High Fidelity Probe and Mitigation of Mirror Thermal
Fluctuations

Thesis by

Tara Chalermsongsak

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

California Institute of Technology
Pasadena, California

2014(Defended May 22, 2014)

ii

c 2014
Tara Chalermsongsak
All Rights Reserved

iii

Acknowledgments
Each day, after spending strenuous hours of lab work and about to call it a night, I liked to lookaround the setup of the work. Not only did I think about what could be improved or what tobe fixed, but I also reminded myself about how far I had accomplished with a lot of help fromseveral people who taught me all about the optic mounts on the table, electronic equipments, oreven softwares for controlling the setup. I consider myself very fortunate to learn from many of thesmartest people I have ever known over the years.
First, I have to thank Rana Adhikari for taking me under his wing and entrusting me to thisexciting experiment, and for his patient for teaching someone who’s not the fastest learner like me.
I’m also grateful for Eric Gustafson, another boss who always has indispensable advice on my work.
I also thank Yanbei Chen, Kenneth Libbrecht, and Alan Weinstein for serving on my committee.
I learned many crucial lab skills from Frank Seifert whose help contributes significantly to mywork. Also, I am grateful to Koji Arai, Peter King, Vladimir Degarchev, Matt Abernathy, Nicolas
Smith, Steve Vass, Jamie Rollins, Jan Harms, Aidan Brooks, Alastair Heptonstall, Manasa Thirugnanasambandamand, and Larry Price. They were always ready to answer any questions I had.
Without them, I would be like a blind man roaming in a minefield.
The work would not be made possible without all the electronic supports from LIGO engineers,
especially, Daniel Sigg, Rich Abbott, and Charles Osthelder. They have been very kind to help me
through these years.
Also, I like to thank all the TNI members, Riccardo DeSalvo, Eric Black, Akira Villar, and Greg
Ogin from their guidance during my first few years at Caltech. Their researches motivated my study
in the following years.
I appreciate all the help from my grad student fellows, David Yeaton–Massey, Jenne Driggers,
Kiwamu Izumi, Zach Korth, Eric Quintero, Denis Martynov, and Evan Hall. I used to struggle withproblems alone, and it was devastating. They taught me how to help and learn from each others
and the value of teamwork.
SURF students who worked with me, Rafael Cervantes, Megan Daily, Larissa Thorne, Vanessa
Acon, Sarah Terry, Erica Chan, and Chloe Ling helped me with various lab tasks. I’m glad I had
chances to co-mentor these brilliant students.

iv
I’m also indebted to all Physics and LIGO members, especially Donna Driscoll, Julie Hiroto, andGina Salone for their help and support on various works.
I also have to thank Laura F. Kim, Daniel Yolder and Caltech ISP team who have helped mewith all the issues for an international student like me.
I am lucky to know some Caltech Thai students who always share laugh and cry with me. Special
thanks to Tichakorn Wongpiromsarn and Piti Ongmongkolkul for several useful helps and discussions
in maths and physics.
Above all, I would like to thank my family. I am indeed fortunate to have them standing besideme throughout my life. The pursuit of Ph.D. would not have been possible without their unfailinglove and support.

v

Abstract
Thermal noise arising from mechanical loss in high reflective dielectric coatings is a significant
source of noise in precision optical measurements. In particular, Advanced LIGO, a large scaleinterferometer aiming to observed gravitational wave, is expected to be limited by coating thermal
noise in the most sensitive region around 30–300 Hz. Various theoretical calculations for predicting
coating Brownian noise have been proposed. However, due to the relatively limited knowledge of thecoating material properties, an accurate approximation of the noise cannot be achieved. A testbed
that can directly observed coating thermal noise close to Advanced LIGO band will serve as an
indispensable tool to verify the calculations, study material properties of the coating, and estimate
the detector’s performance.
This dissertation reports a setup that has sensitivity to observe wide band (10 Hz to 1 kHz)thermal noise from fused silica/tantala coating at room temperature from fixed-spacer Fabry–Perotcavities. Important fundamental noises and technical noises associated with the setup are discussed.
The coating loss obtained from the measurement agrees with results reported in the literature.
The setup serves as a testbed to study thermal noise in high reflective mirrors from different
materials. One example is a heterostructure of Alx Ga1−x As (AlGaAs). An optimized design tominimize thermo–optic noise in the coating is proposed and discussed in this work.

vi

Contents
Acknowledgmentsiii

Abstractv

1 Introduction to Thermal Noise

1

1.1

1.2

Thermal Noise in Various Measurements . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

The Brownian Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.2

Johnson-Nyquist Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.3

Micro Electromechanical Systems: MEMS . . . . . . . . . . . . . . . . . . . .

3

1.1.4

Torsion Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.1.5

Optical Clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.1.6

Laser Interferometer Gravitational Wave Observatory . . . . . . . . . . . . .

4

1.1.6.1

A Short Review on Gravitational Wave . . . . . . . . . . . . . . . .

5

1.1.6.2

LIGO Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.1.6.3

Thermal Noise in LIGO . . . . . . . . . . . . . . . . . . . . . . . . .

6

Direct Observations of Thermal Noise in High Reflective Mirrors . . . . . . . . . . .

8

1.2.1

Previous Coating Thermal Noise Measurements . . . . . . . . . . . . . . . . .

8

1.2.2

Coating Brownian Noise from Fixed Spacer Fabry–Perot Cavity

9. . . . . . .

2 Thermal Noise in Fixed–Spacer Fabry–Perot Cavity

11

2.1

Fluctuation-Dissipation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.2

Direct Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.3

Two Kinds of Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.3.1

Brownian Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.3.2

Thermodynamic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

Thermal Noise in Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.4.1

Brownian Noise in Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.4.2

Thermoelastic Noise in Substrate . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.4.3

Thermo–Optic Noise in Substrate . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.4

vii
2.5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.5.1

Brownian Noise in Coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.5.2

Thermo–Optic Noise in Coatings . . . . . . . . . . . . . . . . . . . . . . . . .

21

Thermal Noise in Spacer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.6.1

Brownian Noise in Spacer . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.6.2

Thermoelastic Noise in Spacer . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.7

Photothermal Noise: Absorption from Shot Noise and Intensity Noise . . . . . . . .

25

2.8

Noise in the Bonding between a Mirror and a Spacer . . . . . . . . . . . . . . . . . .

26

2.8.1

Optical Contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.8.2

How to Measure Mechanical Loss in Optical Contact . . . . . . . . . . . . .

27

How to Reduce Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.9.1

Lower Temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.9.2

Lower Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.9.3

Different Beam Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.9.4

No Coating Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.6

2.9

Thermal Noise in Coatings

3 Experimental Setup

32

3.1

Optical Cavity as a Frequency Reference. . . . . . . . . . . . . . . . . . . . . . . .

33

3.2

Prototype Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.2.1

8–inch Reference Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.2.2

Setup Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.2.3

Result from the Prototype Setup . . . . . . . . . . . . . . . . . . . . . . . . .

40

Two Laser Setup with 1.45 Inch Cavities . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.3.1

1.45 Inch Reference Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.3.2

Overall Explanation of The Two–Laser Setup . . . . . . . . . . . . . . . . . .

43

3.3.3

Result from 1.45 Inch Cavities . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.3

3.4

4 Technical Noise in the Setups
4.1

49

Seismic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

4.1.1

Coupling from Seismic to Displacement Noise . . . . . . . . . . . . . . . . . .

49

4.1.2

Scattered Light and Seismic Isolation in the Setup . . . . . . . . . . . . . . .

51

4.2

Mechanical Peaks from Opto–Mechanical Components . . . . . . . . . . . . . . . . .

51

4.3

PDH Lock: Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

4.4

PDH Lock: Electronic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

4.5

PDH Lock: Residual Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . .

56

4.6

Noise from Phase Locked Loop Readout . . . . . . . . . . . . . . . . . . . . . . . . .

57

viii
4.7

Photothermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4.8

Noise from Ambient Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

5 AlGaAs Crystalline Coatings

63

5.1

History of AlGaAs Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

5.2

Thermo–Optic Noise and Coatings Optimization . . . . . . . . . . . . . . . . . . . .

64

5.2.1

Thermal Noise in AlGaAs Coatings . . . . . . . . . . . . . . . . . . . . . . . .

65

5.2.2

Optimization Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

5.2.3

Uncertainties in Material Parameters . . . . . . . . . . . . . . . . . . . . . . .

69

5.2.4

Errors Due to Manufacturing Process . . . . . . . . . . . . . . . . . . . . . .

70

5.2.5

Optimization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.2.6

Cavity Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

Implications for Advanced LIGO . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

5.3

6 Future Upgrade and Application
6.1

6.2

83

Future Upgrade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

6.1.1

Improvement in Sensitivity at Low Frequency . . . . . . . . . . . . . . . . . .

83

6.1.2

Improvement in Sensitivity at High Frequency . . . . . . . . . . . . . . . . .

85

Application: Frequency Stabilized Light Source . . . . . . . . . . . . . . . . . . . . .

87

6.2.1

Laser Gyroscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

6.2.2

Crackle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

6.2.3

External Cavity Diode Laser . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

A Comparison with Other Experiments

91

B Frequency Stabilization Servo95

C Material parameters for SiO2/Ta2O5 and AlGaAs

99

C.1 Table of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

C.2 Uncertainties in Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

C.3 Note for AlGaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
D High Reflective Coating Structure and its Average Properties

101

D.1 Why Half Wave Cap? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
D.2 Averaged Material Parameters for Coatings . . . . . . . . . . . . . . . . . . . . . . . 102
E 1.45 Inch Reference Cavity: Design and Assembly

104

E.1 Choosing Cavity Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
E.2 Assembling Reference Cavity with Optical Contact . . . . . . . . . . . . . . . . . . . 105

ix
E.3 Eigenmode study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
E.4 Searching for Optimum Support Point with COMSOL . . . . . . . . . . . . . . . . . 111
E.5 Mount Design and Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
F Pre-Mode Cleaner Cavity

117

F.1 Eigenmode Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
G Mathematical Note

119

G.1 Note About Fabry–Perot Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
G.2 Temperature Fluctuations for TO Calculation . . . . . . . . . . . . . . . . . . . . . . 120
G.3 Error Analysis of the Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
G.4 Bayesian Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Bibliography

127

1

Chapter 1

Introduction to Thermal Noise
For a system in thermal equilibrium at temperature T, the Equipartition Theorem tells us that the
average kinetic from each degree of freedom will be equal to 21 kB T . For example, a 1-D mass-spring
system with only one degree of freedom, will oscillate with an average energy of 12 kB T . At first,
it seems that the only movement will be x(t) = x0 sin (2πf t), a perfect sinusoidal motion with its
frequency f determined solely by the mass m and the spring constant k. However, if the spring has
a mechanical loss which can be written as an imaginary part of the spring constant k = k0 (1 + iφ),where φ  1, this loss will cause an off-resonant motion that creates noise in other frequencies. In
the next chapter, we will discuss more about Fluctuation-Dissipation theorem, which explains how
loss and fluctuation in a system are related (for this chapter, we can just remember that a systemthat can dissipate the input energy, can exhibit fluctuations as in the mass-spring system).
This minute motion manifests itself in a larger scale and has been observed in various experiments,
and becomes one of the fundamental limiting sensitivity for sensitive measurements. This chapter
will give some examples of thermal noise in measurements.

1.1

Thermal Noise in Various Measurements

1.1.1

The Brownian Movement

When small particles (e.g., pollen grains) are suspended in still water with no influence from the surrounding, the particles will move randomly. This motion was observed in 1827 by Robert Brown [1]and it was named as Brownian motion. The particle’s movement originates from continuous random
collisions from the surrounding water molecules that move due to their thermal energy kB T . It had
been left as a mysterious phenomenon until explained by Einstein and verified by Jean Perrin [2, 3],almost a century later. This explanation of Brownian motion, which won the Nobel Prize in Physics
in 1926 for Perrin, also confirmed the existence of atoms and molecules. The study also served as
one of the proofs of the Equipartition Theorem [4].

2

1.1.2

Johnson-Nyquist Noise

Voltage noise in resistors relating to the temperature had been observed by Johnson in 1928 [5](Fig. 1.1), and explained by Nyquist in the same year [6]. The cause is electrons’ movement insidea resistor due to thermal agitation. Nyquist proved it using an argument that certain wave pattens
can be formed in a system of two resistors joined by two transmission lines together with the
Equipartition Theorem stating that each degree of freedom has an energy equal to kB T . In general,the power spectral density of the voltage noise is
V 2 (f ) =4hf ∆fexp[ khf
]−1
BT

(1.1)

Usually, in most cases when hf  kB T ( For example, T = 300K, f can be up to 100GHz),
Eq. 1.1 is reduced to a more familiar form,
V 2 (f ) = 4RkB T ∆f.

