inch diameter size, and the radius of curvature is 50 mm. The reflectanceof these fused silica mirrors was chosen to R = 0.9998 in order to set the cavity
finesse as F ≈ 15000. These mirrors are attached to the mirror holders bygluing.
Initially, we planned to install silicon mirrors for both the input and output
mirrors. However, the silicon mirrors we planned to use turned out not tosatisfy the requirement, and was almost impossible to compose an opticalcavity due to the large optical loss. This drove us to make new mirrors, but
due to the limited budget we decided to use fused silica.
75

4.3 Folded Cavity

4 Development of Cryogenic Folded Cavity

Figure 4.6: Picture of fused silica and silicon mirrors.

4.3.3

Folding Mirror

The geometry of the cavity is folded by a mirror, which is located at theapex of the cavity spacer. The size of the folding mirror is also half inch indiameter and its substrate is made of silicon which is the candidate material for
future GWDs. The temperature sensor, DT-670, manufactured by Lake Shore
Cryotronics Inc. is attached on the mirror holder of the folding mirror by using
the indium sheet. We regard this measured temperature as the temperature of
the folding mirror. The sensor has high stability for cryogenic operation [78].
Typical SiO2 /Ta2 O5 coating, which are also employed in KAGRA test masses,
is applied on the surface. The total pairs of coating layers are N = 20 in orderto achieve the high reflectivity at 1550 nm.

4.3.4

Geometrical Parameters

Here we introduce geometrical parameters of the folded cavity. Fused silica
mirrors with the radius of curvature of R = 50 mm are employed as the input
and output mirrors. The cavity length L = L1 + L2 is chosen as L1 = L2 =41 mm, hence 82 mm. Some parameters which characterize the folded cavity
76

4 Development of Cryogenic Folded Cavity

4.4 Optical Layout

Table 4.1: The parameters of the folded cavity. The reflectivities of mirrors
are the value from their spec sheets.
Parameters
Value
Designed finesse∼ 1.5 × 104
Cavity length
L1 = 41 mm
L2 = 41 mm
L = L1 + L2 = 82 mm
Free spectral range
1.83 GHz
Beam size on the input/output mirror
230 µmBeam size on the folding mirror
97 µm
Radius of curvature
50 mm
R1 = r12 , r22
0.9998 ± 0.0001∼ 0.00017
T1 = t21 , t22
R = r20.999998
Folding angle
15 degcan be calculated as bellow[76].
v s
u
u Lλ
1+g
w0 = t,π4(1 − g)c,fFSR =
2L
c arccos(g 2 )fTMS =,πL
π(4.31)(4.32)(4.33)

where g = 1 − L/R is a g-factor of the cavity and λ = 1550 nm is wavelengthof the laser beam. When the cavity length L = 82mm, the optical parameters
are calculated as below.
w0 = 97 µm,
fFSR = 1.83 GHz,
fTMS = 1.34 GHz.

(4.34)(4.35)(4.36)

Here fTMS denotes the frequency separation between two neighbouring spatial
modes. The beam spot size on the mirror becomes wm = 230 µm. Relatedparameters are listed in Tab. 4.1.
77

4.4 Optical Layout

4 Development of Cryogenic Folded Cavity

Figure 4.7: The overview of the experimental setup. The AOM is used forinterrupting the laser injection to the cavity, and 1st diffracted beam is used
for injection. The reflected beam for PDH locking is extracted by the Faraday
isolator. The laser is locked to the cavity by feeding back the error signal to
the laser through its current tuning. The output from the folded cavity ispicked off by the mirror and detected by a PD.

4.4

Optical Layout

Fig. 4.7 shows the optical layout. The laser is phase-modulated by an EOM
and the reflcted beam is picked-off by a Faraday-isolator which is used for PDH
locking. The obtained error signal is filtered by a servo, and then fed back to
the laser frequency through current tuning.
An AOM is used as a switch to turn off the injection of the laser beam into
the cavity. Here, the first-order diffracted beam is injected to the cavity. By
interrupting the AOM drive, the cavity input light could be extinguished ona timescale of ∼ 10 ns which is much smaller than the timescale of the cavity
response. The PD also has enough fast response to detect the transition of
the transmitted beam. Therefore, their influences are enough small for the
ringdown measurement.

4.5

SpacerThree mirrors which form the folded cavity are attached to the spacer madeof an invar. The invar IC-DX employed here can be used even at 4 K cryo78

4 Development of Cryogenic Folded Cavity4.6 Vacuum and Cryogenic System

Figure 4.8: 3D CAD of exploded view of the folded cavity. The color of eachcomponent is displayed just for distinguishing the parts. The input and output
mirror is glued on the mirror holders

genic temperature where typical invars show martensitic transformation, andis suited to this application.
The main purpose to use the spacer is to mitigate the misalignment causedby cooling process. Usual mirror holders show temperature drift in their alignment which can ruin the axis of optical cavity. Especially, the folded cavity
used in this measurement is near concentric, and require strict alignment.
Therefore, one have to reduce the thermal drift of three mirrors as much as
possible. The invar used here has small enough thermal expansion coefficient
down to 4 K to maintain the initial alignment, and this is why we employed theinvar-made spacer. Also the use of the invar has additional advantages that
one can save the cost to develop the optical cavity compared to the silicon or
fused silica substrate spacer, and that one can replace the folding mirror mucheasier compared to the optical contact. These features drive the developmentof the coatings efficiently.
Fig. 4.8 shows the assembled folded cavity.

4.6

Vacuum and Cryogenic System

A cryostat is used for housing the cavity. As shown in Fig. 4.9, the cryostat
chamber has three shields — 300 K shield, 80 K shield, and 4 K shield. Twoof them except for the outermost shield (300 K shield) are covered by super
insulator (SI) for efficient shielding of thermal radiations.
79

4.7 Laser Source

4 Development of Cryogenic Folded Cavity

Figure 4.9: Schematic of cross section of cryostat. Two shields are installedinside the cryostat chamber at temperature of 80 K and 4 K, respectively. The
chamber is pumped down by using a scroll pump and turbo pump. The beam
is injected through the view port where AR windows are attached.

The cryostat has two types of pumps for vacuum. One is a scroll pumpwhich is used for initial pumping down to about 10 Pa. The other is a turbo
pump which achieves high vacuum environment. The vacuum pressure in thechamber can reach 6 × 10−4 Pa by using the turbo pump at room temperature.
The viewport window has AR coating at the wavelength of 1550 nm inorder to reduce the power loss. The power loss at this viewport is measured
to be ∼ 0.1%.
A pulse tube cryo-cooler is used for cooling down the cavity. A cooling bar
is connected to the table in the chamber, and it cool down the system. The
folded cavity is installed on the table in the cryostat chamber with aluminum
pedestal which extracts heat from the cavity.
The pressure inside the chamber decreases due to the cryopumping effect.
After reaching about 8 K, the pressure of room temperature part, i.e., between
the room temperature and the 80 K shields, becomes ∼ 2 × 10−5 Pa.
80

4 Development of Cryogenic Folded Cavity

4.7 Laser Source

Table 4.2: The parameters of the laser source.
Parameters
Value
Wavelength 1550 nm
Power
20 mW
linewidth≤ 5 kHz

Figure 4.10: Picture of the laser source. A butterfly-packaged PW-ECL, current drivers, a frequency tuning terminal and a digital control interface arehoused in the module.

4.7

Laser Source

We employed here a commercial laser module (named ORION), built by RIO.
The module houses a butterfly-packaged PW-ECL, current drivers, a frequency
tuning terminal and a digital control interface. This laser has low frequency
noise, intensity noise and high stability for precision measurements with low
cost [79]. Basic specifications of the laser are listed in Table 4.2, and Fig. 4.10
shows the appearance of the laser source.
814.8 Finesse Measurement Methods4 Development of Cryogenic Folded Cavity

Input

1μ0

Normalized power [W]

Trans

0μ8

0μ6

0μ4

0μ2

0μ0

-5

0

5
Time [

10s]

Figure 4.11: A transient response of an optical cavity. We assumed F ≈1.5×104 with cavity length of L = 10 cm. Black dotted line represents the input
beam power which is interrupted at t = 0. The blue line is the transmitted
beam power which decays exponentially.

4.8

Finesse Measurement Methods

In this section, we describe the ringdown method to measure the finesse of thecavity. In this method, we employ the AOM to turn off the input laser. By
interrupting the driver to the AOM, one can cut the input beam and cause thepower stored in the cavity to decay, resulting in transient of the transmitted
beam as shown in Fig. 4.11. By measuring the decay time of transmitted
beam, one can calculate the finesse of the cavity.
The incident field with intensity modulation is expressed as
Ein = E0 eiω0 t (1 + ξeiωm t + ξe−iωm t ),(4.37)

where ξ and ωm are the modulation depth and modulation angular frequency,
respectively.
82

4 Development of Cryogenic Folded Cavity4.8 Finesse Measurement Methods
We consider the response of the folded optical cavity, assuming that thelaser with ω0 + ωm angular frequency is injected to the cavity. One can define
the transfer function of the cavity as
C≡

Et
.
Ein

(4.38)

This can be expressed as
t21 e−iLωm /c
C=.
1 − r12 r2 e−2iLωm /c(4.39)

Here we assumed that the cavity is kept on resonance to the carrier, which has
the angular frequency of ω0 . Then the gain of the transfer function becomesg,|C| = p
1 + (τs ωm )2(4.40)

where g = t21 /(1 − r12 r2 ) is the gain factor of the cavity. Eq. 4.40 represents
the response of the optical cavity to the intensity modulation, and it works as
the 1st order low-pass filter with time constant of τs . From Eq. 4.40, one canderive the step response of the optical cavity as


a0
−1 1
L
= a0 [1 − exp(−2t/τs )].
(4.41)
s 1 + τs s
We consider the transient response of the transmitted beam to the modulated input beam, especially for the case of a step function. The input field isexpressed as(
E0 eiω0 t t < 0,
Ein (t) =(4.42)(E0 − ξ)eiω0 t 0 ≤ t.
The transmitted field becomes(
gE0 eiω0 t t < 0,
Et (t) =g {E0 − ξ[1 − exp(−t/τs )]} eiω0 t

0 ≤ t,(4.43)

Thus the transient of transmitted laser power can be expressed as
Pt = GPin exp(−2t/τs ).

(4.44)

Therefore, the transmitted beam decays with time as expressed as Eq. 4.44
when the input beam is interrupted by the AOM.
83

4.9 Cavity Enhanced Ellipsometry(P-S Splitting)
4 Development of Cryogenic Folded Cavity

4.9

Cavity Enhanced Ellipsometry(P-S Splitting)

As the impinging beam on the folding mirror has a given incident angle,
there is a difference in resonance frequency between the P-polarization andS-polarization. We call such a frequency difference a P-S splitting in this thesis. This frequency shift carries information of coatings on the folding mirror
and can characterize the coating properties.
In this section, we describe the principle of the P-S splitting measurement.
This measurement can be regarded as a type of cavity enhanced ellipsometry.

4.9.1

Characteristic Matrix

A characteristic matrix is a useful tool to calculate the reflectance or transmittance of a multilayer coating thin films [80]. Here we consider the case suchthat the total number of layer is N . For the case of i-th layer’s material witha complex refractive index Ni and mechanical thickness di , the characteristic
matrix is expressed as
cos δi i sin δi /ηiMi =,(4.45)
iηi sin δicos δiwhere δi = 2πNi di /λ is an optical phase shift induced in i-th layer, and ηi isthe effective refractive index of i-th layer which has polarization dependence
described as
ηi,S = Ni cos θi ,Ni.
ηi,P =cos θi(4.46)(4.47)

Here cos θi is expressed as
1/2
cos θi = 1 − (n0 /ni )2 sin2 θ0,(4.48)

where θ0 is the angle of incidence. The total matrix Mtot and its elements mij
are defined as
Mcoating = M1 M2 · · · MNNYm11 im12=
Mi =,im21 m22i=184(4.49)

4.9 Cavity Enhanced Ellipsometry
4 Development of Cryogenic Folded Cavity(P-S Splitting)where mij are the elements of the matrix. Two coefficients B and C are
calculated as
  
 
B
m11 im12
1=,(4.50)
C
im21 m22
ηmwhere ηm is the effective refractive index of the substrate material written as
ηm,S = Nm cos θi ,Nm
ηm,P =,cos θi(4.51)(4.52)

where Nm is the refractive index of the substrate material. Eq. 4.50 gives theFresnel coefficient of the coatings as
ρ=

n0 B − C,
n0 B + C(4.53)

where n0 is the refractive index of vacuum. Thus, one can calculate the reflectance of the coatings by using R = |ρ|2 .
Not only the reflectance, but also the phase shift induced by the coatings
can be calculated from the characteristic matrix. The phase of the reflected
light can be computed by22η0 (ηm
m12 m22 − m11 m21 )∆φS,P = arctan,(4.54)
2 m2 + η 2 η 2 m2 − m2
η02 m211 − ηm
22
0 m 12
21 S,P
where subscript S and P represent the polarization. As the phase is slightly
different between the two polarization, we can define the phase difference as∆φ ≡ ∆φS − ∆φP .

(4.55)

When the angle of incidence is 0 deg, η is same for both polarization. Therefore, the phase difference does not exist as long as the beam is injected vertically.
When a CML is formed on a mirror surface, the characteristic matrix ofthe mirror is modified as
Mmod = MCML Mcoating ,(4.56)

where MCML is the characteristic matrix of CML. The elements of MCML aredefined in the same manner as Eq. 4.45. CML formation on a mirror surfacechanges the characteristic matrix, hence the phase difference ∆φ.
85

4.9 Cavity Enhanced Ellipsometry(P-S Splitting)
4 Development of Cryogenic Folded Cavity

Figure 4.12: Equivalent block diagram of folded cavity to explain the phaseshift between P- and S-polarization. Each plant represents the propagator,transmission or reflection matrix.

4.9.2

Frequency Shift

We derive the frequency shift induced by the phase shift in the folding mirror.
Here, we calculate the field using an equivalent block diagram shown in Fig.
4.12 [81, 82].
First, we define the vector which denotes the field as


E
j,S
~j =
,
(4.57)
E
Ej,P
where j represents the location of the fields, in, cav, or trans. Propagator,transmission, and reflection matrices are defined as −ikLi
0eP̂Li =,(4.58)
0e−ikLi


ti 0
t̂i =
,
(4.59)
0 ti


ri 0
r̂i =
,
(4.60)
0 ri


r
0
r̂ =
,
(4.61)
0 re−i∆φwhere i = 1, 2. It should be noted that the phase shift is introduced in thereflection matrix of folding mirror as e−i∆φ .
From Fig. 4.12, the intra-cavity field is written as−1~ cav = t1 I − r2 r2 P̂rtp~ in ,
EE(4.62)
186

4.9 Cavity Enhanced Ellipsometry
4 Development of Cryogenic Folded Cavity(P-S Splitting)where I is the identity matrix and P̂rtp represents the round trip propagator.
Solving Eq. 4.62, one can obtain~ cav = t1 diag 1 − r2 r2 e−iΦrtp , 1 − r2 r2 e−i(Φrtp +2∆φ) E~ in ,
E(4.63)
1
1~ cav , the transmitted field canwhere Φrtp is the round trip phase shift. Using E
be expressed as~ trans = t̂2 P̂L2 r̂P̂L1 E~ cav ,
E
= t2 rdiag e−iΦrtp /2(4.64)−i(Φrtp /2+∆φ),e~ cav .
E

(4.65)The matrices introduced here are diagonalized, and one can obtain the field bysimple math. Computing each element, the transmitted field can be writtenast21 re−iΦrtp /2
Ein,S ,
1 − r12 r2 e−iΦrtp
t21 re−i(Φrtp /2+∆φ)
Ein,P .
Etrans,S =1 − r12 r2 e−i(Φrtp +2∆φ)
Etrans,S =(4.66)(4.67)

Therefore, each transmitted power normalized by the incident power is expressed as
Ptrans,S
T12 R

,
=
Pin,S
(1 − R R)2 + 4R R sin2 Φrtp

(4.68)

T12 R
Ptrans,P,=
2 Φrtp +2∆φPin,P2
(1 − R R) + 4R R sin(4.69)

1

1

1

1

2

2

These equations show that there exists a difference in resonance frequency
between S- and P-polarization.
We derive the amount of resonant frequency shift induced by ∆φ phase shift
in folding mirror. The round trip phase shift of S-polarized beam becomes
φrtp =

4πνL.
c(4.70)As shown in above, the resonant peak of P-polarization beam is shifted by−2∆φ compared to that of S-polarization. We define the resonant frequency
split between P- and S-polarization as ∆ν. Then the relation between theround trip phase shift and the resonant frequency split can be written as
φrtp − 2∆φ =4π(ν + ∆ν)L.
c

87

(4.71)

4.9 Cavity Enhanced Ellipsometry(P-S Splitting)
4 Development of Cryogenic Folded Cavity

Transmitted Po er [a.u.]

1.0
0.8

S-pol
P-pol

0.6
0.4
0.2
0.0
−10

−8

−6
−4
−2
Frequency [MHz]02Figure 4.13: Example of the resonant frequency spit between S-polarization(solid line) and P-polarization (dashed line) . Here we assumed ∆φ = 10 mrad
phase shift, F ∼ 1.5 × 104 , and L = 82 mm case.

From Eq. (4.70), one can arrive at∆ν = −

fFSRc2∆φ = −
∆φ.
4πLπ(4.72)

Thus, the resonant frequency shift is induced by the folding mirror. The
example is shown in Fig. 4.13.
As the CML formation changes the elements of characteristic matrix of themirror, it also changes ∆φ. Therefore, the resonant frequency split ∆ν alsochanges as the CML is formed on the mirror surface. By measuring the change
the in resonant frequency split, one can compute the CML thickness.
The amount of resonant frequency split ∆ν is determined by the FSR fFSR
and the phase shift ∆φ. The phase shift ∆φ depends on the refractive index
NCML and the angle of incidence to the folding mirror θ0 . In our setup, thefolding angle γ corresponds to the angle of incidence. Thus, the resonant frequency split depends on the FSR, folding angle of the cavity and the refractive
index of CML. It should be noted that the contribution of imaginary part ofNCML to the phase shift is extremely small. Assuming that the folding angle
88

4 Development of Cryogenic Folded Cavity

4.10 Summary of this chapter

γ = 22.5 deg and NCML = 1.2 − 104 i with 1 nm CML thickness, the imaginary part of NCML generates only 10−8 % difference in the frequency split ∆νfrom the case of NCML = 1.2. Therefore, we ignore the contribution from theimaginary part of CML refractive index in the following discussion.

4.9.3

Errors in CML Thickness Measurement

In this subsection, we introduce the errors which involves in CML thickness
estimation. Firstly, resonant width of the folded cavity affects measurement
accuracy of the frequency difference between the two-peaks. Here we define
the resonant width as the FWHM of the cavity
fFSR.
(4.73)
fFWHM =
F
In our setup, the FWHM is calculated as
1.84 GHz= 0.09 MHz.
(4.74)
fFWHM =
2.0 × 104
In reality, the accuracy of determining the peak frequency can be smaller thanthe FWHM. However, as an upper bound, we assume that the error introducedby the resonant width to be ±0.09 MHz in ∆ν.
Secondly, a systematic error is induced by the CML refractive index value.
In the following chapter, we assume the refractive index of CML as nCML = 1.19
which is estimated by the Lorentz-Lorenz equation 3.23. As we employ theestimated value, there is an uncertainty in the CML refractive index, and it
introduces a systematic error. The refractive index measured in KAGRA is
1.26 at 1064 nm wavelength [58]. By taking this value into account, we set theuncertainty in the CML refractive index to be ±0.1.
In addition, statistical error is involved in the measurement. We use acurrent tuning of the laser to scan the laser frequency. For each scan, we
include not only the P- and S-resonant peaks, but also the peaks from the RF
sidebands at frequency ±29.1 MHz used to control the length of the cavity.
Because the frequency of the RF sidebands is precisely known, the frequency
scanning speed is calibrated on each scan.
However, there may be a fluctuation of the scanning speed within a single
scan. This fluctuation of the scanning speed is assumed to be random, thus
treated as a statistical error.

4.10

Summary of this chapter• We developed a cryogenic optical cavity setup with folded configuration.
The folded cavity can be cooled down to approximately 10 K.
89

4.10 Summary of this chapter

4 Development of Cryogenic Folded Cavity• A folded cavity enables efficient measurements for characterization of
coating performance.
• Ringdown measurement provides an optical attenuation of mirrors.
• A resonant frequency shift is introduced by a folding mirror. Folded
configuration enables one to measure such frequency shift.
• By measuring a drift in resonant frequency split, one can measure the
CML thickness, which changes the frequency split depending on its thickness.

90

Chapter 5
Experimental Characterizationof Cryogenic Molecular Layers
Using a Folded Cavity
In this chapter, we describe the results of experimental characterization of the
CML using a cryogenic folded cavity. First, we derive the optical loss in the
input and output mirrors using the FP cavity. Then, we install the folded
cavity and measure the optical loss in the cavity. By subtracting the opticalloss of the input and output mirrors, we estimate the optical loss in the folding
mirror at room temperature. At the same time, we measure the P-S resonant
frequency split, and confirm the validity of the cavity enhanced ellipsometry.
After that, we cool down the folded cavity, and monitor how the finesse
and the resonant frequency split change induced by the CML during the cryogenic operation. Through these measurements at cryogenic temperature, weestimate the optical loss generated by the CML and its thickness.
These measurements at cryogenic temperature are conducted with two different spacers. First measurement involves the folded cavity described in theprevious chapter. This measurement is to investigate the validity of the cavity
enhanced ellipsometry to characterize the CML. An additional measurement
is done with the modified spacer which has a slit to expose the folding mirror
to the room-temperature vacuum.
91

5 Experimental Characterization of Cryogenic Molecular Layers
5.1 Optical Loss Characterization
Using a Folded Cavity

Trasnsmitted Power [a.u.]

0.40

data
Fitting

0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00 01

2

34Time [μs]567Figure 5.1: Obtained transient response of FP cavity and fitted result. Red
and blue lines represent the measured data and fitted curve, respectively.

5.1

Optical Loss Characterization

The optical properties of a mirror satisfy the conservation of energy expressedas
R + T + L = R + A = 1,

(5.1)

where L is the optical loss, and A ≡ T + L is the attenuation which describes
the total power lost upon a single reflection. From ringdown measurements,
one can evaluate the reflectivity, hence the attenuation.

5.1.1

Finesse of Fused Silica Fabry-Perot Cavity

First, we measure the finesse of the Fabry-Perot cavity with fuese silica mirrors
which are used for the input and output mirror in the folded cavity. The
mirrors are held with vaccum compatible mirror holders and installed inside
the chamber. The cavity length is set as L = 74 mm whichis determined by
the restriction of the space.
92

5 Experimental Characterization of Cryogenic Molecular Layers
Using a Folded Cavity
5.1 Optical Loss Characterization

Table 5.1: The parameters of the folded cavity.
Parameters
Value
Cavity length
L1 = 41 mm
L2 = 41 mm
L = L1 + L2 = 82 mm
Free spectral range
1.84 GHz
Beam size on the input/output mirror
280 µmBeam size on the folding mirror
97 µm
Radius of curvature
50 mm
Folding angle
15 degFig. 5.1 shows the measured ringdown of the FP cavity. Blue line represents
the fitted curve. Here we assumed the fitting function as
f (t) = a exp(−t/τ ) + b,

(5.2)

where τ is the decay time of the cavity, and a and b are the parameters which
compensate the gain and dark offset of the PD. The finesse can be calculatedusing Eq. 4.13. The obtained finesse is 23153 ± 49, which is higher than thedesigned value of ∼ 1.5 × 104 . The error corresponds to the statistical error.
Assuming the same reflectance Rsilica in each mirror, the attenuation of fusedsilica mirrors, Asilica = 1 − Rsilica , can be calculated as
Asilica = 135.7 ± 0.3 ppm,

(5.3)

Rsilica = 0.999864.3 ± 0.0000003.

(5.4)

where

The obtained reflectance is slightly higher than the designed value, resulting
in a larger finesse.

5.1.2

Optical Loss in the Folding Mirror

The optical loss in the folding mirror can be measured from the differencein the finesse between the FP and the folded cavities. The measured finesseof the folded cavity at room temperature is about 1.7 × 104 . Therefore, thereflectance of the folding mirror can be calculated by√
π R1Ffold =,
(5.5)
1 − R1 R
93

5 Experimental Characterization of Cryogenic Molecular Layers
5.1 Optical Loss Characterization
Using a Folded Cavity
where R1 (= Rsilica ) is the reflectance of the input and output silica mirrors.
Thus the reflectance of the folding mirror can be computed using Eq. 5.5 as
R = 0.9999504 ± 0.0000008.

(5.6)

Thus, the attenuation of the folding mirror can be estimated as Afold = 49.6 ±0.8 ppm. As the designed reflectivity was larger than 0.99998, hence less than
20 ppm attenuation, the coatings on the folding mirror has larger optical loss
than the designed value.
One possible source of this optical loss is a defect in the coating. When sucha defect exists around the center of the mirror where the beam impinges, it
introduces optical losses via scattering or absorption [83]. Insufficient polishingof mirror surface also generates optical loss. In order to identify the origin of
this optical loss, further investigations are needed.

5.1.3

P-S Splitting

We confirm the validity of the cavity enhanced ellipsometry (CEE) from themeasurement at room temperature by comparing the results with the theoretical expectation.
First, we theoretically derive the P-S splitting frequency based on the characteristic matrix. The deposited coating is SiO2 /Ta2 O5 multilayer coating, andthe number doublet is N = 20. Optical thickness of each layer is λ/4 except
for the top layer which has λ/2 thickness, called a λ/2 cap as shown in Fig. 5.2.
The refractive indices are nSiO2 = 1.44 and nTa2 O5 = 2.2 at 1550 nm [31]. Assuming these values, the phase shift between P- and S-polarization beams ∆φcan be computed as∆φ = −0.0068 rad.

(5.7)

This value is obtained by numerical calculations using characteristic matrices
as shown in the previous chapter. Therefore, from Eq. 4.72 the frequency split
becomes as∆νP,S = 3.95 MHz.

(5.8)

Then, we derive the experimental result of the P-S resonant frequency split.
First, we calculate the laser frequency scan efficiency through the current tuning. We inject phase modulated beam into the cavity at 29.1 MHz where laser
frequency is scanned with 3 Vpp at 300 Hz triangle wave. As the frequenciesof sidebands are shifted by 29.1 MHz, one can derive the scanning efficiency
by measuring the shift between the carrier and sidebands. The split time of94

5 Experimental Characterization of Cryogenic Molecular Layers
Using a Folded Cavity
5.1 Optical Loss Characterization

Figure 5.2: An example of dielectric multilayer coatings deposited on a silicon
substrate. SiO2 and Ta2 O5 layers are deposited alternately with optical thickness of λ/4 except for the top layer. The top layer has λ/2 optical thickness
which is called λ/2 cap.

the carrier and the sideband is measured as 450 ± 2 µs. Thus, the scanning
efficiency is calculated as 35.9 ± 0.2 MHz.
Then, we measure the P-S splitting in the same manner. The measured
time difference between P- and S-polarizations are ∆ = 61.2 ± 0.4 µs with
3 Vpp at 300 Hz triangle wave scan. As the laser frequency scanning efficiency
is 35.9 ± 0.2 MHz, the split frequency can be calculated as∆ν = (35.9 ± 0.2) MHz/V × 3 V × 300 Hz × 2 × (61.2 ± 0.4) µs,= 3.96 ± 0.03 MHz.

(5.9)
(5.10)

This value is consistent with the theoretical value, 3.95 MHz. Therefore, theP-S splitting measurement using the folded cavity can be considered to bevalid for characterization of coatings. Combining the ringdown measurement,
this device can facilitate the development of low absorption and high reflective
coatings for the future GWDs.
95

5 Experimental Characterization of Cryogenic Molecular Layers
5.2 Optical Loss at Cryogenic Temperature
Using a Folded Cavity

7.5

mirrorboard

Temperature [K]

7.0
6.5
6.0
5.5
0.0

2.5

5.0

7.5 10.0 12.5 15.0
Time [min.]Figure 5.3: The temperature stability during the ringdown measurement.
Blue, and red lines represent the temperature of the mirror, and bread board,
respectively.

5.2

Optical Loss at Cryogenic Temperature

In order to estimate the impacts of the CML, the finesse at cryogenic temperature was measured followed by the room temperature measurement. First,
we conducted the measurements with closed environment, i.e., there were no
apertures to room temperature environment except for small through holes forthe cables. In this scheme, each mirror was exposed not to room temperature,
but to cryogenic environment. Through these measurements at cryogenic temperature, we investigated the validity of the cavity enhanced ellipsometry toestimate the thickness of the CML.
Then, we replace the spacer to the modified one, and perform another setof measurements to characterize the CML. In this additional measurements,
the folding mirror is exposed to the room-temperature vacuum to ensure the
CML formation on its surface. The details are written in the following section.
96

5 Experimental Characterization of Cryogenic Molecular Layers
Using a Folded Cavity
5.2 Optical Loss at Cryogenic Temperature

Figure 5.4: Schematic of formed CML on the mirror. The CML can be regarded as a kind of coating layer.

5.2.1

Cavity Length Change

Here we discuss the cavity length change due to the cooling down. The system
is cooled down to approximately 7 K. Generally, cooling down the material
involves shrinking, which can introduce cavity length change. Here we employ
an invar as the spacer which has low thermal expansion coefficient, and thecavity length change is assumed to be less than 100 µm.
In addition, temperature fluctuation at cryogenic temperature is within±0.002 K as shown in Fig. 5.3. Therefore, the cavity length change is also
negligible during the measurements. Hence the impacts on the finesse measurement due to the cavity length change can be negligible.

5.2.2

Reflectance of the Mirror

Here we discuss how the reflectance of mirror changes due to the molecularlayer formation on the top of the coating as shown in Fig. 5.4. Formed CML
can be regarded as a coating layer which changes the reflectance of the mirror.
The CML, which grows over time, changes the reflectance of the mirror, andthe reflectance is calculated using a characteristic matrix [58]. Fig. 5.5 shows
the calculated result of the reflectance. When the thickness is comparable
to the wavelength, the reflectance oscillates due to the interference of the97

5 Experimental Characterization of Cryogenic Molecular Layers
5.2 Optical Loss at Cryogenic Temperature
Using a Folded Cavity

100.0099.95

Reflectivity [%]

99.90
99.85
99.80
99.75
99.70
99.65
99.60
100

Reflectivity
101

CML thickness [nm]

102

103

Figure 5.5: How the reflectivity of the mirror changes due to the CML growth.
The reflectance can be considered as constant where the thickness of the CML
is much thinner than the wavelength. When the thickness of CML becomescomparable to λ/4, it starts oscillation due to the interference.

reflected field. As long as the thickness is much thinner than the wavelength,
the interference can be negligible. Therefore, we assume that the change inreflectance is caused by the optical loss of CML.