(1.2)

Figure 1.1: Apparent power[V 2 /R] vs. temperature, for Advance wire resistances [5]Aside from this flat Johnson-Nyquist noise, there is another known noise in a resistor. It is
directly proportional to the current flow and its power density varies with 1/f . It has been observedin various metal films and found to be temperature dependent [7, 8]. This excess current noise is
the bunching and releasing of electrons associated with current, e.g., due to fluctuating conductivity
based on imperfect contacts within the resistive material. Hence the amount of this current–noise
depends largely on the methods used to manufacture resistors [9, 10]. For metal films, this noise
was found to be originated from the substrate on which the thin film is deposited [11, 12].

3

1.1.3

Micro Electromechanical Systems: MEMS

Developments in micro-machining process make small tools at micron scale possible. Typically, they
consist of simple mechanical devices (e.g., a mass-spring system, a cantilever blade) with sizes ofaround 100 to 1000 microns, (see Fig. 1.2). These devices are suitable for space-limited applications
because they are small and light. Similar to any mechanical systems, they are also subject to thermal
noise due to intrinsic mechanical losses in the material [13]. Although the devices are usually limitedby the related electrical noise and systematic errors, mechanical thermal noise sets the fundamental
limit of the devices and it is important to characterize them for the improvement of the higher
accuracy devices. Some examples of MEMS devices and their thermal noise considerations can befound in Ref. [14] for a frequency selective device using the mechanical resonance of a small cantileverblade, Ref [15] for a gyroscope, and Ref. [16] for a Fabry–Perot tunable etalon.

Figure 1.2: Left, a schematic diagram of a capacitive MEMS accelerometer. Right, a micrograph of
the ADXL50 MEMS accelerometer’s structure. Credit: www.sensorsmag.com

1.1.4

Torsion Pendulum

A torsion pendulum is another sensitive tool for determining the magnitude of a small force. Oneof the most famous experiment using this technique is Cavendish experiment [17] from which thegravitational constant value G can be derived. The torsion pendulum technique has been used forvarious experiments. For instance, the equivalence principle assumes that the inertial mass andthe gravitational mass are the same, but this might not be true at all time. The Standard Model
predicts the violation of the equivalent mass during a short period of time [18]. An experiment using
a torsion pendulum helps improving a limit on the equivalence-principle violations [19]. Anotherexample is a prediction from quantum electrodynamics (QED), the Casimir effect. It states thattwo parallel conduction plate placed closely to each other will be mutually attracted [20]. This effectcan be demonstrated using a torsion pendulum [21].

4Torsion pendula can be used to study the influence of solar eclipses on some mechanical processes
on earth [22, 23], although, in this case, they are not used for determining any small forces. The
oscillation period of the torsion pendulum increases during the solar eclipse. This phenomenon has
not been properly explained.
Similar to other sensitive measurements, a torsion pendulum is subject to thermal noise due toloss in the torsion spring. A direct measurement to observe thermal noise in a torsion pendulum
was done with a polyamide Nylon wire, and the result was in good agreement with the FluctuationDissipation theorem [24].

1.1.5

Optical Clock

The optical clock [25] has been quickly developed as another frequency standard, which is importantin various fields (e.g., primary standards, time distributions services, global navigation satellite systems (GPS)). Three main components of an optical clock are (1) a highly stable reference frequency(a narrow optical absorption line in an atom or ion)1 , (2) a laser (local oscillator), which is locked
to that reference frequency, and (3) a femtosecond comb for measuring the frequency of the laser.
The laser has two important tasks; it must probe the reference transition (the atomic system) while
introducing minimal noise, and it must act temporarily as stable frequency reference to maintain the
short to medium term stability of the frequency reference [27]. To do this, the laser is also frequency
locked to a reference cavity. Thermal noise in that reference cavity, which will be studied in details
in this dissertation, becomes one of the main limiting noise source of the laser’s performance.

1.1.6

Laser Interferometer Gravitational Wave Observatory

The existence of gravitational waves was predicted as a consequence of the theory of general relativity [28, 29, 30]. As a gravitational wave propagates through space, it contracts and expands the space
periodically. Although observations on the displacement due the warped space has never been madebecause of their extremely weak signals, there is strong evidence suggesting their existence. In 1974,
a piece of evidence of gravitational waves was observed from the shrinking orbit of a binary systemof two neutron stars, PSR1913+16 (one of which is also a pulsar) due to energy loss in the formof gravitational waves [31, 32]. The prediction of general relativity agrees with the measurement toaccuracy of about 0.1 percent, after 30 years of data [33].
Another example is recent news at the time of writing this dissertation, researchers from the
BICEP2 collaboration [34] have announced they found evidence of cosmic inflation and reported theimage of gravitational waves from the Big Bang by observing the B-modes polarization of the cosmic
1 For optical lattice clock [26] the atoms are kept into an optical lattice instead of trapped by laser cooling. The
method results in a better precision.

5
microwave background (CMB) [35]. This new finding is currently under peer review for scientificconfirmation. Nevertheless, gravitational wave is real.
In 1990s, scientists proposed a large scale interferometer to measure gravitational wave (LIGO2 [36]).
The configuration based on a Michelson interferometer with 4–km arms. Later, a network of groundbased observatories (VIRGO3 [37], GEO4 [38], KAGRA5 [39])have been built around the world toincrease the confidence and to pinpoint a location of an event. Some possible GW sources thatwill be detectable by the observatories are: signals from binaries, coalescing binaries, supernovae, or
stochastic signals.
1.1.6.1

A Short Review on Gravitational Wave

In General Relativity, space-time is described by the mass-energy following the Einstein’s field
equation:
18πGRµν − Rgµν = 4 Tµν .
2
c

(1.3)

Rµν is the Ricci curvature tensor, R is the scalar curvature, gµν is the metric tensor, which explains
the structure of the space-time. The stress-energy tensor Tµν describes the density and flux ofenergy, mass, and momentum in space-time. See Ref. [40] for more details. In a place far away fromthe presence of mass or energy, the space time metric gµν can be approximated asgµν = ηµν + hµν with |hµν |  1,

(1.4)

where ηµν is the flat space-time (Minkowski) metric and hµν is a small perturbation due to thegravitational wave. In free space (Tµν = 0), substituting this space time metric to Einstein’s field
equations and using an appropriate gauge choice (transverse traceless gauge), the Einstein’s field
equation reduces to a wave equation for h:∇2 hµν =∂ 2 hµν∂t2(1.5)

Due to the constraints of the choice of gauge and the field equation, hµν can be decomposed into
two linearly independent modes, which are h+ and h× . Conventionally, h+ squeezes and stretches
along the x̂ and ŷ directions while h× will stretch and squeeze along x̂ + ŷ and x̂ − ŷ directions.
If a gravitational wave with h+ polarization propagates along the z-axis, an object with length L
oriented along x or y axes will have the maximum displacement due to the strain equal to∆L = Lh+ .
2 http://www.ligo.org
3 http://www.ego-gw.it/virgodescription/pag_4.html
4 http://www.geo600.org
5 http://gwcenter.icrr.u-tokyo.ac.jp/en/

(1.6)

6
For a binary system with its orbital plane being parallel to x-y plane, the strain h is [41]:h(z, t) =−4G2 M1 M2cos(2πf (z − ct)/c),
c4 rz

(1.7)

M1 , M2 are the masses of the two bodies separated by distance r. For the binary system PSR1913+16,
M1 = 1.44M , M2 = 1.39M , and M

is the solar mass, r = 2.3 light seconds, z = 20, 000 lightyears. Substituting these numbers in Eq. 1.7, we get the strain around 1.3 × 10−22 . However, thewave frequency is around 0.1 mHz, making it undetectable for the current sensitivity of the groundbased observatory, see Fig. 1.4.
1.1.6.2

LIGO Setup

Due to the nature of a gravitational wave that warps space-time as it propagates, a Michelsoninterferometer can be a suitable tool for detecting the wave. A schematic diagram of LIGO is shownin Fig. 1.3. The input laser is separated into two directions orthogonal to each other. They bounce
off of the high reflective mirrors at the end of each arm and recombine at the output. The differential
length between the two arms, induced by the passing wave, results in the varying power of the output
monitored by a photodiode.

Figure 1.3: A simplified schematic diagram of LIGO. It is a Michelson interferometer with a
Fabry–Perot cavity in each arm. The test masses are suspended for seismic isolations.

1.1.6.3

Thermal Noise in LIGO

Since the output signal of the interferometer is proportional to the differential length between the
two arms, any displacement noise will corrupt the sensitivity of the detector. For example, seismic

7
motion on the ground that shakes the test masses’ positions will prevent LIGO from detecting
gravitational waves at frequencies below 10 Hz. The estimated noise budget of Advanced LIGO isshown in Fig. 1.46 .
Quantum noise
Seismic noise
Gravity Gradients
Suspension thermal noise
Coating Brownian noise
Coating Thermo−optic noise
Substrate Brownian noise
Excess Gas
Total noise−21

Strain [1/√Hz]

10

−22

10

−23

10

−24

10

1

10

2

10
Frequency [Hz]3

10

Figure 1.4: Baseline Advanced LIGO’s noise budget (GWINC v2.0). 125 W input power [43].
Coating Brownian noise is plotted in confidence interval where uncertainties of coatings’ mechanicalloss angles and Young’s moduli are taken into consideration.
Thermal noise comes into LIGO sensitivity in two places, the suspensions and the test masses.
• Thermal noise in suspension
To reduce seismic noise due to the ground’s motion, LIGO test masses are suspended withmultiple stages for seismic isolations [44]. Similar to thermal noise in a torsion pendulum, thesuspension wires joining each stage also come with thermal noise because of the loss in thewires [45]. Another example of thermal noise measurement in a mechanical flexure suspension
can be found in Ref. [46]. Not only the losses in the suspension wires that contribute to thedisplacement noise, but also the dissipation associated with the whole suspension system, e.g.,
the magnet-coil actuators, joints between a suspension wire and the test mass, maraging blade
springs, gas damping due to residual gas molecules, or electrostatic damping due to currents
induced in structures and charges. These noise sources must be minimized to achieve thetarget sensitivity.
It is worth mentioning that dislocations that cause mechanical losses in materials may also
cause displacement noise in the audio band from sub Hz-frequency driving force. This noise,called crackle noise, shows up in other nature phenomena, e.g., earthquakes, or the sound of a
6 For a comprehensive review of relevant noise sources in Advanced LIGO, see Ref. [42].

8crumbling paper. An experiment to test this phenomenon in maraging steel blades is on goingat Caltech [47]. The device under test will be driven at sub hertz frequency in order to observe
the noise in audio band interferometrically.
• Thermal Noise in Test Masses
As the output from the asymmetric port of the beam splitter is directly related to the differential arm lengths, thermal noise in the test masses changes the test mass surfaces’ positions anddirectly couples to the output signal. It is estimated that the sensitivity of Advanced LIGO
in the most sensitive band (30 Hz to 300 Hz) will be limited by thermal noise in the highreflective coatings, see Fig. 1.4. Although quantum noise is another dominating noise sourcethat covers the whole measurement bandwidth, it can be reduced below coating thermal noise
by frequency dependent squeezed light [48, 49]. The radiation pressure noise at low frequencies
can be reduced by amplitude squeezing, while the shot noise limit at high frequencies can be
reduced by phase squeezing.
Since coating thermal noise has been identified as an important limiting noise source, researchon coatings has rapidly grown. Many groups have developed mathematical models to calculatecoating thermal noise [50, 51, 52, 53]. However, due to the coating’s multilayer structure anduncertainties in the thin film material parameters (e.g., Young’s moduli, Poisson ratios, andloss angles), thermal noise in coatings is not thoroughly understood. For this reason, a test
facility, which can measure coating thermal noise with high signal-to-noise ratio across a wide
frequency band relevant to Advanced LIGO, is necessary for a comprehensive verification. The
setup for this coating thermal noise measurement will be the one of the main topics in thisdissertation.

1.2

Direct Observations of Thermal Noise in High Reflective
Mirrors

1.2.1

Previous Coating Thermal Noise Measurements

Direct observations of thermal noise in high reflective mirrors were made by Numata et al. andBlack et al. [54, 55]. The results, measured between 1 kHz and 10 kHz band, were in good agreementwith theoretical predictions. These experiments used four independently suspended mirrors to form
two Fabry–Perot cavities that were locked to a frequency stabilized laser and sensed the coating
thermal noise via feedback signals to the cavities. The sensitivity of this method, however, is limited
by seismic noise at frequencies below 1 kHz, even with the passive seismic isolation of the suspended
large test masses (4 inches in diameter, 3 inches thick). Therefore, the noise level between 10 Hz to300 Hz has yet to be verified experimentally. The measurement will be important for characterizing

9Advanced LIGO performance since it is predicted to be limited by coating thermal noise in thisbandwidth. Additionally, the setup similar to that of TNI requires larger, heavier substrates toovercome seismic noise for measurements at lower frequencies. Big custom made substrates are
expensive for testing several samples. A setup that uses smaller commercially available substrates
is preferred. An alternative setup is needed to overcome such issues.