5.2.3

Optical Loss Measurements

We monitored the finesse trend for 15 days in order to investigate the optical loss growth rate induced by the CML. To obtain the statistical errors, we
performed the ringdown measurements 5 times and calculated the standard deviation. Fig. 5.6 shows the obtained data. Finesse of the cavity monotonicallydecreased during the cryogenic condition, and recovered after temperature rising. Therefore, it is natural to consider that the optical loss inside the cavity
grew with time due to the continuous CML formation.
As discussed in the previous subsection, we assume that the reflectancechange is caused by the CML formation as
R0 = 1 − T − L = R − L,98

(5.11)

5 Experimental Characterization of Cryogenic Molecular Layers
Using a Folded Cavity
5.2 Optical Loss at Cryogenic Temperature

Figure 5.6: The obtained decay time by the ringdown measurement. The
finesse was monitored for 15 days at temperature of 7 K. The finesse decreased
with time, and recovered after increasing the temperature.

where T is the transmittance and L denotes the optical loss induced by the
CML. Therefore, the contaminated amplitude reflectivity is written as√
r0 = R − L.
(5.12)
Consequently, the finesse of the folded cavity with additional optical loss canbe expressed asp√
π R1 − L
π r0 21=.
(5.13)F =
1 − (R1 − L)(R − L)
1 − r0 21 r0 2
By monitoring the finesse, one can estimate how the optical loss grows.
In order to estimate the optical loss growth rate, we use the fitting function
expressed asq
π (R1 − L̃t − A0 )F (t) =,
(5.14)
1 − (R1 − L̃t − A0 )(R − L̃t − A0 )99

5 Experimental Characterization of Cryogenic Molecular Layers
5.2 Optical Loss at Cryogenic Temperature
Using a Folded Cavity

Figure 5.7: The obtained decay time by the ringdown measurement and fitted
curve, blue dots and red line, respectively.

where L̃ is the optical loss growth rate and A0 is the initial optical attenuation
which contributes to the slight finesse drop at the beginning of the measurement at cryogenic temperature, and t is the time from the cavity reached 7 K.
Here we assume that the reflectivity of each mirror is constant because theformed CML considered to be very thin. We fitted with Eq. 5.14 using theleast squares method. Here we used the relation between the finesse and the
decay time τ expressed as
τ=

L
F.
πc

(5.15)

Fig. 5.7 shows the obtained results and fitted curve. From the fitting, theoptical loss growth rate can be estimated as
L̃ = 0.13 ± 0.02 ppm/day,(5.16)and the initial optical attenuation is estimated as
A0 = 1.4 ± 0.1 ppm.
100

(5.17)

5 Experimental Characterization of Cryogenic Molecular Layers
Using a Folded Cavity
5.2 Optical Loss at Cryogenic Temperature

Figure 5.8: How the resonant frequency split changes over the thickness of
the CML (solid line), and linearly approximated line (dashed line) when thefolding angle is γ = 15 deg.

Besides the optical loss growth rate, we estimated the thickness of theformed CML by cavity enhanced ellipsometry. The initial P-S split is measured
as ∆ = 57.1 ± 0.7 µs. Corresponding frequency split is calculated as∆νint = −3.69 ± 0.05 MHz.

(5.18)We measured the frequency split after 15 days cryogenic operation. The frequency split at the end of cryogenic operation is measured as ∆ = 55.6 ± 0.5,
which corresponds to∆νfin = −3.58 ± 0.04 MHz.

(5.19)

Here we assume that the CML is formed uniformly on each mirror withthickness of dCML . The frequency split changes linearly where the thickness of
CML is much thinner than the wavelength as shown in Fig. 5.8. This frequency
shift can be approximated as∆ν [MHz] = 0.26 [MHz/nm] × dCML [nm] − 3.95 [MHz].
101(5.20)

5 Experimental Characterization of Cryogenic Molecular Layers
5.3 Measurements with a Modified Spacer
Using a Folded Cavity
Here we assumed that the refractive index of the CML is nCML = 1.19 whichis theoretically derived by the Lorentz-Lorenz equation expressed as Eq. 3.23.
From Eqs. 5.18 to 5.20, the optical thickness of CML can be estimated as
dCML,int = 1.0 ± 0.2 nm,(5.21)

dCML,fin = 1.4 ± 0.2 nm.

(5.22)

and

Therefore, the CEE has a possibility to probe very thin CML with a nanometerlevel. In order to confirm the validity of this method, we performed further
measurements with a modified spacer and further considerations of uncertainties.

5.3

Measurements with a Modified Spacer

In order to ensure the CML formation on the mirror, we conducted additional
measurements with a modified spacer. The modified spacer has a slit on its sidein order to expose the folding mirror to molecular flow from room temperatureenvironment as shown in Fig. 5.9. In this configuration, the molecular flowfrom room temperature environment hits the folding mirror through the slit,
resulting in the CML formation on the folding mirror surface. In other words,
the folding mirror corresponds to the cryogenic mirror in a GWD, which isalso exposed to the room-temperature vacuum. The input and output mirrors
are surrounded by the cryogenic spacer and cover, and conductance to thesemirrors are much smaller than that of the folding mirror. Therefore, we assume
that the CML is formed only on the folding mirror in this experiment. With
this configuration, we did ringdown and P-S splitting measurements to confirm
the validity of the methods to characterize the CML.

5.3.1

Geometrical Parameters

Geometric parameters are same as the previous measurement except for thefolding angle as shown in Tab. 5.2. The modified spacer has larger folding
angle, 22.5 deg in order to expose the folding mirror to the room temperatureshield. This angle is determined by the restriction of available size of thematerial and space of the slit. The setup of input and output optics is sameas the previous one. Mirrors for the folded cavity are also same — fused silica
mirrors for input and output, and a silicon mirror for the folding mirror.
Fig. 5.10 shows how the frequency split changes with the thickness of the
CML. Here, we used the refractive indices of SiO2 and Ta2 O5 as nSiO2 = 1.44
102

5 Experimental Characterization of Cryogenic Molecular Layers
Using a Folded Cavity
5.3 Measurements with a Modified Spacer

Figure 5.9: Overview of the setup of the additional measurement. The spacer
has a slit on its side in order to expose the folding mirror to the room temperature environment, hence the molecular flow. Other two mirrors are notexposed to the room temperature. This configuration enables to form the CML
only on the folding mirror as it is exposed to the molecular flow from the room
temperature vacuum.

and nTa2 O5 = 2.2, respectively [31], and the number of SiO2 /Ta2 O5 pairs are
N = 20. When the thickness of the CML is much thinner than λ/4, thefrequency split changes linearly with respect to the thickness, and the relation
between the frequency split and the thickness of the CML is expressed as∆ν [MHz] = 0.56 [MHz/nm] × dCML [nm] − 20.28 [MHz],(5.23)

which is shown in Fig. 5.10 as dashed line. The offset, −20.28 MHz, is induced by the SiO2 /Ta2 O5 layers. The measurements at room temperature,i.e., dCML = 0, are consistent with this model. The refractive index of the
CML is assumed as nCML = 1.19 in the same manner as the previous measurement.
103

5 Experimental Characterization of Cryogenic Molecular Layers
5.3 Measurements with a Modified Spacer
Using a Folded Cavity

Table 5.2: The parameters of the modified folded cavity.
Parameters
Value
Cavity length
L1 = 41 mm
L2 = 41 mm
L = L1 + L2 = 82 mm
Free spectral range
1.84 GHz
Beam size on the input/output mirror
280 µmBeam size on the folding mirror
97 µm
Radius of curvature
50 mm
R1 = r12 , r22
0.999864
Folding angle
22.5 deg

5.3.2

Cooling of the Cavity

Fig. 5.11 shows the cooling curve of the folding mirror. It takes about 3 days
to reach to 10 K. As the folding mirror is exposed to the room temperature
vacuum, the cooling time becomes longer than the closed case. After reaching
10 K, the temperature becomes stable, and the temperature fluctuation during
each measurement is less than 0.01 K. Though the temperature fluctuation
becomes larger than that of closed case, it is still stable enough so that one
can ignore the impacts of cavity length fluctuation on the measurements.

5.3.3

Finesse and P-S Splitting Measurements

We monitored the finesse degradation and the drift of the resonant frequency
split for ∼ 10 days by the ringdown and the CEE measurements. The cavity ringdown was measured with S-polarization beam. Fig. 5.12 shows theresults of these measurements. The decay time monotonically decreases during the cryogenic operation period, and recovers after raising the temperature.
The drift in the resonant frequency split also changes over time at cryogenic
temperature, and recovers after raising temperature. Therefore, these changes
in the finesse and the resonant frequency split are considered to be inducedby the formation of the CML. The error bars represent the statistical errors
and measurement errors induced by the resonant width. To obtain the statistical errors, we performed the CEE measurements 10 times and calculated
the standard deviation. The uncertainty caused by the resonant width is assumed to be ±0.09 MHz. This causes the error in the CML refractive index as(±0.09 MHz)/(0.56 MHz/nm) = ±0.16 nm.
104

5 Experimental Characterization of Cryogenic Molecular Layers
Using a Folded Cavity
5.3 Measurements with a Modified Spacer

Figure 5.10: How the resonant frequency split changes over the thickness of
the CML (solid line), and linearly approximated line (dashed line) when thefolding angle is γ = 22.5 deg.

5.3.4

Optical Loss and Thickness

As the only folding mirror is exposed to the room temperature vacuum, thefinesse of the modified cavity is written as√
π R1
F (t) =,
(5.24)
1 − R1 (R − LCML )where LCML is the optical loss on the folding mirror by the CML. At room
temperature, the optical loss of the CML is equal to zero, and one can estimate
R. From the obtained results at cryogenic temperature, the optical loss of the
CML can be estimated.
In order to obtain LCML , we calculated the attenuation A of the folding
mirror from the result taken at room temperature in advance of the cryogenic
measurement. The attenuation, A, was calculated as A = 4.7 ± 0.7 ppm,
hence R = 0.999995.3 ± 0.7. The optical attenuation is much smaller than theprevious measurement.
The uncertainty in the refractive index of the CML introduces an error in
the calculated CML thickness. We assume the CML refractive index to be105

5 Experimental Characterization of Cryogenic Molecular Layers
5.3 Measurements with a Modified Spacer
Using a Folded Cavity

300

folding mirror

250

Temperature [K]

200
150
100
50
00

500

1000 1500 2000 2500 3000 3500 4000 4500

Time [min]Figure 5.11: Cooling curve of the folding mirror.

nCML = 1.19 ± 0.1. When the refractive index of the CML is nCML = 1.29, therelation between the frequency split and the thickness of the CML is expressedas∆ν [MHz] = 0.57 [MHz/nm] × dCML [nm] − 20.28 [MHz].

(5.25)

Also, the relation becomes∆ν [MHz] = 0.55 [MHz/nm] × dCML [nm] − 20.28 [MHz],(5.26)

for nCML = 1.09. Therefore, we consider that the uncertainty in the CML
refractive index induces the error in the sensitivity as (0.56 ± 0.01) MHz/nm.
The uncertainty in the CML thickness due to this error depends on the amountof PS split frequency. In this measurement, it becomes maximum when ∆ν isminimum, and can be calculated as(20.28 − 17.95) MHz= (4.13 ± 0.08) nm.
(0.56 ± 0.01) MHz/nm
106(5.27)

5 Experimental Characterization of Cryogenic Molecular Layers
Using a Folded Cavity
5.4 Summary of This Chapter
Therefore, the errors induced by the CML refractive index is ±0.08 nm at most,which is smaller than the errors due to the resonant width. We calculated thiserror for each measurement.
From the measurements, we estimated the optical loss by the CML, LCML
by using Eq. 5.24, and thickness of the CML, dCML , by using Eq. 5.23. Fig.
5.13 shows the growing optical loss LCML , and thickness of the CML dCML .
The error bars in CML thickness include the statistical errors, measurement
errors induced by the resonant width and systematic errors due to the CML
refractive index.
The loss, 7.3 ± 1.1 ppm, and thickness 2.38 ± 0.27 nm, measured at the firstday are generated by the adsorption during the initial cooling down. In thefollowing 10 days, the optical loss and the thickness of the CML grow overtime, and become LCML = 17.4 ± 1.3 ppm and dCML = 4.13 ± 0.36 nm. Afterraising the temperature, the optical loss and the thickness of the CML became
zero. Therefore, a thin CML is formed on the folding mirror, and it can beprobed by not only the ringdown, but also the CEE measurement.
The uncertainty in the estimation of the CML thickness is introduced by
the statistical error, measurement error related to the resonant width and systematic error due to the CML refractive index. Taking these into consideration,
the uncertainty of the CML thickness is (±0.38 ± 0.16 ± 0.07) nm = ±0.42 nm
at most and less than 0.5 nm. Therefore, our setup enables us to measure the
CML thickness at a nanometer level. Further considerations on a design of the
setup will enable more precise characterization of the CML.
It should be noted that the folding angle of the cavity is considered to beconstant through these measurements. Due to the cooling, the folded cavity
shrinks, which can lead to the deformation. Here, we ignore the impacts of thedeformation due to the cooling by assuming that the spacer shrinks uniformly,
and the change in the folding angle is negligible. Especially, during 10 days
cryogenic operation at 10 K, the deformation of the spacer is much small as
the thermal expansion coefficient is less than approximately 10−6 1/K, andthe temperature drift is less than 1 K. Therefore, the relative change in theresonant frequency split is not induced by the deformation of the cavity during
the measurements at cryogenic temperature.

5.4

Summary of This Chapter• We have developed the cryogenic folded cavity, which is the first timeever.
• The optical attenuation of silicon mirror was measured by using the107

5 Experimental Characterization of Cryogenic Molecular Layers
5.4 Summary of This Chapter
Using a Folded Cavity
folded cavity.
• The frequency shift between P- and S-polarization reflects the information of the coatings. By measuring the frequency shift, one can estimate
the status of mirror surface.
• We monitored the finesse during the cryogenic environment and it showedfinesse degradation due to the CML growth, and drift in the resonant
frequency shift.
• In order to confirm the validity of the CEE, the additional measurement
was conducted with modified spacer. The results indicate the CML formation on the folding mirror surface, and the CEE can probe the CML
at a nanometer level.

108

5 Experimental Characterization of Cryogenic Molecular Layers
Using a Folded Cavity
5.4 Summary of This Chapter

Figure 5.12: Trend of measured decay time (upper) and the resonant frequency
split (lower).

109

5 Experimental Characterization of Cryogenic Molecular Layers
5.4 Summary of This Chapter
Using a Folded Cavity

Figure 5.13: Trend of accumulated optical loss on the folding mirror, LCML
(blue circle), and CML thickness, dCML (red triangle).

110Chapter 6
Discussions and Implications
Cryogenic operation of a GWD is a promising way to improve the sensitivity ofa detector by reducing the thermal noise. A cryogenic mirror, however, has atechnical problem caused by the adsorption of residual gas molecules. We havepresented the possible impacts of the CML on the optical losses introduced forthe case of cryogenic GWDs in chapter 3. The effect of scattering loss canbe negligible as long as the thickness of the CML is less than 1 µm. On theother hand, the optical absorption becomes a critical problem for cryogenic
operation because of an additional heat load to the cryogenic system. Even
though the thickness of CML is about 1 nm, the absorbed heat exceeds thecooling capacity for both ET and LIGO Voyager.
In order to investigate the properties of very thin CML, we have developed acryogenic folded cavity and ellipsometric measurement. The obtained results
indicate that the ellipsometric measurement can probe the thickness of the
CML with a certain accuracy.
In this chapter, we estimate the heat input to the test mass based on theexperimental results. Also, we compute the CML formation rate by using aMonte Carlo (MC) simulation. Finally, we discuss the possible countermeasures for the CML.

6.1

Heat Input due to the CML

In this section, we discuss the heat input induced by the CML based on theexperimental results. As the laser wavelength of the experiment is 1550 nm,
we focus on the case of the ET.
111

6.1 Heat Input due to the CML

20.0

6 Discussions and Implications

Measured

OPtical loss of folding mirror [ppm]

17.5
15.0
12.5
10.0
7.5
5.0
2.5
0.0

0

1

2
3
CML thickness [nm]45Figure 6.1: The CML thickness and the optical loss of the folding mirror.

6.1.1

Optical Loss and Thickness of the CML

Fig. 6.1 shows the optical loss and thickness of the CML. The optical loss of
the CML increases as it becomes thicker. In this subsection, we investigate thevalidity of the Lambert-Beer law at very thin region from the obtained results.
As shown in Chapter 3, the optical loss generated by the scattering can be
negligible as long as the thickness of the CML is very thin. We assume thatthe obtained results of the optical loss is induced by the optical absorption of
the CML.
Assuming the Lambert-Beer law, the heat input to the test mass inducedby the CML, QCML , is expressed as
QCML = PCML {1 − exp(−2αCML dCML )},(6.1)

where PCML is the laser power incoming to the CML, αCML is the absorption
coefficient, and dCML is the thickness of the CML. Therefore, the optical loss
due to the absorption of the CML, LCML , is written as
LCML = QCML /PCML = 1 − exp(−2αCML dCML ).
112

(6.2)

6 Discussions and Implications

20.0

Model
Fit 1
Fit 2
Measured

17.5

OPtical loss of folding mirror [ppm]

6.1 Heat Input due to the CML

15.0
12.5
10.0
7.5
5.0
2.5
0.0

0

1

2
3
CML thickness [nm]45Figure 6.2: The thickness and the optical loss of the CML. Red dashed line
and blue dashdot line represent the fitted curves. The shaded bands indicate
the 1σ uncertainty. Black solid line is the amorphous ice model.

Assuming that the CML is very thin, Eq. (6.2) is approximated as
LCML ≈ 2αCML dCML .

(6.3)

Therefore, the optical absorption scales linearly to the absorption coefficient
at very thin region.
Fig. 6.2 shows the obtained results, amorphous ice model, and the fitted
curves. We assume the following function as a fitting curve labeled Fit1 toestimate the absorption coefficient,
LCML =

2αCML dCML
+ α0 .
cos γ 0(6.4)

Here, dCML / cos γ 0 is the effective path length because of the folding angle γ.
1136.1 Heat Input due to the CML

6 Discussions and Implications

Table 6.1: Results of fitting.

α0
α1
α2

Fit 1
−0.49 ± 0.74 ppm
1.91 ± 0.10 ppm/nm

Fit 2
0.20 ± 0.14 ppm
0.72 ± 0.14 ppm/nm
0.14 ± 0.02 ppm/nm2

The angle of incidence, γ 0 , is calculated by Snell’s law ass
0cos γ =



1−

n0

2

nCML

sin2 γ,(6.5)

where n0 = 1.0, and nCML = 1.19 are the refractive indices of the vacuum
and the CML, respectively. The red dashed line labeled Fit1 shown in Fig.
6.2 represents the results of the fitting assuming Eq. 6.5, and shaded band
represents the 1σ uncertainty. The obtained parameters from the fitting are
αCML = 1.91 ± 0.10 ppm/nm and α0 = −0.49 ± 0.74 ppm. A discrepancy fromthe fitted line can be seen at both thin and thick ends of the measurements,
i.e., a discrepancy from the Lambert-Beer linear model.
Here, we fitted the data by another model, which is expressed asLCML = α22dCML
cos γ 0

2
+ α1

2dCML
+ α0 ,
cos γ 0(6.6)

where αi (i = 0, 1, 2) are fitting parameters. Red dashed line labeled Fit2 is
the fitting curve by assuming Eq. 6.6. The obtained parameters are α2 =0.14 ± 0.02 ppm/nm2 , α1 = 0.72 ± 0.14 ppm/nm and α0 = 0.20 ± 0.14 ppm. By
assuming the non-Lambert-Beer model Eq. 6.6, the fitted curve well matcheswith the measured data.
A possible explanation is that the Lambert-Beer law does not hold in verythin region and the optical loss at very thin region may be proportional to
the second order of dCML . Recent simulation study by LIGO Voyager groupindicates that the optical absorption by a CML can deviate from the LambertBeer law and becomes two orders of magnitude smaller than the Lambert-Beerlaw below 10 nm, i.e., the order of sub-ppm [84]. Further investigations areneeded to know the behavior of the optical loss for very thin layers, and will beof importance for the future GWDs, in which the heat input due to the CML
to the test mass becomes the problem even at a few nanometer thickness.
114

6 Discussions and Implications

0.45
0.40
0.35

6.1 Heat Input due to the CML

ModelCooling capacity
Projection

Heat input [W]

0.30
0.25
0.20
0.15
0.10
0.05
0.000

1

2

3

CML thickness [nm]45Figure 6.3: Heat input to the test mass induced by the CML for the case of
the ET. Solid line shows the theoretical expectation, and blue dashed-dottedline represents the cooling capacity of the test mass mirror in the ET. Purple
circles are projections of the measured data.

6.1.2

Heat Input to the Test Mass by the CML

Fig. 6.3 shows the heat input to the test mass for the case of the ET. We project
the obtained results shown as purple circles by assuming that the measured
optical loss is induced by absorption. The intra-arm-cavity power of the ET iscalculated based on the parameters shown in Table 3.2, and constant as longas the thickness of the CML is very thin (< 10 nm). Black solid line represents
the theoretical prediction calculated in chapter 3, and blue dash-dotted line
represents the cooling capacity of the ET. The heat input of expected fromthe measurement shows smaller value than that of the theoretical expectation.
This might be because that the Lambert-Beer law does not hold at such small
thickness. However, the experimental results indicate that the heat absorption
can still be large, and the heat input exceeds the cooling capacity of the mirror,∼ 100 mW, even if the thickness is approximately 2 nm. Thus, this experiment115

6.2 Sensitivity of CEE

6 Discussions and Implications

indicates that the CML on the test mass mirror can prevent the cryogenic
operation of future cryogenic GWDs due to its large heat absorption.
Here, we assumed that the optical loss is induced by the optical absorption. In order to confirm the validity of this assumption, further studies areneeded such as the photothermal self-phase modulation technique [47]. Also,
further investigations on both thinner and thicker regions are required to fully
understand the properties of the CML including the refractive index.

6.1.3

Other Heat Loads

Not only the absorption of the laser power by the CML, but also the radiationfrom the room temperature environment and the absorption by the coatings
contribute to the heat input.
For the case of the ET, heat load induced by the radiation from the room
temperature environment is estimated to be approximately 70 mW. In addition, the absorption by the coatings generates approximately 20 mW heat load.
On the other hand, the cooling capacity is 100 mW. Therefore, the tolerable
heat input by the CML is only 10 mW. Assuming the Lambert-Beer law, thetolerable CML thickness becomes approximately 0.2 nm.
In LIGO Voyager, whose cooling capacity is 10 W, the heat load inducedby the absorption in coatings is estimated as 3 W [35]. The heat load due tothe radiation is reduced to only 6 mW thanks to the presence of the cryotrap.
Thus, the tolerable heat load by the CML is less than 7 W. As the intra-cavity
power becomes 3 MW, a few ppm absorption causes larger heat input than thetolerable value. In the same manner as the case of the ET, the tolerable CML
thickness become less than 0.1 nm by assuming the Lambert-Beer law.

6.2

Sensitivity of CEE

In this section, we discuss the sensitivity of CEE to CML thickness. Here we
define the sensitivity of CEE asζ [MHz/nm] =∆ν [MHz].
dCML [nm](6.7)

Assuming that the refractivie index of CML is constant, the sensitivity ζ isdetermined by the geometrical parameters — folding angle γ and FSR fFSR
of the folded cavity. Red curves in Figs. 6.4 and 6.5 show how the sensitivity
changes over the folding angle and FSR, respectively. Blue curves in Figs.
6.4 and 6.5 represent the amount of P-S split frequency ∆ν without CML. In
116

6 Discussions and Implications

6.2 Sensitivity of CEE

0

−100

Sensi ivi y [MHz/nm]

Spli frequency [MHz]

1.2

0.9

Spli frequency
Sensi ivi your se up

−200

0.6
−300

0.3
−400
0

5

10

15
20
25
30
Folding angle [deg]

35

40

0.0
45

Figure 6.4: How splitting frequency and sensitivity of CEE change over thefolding angle. fFSR = 1.84 GHz is assumed. Black dots correspond to our
setup.

our setup, the folding angle and FSR are set to γ = 22.5 deg and 1.84 GHz,
respectively, and the sensitivity becomes ζ = 0.56 MHz/nm.
From Figs. 6.4 and 6.5, larger folding angle and larger FSR enhance thesensitivity to the CML thickness. Therefore, the sensitivity of the CEE can be
improved by optimizing the geometrical parameters of the folded cavity. On
the other hand, the amount of resonant frequency split also becomes larger
as the sensitivity increases. For example, when the folding angle becomesγ ≈ 40 deg with fFSR = 1.84 GHz, the sensitivity is about 1.3 MHz/nm and theresonant frequency split is enhanced to be −230 MHz. Such a large frequency
split makes it difficult to scan the laser frequency by a current tuning because
the scan range is about 200 MHz for our setup. Therefore, the improvementof the sensitivity accompanies difficulties.
One possible approach to measure the CML thickness with improved sensitivity is to add another P-polarization beam path with double-pass AOM configuration [85]. Fig. 6.6 shows the overview of possible upgrade configuration.
Here, we set the folding angle and FSR to be γ = 33 deg and fFSR = 3.00 GHz.
With these parameters, the sensitivity and resonant frequency split becomeζ = 1.83 MHz/nm and ∆ν ≈ 160 MHz, respectively. By using an AOM which
has 80 MHz central frequency, double-pass AOM configuration enables us to117

6.3 Formation Speed

6 Discussions and Implications

Figure 6.5: How the sensitivity of CEE change over the FSR. Folding angle
22.5 deg is assumed. Black dots correspond to our setup.

shift the laser frequency about 160 MHz without beam jitters. By tuning thelaser frequency shift introduced by a double-pass AOM, one can determine thefrequency split between P- and S-polarization beams. Thus, the CML thickness can be probed with improved sensitivity. It should be noted that theresonant width, which introduces an error, becomes large depending on the
FSR. In order to maintain the resonant width less than 0.1 MHz, the finesseof the folded cavity should be increased to approximately 3 × 104 . However,with a large folding angle, realizing a coating with very high reflectivities forboth S- and P-polarizations at the same time becomes increasingly difficult.
Therefore, it may be difficult to increase the finesse so much.

6.3

Formation Speed

6.3.1

Formation Speed Degradation

As shown in Fig. 5.13, the CML growing rate decreases with time, i.e., theCML formation rate is not constant. This can be caused by the molecularflow onto the mirror gradually reduced as a consequence of improvement in118

6 Discussions and Implications

6.3 Formation Speed

Figure 6.6: A possible upgrade configuration. Additional beam path is introduced with double-pass AOM to shift the laser frequency about 160 MHz.

the vacuum level. Fig. 6.7 shows the pressure between the room temperatureand the 80 K shields and the CML formation speed. Here the CML formation
speed is obtained by taking the derivative of CML thickness shown in Fig.
5.13. During the cryogenic operation, the pressure and the CML formation
speed decreases as shown in Fig. 6.7. The CML formation speed, η, scaleswith the pressure difference, ∆P , between the room temperature and cryogenic
vacuum as [58]η ∝ ∆P.

(6.8)

Here, we assume that the pressure difference is the same as the pressure at theroom temperature vacuum because the cryogenic vacuum level is expected to
be much lower than the room temperature part.
The improvement in the room temperature vacuum, hence the reductionin ∆P leads to a smaller formation speed. Thus, the decreasing pressure could
cause the thickness trend as shown in Fig. 5.13. However, the CML formation
speed decreased about one order of magnitude though the improvement invacuum level is about a factor of 2. Therefore, the CML formation speed may
not be simply proportional to ∆P . Further investigations are needed to reveal
the relation between the CML formation speed and the vacuum level.

6.3.2

Comparison with the Measurement in KAGRA

Tab. 6.2 shows the comparison of our results with the previous measurementin KAGRA. The CML formation speed in KAGRA was 27 nm/day, which1196 Discussions and Implications

4.0

0.75
Pressure
CML forma ion speed

Pressure (×10−5) [Pa]

3.2

0.50

2.4

0.25

1.6

0.00

0.8 0

2

4

6
Time [day]8

10

CML forma ion speed [nm/day]

6.3 Formation Speed−0.25

Figure 6.7: Trend of the pressure at room temperature vacuum and CML
formation speed during the cryogenic operation. The error bars of pressure
are due to the uncertainty of the vacuum gauge.

was obtained by measuring the finesse oscillation induced by CML formation[58]. Here, CML formation speed is assumed to be proportional to the vacuum
pressure at room-temperature part ∆P , and a solid angle of Ω from the centerof the mirror to the room temperature part. The solid angle is calculated as


z
,
(6.9)
Ω = 2π 1 − √z 2 + r2
where z is the distance from the mirror to the edge of cryogenic duct and r
is the radius of the duct. For the case of KAGRA, z = 5 m and r = 270 mm
are assumed, and the solid angle becomes 2 msr. In our setup, substituting z = 300 mm and r = 20 mm to Eq. 6.9, the solid angle to the roomtemperature vacuum becomes 14 msr, which is 7 times larger than that of
KAGRA. Moreover, the vacuum level of our setup is about 6 times worse
than the previous study [58]. Considering these difference, the expected CML
formation speed in our setup from measurement in KAGRA becomesηexpect = 27 nm/day × 7 × 6 ≈ 1.1 × 103 nm/day.

(6.10)On the other hand, the experimentally obtained formation speed in our setup
is only 0.6 nm/day at most, which is about 2 × 103 times smaller than the theoretically expected value from the KAGRA case. Such a large discrepancy in120

6 Discussions and Implications

6.4 Estimation of the CML Formation Rate
in Future GWDs

Table 6.2: Comparison with the previous measurement in KAGRA [59].
in KAGRA
Configuration of cavity
Fabry-Perot
Wavelength1064 nm
Solid angle
2 msr
Vacuum level
5 × 10−6 Pa
Methodfinesse oscillation
Formation speed(27 ± 2) nm/day

our setup
Folded1550 nm
14 msr3 × 10−5 Pacavity enhanced ellipsometry(0.6 ± 0.1) nm/day

the CML formation speed may be generated by the difference in the geometryof the vacuum system between KAGRA and a table top experiment. Further
studies are needed to understand this difference in the CML formation speed.
It should be noted that the measurement in KAGRA assumed the constant
CML formation though the formation rate may not be constant in practice [58].
The cavity enhanced ellipsometry enables the measurement of the formation
rate even if the rate is not constant, which is one of the advantages of thistechnique. Further measurements in GWDs will help understand the CML
formation rate combined with the simulation presented in the following section.

6.4

Estimation of the CML Formation Rate
in Future GWDs

So far, we have revealed that the CML can hinder the cryogenic operation of
future GWDs even though its thickness is less than 1 nm. Here, we compute
the CML formation rate, in the same manner as in a previous study [59], toestimate the timescale that the heat input due to the CML becomes a problem.
The timescale can be regarded as an index for designing the cryogenic systemsof future GWDs. We assume a simple beam duct model for order estimationof CML formation rate during the stationary state.
In order to evaluate the formation rate, first, we derive the sticking probability of residual molecules, Pst , which describes the ratio between the numberof molecules that input to the cryogenic surface and that adhere to the surface.
The sticking probability can be expressed as [86].
Pst = α{1 − tanh[β(T − γEdes )]},

(6.11)

where α, β and γ are the parameters to reproduce the experimental resultsand Edes is the desorption energy. These values are listed in Table 6.3.
121

6.4 Estimation of the CML Formation Rate
in Future GWDs

6 Discussions and Implications

100
10−1
Sticking Probabilit

10−2
10−3
10−4
10−5
10−6

N2
O2
H2 O
50

100
150
200
Tempereture [K]

250

300

Figure 6.8: Sticking probability of several molecules. The parameters to calculate the sticking probabilities are listed in Tab. 6.3. At the cryogenic temperature of 10 K, almost all residual gas molecules can stick to the cryogenic
surface.