Figure 1.5: Inside of TNI’s main vacuum chamber, ca. 2007. The suspended test masses are on theleft, the triangular Mode Cleaner cavity is on the right. Picture credit: Akira Villar

1.2.2

Coating Brownian Noise from Fixed Spacer Fabry–Perot Cavity

In the field of optical frequency metrology, a fixed–spacer Fabry–Perot cavity (see Fig. 1.6) is typicallyused as a stable reference for locking laser frequency. By designing the shape of the spacer, andsearching for vibration insensitive support points, several groups have demonstrated that the total
displacement noise of a rigid cavity can be very close to the thermal noise limit from the highmirror coating at frequencies around 0.01 - 1 Hz [56, 57, 58]. However, none have reported Brownian
thermal noise in the frequency band relevant to ground based GW detectors.7
This motivates us to develop a testbed that can directly observe coating thermal noise in the
10 Hz to 1 kHz bandwidth using fixed spacer Fabry–Perot cavities and standard 1 inch diameter
substrates. Fortunately, the spacer does not introduce a lot of excess displacement noise. Thermal
noise associated with a spacer made from fused silica is predicted to be much smaller than that indielectric coatings [59, 60]. More details are discussed in chapter 2.
Another advantage of the fixed spacer setup, besides the lower cost for the samples, is the shorter
turnaround time because of commercially available substrates and the less complicated suspension
7 See the comparison among the experiments in appendix A

10system. It takes approximately a month for each set of samples to be installed and measured. In
addition, as the measurement can be done optically from the transmitted beams, the calibration isrelatively simple (cf. section 4.6) compared to the readout technique used in TNI measurement [55].
Furthermore, an error related to spot size and cavity’s length in a fixed–spacer cavity will be less thanthat of a suspended cavity. Better knowledge on the spot size leads to more accurate coating thermal
noise calculation. In the next chapter, thermal noise in all parts of a fixed-spacer Fabry–Perot cavity
will be discussed in detail.

Figure 1.6: An 8 inch reference cavity used in LIGO as a low noise reference cavity, the spacer andmirror substrates are made from fused silica. The cavity is suspended with two spring wires fitted
around the Airy point. The eddy current damper can be seen on the top of the plate.

11

Chapter 2

Thermal Noise in Fixed–Spacer
Fabry–Perot Cavity
Typically, a Fabry–Perot cavity consists of three parts: two mirror substrates, high reflective coatings
deposited on the substrates, and a spacer with a bore hole along the beam line axis for the beam topropagate in vacuum. The substrates are usually optically bonded to the spacer on both ends, as
shown in Fig. 2.1. To use a fixed-spacer Fabry–Perot cavity in a coating thermal noise measurement,
all fundamental noises related to the cavity have to be considered.
This chapter will start with the Fluctuation-Dissipation theorem where the coupling between
the dissipation and the fluctuation in a system is derived. Then the discussion will move to explain
how a Gaussian beam resonating inside the cavity senses thermal noise contributed from each partof the cavity. All assumptions and formulas used to generate the thermal noise contributions will be
explained. Numerical values of the relevant parameters and symbols are given in Tab. 2.1. A brief
discussion on several efforts to reduce thermal noise is included at the end of the chapter.

12

Figure 2.1: Assembly of a 1.45-in spacer used in this dissertation. The substrates are opticallycontacted on both ends of the spacer. The vent hole for using the cavity in a vacuum system canbe seen.

Symbol

Name

unit

f

temporal frequency

Hzclight speed in vacuumm/s

Sx (f )

single-sided power spectral density

m2 /HzkB

Boltzmann constant

J/K

T

Mean temperature

K

C

Heat capacity per volume

J/Km3

κ

Thermal conductivity

W/mK

Y

Young’s modulus

N/m2

σ

Poisson’s ratioαThermal expansion coefficient

1/K

βThermorefractive coefficient, ∂n/∂T

1/K

n

Refractive index

λBeam wavelengthmνcarrier frequency

Hzω0Beam radius (1/e2 power)md

Coating thicknessmφMechanical loss angleLcavity lengthm

Table 2.1: Symbols, constants, and material parameters frequently used in this chapter. The subscripts s and c following a parameter will be used to denote a property of the substrate or thecoating.

13

2.1

Fluctuation-Dissipation Theorem

Let’s look back to Brownian motion of a particle in water. When the particle is left alone, watermolecules randomly and continuously collide with it and move the particle around. On the other
hand, if the particle is dragged through the water, these random collisions from the water will act asa frictional force trying to dissipate the particle’s kinetic energy. From these two situations, we seethat the fluctuation of the particle’s position (or velocity) and the dissipation of the particle’s energy
under the influence of an external force are both caused by the same phenomenon: the collisions withthe surrounding water molecules. This implies that the fluctuation and the dissipation are related,as they share the same origin. The coupling between them given by the Fluctuation-Dissipation
Theorem (FDT) [61] is 1

Sx (f ) =kB T
Re[Y (f )],π2 f 2(2.1)

where f is Fourier temporal frequency, Y (f ) is the mechanical admittance associated with x(t). The
admittance can be computed by looking at a generalized force F (t) that drives the system in one of
the generalized coordinates, x(t). The admittance is the Fourier transform of the derivative of x(t)divided by the applied force which is
Y (f ) =2πif x(f )ẋ(f )=.
F (f )F (f )(2.2)

Let’s see an example in a mass-spring system. A mass m is attached to a spring with spring
constant k(1 + iφ). The equation of motion can be written as
F = k(1 + iφ)x + mẍ.

(2.3)

The equation can be transformed to the frequency domain via a Laplace transformation. Rearrange
the equation to get,

ẋ
iω
=
F
k − ω 2 m + ikφ(iω)(k − ω 2 m − ikφ)=, and(k − ω 2 m)2 + k 2 φ2kφωRe[Y ] =,(k − ω 2 m)2 + k 2 φ2

Y =where ω = 2πf . Substituting this back into Eq. 2.1, in the limit of 2πf (2.4a)(2.4b)(2.4c)qkm we get

1 A comprehensives review about the Fluctuation–Dissipation can be found in Ref. [62], and an introduction to amodern aspect of the fluctuation-dissipation theorem can be found in Ref. [63]

14

Sx (f ) =2kB T φ
.
πf k

(2.5)

This equation captures all significant characteristics of thermal Brownian noise. It is directly proportional to the temperature and the loss of the system. It is inversely proportional to the frequency
and the stiffness of the material (spring constant). In general, φ can be frequency and temperaturedependent.

2.2

Direct Approach

Thermal noise in a beam-optic system is not as simple as the mass-spring example. Previous
calculations to determine thermal noise were done by a modal expansion, for example, in a suspension
system [64], or in a mirror of a laser interferometer gravitational-wave detector [65]. This method
calculates the admittance and sums up contributions from all vibration modes contributing to thestrain of the system. However, it is computationally expensive, complicated, and the result is
not guaranteed to converge [66]. Additionally, this method cannot handle non-uniform loss in thesubstrate. The dielectric coatings turn out to be more lossy than the mirror substrate. As the coating
is closer to the beam than the substrate, its effect contributes to the surface fluctuation even more.
Instead of using modal expansion, one can use the so-called “direct approach” to compute Re[Y (f )].
This was introduced by González and Saulson [67] for computing thermal noise in suspensions, andwas later applied to a laser mirror by Levin [66]. In this approach, one calculates the thermal noise
by applying a cyclic force which causes power dissipation in a lossy system. With the FDT, thedissipated power Wdiss and the PSD Sx are related by
Sx (f ) =2kB T Wdiss,
π 2 f 2 F02(2.6)

where F0 is the magnitude of the applied force to calculate the dissipated power. In the case of
mirror whose position is interrogated by a laser beam, the cyclic “force” applied is a pressure withthe same profile as the intensity of the beam.

2.3

Two Kinds of Thermal Noise

As we can see from Eq. 2.6, any dissipation due to the applied force creates thermal noise in asystem. For a solid system, there are two known sources of dissipation: mechanical loss and thermal
dissipation. Mechanical loss is caused by impurities or dislocations in the material, and displacement
noise associated with this mechanism is usually referred to as Brownian noise. Thermal dissipation,
often called Zener damping [68], occurs due to irreversible heat flow down temperature gradients

15
which create temperature fluctuations in a system.

2.3.1

Brownian Thermal Noise

Mechanical loss arises from the internal structure of a material, such as impurities or dislocations [69].
It is represented by introducing an imaginary part to the Young’s modulus of the material: E =E0 (1 + iφ). The quantity φ is referred to as the loss angle, and may be temperature and frequency
dependent. When a sinusoidal force is applied to a system subject to mechanical loss, the dissipatedpower due to the applied force is given as [66]Wdiss = 2πf U0 φ,(2.7)

where U0 is the maximum energy of elastic deformation. The loss angle φ is equal to 1/Q, where Q
is the quality factor. In general, Q can be measured from a ring down measurement: an amplitude ofan oscillation with resonant frequency f0 decays over time as A(t) = A(0)Exp[i2πf0 t]Exp[−tπf0 /Q].
If one is interested only in frequencies f below the first mechanical resonance frequency of thesystem (as is the case with our reference cavities), it is sufficient to compute the stored energy U0 in
the presence of a static force. The problem of evaluating Wdiss then reduces to a single elastostaticcomputation, which can be carried out with finite element analysis (FEA) if necessary. Together
with Eq. 2.6, one can then calculate the Brownian contribution to the apparent position fluctuationof the mirror as sensed by a laser beam interrogating the mirror surface.

2.3.2

Thermodynamic Noise

In contrast to Brownian noise, thermo–optic noise is related to thermal, rather than mechanical,
dissipation; it arises from fluctuations in the temperature field T (r, t) throughout the mirror [70].
In classical solid state thermodynamics, these fluctuations are not correlated with the volume fluctuations driven by Brownian noise. The two effects have to be summed incoherently.
In general, for a volume V, the total variance of temperature fluctuations can be described by [71]< δT 2 >=kB T 2.
CV(2.8)

With non-zero thermal expansion α, and thermorefractive β, in coatings, a mirror and a spacer,
the thermal fluctuation is converted into displacement noise or phase noise as the beam is reflected
off of the mirror surface.
To compute thermo–optic noise using the direct approach, one can apply either an imaginary
force [52, 72] or imaginary heat [73, 74] to the mirror’s surface; the results will be the same if thestress inside the coating is uniform [52]. The applied force will cause temperature gradients inside

16
the mirror through the equation of static stress balance [75]∇(∇ · u) + (1 − 2σ)∇2 u = −2α(1 + σ)∇δT,(2.9)

where u is the displacement vector of the mirror body due to the applied pressure. Then, thetemperature perturbation evolves according to the thermal diffusion equation as [75]∂(δT )
κ−αY T ∂(∇ · u)
− ∇2 (δT ) =.
∂t
C
C(1 − 2σ)∂t(2.10)

Finally, the power dissipation due to the heat flow caused by the temperature gradient is given by
the expression [75, Eq. 35.1]
Wdiss =

T

dS
dt



Z
=


κ
d3 r (∇δT )2 .
T

(2.11)

Here T is the unperturbed temperature of the system and δT is the temperature perturbation due
to the applied force F0 . The entropy S of the system changes due to the heat flux −κ∇(δT ), andthe angle brackets denote an average over the period of oscillation of the force. By substituting
Eq. 2.11 into Eq. 2.6, we can obtain the temperature fluctuation on the mirror sensed by a Gaussianbeam. This fluctuation couples into the electromagnetic response of the mirror via α and β.
In the literature, the term “thermoelastic noise” refers to physical displacement noise of themirror’s surface position due to thermal expansion of the substate and the coating [76, 72, 77, 78].
In a beam-optic system, the change of the mirror position causes the phase shift of the reflected
beam. On the other hand, “thermorefractive noise” refers to the phase fluctuation of the beam as
it propagates through or reflects off the mirror (the path length of the beam changes even though
the mirror surface position is fixed), and it is a combined effect of both α and β. This is a slightmisnomer, and it will be emphasized again.Since both thermoelastic and thermorefractive noises
arise from a common origin, they are computed in a coherent fashion and the combined effect iscalled thermo–optic noise [74].

2.4

Thermal Noise in Substrate

A substrate is the bulk of a mirror on which the coating is deposited. In a Fabry–Perot topology, thelight goes through the input substrate, and bounces back and forth on the surfaces before leaving theend substrate. The displacement noise on the substrate from both Brownian noise and thermoelastic
noise will change the cavity length. Thermo–optic effects in the substrate will be negligible comparedto the other two. We will start the discussion with the substrate because of its relative simplicity
compared to that of the coating.