Fig. 6.8 shows the sticking probabilities of H2 , N2 , O2 , and H2 O molecules
as functions of temperature. The sticking probability of water molecules can betreated as equal to 1 below about 250 K. In addition, the sticking probabilitiesof other molecules are also almost unity at 10 K. Therefore, the measured
finesse fluctuation can be induced by the mixture layer of these molecules.
In order to evaluate the molecular layer formation rate, an MC simulator
package developed at CERN, Molflow+ [87], is used. A previous study showsthat the numerical simulation using Molflow is valid for order estimation [58].
Here we make 3D CAD models of cryogenic duct of the ET and LIGO Voyager
based on the Refs. [30, 88]. By using these models for MC simulation, one canobtain the formation rate of CML as following. Assumed parameters for the
cryogenic duct are listed in Tab. 6.4.
122

6 Discussions and Implications

6.4 Estimation of the CML Formation Rate
in Future GWDs

Table 6.3: The parameters used to calculate the sticking probability [86]. Edes
denotes the desorption energy.
Molecule
β
γ
Edes
N2
0.12 0.043 1250
O2
0.17 0.042 2320
H2 O
0.11 0.042 6000

Figure 6.9: Model for the conductance simulation of the space between the ductshield and the test mass. The particles are injected from the left-end facet,
and the number of particles which reach to the right-end facet are counted.

First, we define the volumetric flow rate Q as
Q = C∆P,(6.12)

where C is the conductance of the system and ∆P is the pressure difference.
Here we consider the cylindrical beam tube. MC simulation is one of the best
approaches to estimate the conductance. By using the transmission probability(or Clausing factor), K, the conductance of the beam tube can be expressedas
C = C0 K,(6.13)

where C0 is the opening conductance of the system which can be expressed as
r
πkB T
2
.
(6.14)
C0 = r2mHere r is the radius of the duct, kB is the Boltzmann constant, T is thetemperature and m is the mass of the molecule. The CML grows with time dueto the continuous residual molecular flow from the beam tube. The formation
rate can be calculated asmQη=[m/s],(6.15)SρkB T123

6.4 Estimation of the CML Formation Rate
in Future GWDs

6 Discussions and Implications

Table 6.4: The assumed parameters of cryogenic duct of the ET and LIGO
Voyager.
Name
DiameterTemperature
LengthThe ET
0.75 m4 K and 77 K
5 m / 4 K + 50 m / 77 K

Voyager
0.50 m80 K∼ 10 mwhere S is the area of the cryogenic mirror surface and ρ is the density of themolecule. Assuming the pressure difference between the beam duct and thecryostat, one can estimate the formation rate of the CML from Eq. 6.15.

The Einstein Telescope
We consider the case of the ET. The cryogenic beam duct of the ET consistsof two parts — 50 m long 77 K cryotrap, and 5 m long 4 K cryotrap. The
conductance can be calculated as
CET =

1
,
1/C77K + 1/C4K(6.16)

where C77K , and C4K are conductance of each cryotrap.
The transmission probability of 77 K cryotrap, Ktrans,77K , for each moleculeis simulated as
−5
4.9 × 10
Ktrans,77K = 2.0 × 10−4

5.0 × 10−3

(H2 O),(N2 ),(O2 ).

(6.17)(H2 O, N2 , O2 ).

(6.18)

That for 4 K cryotrap also is calculated as
Ktrans,4K = 5.5 × 10−3

Here we assumed that the sticking probability of these molecules are the unity
at 4 K.
Then we compute the conductance. The opening conductance of each cry124

6 Discussions and Implications

6.4 Estimation of the CML Formation Rate
in Future GWDs

otrap is calculated as
−33.6 × 10
C0,77K = 6.0 × 10−3

3.3 × 10−1
−1
3.6 × 10
C0,4K = 1.6 × 10−1

1.5 × 10−1

(H2 O),(N2 ),(O2 ).

(6.19)(H2 O),(N2 ),(O2 ).

(6.20)

Substituting these values into Eqs. 6.13 and 6.16, one can obtain the conductance of cryotrap as
−3
(H2 O),
3.6 × 10−3(6.21)
CET = 5.8 × 10
(N2 ),

1.1 × 10−1 (O2 ).
The residual gas inside the room temperature beam duct is estimated as
5 × 10−9 Pa for H2 O, and 1 × 10−10 Pa for the others [30]. As the vacuum levelin the cryostat chamber is much better than the room temperature beam duct
because of the cryopumping, it is natural to adopt these values as pressure difference between the beam duct and the cryostat chamber. Thus the molecularflow can be estimated as
−11
(H2 O),
1.8 × 10−13(6.22)QET = 2.9 × 10
(N2 ), Pa · m/s
5.3 × 10−12 (O2 ).
It should be noted that we assumed ∆P = 5 × 10−11 Pa for N2 and O2 . From
Eq. 6.15, the formation rate is calculated as
−5
(H2 O),
6 × 10−6
ηET ∼ 4 × 10
(6.23)(N2 ), nm/day

−5
6 × 10(O2 ).
Assuming that the CML is a mixture of these molecules, the formation rate of
CML is approximately 1.2 × 10−4 nm/day ∼ 0.05 nm/yr for the ET. Therefore,
the CML formation can be tolerable for about 4-year operation of the ET
as long as it achieves the designed vacuum level with approximately 50 m
cryotraps.
1256.4 Estimation of the CML Formation Rate
in Future GWDs

6 Discussions and Implications

Figure 6.10: Schematic of CAD model of Voyager cryogenic duct. The lengthis 10 m with 500 mm diameter. Baffles are located at 1 m intervals. The yellow
part represents the space between the end of the duct and mirror surface.

LIGO Voyager
LIGO Voyager is planning to employ cryogenic ducts at temperature of 80 K
to isolate thermal radiation [88]. The length of cryogenic shield is about 10 m
as LIGO Voyager utilize the facility of LIGO.
Temperature of LIGO Voyager test mass is designed as 123 K where only
H2 O can be adsorbed, i.e., the adsorption of other species are negligible. Therefore, here we only consider the adsorption of H2 O molecules. Fig. 6.10 shows
the CAD model of Voyager cryogenic duct. The transmission probability Ktransof the LIGO Voyager cryogenic duct is simulated as
Ktrans,80K = 4.5 × 10−4 .

(6.24)It should be noted that the transmission probability without baffles is computed as 6.0 × 10−4 . Therefore, the baffles reduces the transmitted molecules
approximately by 25 %. Opening conductance is also calculated as
C0,Voyager = 23.37.

(6.25)

Then, the conductance of the Voyager cryogenic duct becomesCVoyager = 1.3 × 10−2 .

(6.26)For the case of LIGO Voyager, residual water molecules inside the beamtube is estimated as ∼ 1×10−10 Torr = 1.33×10−8 Pa [35]. Thus, the molecularflow is calculated as
QVoyager = 1.7 × 10−10 Pa · m3 /s.

(6.27)

Finally, the formation rate is estimated asηVoyager ∼ 7 × 10−4 nm/day.
126(6.28)

6.5 Possible Solutions
to Mitigate the Impacts of the CML

6 Discussions and Implications

Table 6.5: The estimated formation rate with designed vacuum level. The
values for the KAGRA case are derived from the reference [59].
Detector
KAGRA

The ET
VoyagerMolecules
H2 O
N2
O2
H2 O
N2
O2
H2 O

Formation Rate [nm/day]
6 × 10−1
3 × 10−1
2 × 10−1
6 × 10−5
4 × 10−6
6 × 10−5
7 × 10−4This result indicates that the thickness of CML can become approximately
0.2 nm after one year run, which can deteriorate the detector performance. It
should be noted that the actual diameters of cryotraps can become larger thanassumed here to avoid beam clippings. If the diameter of the beam duct islarger, the formation rate is also enhanced, and the CML becomes a problem much faster than the estimation shown above. Moreover, the unexpectedleakage or degas can happen during the operation, resulting in the CML formation. Therefore, the formation rate calculated here is an optimistic estimation.
Careful treatments on vacuum system and cryogenic mirror are still necessary.
The estimated results including KAGRA case are summarized in Tab. 6.5.
Table 6.6: Assumed parameters to compute the CML formation rate.
Molecule
H2 O
N2
O2

6.5

Density [kg/m3 ]Mass [kg]940
2.99 × 10−26
1026
4.65 × 10−26
1520
5.32 × 10−26

Possible Solutions
to Mitigate the Impacts of the CML

The optical absorption induced by the CML can become a critical problem
within one or two years operation with current proposed design. In this section,
1276.5 Possible Solutions
to Mitigate the Impacts of the CML

6 Discussions and Implications

we propose several approaches, which can mitigate the impact of the CML.

6.5.1

Longer Cryotrap

Cryotraps mitigate the CML formation as it reduces the molecular flow fromthe room temperature vacuum. Not only the CML formation, but also the heat
input to the test mass by thermal radiation is reduced by longer cryotrap. Ina cryogenic system, unknown heat sources may exist, which prevent one tocool down the system to the target temperature. Therefore, the designing of
the cryogenic system with extra cooling capacity will be desirable in case of
unforeseen heat sources. This can be achieved by employing longer cryotraps
for the case of the ET, in which a radiation heat from room temperature
vacuum is critical.
Current design of the ET adopts 5 m long for the 4 K cryotrap and 50 m
for 77 K, and they reduce the thermal radiation from the room temperature toapproximately 70 mW. Furthermore, by making the 4 K cryotrap 3 m longer,i.e., by employing 8 m long 4 K cryotrap, one can achieve less thermal radiation. It is estimated that such cryotrap can reduce the thermal radiation toapproximately 10 mW. Thus, the heat input due to the absorption of coatings
and thermal radiation becomes ∼ 30 mW. This smaller heat input relieves thetolerance on the CML, and can handle with unexpected heat source. Moreover, the formation rate of the CML is also reduced. The conductance of 8 mlong 4 K cryotrap becomes−2
(H2 O),
7.2 × 10
C4K,8m = 3.3 × 10−2 (N2 ),(6.29)

−2
3.1 × 10
(O2 ),and the formation rate is
−56 × 10η ∼ 4 × 10−6

2 × 10−5(H2 O),(N2 ), nm/day(O2 ).

(6.30)

Thus, the formation rate of the CML becomes approximately 8×10−5 nm/day ∼0.03 nm/yr.
As shown above, longer cryotraps can mitigate the impacts of the CML.
By employing 8 m long 4 K cryotraps, heat input due to the thermal radiation
and absorption of coatings becomes approximately 30 mW. In other words,
the ET can tolerate about 70 mW unexpected heat input, which corresponds
to the heat input caused by 1 nm thick CML. The CML formation also can be128

6.5 Possible Solutions
to Mitigate the Impacts of the CML

6 Discussions and Implications

mitigated as 0.03 nm/yr. Thus, the ET can be operated with 10 K test masses
for about 30 years.
For the case of LIGO Voyager, as the current LIGO facility will be used,
longer cryotraps are difficult to be installed. Cosmic Explorer (CE), a next
generation GWD planned in the US, may better employ 50 m-scale cryotraps
from the view point of the CML formation though the 10 m long cryotrap isenough from the view point of the thermal radiation.

6.5.2

Cooling Order

One possible approach to prevent the CML formation during the cooling period
is to cool down the mirror at the last stage of the cooling process, i.e., cool
down the surrounding components first, then start to cool down the test mass.
During this process, the residual gas molecules are adsorbed by the surrounding
cryogenic components, which mitigates the CML formation on the test mass
mirror surface. LIGO Voyager will employ radiative cooling system in whichcryotraps are cooled down prior to the test mass, resulting in the adsorptionof residual molecules to the cryotraps. Therefore, the CML formation on the
mirror during cooling process can be mitigated for LIGO Voyager. For thecase of the ET, long cryotraps are installed in the beam tube. By cling down
the cryotraps first, they can adsorb the residual gas molecules and the CML
formation on the test mass mirror is much reduced.
However, still the unwanted leakage or degas can happen accidentally. They
may cause unexpected CML formation, which cannot be predicted by thesimulation. The monitoring system, which can witness the CML, may benecessary to check the status of the mirror.

6.5.3

Monitoring and Desorption System

The CML formation can degrade detector’s performance, and the system which
can witness the CML may be needed to identify the optical loss source. As astarting point, we propose a conceptual design of a monitoring system whichis inspired by the P-S splitting measurement.
The schematic picture of the monitoring system is shown in Fig. 6.11. In
this scheme, the auxiliary cavity is employed in which the test mass mirror
takes the role of folding mirror. One can probe the cryogenic mirror surface
by monitoring the resonant frequency shift between two polarizations. This
enables to confirm the status of the mirror surface; whether the CML is formed
or not.
Tab. 6.7 describes the assumed parameters. Here we consider the caseof the ETM in the ET, and assume the mirror coating as proposed in the129

6.5 Possible Solutions
to Mitigate the Impacts of the CML

6 Discussions and Implications

Figure 6.11: Conceptual design of cryogenic mirror surface monitoring system.
Two auxiliary mirrors compose input and output mirrors, and the test mass
mirror takes the role of folding mirror. Hence these three mirrors consist afolding cavity. By detecting transmitted beam using a PD, one can measure
the two resonant peaks depending on the polarization.

reference [32]. We assume the finesse of the auxiliary cavity as 5 × 103 . Thenthe frequency shift can be calculated by using a characteristic matrix as∆ν(dCML = 0 nm) = −7.547 MHz.

(6.31)As the FWHM is approximately 10 kHz and the transverse mode spacing is28.7 MHz, the shifted peak is not contaminated by other modes. Therefore,
one can detect the both P- and S- polarization transmitted peak.
Fig. 6.12 shows the response of frequency shift against the CML formation.
The frequency shift changes linearly against the CML formation at very thinregion. Such response can be fitted by∆ν [MHz] = 0.022 [MHz/nm] × dCML [nm] − 7.547 [MHz].

(6.32)

By using this relationship, one can estimate the optical thickness of the CMLat thin region.
For the case of the ITM, the finesse of the auxiliary cavity becomes smaller
than that of the ETM due to lower reflectance of the ITM. The finesse of the130

6.5 Possible Solutions
to Mitigate the Impacts of the CML

6 Discussions and Implications

Table 6.7: Assumed parameters for the monitoring system.
Parameters
Value
Wavelength1550 nmFinesse∼ 5 × 103
Cavity length
L1 = 1.5 m
L2 = 1.5 m
L = L1 + L2 = 3 m
Free spectral range
50 MHzTransverse mode spacing
33.7 MHz
FWHM
7.5 kHz
Beam size on the input/output mirror
1.7 mmBeam size on the folding mirror
1.4 mmRadius of curvature
10 m
Folding angle
45 deg

ITM auxiliary cavity is 500 at most, hence νFWHM = 75 kHz. The resonant
frequency split with very thin CML is approximated as∆ν [MHz] = 0.022 [MHz/nm] × dCML [nm] − 7.487 [MHz].