17

2.4.1

Brownian Noise in Substrate

By using the direct approach, Brownian noise from a substrate as sensed by a Gaussian beam isgiven as [66],Sx (f ) =2kB T 1 − σs2
φs .
π 3/2 f w0 Ys(2.12)The result here is obtained from a half-infinite model, where the spot size is much smaller thanthe mirror size. Corrections for a finite size mirror are given in Ref. [72]. For our case, where spot
sizes will be on the order of 200-300 µm, with 1 inch diameter and 0.25 inch thick substrates, thecorrection is negligible.

2.4.2

Thermoelastic Noise in Substrate

For Initial LIGO, the test masses were made of fused silica. In hopes of improving the sensitivityof the detector, scientists had proposed to switch test masses’ material from fused silica to sapphiredue to its lower mechanical loss. However, Braginsky et al. [76] pointed out that thermoelastic noise
had to be taken into consideration. As a result, a sapphire substrate was predicted to have higher
total thermal noise than that of a fused silica substrate.
For the case of a half-infinite substrate in the adiabatic limit lth  w0 , where lth =pκs /(2πCs f )is the thermal diffusion length at frequency f , and κs and Cs are, respectively, the thermal conductivity and the heat capacity per unit volume of the substrate, thermoelastic noise in substrate canbe expressed as
4kB T 2 αs2 (1 + σs )2 κs√.
π 5 Cs2 w03 f 2

SxT Esub (f ) =(2.13)

Later, a non-adiabatic correction for low frequencies and small beam sizes was computed by
Cerdonio et al. [77, Eq. 20], with some minor corrections as stated in Ref. [54, 79] :4kB TSx(subTE) (f ) = √
π

2 α2
s

2

1 + σs w0
J f /fT ,κs(2.14)p
where fT = κs /πw2 Cs , and J(f /fT ) is a non-elementary function whose asymptotes are 2/ 3 πf /fT
2
for f /fT  1 and 1/ f /fT for f /fT  1; the full expression is 1/2 Z∞Z∞
2u3 e−u /2
2

.
dv 2
J(f /fT ) =duπ(u + v 2 ) (u2 + v 2 )2 + (f /fT )20(2.15)−∞

At f  fT , or in the adiabatic limit, Eq. 2.14 reduces to Eq. 2.13. The simplified equation
presents how thermoelastic noise level changes explicitly with the spot size and frequency.
An experiment at the TNI measured thermoelastic noise in sapphire mirror at room tempera-

18
ture [79], and found a good agreement with the theoretical calculation, showing that sapphire, evenwith lower mechanical loss than that of fused silica, has a higher total noise level due to thermoelastic
noise.

2.4.3

Thermo–Optic Noise in Substrate

In a Fabry–Perot cavity topology, the beam passes through each substrate only once. Hence,
thermo–optic noise in the substrate is usually much smaller than other thermal noise in the cavity.
An analytical result for thermo–optic noise in a finite–sized cylindrical substrate sensed by a Gaussian beam is given in Ref. [80]. The frequency noise on the signal due to this effect is on the order√of 10−7 Hz/ Hz which will be negligible for our setup.

Frequency noise [Hz/rtHz]

10

10

10

−7

TR noise in substrate, fused silica 300um beam size

−8

−9
0

10

1

10

2

10
10
Frequency [Hz]

3

4

10

10

5

Figure 2.2: Frequency noise of the beam passing through a quarter inch thick fused silica substrate.
It is negligible compared to other noise source as shown in Fig. 3.5 and Fig. 3.9.
Note that the “thermo–optic” noise calculation in Ref. [80] only takes the phase noise due to β(∂n/∂T ) into account (it is also called thermoref ractive noise in the original paper). In reality, theoptical path of the output beam is also changed by thermal expansion of the substrate due to thetemperature change. For example, see the calculation for thermal noise in optical fiber where thethermal noise from β and α in fiber optic are calculated coherently [81]. By using β + nα insteadof just β, the effect from thermal expansion will be included. As the calculation here includes thecontribution from α, the effect is called thermo–optic noise. For fused silica, the effect will not bealtered by much because the contribution from thermal expansion is an order of magnitude smaller.

19

2.5

Thermal Noise in Coatings

One of the most important conclusions derived from the direct approach calculation is that the loss
from the reflecting surface contributes to the noise significantly [66]. As a result, attention shiftedto study more about the dielectric coatings. The coatings are usually more lossy than the bulk
substrate, so they are expected to have higher thermal noise. Additionally, because of the multilayer
structure and how light interacts within the coatings, thermal noise inside the coatings becomes
more complicated, as we will see in this section.

2.5.1

Brownian Noise in Coatings

Brownian thermal noise in a thin film on a half-infinite substrate can be expressed as [50]Sx(cBR) (f ) =4kB T (1 + σs )(1 − 2σs ) d
φc ,
π2 f
Es
w02

(2.16)

where d is the total thickness of the coating, and φc is the coating’s loss angle.
This equation assumes that the elastic properties of substrate and the thin coating are the same,and that all the coating properties are isotropic. Due to the multilayer structure of the amorphousmaterials, the coating loss and elastic properties may be anisotropic. For this reason, authors such as
Harry et al. [51] decompose coating loss and elastic deformation into parallel (k) and perpendicular(⊥) directions relative to the mirror normal. Then, in accordance with Eq. 2.7, the total dissipatedenergy can be written as Wdiss = 2πf (U⊥ φ⊥ + Uk φk ). The stored elastic energy U⊥ and Uk can becalculated in the cylindrical coordinate. The spectral density is given as

Sxcoating (f ) =d
2kB T 1 − σ 2×
π 2 f ω02 Y
Y Yc (1 − 2σc )(1 − σ 2 )× Yc2 (1 + σ)2 (1 − 2σ)2 φk(2.17)+Y Yc σc (1 + σ)(1 + σc )(1 − 2σ)(φk − φ⊥ )
2
2
2
+Y (1 + σc ) (1 − 2σc ) φ⊥ .
This equation assumes that the coating’s Young’s modulus is isotropic: the modulus is the same
in both parallel and perpendicular directions. In the case Yc = Y , σc = σ and φ⊥ = φk , Eq. 2.17
agrees with Eq. 2.16.
For coatings with multiple layers of high and low refractive index materials, Young’s moduli andPoisson’s ratios have to be averaged (this is similar to the calculation of connected springs) and20
Eq. 2.17 can be rewritten as [82]2kB T 1 − σ 2×d
π 2 f ω02 Y Y⊥(
2σk2 Y YkY×−φ⊥
1 − σ⊥
Y⊥ (1 − σ 2 )(1 − σk )Sxcoating (f ) =Yk σk (1 − 2σ)(φk − φ⊥ )(1 − σk )(1 − σ))Yk Y⊥ (1 + σ)(1 − 2σ)2
φk ,+Y (1 − σk2 )(1 − σ)(2.18)+

with averaged coatings parameters:Y⊥ = (d1 + d2 )/(

d1
d2
+ ),
Y1
Y2

Y1 d1 + Y2 d2,
d1 + d2
σ1 Y1 d1 + σ2 Y2 d2σ⊥ =,Y1 d1 + Y2 d2
φ1 d1 φ2 d2
φ⊥ = Y⊥ (
+
),
Y1
Y2
Y1 φ1 d1 + Y2 φ2 d2,
φk =Yk (d1 + d2 )Yk =(2.19a)(2.19b)(2.19c)(2.19d)(2.19e)where subscripts 1 and 2 refer to the two coating materials. For σk the equivalent equation is more
complicated. In general, an averaged values between σ1 and σ2 is good within 5 %.
However, as argued by Hong et al. [53], φ⊥ and φk are not a suitable choice to be consistently
used as the loss angles of a material, since the corresponding energies U⊥ and Uk can be negative incertain cases. Instead, Wdiss should be decomposed into bulk (“B”) and shear (“S”) contributions:
Wdiss = 2π(UB φB + US φS ). If the effect of the light propagating inside the coating layers is omitted,
the power spectral density for a duplex structure of material with high reflective index and low
refractive index can be written as,

B B
S S
B B
S S
SxcoatBR = qH
SH + qH
SH + qL
SL + qL
SL ,

with
SjX =

4kB T λj φjX (1 − σj − 2σj2 ),3π 2 f Yj (1 − σj )2 w02

X = B, S, and j = H, L

(2.20)(2.21)The subscripts H and L denote the material with high and low indices respectively, while thesuperscripts B and L denote the bulk and shear modes.
For SiO2 /Ta2 O5 coatings, the individual loss angles (either φ⊥ and φk , or φB and φS ) are notwell known, and knowledge of the individual material properties is also limited. These uncertainties

21
will propagate forward toward the estimate of the loss angle [53].
It is remarkable that if we assume that of φB = φS , and elastic parameters of the substrate andthose of the coating are the same, then the results of Harry et al. [51] and Hong et al. [53] reduce to
Eq. 2.16.
In any case, since there is no fundamental reasons for the assumptions, the “coating loss angle”
φc as defined in Eq. 2.16 should be viewed not as a physical parameter, but as a figure of meritwhich is related to the various loss angles and material parameters of each coating material.
More details about Brownian noise in coating with finite size substrate is done in Somiya andYamamoto [52]. Although the loss angles are assumed to be isotropic in the coatings, this mathematical treatment explains how the finite size causes the noise level to increase. For our case, thespot size is usually much smaller than the substrate, and the finite size correction is negligible.

2.5.2

Thermo–Optic Noise in Coatings

The term thermo–optic noise includes two effects that occur in the coating, thermoelastic noise andthermorefractive noise. Each noise was calculated independently [78, 83, 84]. Later, when scientists
realized that both effects originated from the same temperature fluctuation, they were computed
coherently [74].
Thermoelastic Noise in Coatings
Temperature fluctuations in coatings directly change the position of the mirror surface via thermal
expansion coefficient. As the coating expands, the total length of the cavity contracts. This is very
similar to thermoelastic noise in substrate discussed in section 2.4.2.
Thermorefractive Noise
For a highly reflective mirror with multilayer dielectric coatings used in an interferometric measurement, the output phase of the beam reflected from the mirror carries information about the mirror
position. However, not all of the electric field of the beam just reflects off the top surface of themirror. A certain amount of electric field propagates inside the coating layers and reflects off fromevery coating interface. Ideally, for a quarter wavelength design, all the reflected beams will interfere
constructively at the air-coating surface, see Fig. D.1. Because of the non-zero thermo refractive
coefficient, the temperature fluctuation adds the random path length in each layer, resulting in thephase noise summed up at the total reflected beam. Even though the surface position is kept stationary, the reflected phase is still varied due to the thermorefractive mechanism resulting in thedisplacement noise on the mirror surface as interpreted by the probe beam.
It should be emphasized that the term “thermorefractive” used in this context refers to themechanism that alters the optical path length inside the coatings. As the temperature changes, notonly ∂n/∂T that changes the optical path length of the beam propagating inside the layers, but alsoα, as seen in Eq. D.8 which is taken from B15 of Ref. [74].

22
Thermo–optic Noise in Coatings: The Combined EffectSince both thermoelastic noise and thermorefractive noise originate from the same thermal fluctuations inside coatings, the effects are summed coherently, and called thermo–optic noise. The
calculation is done by injecting Gaussian profile heat (entropy) on the mirror then calculating theheat flow loss [73]. Then the FDT is applied to relate loss to the spectral density of noise in temperature. Finally, the temperature fluctuation is converted to displacement noise and phase noise
via the thermoelastic coefficient and the thermo refractive coefficient. The unified result is [74]2Cc
α¯c d − β̄λ − α¯s d,Cs(2.22)

kB T 2√,
π 3/2 ω02 κCf(2.23)TO
SxT O (f ) = S∆T(f )ΓtcandTOS∆T(f ) =

2

where α¯c and β̄ are the effective thermal expansion coefficient of the coating and the effective
thermorefractive coefficient respectively. Their full expressions are shown in Appendix D.
TO(f ) is the power spectrum of the coating thermal fluctuations sensed by a Gaussian beam.
S∆TThe parameters used in that factor are those of substrate, whereas the contribution from the coatings
are included in the thick coating correction Γtc . The calculation above uses the adiabatic-assumption,
where δT in the material due to the applied heat does not flow in or out in the orthogonal direction(thermal diffusion length lth  spot size w0 ) and solves the heat equation in 1 dimension along
the beam line direction [84, 73, 74]. However, this is not true at low frequency, or when the spot
size becomes comparable to the thermal length. At such regimes, the heat flow in the orthogonal
direction becomes large, causing ∇δT to be smaller and the noise is overestimated. The correction
for small spot size-low frequency regime is done in Ref. [27], by extending the calculation from
Braginsky et al. [84]. The outline for mathematical derivation is shown in appendix G. Hence, inthe general case, the temperature fluctuation sensed by a Gaussian beam isTOS∆T(f ) =

with"Z
K(Ω) = R√
2 2kB T 2
K(Ω)
πκω0

∞

du ue
0

−u2
2

#
√
u2 + iΩ√
u4 + Ω2(2.24)(2.25)

The correction is conceptually similar to what is done in Ref. [77] for thermoelastic noise in thesubstrate. However, the thick coating correction Γtc from Ref. [74] treats the heat flow in the coating
only in the beam line direction and must be modified. To my knowledge, the correction where theheat flow in the coating is solved in 3 dimensions does not exist in the literature yet. This will leadto some error in the thermo optic noise estimation for materials like AlGaAs coatings where lth can

23
be quite large compared to the spot size, see Tab. 2.2.
Parameters

Thermal conductivity,

SiO2 Sub-

SiO2 /T a2 O5 optimized

strateQWL

AlGaAs

1.38

2.23

61.5

1.63

1.82

1.72

κ [W/mK]
Heat

capacity,
3

C

6[J/m ] ×10
Thermallength,lT ,diffusion
q

37

q

100Hz
f

44

q

100Hz
f

238

q

100Hz
f

κ
C2πf

[µm]
beam radius, ω0 [µm]-182-292

215

coating

-

4.5

4.6

thickness,d,[µm]
Table 2.2: Some parameters for TO noise calculation. Parameters in QWL and optimized AlGaAs
columns are averaged based on the coating structure and the material parameters. QWL here is a
27 layer quarter wave thickness stack with a half wave cap of SiO2 on top (a total of 28 layers). The
Optimized structure for AlGaAs can be found in chapter 5.

2.6

Thermal Noise in Spacer

2.6.1

Brownian Noise in Spacer

Another part of a fixed-spacer Fabry–Perot cavity that has thermal noise is the spacer. Brownian
noise in spacer was first calculated by Numata et al. [59] and later refined by Kessler et al. [60] withthe effect of the size of the bore hole along the spacer length.
Sxspacer (f ) =4kB TLφspacerπf 2π(R2 − r2 )Y(2.26)

Where R is the radius of the spacer, r is the bore hole radius, and L is the length of the spacer.
However, the analytical result assumes that the mirror and the spacer have the same outer radii andare fully attached together. Typically, the mirror radius is smaller than that of the spacer, and thecontact area is only an annulus with thickness around 2 mm, see Fig. 2.3. To calculate the effectcorrectly, a finite element analysis (FEA), by COMSOL, is used together with the direct approach.
By computing the elastic energy stored in the body of the spacer, the displacement noise in spacer
can be calculated. First, the FEA result where the force is applied on the whole surface area of thespacer is compared with the analytical result in Eq. 2.26. Both results agree well within 2 %. Oncethe model is verified, the contact surface is changed to match the real situation, see Fig. 2.3, and24the elastic energy is calculated. The dependence of spacer Brownian noise on the size of the contact
surface is shown in Fig. 2.4. The smaller area creates more elastic energy and increases the thermal
noise level. This explains why the analytical model which assumes the full mirror-spacer contact
area underestimates the noise level.
For a fused silica spacer, its Brownian noise will be smaller than the coating noise by at least anorder of magnitude.

Figure 2.3: COMSOL model for thermal noise calculation. Left, 1/8 of the model is used withsymmetric boundary condition on three planes to reduce the computation time. Since most of thedeformation will occur close to the applied force, to further minimize the calculation time, only thesmall volume at the center of the mirror has very fine mesh size while the mesh size is larger far awayfrom the beam. Right, the deformation on the spacer because of the applied force on the mirror
contacted on the spacer (not shown), and the elastic energy in the spacer can be calculated.

2.6.2

Thermoelastic Noise in Spacer

To estimate the level of thermoelastic noise in the spacer, one can follow the method outlined by
Liu and Thorne [72]:2kB T
Sx(spTE) (f ) = 2 2 κsp T
π f



Ysp αsp(1 − 2σsp )Csp

2 Z[∇(∇ · u)]2 3
d r,
F02

(2.27)

where u(r) is the displacement field of the spacer in response to a static pressure of total force F0
applied to the mirror faces. The subscript sp denotes the material parameters of the spacer. For
our case, the spacer is made from fused silica similar to the substrates.
We use the same FEA model as described above for computing the spacer Brownian noise toevaluate the integral in Eq. 2.27. The calculation is performed under the adiabatic assumption, since
the diffusion length lth is much smaller than the width of the contact area between the spacer andthe mirror. Due to the thickness of the annulus (≈ 2 mm), the assumption lth  ω0 should be valid
down to a few millihertz. At very lower frequencies, where the assumption on lth is not satisfied,the expected thermoelastic noise is smaller than the adiabatic prediction [77].

25
5.9

data 1
cubic

Stored Energy x10−11 [Joule]

5.8
y = p1*x3 + p2*x2 +p3*x + p4

5.7

5.5

Coefficients:
p1 = 0.030756
p2 = −0.11503
p3 = −0.35787
p4 = 6.4391

5.4

Norm of residuals =0.01447

5.6

5.3
5.2
5.1
1.5

2

Thickness [mm]2.5

3Figure 2.4: Maximum elastic energy due to the applied pressure in 1/8 of the cavity vs the annulus
thickness. The outer radius of the annulus is fixed at 0.5 inch which is the radius of the mirror. The
fit parameters are shown in the plot. The cavity is 1.45 inch long, with outer and inner radii of 0.75
inch and 0.1875 inch, respectively
To validate the result from FEA, the model is used to calculate thermoelastic noise in substrate.
This is done by applying a force with Gaussian profile on the mirror and solving Eq. 2.27 only inthe substrate domain. The FEA result agrees well with the adiabatic calculation within 20 percent.
It is certain that both Brownian thermal noise and thermoelastic noise in the spacer will not be alimiting noise source in our cavities.

2.7

Photothermal Noise: Absorption from Shot Noise andIntensity Noise

Fluctuation in laser power from shot noise, classical intensity noise, or squeezed light, induces a local
temperature change in both coatings and substrate because of the absorption on the mirror. Due tothe thermal expansion and thermorefractive coefficients of the mirror substrate and the coating, thetemperature gradient caused by the absorbed laser power couples into cavity’s displacement noise,and this is called photothermal noise. Similar to thermo–optic noise, the effect in the substrate ismostly thermoelastic. This noise source was first considered in a restricted regime by Braginsky

26
et al. [76], and later refined for small beam size and low frequencies, can be written as [77]Sx(PT) (f ) =2 (1 + σs )2
Sabs K(f /fT ),
π2
κ2s

where
1
K(f /fT ) =
π

Z∞

Z∞
du

0

−∞

(2.28)22u2 e−u /2
dv 2(u + v 2 )(u2 + v 2 + if /fT ),(2.29)

andSabs = δP (f )2F/πχabs ,
1 + (f /fcav )2(2.30)

where F is the finesse of the cavity, δP (f ) is the input power fluctuation, χabs is an absorption
coefficient of the mirror, and fcav = fFSR /(2F) is the cavity pole.
The effects from the coating (both thermoelastic and thermorefractive effects) were later includedin the work of Farsi et al. [85], who treated all the contributions from substrate and coating coherently. The effect can be measured directly by modulating the power of the laser and observing thedisplacement noise of the cavity. The results are complicated and will not be repeated here.
Generally, classical intensity noise in a laser is much higher than the shot noise limit. The relative
intensity noise due to shot noise is
δP
=
P

√

2hνP
= 2 × 10−8
P

"r

1mW
P

#
√

1
,
Hz

(2.31)

where h is the Planck’s constant. On the other hand, other sources of power fluctuations in a laser
are relaxation oscillations, power fluctuations in the pump laser, and beam pointing fluctuations 2 . A
measurement of the relative intensity noise due to these classical effects usually range from 1 × 10−61
to 1×10−3 √Hz
between 0.1 Hz and 10 kHz. Since they can be suppressed with intensity stabilization

servo (ISS), we will revisit the issue in section 4.7, during technical noise discussion.

2.8

Noise in the Bonding between a Mirror and a Spacer

For a fixed–spacer Fabry–Perot cavity, the interfaces between the spacer and the mirrors are anotherparts that might introduce displacement noise as well. The level of the noise highly depends on thebonding method. Several bonding methods, for example, epoxy, hydroxide bonds, or optical contact
2 When the beam is directed to a reference cavity, the misaligned beam leads to fluctuations in the power coupled into
the cavity. For example, power fluctuations due to beam translation dx, and the angular tilt dα can be approximatedas [86]
"


 #
dα 2
δP
dx 2
≈ exp
+
,
(2.32)
P
w0
ΘD

where w0 is the spot size, and ΘD = λ/πw0 . Several causes of pointing fluctuations are thermal lensing, vibrations
and thermal expansion on opto–mechanical components and thermal lensing.

27
can be used to make a fixed-spacer cavity. For this experiment, noises associated with the bond has
to be low, so the signal from the coating thermal noise is not masked. Plus, the bond has to be
vacuum compatible since the cavities are placed in a vacuum chamber. The simplest solution is to
optically contact the pieces together. It required no bonding agents, and, if executed correctly, has
no noise associated with the bond as the assembly becomes a monolithic piece [87].

2.8.1

Optical Contact

When two objects with flat surfaces are brought to contact, intermolecular force, i.e., van der
Waals force becomes large enough to keep the two objects together. In general, different kind of
materials can also be bonded [88]. As time goes by, the van der Waals bonds (hydroxyl bonds) turninto covalent bonds (siloxane bonds) making the bond stronger, see Fig. 2.5. Annealing can help
accelerating the process. However, care should be taken when the object is heat up or cooled down.
The thermal expansion might bend the objects, especially, if the attached pieces are of different
material with unequal thermal expansion coefficients, causing a rip in the joint-surfaces that breaks
the bond. For this reason, optical contact may not be the most suitable solution for bonding objects
used in a space project, (e.g., LISA3 ), where a large thermal change occurs. Nevertheless, opticalcontact is still proved to have the lowest noise compared to other bonding methods [89]. More details
about how to optically contact a mirror to a spacer can be found in appendix E.2.
In this experiment, all the cavities were assembled by optical contact. As discussed above, noisein the optical bonds are assumed to be negligible as the spacer and the mirror become monolithic.
Furthermore, from the direct approach calculation, the beam is far away from the bonding area, andits noise contribution should be negligible. This assumption, as shown in the measurement resultsin the next chapter, is consistent with the experimental results.

2.8.2

How to Measure Mechanical Loss in Optical Contact

To measure noise introduced by the optical contact, we can use a setup similar to the setup formeasuring loss in KOH bond [90]. First, we measure a mechanical loss from a control sample withlength L. Then we can compare it with the mechanical loss of a test sample, which is formed by
optical contacting two pieces with length ∆L and L − ∆L. Since we want to measure the loss
associated with perpendicular direction of the bonded surface, the measurement must be done on
longitudinal modes. The first longitudinal mode will have a node at the center of the rod, so, in
general, ∆L should be shorter than L/2. The thickness of the bond can be determined by the surface
grinding specification or it can be measured by optical scattering from the surface [91]. Once themechanical loss is determined, the loss per area can be estimated and applied to the cavity-substrate
3 http://lisa.nasa.gov

28

Figure 2.5: Measured bond energy vs time after optical bonding: The bond strengthens as time
passes and more van der Waals bonds become covalent bonds. This settling process might take up
to several days [87].
situation.

2.9

How to Reduce Thermal Noise

From the discussion above, thermal noise sensed by a Gaussian beam on a dielectric mirror has been
explained. For fused silica/tantala coatings, coating Brownian noise will be the dominating noise.
Let’s look at Eq. 2.16 again,
Sx(cBR) (f ) =4kB T (1 + σs )(1 − 2σs ) d
φc .
π2 f
Es
w02

(2.33)

This simplified formula captures all significant parameters contributing to coating thermal noise. To
reduce thermal noise, we can either decrease the temperature T , reduce loss in the coating φc , orincrease the spot size w0 . Most methods evolving around these concepts are explained below.

2.9.1

Lower Temperature

As thermal noise is driven by the ambient temperature, reducing the temperature seems to be themost straight forward way to tackle the problem. But it is not as easy as one might hope. Some of
the issues are29• Cooling down: The first task is to cool down the setup. Suppose that the goal is to reduce
the power of the noise by a factor of 10. The required temperature will be around 30 Kelvin.
Liquid Nitrogen (LN) can cool the setup to 77 K, liquid Helium can cool the setup down to
4 K. But a well designed cryostat is required.
• Material Properties: Most of material parameters are temperature dependent.For fusedsilica and tantala deposited by ion beam sputtering process (IBS), their losses are temperaturedependent. At temperature between 10 K and 300 K, tantala’s mechanical loss can be varied
from 3 × 10−4 to 10 × 10−4 [92, 93], and from 1 × 10−4 to 9 × 10−4 for fused silica [93]. Thesevariations due to temperature are not monotonic and usually have peaks around 20–30 K.
More importantly, fused silica substrates will have higher loss at lower temperature around
30–40 K [94, 95], making it an unsuitable substrate for a mirror. Other materials that have
low loss at cryogenic temperature and have good optical properties are, for instance, sapphireand silicon.
One example of a large scale gravitational wave detector operated at cryogenic temperature will
be KAGRA. It will use sapphire test masses during a science run at 20 K. Although sapphire is
not a suitable choice for substrates at room temperature because of its high thermoelastic noise, atcryogenic temperature, thermal expansion of sapphire can be very low resulting in the lower level ofthermoelastic noise.