(6.33)As the resonant frequency separation is still larger than the FWHM, the monitoring system can be valid for the ITM though the resolution of the frequency
split gets worse due to the lower finesse. Further tuning such as slightly different laser wavelength may improve the performance of the system.
In order for precise P-S measurement using such monitoring system, calibration lines around the split peak is useful.From the separation of the calibration line and resonant peak, one can determine the scanning efficiency ofthe laser frequency. By using the obtained scanning efficiency, the resonant
frequency split is calculated, and the thickness of the CML can be estimatedfrom Eq. 6.32. Also, the reduction of the mirror motion is important factor
for precise measurement. The beam jitter generated by the mirror motion
leads to the fluctuations in the split frequency. When the beam jitter is lessthan 200 µrad, the fluctuations in the split frequency becomes less than 7 kHz.
Then, one can measure the thickness of the CML with an accuracy of 0.3 nm.
A single or double pendulum can reduce the vibration contamination to themirrors, and enable the estimation of the CML thickness.
One advantage of such monitoring system is the linearity of response to themolecular layer formation. In addition, this system does not require a robust
control system to maintain the resonant condition for the P-S measurement.
131

6.5 Possible Solutions
to Mitigate the Impacts of the CML

Frequency split [MH ]6

6 Discussions and Implications

Model

4
2
0

−2
−4
−6
−80

50

100
150
CML thickness [nm]

200

250

Figure 6.12: Example response of the resonant frequency shift of assumed
monitoring system. Dashed line is a linear approximation at the thin region.

However, the sensitivity of monitoring system becomes about one orderof magnitude smaller than our setup though the folding angle is larger which
improves the sensitivity. This is because of the longer cavity length of themonitoring system. In other words, the FSR of the monitoring system becomes about two orders of magnitude smaller than our setup, resulting in lesssensitivity as shown in Fig. 6.5. Here we assumedthe cavity length to be 3 m
in order not to clip the main laser of the GWD.
The resonant width becomes small as the FSR decreases. Therefore, theerror due to the resonant width also is unchanged in terms of the CML thickness. The systematic error induced by the CML refractive index also does notchange because the error manifests itself as an error in the phase shift ∆φ,which is scaled by the FSR as shown in Eq. 4.72 when converted to the frequency shift. Therefore, the only error which increases as the FSR decreases
is the statistical error. This error can be reduced by increasing the numberof averaging to a large number like 104 . Manually performing such a large
number of measurements is impractical. Therefore, it is necessary to develop132

6 Discussions and Implications

6.6 Future Prospects

an automatic measurement system to perform the scan, data acquisition andpeak finding by a fit without human intervention.
The monitoring system proposed here should take an important role in
not only operation period, but also commissioning phase. During the commissioning, it is important to identify the noise sources and technical problems
which hinder the cryogenic operation. The cryogenic mirror monitoring system can provide significant information to determine the excess optical loss
source. Such monitoring system assists the commissioning work and achieve
the design sensitivity, resulting in discoveries in the Universe.
Towards the future GWDs, implementation in prototypes and optimizationof the geometric parameters will be of importance. Also, the obtained resultsby the measurements in the prototypes can provide insights into the design ofthe vacuum system such as length of the cryotrap and vacuum level. Further
studies will help developing the vacuum and cryogenic systems, and increase
the feasibility of future cryogenic GWDs.

6.5.4

Laser Induced Desorption System

The development of the system, which can actively remove the CML will also
be important. Previous study shows the possibility to desorb the CML byilluminating an auxiliary CO2 laser [59]. A CO2 laser can increase the temperature of a mirror. A thermal compensation system which employs a CO2 laser
is installed in current GWD such as aLIGO [89]. This previous study indicatesthat the adsorbed water molecules can be desorbed by increasing the temperature to about 170 K. However, such desorption has not been demonstrated
with an optical cavity. The cryogenic folded cavity and the cavity enhanced
ellipsometry, which we have developed, can be used as a test bench for thelaser induced desorption system. Further studies utilizing the folded cavity
can pave the way to develop a desorption system for future cryogenic GWDs,
which will be able to solve the problem generated by the CML.

6.6

Future Prospects

The cryogenic folded cavity developed in this study can be utilized for development of low thermal noise coating toward cryogenic GWDs. By locking
two higher-order modes to this cavity and taking beat note of them, one canmeasure the coating thermal noise of folding mirror [90, 91]. The device we developed here will enable the direct measurement of coating thermal noise evenat cryogenic temperature in addition to the optical loss. Combining thesemeasurements, one can develop high-end coatings for cryogenic GWDs.
133

6.6 Future Prospects

6 Discussions and Implications

In order to achieve the direct measurement of coating thermal noise, a vibration isolation system is necessary as the vibration contamination preventthe precision measurements of coating thermal noise. Developments of additional optical and readout systems are also required. Further studies utilizing
the cryogenic folded cavity will be able to contribute to the future cryogenic
GWDs.

134

Chapter 7
Summary and Conclusion
The detection of GW opened a new window to the Universe, which is called
GW astronomy. Future cryogenic gravitational-wave detectors will be able toprovide further discoveries such as the formation scenarios of massive black
holes by improving the sensitivity. The development of cryogenically cooled
mirror is an important key to achieve the future cryogenic GWDs. A number ofstudies were conducted to realize cryogenically cooled mirrors in future GWDs.
CLIO and KAGRA are pioneers toward the development of a cryogenic GWD.
For the construction and cryogenic operation of KAGRA, experimental studies
were performed to estimate the degradation time of PRG and to estimate the
CML formation speed. Based on these studies, KAGRA was designed andconstructed.
In future GWDs such as the ET or LIGO Voyager, employment of cryogenically cooled mirrors is an indispensable technology. These GWDs are planing
to employ longer wavelength laser than that of KAGRA to utilize silicon mirrors. At these longer wavelength, amorphous ice, which is a main component ofa CML, has two orders of magnitude larger absorption coefficient. Therefore,studies to investigate the impacts of CML at these wavelength is importantfor future GWDs.
We have theoretically estimated the impacts on optical loss induce by the
CML for future GWDs, which have not taken into account for their current
design. The results show the optical absorption by the CML can deteriorate
the performance of GWDs. The PRG, which is a merit of optical performanceof GWD, decreases due to the CML formation. We estimated the tolerable
CML thickness at which the degradation of PRG becomes 10%. The tolerable
thicknesses for the ET and LIGO Voyager are 60 nm and 25 nm, respectively,
while that of KAGRA is about 750 nm. Therefore, the optical performance of
future GWDs can be easily deteriorated by the CML formation.
Moreover, as the CML will be exposed to high power beam, it absorbs a135

7 Summary and Conclusionlarge amount of heat. Assuming the Lambert-Beer law, the author revealed
that heat input to the cryogenic mirror can exceed the cooling capacity even
if its thickness is about 1 nm, which can hinder one to reach the target temperature, resulting in worse sensitivity.
The heat input to the cryogenic mirror can exceed the cooling capacity
even at a ppm level optical loss and a nanometer level thickness for the case of
the ET because the absorption coefficient of CML is expected to be 2 ppm/nm
from the literature value of amorphous ice absorption coefficient. Therefore,
the optical loss characterization at a ppm level and a nanometer thickness levelis necessary to investigate the possible impacts on GWDs. Previous studies
investigated the impacts of CML on KAGRA, which employs shorter wavelength laser than the ET or LIGO Voyager. However, the impacts of CML atlonger wavelength such as 1550 nm or 2000 nm had not been studied so far. In
order to characterize the optical loss and thickness of the CML with a ppm
and nanometer levels at 1550 nm wavelength, we have developed a cryogenic
folded cavity which consists of three mirrors — input and output mirrors,
and one folding mirror. The device enables optical loss measurement by ringdown method at broadband temperature range, i.e., from room temperature to∼ 10 K cryogenic temperature. One advantage of folding configuration is that
one can characterize the optical loss of very high reflective mirror. Moreover,the folding mirror enables cavity enhanced ellipsometry measurement which
utilizes the phase shift induced by the folding mirror. The device and experimental method developed here enables us to characterize the optical loss andthickness of CML at longer wavelength, 1550 nm, where the impacts of CML
has not been studied so far.
The measurement of frequency shift between P- and S-polarization, called
P-S splitting measurement here, turned out to be a useful method to monitor the cryogenically cooled mirror surface. As the frequency split shifts with
CML growth, one can estimate the CML thickness. The sensitivity of thismeasurement to the CML thickness is mainly determined by the geometrical parameters, folding angle and FSR of the cavity. In our setup, folding
angle is set to 22.5 deg and FSR is 1.84 GHz. Also we assumed the CML refractive index as nCML = 1.19. From these values, the sensitivity to CML
thickness can be calculated as 0.56 MHz/nm. The uncertainty in the measurement of the CML thickness is introduced by the statistical error, measurement
error related to the resonant width and systematic error due to the CML
refractive index. Measurement error related to the resonant width and systematic error due to the CML refractive index are calculated as ±0.09 MHz
and ±0.01 MHz/nm, respectively. In our setup, the error due to the resonant width is converted as (±0.09 MHz)/(0.56 MHz/nm) = ±0.16 nm. Tak136

7 Summary and Conclusion
ing these into consideration, the uncertainty of the CML thickness becomes(±0.38±0.16±0.07) nm = ±0.42 nm.Therefore, the CML thickness can be measured by the cavity enhanced ellipsometry with an accuracy of about 0.5 nm.
By increasing the folding angle γ and the FSR of the cavity fFSR , a sensitivity
to CML thickness can be enhanced. Further optimizations of these geometrical parameters may enable CML thickness measurements with higher accuracy.
Through these measurements, we have achieved the optical loss characterization of CML at a ppm level and a nanometer level.
The measured optical loss of the CML shows large values even if its thickness is a few nanometer. When the thickness of the CML is 2.38 ± 0.27 nm, theoptical loss becomes 7.3 ± 1.1 ppm. This result indicates that heat input to thetest mass in the ET becomes larger than 100 mW with 2.4 nm thickness, i.e.,the heat input exceeds the cooling capacity. On the other hand, the obtained
result indicate that the Lambert-Beer law does not hold in very thin region
as it shows the discrepancy from the fitted curve assuming the Lambert-Beerlaw. A recent simulation study by the LIGO Voyager group indicate that the
optical absorption of a CML can deviate from the Lambert-Beer law at verythin region. However, the amount of deviation is too large compared to our experimental results. Further investigations are needed to precisely characterize
the optical loss for a very thin layer.
CML formation speed is an important factor from the view point of maintaining a performance of cryogenic GWD. The study performed in KAGRA
shows the CML formation speed of 27 nm/day. Our setup has worse vacuum level and larger solid angle to room temperature part compared to thoseof KAGRA. Therefore, large CML formation speed can be expected in our
measurement, about 1.1 × 103 nm/day. However, the CML formation speed
measured in our setup is only 0.6 nm/day, and about 2 × 103 times smaller
than the theoretical expectation from previous study in KAGRA. One possible reason is that this difference may come from the difference in geometry of
vacuum system between KAGRA and a table top experiment. Further studies
are needed to understand the CML formation speed both in GWDs and table
top experiments.
Besides the experimental study, we conducted numerical simulation to estimate the order of CML formation rate in the future GWDs. For the case of
the ET, the CML formation can be tolerable for 4 years with current design.
By employing 8 m long 4 K cryotrap instead of 5 m long one, the heat input
is reduced and the CML formation can be tolerable for 30 years. LIGO Voyager can suffer the formation of CML after a half year of operation even if it
achieves designed vacuum level. As the length of the cryotrap in LIGO Voyager is limited to about 10 m, it is difficult to mitigate the CML formation by137

7 Summary and Conclusion
employing longer cryotraps. Though they achieve the designed vacuum level,
unexpected degas or leakage can suddenly happen during long term operation,
and a certain amount of CML might be formed. Such accidental contamination reduces the detector’s performance and cannot be easily identified. Basedon the cavity enhanced ellipsometry measurement, we propose a conceptual
design of monitoring system for cryogenic mirror surface which can detect the
CML formation with high sensitivity. Such monitoring system will be able toidentify the CML formation and be useful not only for operation but also forcommissioning.
In the future, the device and ellipsometric measurement, developed in thisstudy, cab be used to develop the vacuum and cryogenic systems toward future
cryogenic GWDs. One possible application is a measurement in the prototype
detector such as Mariner, a prototype for LIGO Voyager [92]. Measurements
for characterization of the CML in such prototypes will provide the insights
into the formation rate or optical loss of the CML. Moreover, the cryogenic
folded cavity can be utilized for the R&D of the CML desorption system andthe direct measurement of coating thermal noise.
Our study revealed that the cryogenic molecular layer can become a criticalproblem in future cryogenic GWDs. The developed cryogenic folded cavity andcavity enhanced ellipsometry will help the design and the construction of future
cryogenic gravitational-wave detectors.

138

Appendix A
Direct Measurement of Thermal
Noise
The mirrors and their suspensions are thermally fluctuated because they areat a finite temperature. Thermal noise can be estimated by the fluctuationdissipation theorem (FDT) and relates to the mechanical loss materials. As
the mechanical loss of coating materials is much larger than that of substrates,
thermal noise from dielectric coatings, called coating thermal noise (CTN), sets
a limit in precision measurements such as gravitational-wave detection [20]. In
order to realize future cryogenic GWDs, the development of low thermal noise
coatings are necessary. Direct measurement of CTN at cryogenic temperature is important to develop new coatings. In this appendix, the direct CTN
measurement technique and possible upgrade plans toward the direct CTN
measurement are introduced.

A.1

Coating Thermal Noise

In order for a high reflective mirror such as a test mass in interferometric
GWD, multilayer coatings are indispensable though which introduced thermal
noise — this noise is called coating thermal noise (CTN). Thermal noise incoatings limits the sensitivity of the current gravitational wave detectors, thefrequency reference and macroscopic quantum measurements. In other words,
the improvement in the sensitivity of GWDs can be achieved by reducing the
CTN with low mechanical loss coating, resulting in larger number of detectionof GWs.
The fluctuation-dissipation theorem (FDT) describes the relationship between a fluctuating force and mechanical dissipation [93]. The single side power139

A.2 Reduction of CTN

A Direct Measurement of Thermal Noise

spectral density (PSD) of the observable is given by [94]Sx (f ) =8kB T Wdiss,(2πf )2 F02(A.1)

where kB is the Boltzmann’s constant, T is the temperature and f is the frequency. And Wdiss is the averaged dissipated energy in the coatings whensubjected to a sinusoidal force F (t) = F0 cos 2πf t, where F0 is the magnitude of the applied force to the system. The main mechanism which causes
the dissipation in coatings is the mechanical loss, and the dissipation can be
expressed as
Wdiss (f ) =2F02 d(1 + σ)(1 − 2σ)φc
f,
w02 Y

(A.2)

where σ and Y are the Poisson ratio and Young’s modulus of the coatings,
respectively, and w0 is the beam size.

A.2

Reduction of CTN

In order to reduce the thermal noises, several approaches are taken or being
developed. Thermal noise reduction methods are explained below.

A.2.1

Cryogenic Mirror

Thermal noise can be reduced by employing a cryogenic test mass because
thermal noise is proportional to the temperature. KAGRA employs cryogenic
sapphire mirrors for test masses and they are cooled down to 20 K [20]. The
future gravitational-wave detectors such as Einstein Telescope (ET) in Europe
is also planning to use cryogenic mirrors to improve the sensitivity [69].
Cryogenic mirror, however, can induce vibration contamination through theheat links which in turns an undesirable noise. In order to achieve very low
test mass temperature i.e., below 100 K where the radiation is not dominant,heat links are indispensable to extract heat from test masses. Heat links carry
not only heat but also vibration and become noise sources. Moreover, residual
gas molecules form molecular layers on cryogenic mirror surfaces which introduces optical loss i.e., scattering and absorption and reduce the finesse of thecavity [16, 58]. Therefore, one has to develop sophisticated cryogenic systems
to use cryogenic mirrors in gravitational-wave detectors.
140

A Direct Measurement of Thermal Noise

A.3 Direct CTN Measurement
by Using a Folded Cavity

Figure A.1: A schematic view of planned input optics for direct measurement of
coating thermal noise. There are three optical paths — one is to lock TEM00
mode, and other two are to lock TEM02/20 modes. The two higher-ordermodes are generated by shifting the laser frequency using AOMs. By taking
the beat note between TEM02 and TEM20 modes, one can measure the CTN.
The details of measurement scheme are reported in the reference [90, 91].