2.9.2

Lower Loss

Since Brownian noise is also directly proportional to the material loss, reducing the loss in thecoatings can lower the noise, but material with lower mechanical loss is not always the best solution(as in the fused silica/sapphire cause). In addition, the materials must have good optical properties.
A few ways to improve the loss without compromising the optical properties are:
• Doping T a2 O5 with T iO2 : Loss in tantala is usually higher than that of fused silica. Lower
loss in tantala will result in the reduction of the overall loss of the coating. One way to reduce
loss in tantala is to mix small amount of T iO2 into it. This method can reduce tantala’s loss
by 40 percent [96] without weakening its optical properties.
• Heat treatment after coating process Ion-beam sputtered coatings are often heat treatedafter deposition to reduce the stress in the film and to reduce the optical absorption. This
heat treatment also reduces loss in both SiO2 [97] and T a2 O5 [98], as a result of a reduction ofinternal stresses due to the altered distribution of the potential barrier heights. But the coating
can be annealed to around 800 Celsius. At higher temperature T a2 O5 starts to crystallize andits optical properties change.

30• Optimized coating structure: Most of the loss in the dielectric coatings come from T a2 O5 .
Ring down measurements show that coatings with more T a2 O5 have higher loss [99, 100, 101].
If the overall physical thickness of T a2 O5 in the coatings can be decreased, the overall loss will
be smaller. By adjusting the thickness of SiO2 and T a2 O5 and the total number of layers [102],
TNI team has shown that the effective loss in the optimized coating can be reduced by about
twenty percent while preserving the reflectivity of the mirror [103].
• New Material with lower loss: Recently, thin film of GaAs and AlGaAs and AlGaP has
been researched in hope of new low loss coating materials. As crystalline structure has muchsmaller mechanical loss due to less dislocations and impurities in the material. Its mechanicalloss is expected to be one order of magnitude smaller than the conventional fused silica/tantala
coatings. More about AlGaAs as coating materials can be found in chapter 5

2.9.3

Different Beam Shape

Using a larger beam size can also reduce thermal noise as the averaged surface is increased, but thismethod requires larger substrate and coatings to keep the loss from clipping below the acceptablelevel. By using a different beam shape, the substrate size can be maintained while the effective beam
area is larger. Other beam shapes that might be possible to use are mesa beam (flat–top) [104],
conical beam [105] or Lagurre–Guass 33 [106]. However, these beam requires special mirror shapeand more unstable because of the parametric instability4 [109]. Compared to a Gaussian beam
shape, they are more sensitive to angular misalignments [110]. Because of these drawbacks, using
different beam shape to reduce coating thermal noise might not be a practical solution for the current
technology.

2.9.4

No Coating Cavity

Another way to reduce coating thermal noise is to get rid of the coating. This raises the question of
how to get a high reflective surface. Two examples are:
• Total internal reflection Cavity: Instead of a dielectric mirror, another type of cavity
employs a basic principle of total internal reflection to bounce the beam [111]. Although
Brownian noise from coatings is completely removed from the system, thermo–optic noise still
persists [112]. However, this can be minimize by choosing materials with low thermal expansion
coefficient or tuning the temperature such that α is minimized.
• Grating cavity: The input mirror of a Fabry–Perot cavity can be replaced by a grating [113].
But a lot of power is lost to other diffraction orders.
4 Parametric instability occurs when the beat between two optical modes is close to the mechanical frequency ofan acoustic mode of the mirror [107, 108].

31

Figure 2.6: Different kinds of Beam profile. Red: Gaussian, Blue: conical, Green: Mesa [105]A coating free cavity with high mechanical quality factor and a small optical linewidth has not
yet been demonstrated. As these techniques are relatively new compared to a conventional cavity
formed by dielectric mirrors. More researches have to be done to improve their performances before
they can be used as a high finesse reference cavity.

32

Chapter 3

Experimental Setup

Figure 3.1: A picture taken of the setup for this work. The thermally insulated vacuum chamberthat houses the two cavities is at the further end of the optical table.
In this chapter, two experimental setups for observing coating thermal noise are explained indetail. Technical noise sources associated with the setups will be discussed in the next chapter. The
first setup measures the noise from 8 inch reference cavities. The second setup which is conceptuallysimilar to the first one, but measures coating thermal noise from 1.45–inch reference cavities.

33

3.1

Optical Cavity as a Frequency Reference

A Fabry–Perot cavity is formed by two reflecting mirrors facing each other, separated by a distance
L. One useful property of the Fabry–Perot cavity is frequency selection. Only light with certain
frequencies can pass through the cavity. The cavity’s Free Spectral Range (FSR) frequencies are
given by
ν=n

c
,
2L

(3.1)

where n is an integer denoting the axial mode number. Basically, it is a condition that the laser must
form a standing wave pattern in the cavity. This property allows a Fabry–Perot cavity be a useful
tool for laser frequency stabilization. One important property of a laser is its highly coherence
output, i.e., a single frequency light, but it has frequency noise which can be measured in the
spectral density of noise power, Sν (f ), in Hz 2 /Hz or the linear spectral density, δν(f ), with units√
of Hz/ Hz.
The fundamental frequency noise in a laser is placed by the Schawlow-Townes limit [114, 115],r
SfS−T = δνL

2hν
Pout

(3.2)

where δνL is the laser cold cavity line width and Pout is the laser output power. This is the random
frequency noise from spontaneous emission inside a laser. The distribution is flat and can be as low√
as 0.5Hz/ Hz for 1mW output because of the narrow laser cavity linewidth [116]. However, theactual noise from a laser is far higher than this. Anything that changes the optical path length of thelaser cavity affects the laser frequency. Frequency noise in a Non-Planar Ring Oscillator (NPRO)hi√Hz from 1 Hz to 10 kHz. This estimatedlaser can be roughly estimated to be 10 kHz 1HzfHz
frequency noise mostly originates from power fluctuations in its pump laser [117].
If the laser frequency is forced to follow the length of the cavity by means of frequency discrimination techniques e.g. Pound–Drever–Hall locking (PDH) [118, 119] (examples for comprehensive
reviews of PDH technique can be found in Ref. [117, 120]). The laser frequency noise can be suppressed significantly until it matches the displacement noise of the reference cavity. The relation
between the two is
δL
δν
=− .
ν
L

(3.3)

If the dominant displacement noise in a reference cavity is coating Brownian noise, this meansthat the frequency noise of the laser locked to the cavity contains information about the coating
noise in the cavity. By measuring the stabilized frequency noise we can get information about thecoating noise. The frequency locking technique can be cast as a linear control theory problem asshown in Fig. 3.2. The laser has free running noise δν. The frequency discriminator D, electronic
servo gain G, and actuator response A combine to produce the open-loop gain H = DGA. When34
the loop is engaged, the suppressed frequency noise δνs of the laser becomes

δν

δνs

-

δL

D

A

G
Na

Ne

√
Figure 3.2: A block diagram of PDH lock. δν is the laser free running noise in Hz/ Hz, δνs isthe suppressed frequency noise, or the frequency noise of the transmitted beam behind the cavity,
δL is the cavity’s displacement noise which converts to the frequency noise via PDH lock. D is thefrequency discriminator gain in V/Hz, G is the servo gain in V/V, A is the actuator gain in Hz/V.
Ne and Na are the error signal and the control signal, respectively. The minus sign from the actuator
A, represents the negative feedback. The minus sign from δL means the displacement noise of thecavity is compared to the laser frequency.

δν
H
c
+
×
δL
1+H
1+H
Lλ
δν
c
≈
+
δL
for |H|  1.
H
Lλ

δνs =

(3.4a)(3.4b)

Within the loop bandwidth, where the magnitude |H| of the open-loop gain is large, the displacement
noise δL of the cavity is impressed onto the frequency noise of the laser: δνs ≈ (c/Lλ)δL. The power
spectral density of the frequency noise is given by Sν (f ) = |δνs |2 .
It is impossible to measure δL inside the control loop, as Ne is an in-loop sensor, NA will contain
only the information about the free running noise of the laser which is much larger than δL. To
measure δL, we have to measure it optically from δνs as an out-of-loop witness. The transmitted
beam can be compared with a very stable reference, or beat against another identical setup. The
latter method is chosen since it offers common mode rejections for disturbance from seismic noise,
acoustic noise, ambient temperature noise, and other noises that are coherent between the two
setups.
In other words, thermal noise can be measured by having two setups of a laser frequency locked
to a reference cavity with the two transmitted beams recombined at the output. Because of theslightly different lengths of the two cavities, the two beams have different frequencies, ν1 and ν2 .

35
When directed onto an RF photodiode, the combined beam results in a beat note with frequencyν̂ = ν1 − ν2 (the high frequency portion ν1 + ν2 is filtered out by a low pass filter). The frequency
noise of this beat note has a power spectral density Sν̂ = Sν1 + Sν2 which can be read out using aphase-locked loop (PLL) technique1 which will be discussed in section 4.6.

3.2

Prototype Setup

Instead of using two similar setups of a laser locked to a reference cavity as suggested at the beginningof this chapter, the first setup, serving as a prototype for thermal noise experiment, requires only
one laser, one high performance frequency stabilizing servo (FSS), and two references cavities.

3.2.1

8–inch Reference Cavities

These reference cavities were originally designed as reference cavities for Initial LIGO. We used themlargely due to their immediate availability.
• Mirrors: The mirror substrates are commercially available fused silica with a 1 inch diameter,
0.25 inch thickness, and a 0.5 m radius of curvature (ROC). The coatings were fabricatedby Research Electro–Optics Corporation (REO) of Boulder, Colorado, USA, via ion-beam
sputtering in 1998. The coating was annealed at 480 C after deposition. They consist of
28 alternating layers of fused silica (SiO2 ) and tantala (T a2 O5 ). The first 27 layers are eachdeposited to a thickness of λ/4n, where n is the refractive index of the layer material. Due to its
hardness, the final layer is fused silica, and in order to give the correct interference condition, it
is deposited to a thickness of λ/2n. The transmission of each mirror is approximately 300 ppm.
• Cavities: Using these mirrors, we initially constructed two symmetric cavities using fusedsilica spacers with length L = 8 inches. The cavities were formed by optically contacting
mirrors to cylindrical fused-silica spacers. Both substrates and spacers are made of fused silica
because of its low mechanical loss and small thermal expansion coefficient.
• Cavity mount: Each cavity is fitted with a pair of O-rings close to the cavity’s Airy points.
These O-rings then sit upon a pair of teflon blocks with a semicircular cut. The block has atransverse V-shaped groove to keep the O-ring in place. The cavities are placed side by side on
a double-stack seismic isolation platform, as shown in Fig. 3.3. The resonances of this platform
all lie below 10 Hz. The use of a single platform and chamber endows the beat measurementwith some amount of common-mode rejection of seismic and ambient temperature related
noises.
1 For example, see [121].

36• Vacuum chamber: The cavities and the platform are installed inside a temperature-stabilizedvacuum chamber with the pressure below 10−7 torr (using an ION pump). Heater jackets areattached around the chamber and a software control loop stabilizes the overall temperatureof the vacuum chamber to within 1 mK. The chamber sits on pneumatic legs which have aresonant frequency around 5 Hz.
To tune the beat frequency between the two cavities, each cavity has a separate radiative shieldwrapped with a heater wire. The length of the cavity is controlled by thermal expansion via heatingof the shield, and the resonant frequency of the transmitted laser is changed accordingly.

Cavity length / FSR

8 inch (20.32 cm) / 738 MHz

Finesse

9,816

Cavity linewidth (FWHM)/ cavity pole

75.2 kHz 37.6 kHz

Beam radius on the mirror

w0 = 292µm

mirror coatings

27 Layers of QWL SiO2 /T a2 O5 with λ/2 cap of SiO2

mirror transmission

300 ± 30 ppm

Loss in each mirror

20 ppmMirror radius of curvature

0.5 m

Table 3.1: Table of parameters for 8 inch cavity and the mirrors.

Figure 3.3: 8 inch cavities on the passive seismic isolation stack. The thermal shields are removed
to show how the cavities are held.

37

3.2.2

Setup Layout

Fig. 3.4 shows a schematic of the prototype experimental setup. A 100 mW Nd:YAG laser is simultaneously stabilized to two identical cavities: a reference cavity and an analyzer cavity. Eachcomponent in the setup is explained below following the operational step.
• PMC: Because the beam profile from the laser does not have a perfect Gaussian profile(TEM00), the beam shape is filtered by a Pre-Mode Cleaner (PMC) triangular cavity [122].
The PMC is locked to the laser frequency via PDH technique, with a sideband frequency of
21.5 MHz. When the beam is close to a purely TEM00 profile, we can reduce the amount oflight that cannot be coupled into the Fabry–Perot cavities because of the mismatched beam
shape. This left-over reflected light causes extra noises in several ways. It can go back all theway to the laser source and cause instability in the laser frequency. It can increase shot noiselevel on the RFPD. Additionally, the scattered light associated with the reflected beam can
add random phase noise in the PDH locking technique.
• Locking the laser to reference cavity: After the beam profile is filtered by the PMC, theside band at 14.75 MHz is added on the carrier by an EOM. The EOM used in the setup is
New Focus 4003 with resonant frequency at 14.75 MHz. Then, the beam is separated into twopaths. The main path is directed to the reference cavity and the laser is frequency stabilized tothe cavity using PDH lock with three actuators acting on the laser. From DC to around 1 Hz,
the temperature actuator on the laser head has a large dynamic range. Although the response
is slow, the tunable range is up to 40 GHz. The range is calibrated at 4 GHz/V. Around
1 Hz–1 kHz, the PZT attached to the NPRO is used to change the laser frequency, and from
1 kHz to 1 MHz, a broadband EOM dithers the beam’s phase, which is similar to changing its
frequency. With all these actuators, the unity gain frequency can reach almost 1 MHz. Moredetails about the frequency stabilization servo (FSS) are discussed in appendix B.
• Locking the secondary beam to analyzer cavity: The second beam path is picked offfrom the main beam behind the PMC and the EOM, so the side bands are already includedin the second beam. An acousto-optic modulator (AOM) is used to lock this beam to thesecond cavity (analyzer cavity). The AOM is driven by a voltage–controlled oscillator (VCO)
operated at the nominal frequency of 80 MHz, and the beam is double passed to reduce any
position jitters associated with the frequency locking since the deflection angle depends onthe driving frequency [123]. First, the AOM shifts the carrier frequency by 160 MHz (double
passed, 1st order beam). The cavity length is tuned (by changing the temperature) so thatit roughly matches the frequency of the beam. Next is to lock the beam to the cavity. The
control loop uses the AOM as a frequency actuator to shift the frequency light going to theanalyzer cavity. The feedback to the VCO varies the driving frequency which changes the

38frequency of the beam to keep it locked to the cavity. Usually, the operational range of the
AOM is around ±5 MHz. The performance of this loop can be less demanding because thefrequency of the main laser is already stabilized to the reference cavity and becomes very quiet.
In general, the servo gain has to be enough to suppress any noise introduced by the servo andthe VCO. See section 4.4 for more details.
• beat path Behind the cavities, the two transmitted beams are recombined (beat) and detectedwith a photodiode. In order to extract the frequency noise of the beat note due to thermal
noise, the beat note is compared to an ultra-stable frequency reference via a phase locked loop
(PLL).

EOM
EOM

14.75 MHz

EOM

14.75 MHz
Resonance Broadband
AOM

Reference Cavity

Analyzer Cavity

VCO 80 MHz

CCD

CCD

PLL

Figure 3.4: The prototype experimental setup for measuring the coating thermal noise of LIGO reference cavities. A Nd:YAG laser is simultaneously
stabilized to two identical cavities and the transmitted beams are used to measure the length noise by comparing their beat signal.

NPRO

21.5 MHz

39

40

3.2.3

Result from the Prototype Setup

The measurement of the noise profile of the prototype setup is plotted together with the noise budget
in Fig. 3.5. The sensitivity at low frequencies is limited by seismic noise and scattered light. At
frequencies above 1 kHz, the frequency is limited by the local oscillator’s phase noise from the PLL
readout technique. Since the measured signal is heavily contaminated by technical noise sources,
the coating loss extracted from the data is not reliable. This is not surprising, as these reference
cavities were not intended for thermal noise measurement. The spot size is fairly large, and thecavities are long. These result in a low frequency noise which is the object of a reference cavity.
However this setup has served to provide proof of principal and expertise in the requirement of thefrequency stabilization servo and readout technique. More importantly, this prototype demonstrated
the achievable sensitivity of the design. Hence the prototype provided confidence that the upgraded
setup would be able to observe coating thermal noise.

1

0

10

0

10

−3

10

−2

10

−1

10

1

10

2

10
Frequency [Hz]

3

10

4

10

Measured noise
Total estimated noise
Coating Brownian noise
Coating thermo−optic noise
Substrate thermal noise
Spacer thermal noise
Photothermal noise
Seismic noise
PLL noise
Residual frequency noise

Figure 3.5: Noise budget and the measurement from 8 inch cavities. The estimated coating thermal noise shown in the plot is obtained from Eq. 2.16
with φc = 4.15 × 10−4 . Seismic related noise is dominating from DC to 80 Hz. Mechanical peaks around 500 Hz to 1 kHz are from optic mounts andposts in the beat setup.

ASD of beat note fluctuation [Hz/Hz1/2]

10

41

42

3.3

Two Laser Setup with 1.45 Inch Cavities

To increase the coating thermal noise, the new setup uses shorter spacers, which are 1.45 inches
long, and two lasers, each independently frequency locked to one of the two cavities.

3.3.1

1.45 Inch Reference Cavities

In order to raise the coating noise level, we subsequently constructed two shorter cavities using
similar mirrors from the same coating run as those used in the prototype. The shorter cavities
will raise the beat frequency noise because of two reasons. First, for mirrors with similar radii ofcurvature (0.5 m), the shorter cavity results in smaller beam size on the mirror surfaces, and thepower spectral density of the displacement noise is inversely proportional to the square of the spot
size (cf. Eq. 2.16). For a symmetrical cavity with length L, the spot size on the mirror surface isgiven by [124]
w04 =



λR
π

2

L
,
2R − L(3.5)

where R is the radius of curvature of both mirrors. The spot size on the mirror vs. the cavity lengthis shown in Fig. 3.6. Second, the observed frequency noise increases with inverse of the cavity length(cf. Eq. 3.3).

350
X: 7.992
Y: 337.4

300

Spot size [um]250

X: 7.992
Y: 292.3

X: 1.457
Y: 215.6

R = 0.5m
R = 1.0 m

200
X: 1.417
Y: 180.9

150
100
50
0
0

1

2

3

4
5
cavity length [inch]

6

7

8

9

Figure 3.6: The spot size on a symmetrical cavity with length L varied between 0 to 8 inches. The
red and blue traces are the spot sizes on mirrors with 1 and 0.5 inch radii of curvature, respectively.
Although the shorter cavity increases the observed frequency noise as discussed above, there areseveral considerations that place a lower limit for the allowable length of the new cavities. First, it
must be possible to use a heater to tune each cavity length by half of a free spectral range, so thatthe beat note frequency can be brought within the bandwidth of the readout photodiode. A cavity

43that is too short would require excessive heating in order to achieve this. Second, the cavity must
form a stable optical resonator. Finally, the length must be chosen so that no higher-order transverse
laser modes resonate simultaneously with the TEM00 mode. The full detailed explanations for thedecision to use 1.45 inch cavities are given in appendix E, and the summary of the cavity properties
is shown in Tab. 3.2.
For a 1.45 inch cavity, the frequency noise of the beat note due to the coating Brownian noise
will be increased by a factor of ∼ 8.7 from that of the prototype. Thus, with the shorter cavity
length, and the already achieved sensitivity in the previous setup, the Brownian thermal noise of
the quarter wave silica/tantala from the 1.45 inch cavities should be high enough to be observed
between 65 Hz and almost 1 kHz.
In order to improve the sensitivity at low frequency which is dominated by seismic noise, the cavities are held at the optimum support positions to minimize the displacement noise due to vibration.
See appendix E for more details about the search for the optimum support positions.
Cavity length / FSR

1.45 inch (3.68 cm) / 4.07 GHz

Finesse

9,816

Cavity linewidth (FWHM)/ cavity pole

414.9 kHz / 207.5 kHz

Beam radius on the mirror

w0 = 182µm

Table 3.2: Table of parameters for 1.45 inch cavity.

3.3.2

Overall Explanation of The Two–Laser Setup

Fig. 3.8 is a schematic of the final setup. It is symmetric; the PDH error signal from each cavity isused to actuate on the PZT of an independent NPRO and on a broadband EOM. For each path, 1
mW of light is incident on each cavity. The visibilities of both cavities exceed 0.9, indicating thatthe incident beams have a nearly Gaussian spatial mode and that the cavities are close to criticallycoupled. The use of two lasers also allows larger possible range for the beat frequency; in the previous
setup, this was constrained by the operational range of the AOM. Typically, an AOM also distorts
the beam and thereby reduces the coupling into the cavity and makes the cavity stabilization scheme
more sensitive to other noise sources. It also limits the loop bandwidth when it is used as an only
actuator for frequency stabilization.
In this setup, the relative intensity noise (RIN) in both cavities becomes uncorrelated, and so an
intensity stabilization servo (ISS) is required. In each path, an electro-optic amplitude modulator(EOAM2 ) is used to suppress the laser’s RIN, and thereby decrease the photothermal noise to beless than the estimated thermal noise of the coatings. Similar to the prototype setup, each cavity is
installed with a thermal shield and a heater wire to control the differential length between the two
2 Newport 4104, http://assets.newport.com/webDocuments-EN/images/410412F_410X%20Manual_RevF.pdf

44
cavities via thermal expansion, see Fig. 3.7.

Figure 3.7: Reference cavities with thermal shields and the mount on double seismic isolation stacks.
Heater wires are wrapped around the shields. The shields have opening on their sides so that thecavities can sit on the supports.

EOM

EOM

EOM

14.75 MHz

EOM

EOAM

EOAM

FFT

ISS

Figure 3.8: Setup for measuring the coating thermal noise using two independent lasers. Each laser is stabilized to one of the two identical cavities.
The readout-scheme would be the same: the transmitted beams are used to measure the length noise by comparing their beat signal. One of the
NPRO has a Pre-Mode Cleaner cavity which is not shown in the diagram

NPRO

NPRO

ISS

45

46

3.3.3

Result from 1.45 Inch Cavities

Is it Coating Thermal Noise?
The result of this two-laser measurement is shown in Fig. 3.9. The technical and environmental
sources which limited the one-laser measurement are now subdominant in the band from 10 Hz
to 1 kHz. The measurement is compared to Eq. 2.16, and the fitted coating loss angles φc is(4.2 ± 0.3) × 10−4 (the error analysis is given in appendix G.3). When this effective loss is applied tothe 8 inch cavity measurement, the estimated result is also comparable to the measurement shownin Fig. 3.5. There are three pieces of evidence that suggest that the observed noise is really thecoating thermal noise sensed by a Gaussian beam.
• The slope of the linear spectral density of the frequency/displacement noise is inversely proportional to square root of frequency, cf. Eq. 2.16.
• The extracted loss angle is comparable to the previous direct measurement. Numata et al. [54]also reported the loss angles of QWL SiO2 /T a2 O5 on fused silica substrate to be 4 × 10−4 .
• The signal scales correctly with the spot size. As seen in the results between 8–inch and
1.45–inch cavities. Had the signal come from other sources (e.g., substrate, spacer Brownian
noise, noise in the bonding between the substrates and the spacer), the scaling would havebeen different. Substrate thermal noise is inversely proportional to spot size. Spacer thermal
noise should be independent from spot size.
This effective loss angle φc obtained from the data here through Eq. 2.16 should be viewed as a
figure of merit and not a real physical loss of the coating, since the equation assumes that elastic
properties of the coatings are similar to those of the substrate. The purpose of the fitting data using
Eq. 2.16 is to compare the result here with Ref. [54] that uses the same Coating Brownian noise
formula.

3.4

Discussion

According to the calculation by Hong et al. [53], coating Brownian noise can be written as a linear
combination of φL and φH 3 Eq. 2.20 can be rewritten as
SxcoatBR = qL φL + qH φH ,(3.6)

3 Here we use two assumptions. First, losses of bulk and shear modes are similar in each material. Second, light is
promptly reflected at the coating-air surface.

47
where subscripts L and H denote fused silica and tantala, respectively. Combining the above measurement and the material parameters in Tab. C.1, Eq. 3.6 becomes,
φH = −1.41 φL + (9.7 ± 0.7) × 10−4

(3.7)

To extract each loss angle, the equation is compared with a ring down measurement from Ref. [51]with a correction noted in Ref. [99]. The ring down measurement is made on a disc sample. The
coating is QWL structure of SiO2 /T a2 O5 with the total thickness of 4.66 microns. Both the ring
down samples and our mirrors were fabricated around 2000 by the same vendor (REO), so thematerial formula, depositing process, and treatment techniques should be very similar. Because thecoating is much thinner than the substrate, the ring down measurement is associated mostly withthe parallel direction of the coating surface [51]. The reported loss angle from such coating is
φk = (5.2 ± 0.3) × 10−4 ,(3.8)

Together with Eq. 2.19, the result can be written as(5.2 ± 0.3) × 10−4 = 0.58 φH + 0.42 φL .