A.2.2

Beam Size and ShapeAs the CTN scales the beam size, the larger beam size leads to smaller CTN.
In order for larger beam size, however, bigger mirror substrates have to be
developed. In addition, high quality polishing of mirror surface and coating
deposition are necessary. Therefore, enlarging the beam size involves several
technical problems.
Another approach is to use a different beam shape. The substrate size canbe maintained in this scheme while the effective beam area is larger. Different
beam shapes can reduce the CTN such as mesa beam [95] or Laguerre-Gaussian
33 mode [96]. On the other hand, such beam shapes require special mirror
shape and can be unstable due to the parametric instability [97].
141A.3 Direct CTN Measurement
by Using a Folded Cavity

A.3

A Direct Measurement of Thermal Noise

Direct CTN Measurement
by Using a Folded Cavity

Direct measurement of CTN by using a folded cavity is achieved at room
temperature [90, 91]. In our experiment, a similar geometric folded cavity isused. Therefore, the cryogenic folded cavity developed here can be utilized
for direct measurement of CTN with several upgrades in the future. Fig. A.1
shows a planned schematic layout of the input optics.

A.3.1

Three Different Spatial Modes

In MIT experiment, they employ three different spatial modes to probe the
CTN.
Hermite-Gauss Modes
Laser beam has spatial modes which is called Hermite-Gauss (HG) modes. The
complete set of transverse distribution of HG modes can be expressed as [81]1
exp[i(m + n + 1)Ψ(z)]umn (x, y, z) =(2n+m−1 n!m!π)−1/2
w(z)
√ !
√ !


k(x2 + y 2 ) x2 + y 2
2x
2x
× Hn
Hm
exp −i
− 2
,
w(z)
w(z)
2RC (z)w (z)(A.3)
where n and m are the mode numbers. The order of the mode is n + m andHn (x) is the Hermite polynominals of order n.
In this measurement, three different spatial modes are used: one is TEM00,
and the other two are HOMs called TEM02 and TEM20. TEM00 mode isused for stabilization of the laser frequency via PDH locking. Then, the CTN
is sensed by two HOMs which have different spatial distribution.
TEM00 mode
TEM00 is the fundamental spatial mode of Hermite-Gaussian beam. In thisexperiment, TEM00 mode is used to stabilize the laser frequency via PDH
scheme. The TEM00 beam is kept on resonance by feeding back the error
signal to the laser.
TEM00 beam is phase modulated using EOM at 29.1 MHz for PDH locking
to a folded cavity, which stabilizes the laser frequency by feedback control.
Then the frequency stabilized laser will be split and shifted their frequencies
by double-pass AOMs in order to generate HOMs.
142

A Direct Measurement of Thermal Noise

A.4 Related Noise

TEM02/20 modes
TEM02/20 modes are the second lowest higher-order of HG beam which isused to sense the CTN in this experiment. There are a number of reasons
to choose the TEM02 and TEM20 modes for the CTN measurement. First,
as they are the second lowest higher order modes, coupling into these modes
has no first order sensitivity to the misalignment of the cavity axis relative tothe input beam axis. Secondly, even order modes of the optical cavity can be
excited by the input beam in the fundamental mode. No special optics are
required to achieve 12% of the power coupling. Lastly, modes of the sameorder are required to maintain a small separation in their resonant frequencies
and keep high common mode rejection to the cavity noises.
These modes are generated by shifting the laser frequency by using AOMs.
In our setup, two additional optical paths with double-pass AOMs are planed
to be employed to lock the TEM02/20 modes, and measure the CTN.

A.4

Related Noise

Direct measurement of CTN involves a number of noises. Here we introduce
several main noise sources which can prevent the measurement.

A.4.1

Seismic Noise

When we do precision measurements on the ground, seismic noise is an inevitable noise source which is dominant at lower frequency range. Seismic
motion causes mechanical vibration of optical table which couples to the readout signal via several mechanisms.
Fluctuations of the cavity length induced by the seismic motion couples tothe readout signal. The coupling can be reduced by common mode rejection.
Residual coupling between the TEM02 and TEM20 modes can be expressedbyN02/20 =∆f02/20
Nlengthf0(A.4)

where f0 = is the laser frequency and ∆f02/20 is the frequency difference between the two HOMs.
Vibrations of the cavity mirrors cause backscattering noise which couples
to the readout signal, which can be expressed as [90]N02/20 (f ) = (α02 cos θ02 − α20 cos θ20 ) N (f ).
143

(A.5)

A.4 Related NoiseA Direct Measurement of Thermal NoiseThe backscattering coefficient α can be determined bysλ2
α = BRDF 2 ,
πw0

(A.6)

where BRDF ∼ 10−6 sr−1 is the bidirectional reflectance distribution functionof the sample mirror. Backscattering coefficients are slightly different fromeach other as TEM02/20 modes sense the different area of the sample mirror
surface.
Mechanical motion of the input mirror induces noise in two ways. First,
the longitudinal motion Linput modulates the phase of the beams according to
the equation
N02/20 (f ) =Linput f
2π L.
λ
f0

(A.7)

Also the angular motion of the input mirrors modulates the intra-cavity power
which couples to the readout signal through the photothermal noise.
These noises induced by vibration are dominant at lower frequency region.
In order to reduce the vibration noise, a vibration isolation system is necessary. Such an isolation system not only reduce the noise, but also vibration
contamination from a cryocooler, which hinders the lock of the laser. On
the other hand, a vibration isolation system reduces not only the vibration
contamination, but also the cooling capacity below ∼ 100 K. Further studies
are needed to implement the vibration isolation system which can work at
cryogenic temperature.
Residual gas noise
The residual gas along the optical path induces fluctuations in the phase oflight through their refractive index. The noise level of the residual gas noiseis expressed as√
Sgas =(n0 − 1)2√8πL
(A0 /V0 )u0 λ



p
p0



T0
T

3/2
,

(A.8)

where n0 is the refractive index of the gas, V0 is the volume of one mole of gas at
the standard temperature, T0 , A0 is the Avogadro’s number and u0 is the mean
velocity of the gas molecule at the standard temperature [98]. At cryogenic
temperature, the vacuum level is enough high due to the cryopumping, andthe residual gas noise can be much smaller than the CTN.
144

A Direct Measurement of Thermal Noise

A.4 Related Noise

Shot Noise
Shot noise is fundamental noise which is derived from the nature of quantum
mechanics. Shot noise can be understood as photon counting error at a PD in aclassical view [99]. In a modern picture, it can be considered as a consequenceof the vacuum fluctuation in the electric magnetic field. It results in white
noise and can be expressed aspr
1 − J02 (Γ)M2hν
λ√Sshot =,
(A.9)
8F Pin 2M J0 (Γ)J1 (Γ)
where Jk (Γ) is the k-th order of the Bessel functions and M is the modematching ratio. As the shot noise is in proportion to the inverse of the squareroot of the laser power, its noise level can be reduced by increasing the laser
power. Larger laser power, however, introduces large heat input to the cryogenic folding mirror. Careful choice of the laser power may be required tomeasure the CTN.

145

A.4 Related NoiseA Direct Measurement of Thermal Noise146

Appendix B
Angular Response
In this appendix, we introduce the analytical calculation method for angularresponse of an optical cavity which reduces the cost for calculation.
Angular drifts directly introduce misalignment in the cavity axis. Such a
misalignment can deteriorate detector sensitivity and stability by increasinga few noise couplings including those to length fluctuation and beam jitter.
Moreover, angular misalignment causes light in the fundamental mode to becoupled into higher-order spatial modes, reducing the amount of power injectedto the main part of gravitational wave detector. This directly decreases thedetector sensitivity. Therefore, it is crucial to maintain the alignment of the
IMC to achieve the detector operation with the best performance. In order toreduce angular misalignment, an alignment sensing and control (ASC) system
is adopted by using the wave front sensing (WFS) technique [100, 101].
The WFS technique is sensitive to small tilts of the mirrors, and the WFS
signals are obtained by demodulating signals detected by quadrant photodiodes(QPDs) [102]. Since the signals derived by the WFSs have a Gouy phasedependence, one has to precisely estimate its behavior and choose the QPD
positions carefully in order to discriminate which mirror causes misalignmentsin either pitch or yaw. Previous report shows the analytical calculation of the
WFS signals [102, 103]. These approaches, however, contain complexities dueto the fact that the non-diagonal matrix elements are fully incorporated in thecalculation. Therefore, the WFS signals of the IMC have been traditionallycomputed by simulation tools such as Finesse [104] or Optickle [105], and it
was not easy to acquire intuitive interpretations.
The x and y axes of the coordinate system are chosen to be transverse to thebeam propagation (z axis) which is perfectly aligned. We additionally assume
that the y axis always points upwards regardless of the propagation direction.
This automatically means that the x axis needs to be mirrored every time thelight is reflected by a mirror. One can expand any paraxially approximated147B Angular Responseelectromagnetic fields of light beams by a set of Hermit-Gaussian (HG) modes
as [76]E(x, y, z) =X

hlm|EiUlm (x, y, z),

Zlm∞ Z ∞hlm|Ei ≡−∞∗Ulm(x, y, z)E(x, y, z)dxdy.

(B.1)
(B.2)

−∞

The coefficients hlm|Ei can be represented as the elements of the vector in
the modal space. For small misalignment due to the mirror tilts, the only
important modes are fundamental TEM00 mode and the second lowest-ordertransverse modes, TEM10 and TEM01 [102]. Angular motions of a mirror in
pitch correspond to rotations of the mirror about x axis and excites a small
amount of the TEM01 mode. Similarly, yaw is a rotation about the r axis,
exciting a TEM10 mode. For simplicity, we assume that an angular misalignment exists only in pitch and leave TEM01 mode only into the considerationin the rest of this section.
We now consider a simple setup as shown in Fig. B.1 and shows how thelinear approximation can be placed. Although a simple example is used, theargument here holds for any optical cavities as long as angular misalignment
are sufficiently small. We define a collection of the field vectors containing thetwo sets of the fields i.e., one set from node 1 and the other from node 2.
h00|Ei1~ 00 ≡ h00|Ei2  .
E
h00|Ei3(B.3)

These fields are related to each other and can be expressed in a matrix form [81],00r1 e−iΦ1
h00|Ei1r2 e−iΦ200  h00|Ei2 0r3 e−iΦ30h00|Ei3 
t1 h00|Ei
h00|Ei1 = h00|Ei2  ,0+
0h00|Ei3
(B.4)

where Φi (i = 1, 2, 3) is a one-way trip phase shift and ri are reflectivities ofeach mirror. The incoming beam is assumed to be perfectly aligned to thecavity. In other words, only a TEM00 mode is pumped into the system.
148

B Angular Response

Figure B.1: The electric fields and nodes of a triangular cavity system. Red
arrows and black dots represent electric fields and nodes, respectively. The
misalignment of input mirror induces perturbation in the promptly reflection.
The sign flip upon reflection of fields is labeled by ”+” and ”−” symbols.
To reduce clutter, we introduce a matrix and vectors as00
r1 e−iΦ100 ,
M̂00 ≡ r2 e−iΦ2
−iΦ3
0
r3 e
0


t1 h00|Ei~.
0E00,in ≡
0


(B.5)

(B.6)

Then, one can rewrite Eq. (B.4) as~ 00 + E~ 00,in = E~ 00 .
M̂00 E
149(B.7)

B Angular Response
Similarly, the TEM01 mode fields satisfy the following relation,~ 01 = E~ 01 ,
M̂01 E

(B.8)

where



h01|Ei1~ 01 ≡ h01|Ei2  ,E
h01|Ei300
r1 e−i(Φ1 +η1 ).
M̂01 ≡ r2 e−i(Φ2 +η2 )000
r3 e−i(Φ3 +η3 )
0

(B.9)

(B.10)

ηi (i = 1, 2, 3) represents a Gouy phase shift.
Small angular misalignments are introduced in pitch which transfers a fraction of the TEM00 modes to TEM01 and vice versa [102]. We will express these
perturbations using normalized rotation angles, Θi , (i = 1, 2). These angles arenormalized by divergence angle of the beam, λ/πw(z), and expressed as
Θi ≡ 2

πw(z)
θi ,
λ

(B.11)

where the θi is a small rotation of each mirror, λ is the wavelength of light andw(z) is a beam size at a position of z [102]. By using this normalized angle,
the angular misalignment matrix is defined as


0
0
0
−ir1 e−iΦ1 Θ10,(B.12)Θ̂ ≡ −ir2 e−iΦ2 Θ3
0
0
0
−iΦ3
Θ2
0
0
−ir3 ewhere Φ0i represents the one-way trip phase shift for TEM00 or TEM01 modes.
Then one can express the fields with pitch angular misalignments as~ 00 + E~ 00,in + Θ̂E~ 01 = E~ 00 ,
M̂00 E~ 01 + Θ̂E~ 00 = E~ 01 .
M̂01 E(B.13)(B.14)

Since the incoming beam is pure TEM00 mode, TEM01 mode only exists when
there is an angular misalignment. The fields of TEM01 mode can be writtenas~ 01 = (I − M̂01 )−1 Θ̂E~ 00
E~ 00,in + Θ̂E~ 01 ),= (I − M̂01 )−1 Θ̂(I − M̂00 )−1 (E150(B.15)

B Angular Response

B.1 Angular Response of a Triangular Cavity

Figure B.2: The block diagram.
~ 00 given by Eq. (B.13).
where I is the identity matrix Here we substituted E
Since the angular misalignment angle is so small that one can ignore higher
order term O(Θ̂2 ). In fact, this is the essence of our linear approximation. With
this linear approximation, TEM01 fields excited by angular misalignments canbe expressed as~ 01 ≈ (I − M̂01 )−1 Θ̂(I − M̂00 )−1 E~ 00,in .
E(B.16)

One can express TEM10 fields introduced by angular misalignments in thesame manner.
Our method, in comparison to the previous study which accompanies higher
orders to compute the angular response as shown in Fig. 4 in Ref. [102], obviously simplifies the resulting solutions at a cost of losing the accuracy inparticular when the amount of misalignment is large. However, in practicalexperiments, the angular drifts are suppressed by active controls and therefore
the amount of misalignment is typically small enough that our approximation
is valid.
Besides, another advantage of the linearization is that the entire analysis
can be expressed by a block diagram as shown in Fig. B.2 similarly to thoseused in the classical feedback control theory [106]. Such block diagrams canhelp ones to acquire comprehensive physical pictures and one can visualize how
the misalignment grows up inside the optical cavity.

B.1

Angular Response of a Triangular Cavity

We now consider the case in which a triangular cavity is in the form of isosceles
triangle with the length of one side much shorter than the other two.The
151

B.1 Angular Response of a Triangular Cavity

B Angular Response

incoming laser beam is assumed to be injected to the shorter side. Fig. B.2
shows a block diagram for the system with the mirrors misaligned by somesmall amounts.
The free-space propagator can be defined as(B.17)
P̂L,l = diag e−iΦL,l , e−i(ΦL,l −ηL,l ) , e−i(ΦL,l −ηL,l ) ,where its subscript, L and l, correspond to the length of propagation. The
matrices t̂ and r̂ denote the transmission and reflection matrix of the mirror
as,r̂i = diag(ri , −ri , ri ),(B.18)

t̂i = diag(ti , ti , ti ),(B.19)

where the subscripts, i = 1, 2, 3, express each mirror of the IMC. It should be
noted that the sign of reflectivity for TEM10 is negative due to the mirroring
effect which flips the coordinates left-to-right in the horizontal plane. Combining these matrices, the cavity round-trip propagator, P̂rt , can be expressedasP̂rt ≡ P̂L r̂2 P̂L r̂3 P̂l r̂1= re−iΦrt diag 1, −eiηrt , eiηrt ,(B.20)

where r ≡ r1 r2 r3 is a total reflectivity of the system which is merely a multiplication of the amplitude reflectivity from all the mirrors and ηrt represents around-trip Gouy phase. Using the round-trip propagator, one can express thefields inside the cavity as~ cav = (I − P̂rt )−1 t̂1 E~ in .
E(B.21)

In practical experiments, the phase modulated laser beam which can be
split into the carrier and the sidebands is widely used in order to lock the laser
to the cavities. Assuming that the cavity is kept resonant for the carrier i.e.,e−iΦrt = 1, then one can define the cavity gain for the carrier in the form ofeach HG modes as
1
,
1−r
1
G10 ≡
,
1 + reiηrt
1
G01 ≡
.
1 − reiηrt

G00 ≡

152

(B.22)(B.23)(B.24)

B Angular Response

B.2 Beam Spot Response]
Figure B.3: The block diagram.
It should be noted that the sign of cavity gain is flipped between TEM01 andTEM10 modes due to the mirroring effect for horizontal axis.
On the other hand, the cavity gain for the sidebands which are kept offresonant can be written as
GSB
00 ≡

1
.
1+r

(B.25)

Here we assumed that the reflectivity for both upper and lower sidebands are
identical. When the reflectivities of mirrors are high i.e., ri ≈ 1, the cavity
gain for the sidebands are much smaller than that of the carrier. Therefore,
ignoring the effect of the sidebands inside the cavity, we will consider only the
carrier field for the intra-cavity fields. Then Eq. (B.21) can be rewritten as~ cav = diag(G00 , G10 , G01 )t̂1 E~ in .
E(B.26)

Thus each HG mode excited by angular misalignment is built up inside thecavity as characterized by each cavity gain.

B.2

Beam Spot Response

The response to the beam spot positions is one of the most important properties to understand the behavior of a triangular cavity. The lateral shift ofthe beam on each mirror can be derived by evaluating the real part of thecoefficients for the 01 and 10 modes. For the purpose, let us expand a field
into real and imaginary parts as,
A (U00 + CU01,10 )153(B.27)

B.2 Beam Spot Response

B Angular Response

where C is a complex number which can be expressed using a small beamtranslation a and a small tilt α [107],
C=

a
kw0
+i
α,
w0
2

(B.28)

where w0 is the beam size at its waist and k is the wavenumber. Eq. (B.28)
tells the real part of C denotes the beam translation and the imaginary part
denotes the beam tilt. Therefore, the beam spot translation can be derived
from the real part of normalized coefficient, C, of TEM01/10. The notationsof angular tilts and axes on the mirrors are as shown in Fig.B.3.
Evalating the real part of the field on each mirror and approximating all
the mirrors to be a perfect reflector i.e., ri = 1 (i = 1, 2, 3), one can find thefollowing relation between the mirror tilts and resulting variation in the beam
spot as
 

 
θ1
g
1 g + 2ly1Ry2  = −R  1
11
θ2  ,(B.29)2l
θ3
y3
g+ R 1gwhere g is the g-factor, defined as
g ≡1−

L+l
,
R

(B.30)and yi (i = 1, 2, 3) describe vertical displacements of the beam along the y-axis.
A detailed calculation for obtaining Eq. (B.29) is presented in Appendix A.
Since MCs 1 and 3 have an incident angle of approximately π/4, the y1
axis and the y3 axis are rotated π/4 around the z axis with respect to the yaxis [104]. Therefore,
an effective misalignment angle in pitch is reduced by√cos(π/4) = 1/ 2, so that1
ψ1,3 = √ θ1,3 .
2(B.31)

The axis y is same as the axis on the mirror v. Hence the beam spot displacement on the mirror can be written as
√

 
 
2l
g
2
g
+
v1
ψ1
R
√
R 
v2  = − √ψ2  .
(B.32)
1
2
1
√
2
2l
v3
ψ3
g+ R
2
g
Eq. (B.32) shows us how the eigenmode of triangular cavity transforms as afunction of the tilt of each mirror.
154

B Angular Response

B.2 Beam Spot Response

Mirror positions may not always be preferable DoFs. They can be converted
into the ones representing the translation and tilt of the beam at the waist,and the spot position on MC2 as    v1 +v3   1
v1
yw
0 12
2
2
θw  ≡  v1 −v3  =  1 0 − 1  v2  .
(B.33)
2l
2l
2l
v2
0 1 0
v3
v2
In addition, we convert the actuation basis into MCs 1 and 3 common anddifferential motions expressed as    ψ1 +ψ3   1
1
0
ψ1
ψc
2
2
2
ψd  ≡  ψ1 −ψ3  =  1 0 − 1  ψ2  .
(B.34)
2
2
2
0 1 0
ψ3
ψ2
ψ2
By using these bases, the beam spot response to pitch angular misalignment
can be summarized as  √ 2(L − R) √0 −Rywψc θw  = .
ψ(B.35)
2 0
d
√0
v2
ψ2
0 −R
− 2R
Eq. (B.35) shows that misalignment in MC2 cannot access to the tilt at the
waist position, and only shift the plane of cavity eigenmode. Common angulartilt rotates cavity eignemode around the center of curvature as shown in figure
19. in [108]. Differential tilt does not change the spot on the curved mirror
and the waist, though the tilt at waist is rotated.
In the same way, yaw misalignment can be computed as
 
  
ξ − gl −ξ
x1
φ1
l 
x2  = − gl L+l
−
φ2 .
g
g
l
φ3
x3
−ξ − g ξ(B.36)

where xi is defined as ξ ≡ L + l(1 − L/R)/g. When the displacements perpendicular to the beam, x1 and x3 , are projected on each mirror, MCs 1 √and3, the displacements on the mirror surface are enhanced by 1/ cos(π/4) = 2.
Therefore, Eq. (B.36) can be written as√ l√    √
2ξ
−
2
−
2ξ
h1
φ1
g

 
l
L+l
l
h2  =  − g−
φ2 .
(B.37)√g l √ g √
h3
φ3− 2ξ − 2 g2ξ
155B.2 Beam Spot Response

B Angular Response

The bases can be converted in the manner same as that for pitch by using therelationship expressed as    (h1 +h3 ) cos(π/4)   1√
0 2√1 2h1xw
2 2
2
1   θw  ≡  (h1 −h3 ) cos(π/4)  =  √1√
0 − 2 2lh2 ,(B.38)
2 2l
2l
h
h2
h2
0 1
0
3
   φ1 +φ3   1
 
1
φc
0 2
φ1
2
2
φd  ≡  φ1 −φ3  =  1 0 − 1  φ2  .
(B.39)
2
2
2
0 1 0
φ2
φ3
φ2
√
It should be noted that the effect of 2 enhancement due to the projection on
MCs 1 and 3 is cancelled. Then the beam spot response to yaw misalignments
can be expressed as
  
 
0
−2L
0
xw
φc
R
 θw  =  2(−L+R)


.
0
−
φ
(B.40)
d
L−R+l
L−R+l
R(L+l)
2Rl
h2
φ20− L−R+lL−R+l
Eq. (B.40) shows that only misalignment in MCs 1 and 3 differential motion
can access to the beam waist shift. The beam waist tilt and MC2 spot positionchange can be expressed by a combination of MCs 1 and 3 differential motionand MC2 motion.
These results are fully consistent with the previous analysis based on thegeometrical argument [108]. Therefore, this linear approximation offers analternative method to compute variations in the spot positions without elaborating the geometrical analysis.

B.2.1

Wave Front Sensing Signals of the IMC

The WFS technique is sensitive to small tilts of the mirrors. The signals
are obtained by demodulating the intensity information detected by quadrant
photodiodes (QPDs) [102] in similar manner to the Pound-Drever-Hall (PDH)
technique [77]. When one of the mirrors is misaligned, it excites the second
lowest-order transverse modes, namely the TEM10 or TEM01 in the carrier.
The WFS signals are essentially the interference between the TEM00 of the
sidebands and the TEM01 or TEM10 modes of the carrier.
We assume that the incoming light is a purely phase modulated TEM00
mode described as
h00|Eiin~ in =  0  .
E(B.41)
0156

B Angular Response

B.3 Implication to Experiment

Phase modulated input laser beam, which is used in PDH scheme, can be
expressed ash00|Eiin ≈ [J0 (m)eiΩt +J1 (m)ei(Ω+ωm )t− J1 (m)ei(Ω−ωm )t ] × h00|Ei(B.42)

where Ω corresponds to the angular frequency of the carrier field, ωm is themodulation angular frequency, m is the modulation depth and Jk (m) is Bessel
functions of the first kind.
By measuring the differential of reflected field using a horizontally and vertically split photodiodes, QPD, for pitch and yaw, one can obtain the intensity
information, SQPD , which can be expressed as
Z ∞ 
 Z 0


†
~ ·E
~ refl −~† · E~ refl .
SQPD ∝
dx E
dxE(B.43)
refl
refl
−∞

0

The fields at reflection port of each mode can be calculated by multiplying
propagators. The WFS signals are obtained by demodulating the signals,
SQPD , with the modulation angular frequency, ωm . After some math, one canderive the WFS signals described as ∗WFS = iP0 J0 (m)J1 (m) U00(G00 G10,01 eiη̄ ΘU10,01 )− U00 (G00 G10,01 eiη̄ ΘU10,01 )∗ ,(B.44)
where P0 denotes the optical power of the input laser beam and η̄ is a spcific
Gouy phase which depends on the misaligned mirror. When we apply parameters of KAGRA IMC, the WFS signals can be calculated as shown in Fig.
B.4.

B.3

Implication to Experiment

The IMC has three DoFs for angular misalignments in both pitch and yaw.
On the other hand, the number of observable DoFs by the WFS technique
is only two since the WFSs sense the difference between the two beams —the incoming and intra-cavity beams. In this section, the observable degrees
of freedom (DoFs) of the IMC are discussed for the case where the incoming
beam is injected to the shorter side of the IMC such as KAGRA [20]. For theother case where the incoming beam is injected to longer side of the IMC suchas aLIGO [23], the results are shown in Appendix B for completeness.
The singular value decomposition (SVD) is one of the useful tools in orderto understand what DoFs can or cannot be sensed. The SVD is widely used
157

B.3 Implication to Experiment

B Angular Response

for data analysis [109]. The WFS signals contain beam tilt and translation
components and are expressed as bellow by using a sensing matrix Ŝ,
 


θ1
tilt
= Ŝ θ2  ,(B.45)
trans.
θ3
where Ŝ is a 2 × 3 matrix and its elements can be written by using the cavity
gain and Gouy phase shift. This sensing matrix can be expressed by geometrical parameters with high reflective approximation (ri ≈ 1) and incoming beamis pure TEM00 mode. Then we will apply SVD to rewrite them to determine
which DoFs are observable by the WFS technique.
In the case of the geometry used in KAGRA [20], one can obtain therelationship between the WFS signals and mirror tilt angles in pitch as
 

 ψ1


√
gγ
0
−gγ  tilt√
ψ2 ,(B.46)= 2γw0trans. Pitch−g − 2 −gψ3where γ ≡ (L + l)/zR , and w0 is beam size at its waist. Adopting the SVD
analysis, one can arrive at



 1
0
−1
√
√
gγ 0 0 −g −√ 2 −g  .
ŜPitch = 2γw0 I(B.47)0 1 0
1 − 2g 1
It is clear that one the DoF shown in the lowest row in the rightmost matrix.
On the other hand, the first and second rows are observable. They correspondto MCs 1 and 3 differential motion and the combination of MCs 1 and 3 common and MC2 motion, respectively. Comparing to Eq. (B.35), one can identify
that the observable DoFs correspond to the beam waist tilt and translation.
For the case of yaw, the sensing matrix can be written as
 

 φ1


√
−g 1 −g  
tilt
φ2 .
(B.48)= 2γw0−gγ 0 gγtrans. Yaw
φ3
The decomposed form can be expressed asŜYaw =√



−1 g1 −1
g 0 0 2γw0 I−1 0 1  .
0 gγ 0
1 2g 1


158(B.49)

B Angular Response

B.4 Conclusion

The observable DoFs are the combination of MCs 1 and 3 common and MC2
motion and MCs 1 and 3 differential motion. Similarly to the pitch case, theseobservable DoFs correspond to the beam waist tilt and translation. Therefore,
when the beam is injected to the shorter side of the IMC, the beam waist tilt
and translation are observable.

B.4

Conclusion

We have presented the linear approximation method that can simplify the
analytical calculation of angular response of optical cavities. The method
enables us not only to simplify the calculation but also to make equivalent
block diagrams which can bring comprehensive pictures.
The method was applied to a triangular cavity called the input mode cleaning cavity. The angular response to the shift in the beam spots and the WFS
signals of the IMC in gravitational wave detectors were explicitly given by thisapproach. The results are consistent with the previous report which conducted
a completely different approach — geometrical analysis.
Finally, the WFS sensing matrices are analyzed via the singular value decomposision scheme in order to study the observable DoFs of the IMC. Whenthe incoming beam is injected to the shorter side of the IMC, the observable DoFs by the WFS scheme are determined to be the beam waist tilt andtranslation.
The linear approximation method presented in this article can be applied to
optical systems which are more involved, including the main interferometer ofgravitational wave detectors. The linear approximation method should be able
to give deeper insight into the fundamental comprehension of the gravitational
wave interferometers.

159

N rmalized WFS signals [1/rad]B.4 Conclusion

B Angular Response

MC1
MC2
MC3

5000

0

−5000

−10000−15000
0

30

60

90

120

150

180

N rmalized WFS signals [1/rad]Accumlated G uy Phase fr m MC1 refl [deg]
MC1
MC2
MC3

20000

10000

0

−10000−20000

0

30

60

90

120

150

180

Accumlated G uy Phase fr m MC1 refl [deg]Figure B.4: The WFS signals of KAGRA IMC. Upper figure shows the pitch
WFS signals and lower figure shows that of yaw.

160

Acknowledgements
First of all, I am grateful to professor Yoichi Aso, who has supervised me. He
gave me a good opportunities to work on the various fields in gravitational
wave detection.
I thank Dr. Kiwamu Izumi for giving me an attractive and instructive talk.
I learned the interferometric techniques from him.
I would like to professor Matthew Evans, and Dr. Slawomir Gras who
supervised me when I visited MIT LIGO Lab.
I would like to thank the people who have helped me out — Ryutaro
Takahashi, Tomotada Akutsu, Matteo Leonardi, Naohisa Sato, Naoatsu Hirata, Yoshikazu Namai, Ayako Ueda, Koki Okutomi, Yoshinori Fujii, KunihikoHasegawa and Naoki Aritomi.
The administrative office members in NAOJ, Mizuho Yoshizumi, Mikiko
Harada, Megumi Ohyama, Mie Ueda, and Eri Sakamoto supported me bytaking care of paperwork when purchasing something, attending meetings,
going on trips, and so on. Thanks to their help, I was able to concentrate myresearch.
This work was supported by JSPS KAKENHI Grant Number JP18K03681
and JSPS Grant-in-Aid for Specially Promoted Research 26000005. This work
was partially supported by Research Fund for Students (2017) of the Department of Astronomical Science, The Graduate University for Advanced Studies,
SOKENDAI.
Finally, I thank my parents, brother, and grandparents for allowing me todo the research.

161

B.4 Conclusion

B Angular Response

162

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