(3.9)

Finally, Eq. 3.7 and Eq. 3.9 yield4 ,φL = (1.1 ± 0.3) × 10−4 ,(3.10a)φH = (8.2 ± 0.3) × 10−4 .

(3.10b)

These numbers are about a factor of 2 higher than the values reported in the literature. The
losses of fused silica and tantala in thin–film form (ion beam sputtered, annealed at temperaturearound 500 C) obtained from ring down measurements [99, 100, 101, 125] are φL = (0.7±0.3)×10−4 ,and φH = (4.4 ± 0.4) × 10−4 . However, our samples were coated by DC sputtering, which might add
carbon, iron to the coatings. The samples reported from the ring down measurements with lower
losses were fabricated after 2003. It is highly possible that these coating samples were fabricated bybetter equipment (RF sputtering technique) that introduced less contaminants to the coatings [126].

4 For more details on extracting φ

L and φH by Bayesian analysis, see appendix G.4.

ASD of beat note fluctuation [Hz/Hz1/2]

1

10

0

10

−3

10

−2

10

−1

10

0

10

1

10

Figure 3.9: result from short cavity

2

10Frequency [Hz]

3

10

4

10

Measured noise
Total estimated noise
Coating Brownian noise
Coating thermo−optic noise
Substrate thermal noise
Spacer thermal noise
Photothermal noise
Seismic noise
PLL noise
Residual frequency noise

48

49

Chapter 4

Technical Noise in the Setups
Both setups discussed in the previous chapter have similar tools (reference cavity, seismic isolation,
vacuum chamber, electronic equipments). They also use PDH method to lock laser frequency to thecavity, and PLL to measure the frequency noise of the beat note. As a result, the technical noise
sources associated with these setups are also the same. They can be roughly categorized into threekinds. Some significant ones are the following:• Mechanical-Related: Cavity sagging due to seismic noise, vibrations from mirror mounts.
• Electronic-Related: Sensing noise from PDH lock, readout noise from PLL.
• Optical-Related: Photothermal noise, ambient temperature changing the cavities’ length via
thermal expansion.

4.1

Seismic Noise

4.1.1

Coupling from Seismic to Displacement Noise

Cavity bending due to vibration has been known as a significant displacement noise for a reference
cavity. Attempting to minimize the effect, some groups reported holding methods by cutting or
drilling into a spacer to hold it at the optimum point where the coupling from seismic noise issignificantly reduced, or even holding the cavity vertically [127, 128, 129]. We decided to leave thespacer intact and to support it on the outer surface to avoid any extra loss due to a machining
process on the spacer. Because of the space limitation, we also hold the cavities horizontally.
The total displacement noise due to seismic is given by
Sx (f ) = v̇table (f ) T Fmount (f ) hcavity L,(4.1)

where v̇table (f ) is the power spectrum of the vertical seismic noise (acceleration) measured on the2√table [ m/s], T Fmount is the transfer function of the seismic isolation mount from the table to theHz50
cavity, hcavity is the coupling from the seismic noise to the cavity strain noise computed by FEA(COMSOL), and L is the cavity length.
2

For the 8 inch cavity held at their Airy position as shown in Fig. 3.3, hcavity = 2 × 10−10 [ sm ].
For the 1.45 inch cavity, as we changed the supporting method, we also searched for the optimum
support points to minimize the coupling. Consequently, the strain due to seismic noise is brought2

down to 6 × 10−12 [ sm ]. Details about how to calculate hcavity and how to search for the optimum
support points for the 1.45 inch cavity can be found in appendix E.4.
This hcavity is the coupling from a single cavity. The actual effect could be different because thefrequency noise in the beat note is affected by the differential displacement between the two cavities.
It also depends on how well both cavities and the support points are positioned, as misplaced support
points lead to different coupling.
To estimate the differential coupling, we added random error on the support positions, limited
to ± 0.5 mm, then recalculated the coupling on a cavity. The results from simulations are shown in
the histogram below.
16
14
12

Count

10
8
6
4
2
0
0.92

0.94

0.96

0.98
1
1.02
1.04
1.06
Cavity Strain per Acceleration [s2/m]

1.08

1.1

1.12
−10

x 10

Figure 4.1: Histogram of the vertical seismic to strain coupling with random variations in the support
positions on 1.45 inch cavity.
Next, we must take common mode rejection into account. According to the simulation result,most of the time, the strain will fall between 0.98 × 10−10 and 1.04 × 10−10 [s2 /m]. We can assume
an upper bound that one cavity has the strain of 1.04 × 10−10 and another has 0.98 × 10−10 . Thus,2the effective coupling for the differential strain is (1.04 − 0.98) × 10−10 = 6.0 × 10−12 [ sm ]. Dueto the common mode rejection, our the strain coupling is comparable to other techniques used
for minimizing seismic coupling noise, see section A for comparisons with other low noise cavity
experiments.

51

4.1.2

Scattered Light and Seismic Isolation in the Setup

Seismic noise not only shakes the cavity length, but also perturbs the phase of the reflected lightconsisting of the carrier and the scattered light. This reflected light adds on random phase noise to
the error signal of the PDH lock. The FSS will interpret this phase noise as the cavity’s displacement
noise causing an extra frequency noise in the beat note. Scattered light occurs at every optic’s surface.
The most important ones lie between the RF photodiode for PDH lock and the cavities because of
their proximity to the RF photodiode, which is the main sensing equipment. It is complicated toquantify or to directly measured the coupling of this effect in the setup [130]. Hence, it is best to
try to minimize the motion of the cavities and the vacuum chamber. A few passive seismic isolation
systems are used to reduce the seismic noise in the setup. They will not only decrease the scattered
light caused by vibration, but also reduce the transfer function T Fmount (f ) in Eq. 4.1. The seismic
isolation stages are• Newport S-2000 Pneumatic Vibration Isolator with Automatic Re-leveling. The table is floatedwith a set of 4 pneumatic legs. The resonant frequency is around 1 Hz on both verticaland horizontal directions. The legs are supplied with a bottle of compressed air during themeasurement.
• Air springs. A set of pneumatic air springs are used to support the vacuum tanks on the
optical table. The attenuation is at least a factor of 10 at 20 Hz and above, see Fig. 4.2.
• Double–stage seismic isolation stack. Each stage rests on a set of RTV silicone. This material
is vacuum compatible [131]. It acts as a spring for the passive isolation stage.
Aside from the seismic isolation, the scattered light can be minimized by dumping all the straybeams appropriately. These stray beams are the reflected beams from optic components (e.g.,
photodiodes, beam splitters, vacuum chamber’s window).

4.2

Mechanical Peaks from Opto–Mechanical Components

Motions on the optic mounts perturb the phase of light as it propagates. This mechanically induced
phase noise is also added up to the laser frequency noise. These mechanical peaks are not a serious
problem if they occur between the laser and the polarizing beam splitter (PBS) for the reflectedbeam used in PDH lock, since they will be similar to the laser frequency noise and can be suppressedwith the FSS. However, if they occur between the reference cavity and the RF photodiode, the effectwill be indistinguishable from the real cavity’s displacement noise. The same problem also happens
behind the cavities where the beams recombine. We identified the mechanical peaks associated witheach optic mounts in the beam path by exciting each one and observing the resulting peak in the

52

Figure 4.2: Top, the air springs supporting the vacuum chamber. Bottom, the transfer function
between the table and the chamber.

53
power spectrum of the beat readout, see Fig. 4.3. Due to the structure of the components, the mostvibration sensitive ones are the periscopes used for directing the beams to the cavities, the beam
splitter for recombining the beams, and the quarter wave plates. These resonant peaks vary from500 Hz to 1.2 kHz. We minimize those peaks by damping them with blocks of rubber. We removed
the output periscopes behind the cavities by installing the beat readout path on a raised platform.
The mirrors are mounted on pyramid-like bases which are screwed on the table, not clamped. The
end beam splitter is mounted in a solid block with no adjustment knobs. These solid bases can
reduce the associated mechanical peaks significantly.

4.3

PDH Lock: Shot Noise

When a reference cavity is used for frequency locking via PDH technique, the reflected light fromthe cavity beats with the sidebands to create the error signal. The slope of the error signal from
PDH locking technique can be written as [120]dP
F inesse= −16Pin J0 (β)J1 (β) ×
,
dL
λ

(4.2)

where Pin is the input power, J0 (β) and J1 (β) are Bessel’s functions of the first kind, β is themodulation depth (= 0.2 rad for the setup). In other words, the power of the error signal is
proportional to the mismatch between the laser frequency and the cavity’s length. In general, thismethod is insensitive to power fluctuations. When Pin changes due to some classical fluctuations,the slope changes, but the servo still knows which direction it has to act follow the cavity. However,
the incident light that carries information about the cavity length exhibits another fundamental
noise, which is shot noise.It arises because of the vacuum fluctuations which occur even though thelaser power, Pin , is constant. The servo will not be able to distinguish it from the real power change
due to the cavity length change and try to compensate for it and create extra frequency noise. The
power fluctuation due to the total power on the photodiode is

dPrefl =q

2hνPin ((1 − η)J02 (β) + 1.5 × 2J12 (β)),(4.3)

where PRF P D is the power falling on the photodiode for PDH lock, η is the visibility (ranging
between 0 and 1). The reflected power contains power from a part of the reflected carrier due tomode mismatch and the two side bands. The factor of 1.5 in the sideband term originates from thenon–stationary nature of the shot noise produced by the sidebands [132] and the fact that we areusing a sine wave demodulation [133]. Combining Eq. 4.2 and Eq. 4.3 together, we get the equivalent

54

Figure 4.3: Mechanical resonant peaks due to opto-mechanical components. Top, the setup of the
input optics. Bottom, the setup behind the cavity where the transmitted beams recombine.

55
length noise due to the shot noise asdL(f ) =dPrefl
.
dP/dL

(4.4)

This is important in the noise optimization aspect. Theoretically, the maximum sensitivity of the
PDH lock is achieved when the modulation index is equal to 1.08 [120]. However, the more power
allocated to the sideband (term with J1 (β)), the higher the shot noise. By tuning the modulationindex β, we can minimize shot noise. Care should be taken, since it also affects the electronic noisein the PDH lock, as we will see in the next section.

4.4

PDH Lock: Electronic Noise

δν

δ νs

D

δs3
A

G
δs2

δs1

Figure 4.4: Noise in PDH lock technique.
Electronic equipment used in PDH lock contributes electronic noise along the process. Fig. 4.4
demonstrates possible noise sources in each component of the control loop. The suppressed frequency(ignoring the cavity displacement noise) can be calculated as followδνs = δν − δνs DGA ± δs1 GA ± δs2 A ± δs3δνδs1 GA
δs2 A
δs3
±
±
±
1 + DGA 1 + DGA 1 + DGA 1 + DGA
δν
δs1
≈
±
, when G  1,1 + DGAD=(4.5a)(4.5b)(4.5c)

where ± signs denote the incoherent sum from the noises due to their random nature. In thelimit of very large servo gain, the noise associated with the frequency discriminator (from RFPD

56
to the mixer out of the demodulated signal) dominates all other terms and the effective noise isδs1 /D. δs1 can be measured from the mixer output when the loop is disengaged and the beam to
the RFPD is blocked. The frequency discriminator gain D, can be calculated according to Eq. 4.2,
if the modulation index, quantum efficiency of the photodiode, and the impedance of the RFPD are
known, or it can be measured directly from the setup by scanning the laser frequency an observing
the error signal at the mixer output on an oscilloscope. The slope will be in units of [Volt/second ] as
we record the time domain of the signal. If the scanning is large enough to include the error signal
due to the sideband, the time during the scan between the carrier and the sideband frequency will
give the conversion from [second] to [Hz], these will provide the gain of the frequency discriminatorin the units of [Volt/Hz ]. The electronic noise associated with the RFPD and the mixer can bedirectly measured at the mixer output while blocking the beam.
To reduce the frequency discriminator noise, we can increase the gain D, by increasing the input
power, the cavity finesse, the modulation index, or the transimpedance of the RFPD. In practice,
the finesse and the transimpedance of the RFPD might not be easily adjusted. So increasing the
input power or the modulation depth will be done most of the time during the optimization. But
care should be taken since these parameters also affect other noise sources as well (e.g., shot noise
limit, and photothermal noise).
In the prototype setup (a single laser with 8-in cavities), where the main beam is already locked
to the reference cavity, the frequency noise of the laser becomes δνs and the servo will lock the beam
to match the analyzer cavity with displacement noise δL2 , which results in the frequency noise δνs2 .
If we follow the diagram shown in Fig. 4.4, the noise calculation for the second loop can be writtenasδνs2 = δνs − δνs2 DGA + δL2 DGA ± δs1 GA ± δs2 A ± δs3δνsδL2 DGAδs1 GA
δs2 A
δs3
+
±
±
±
1 + DGA 1 + DGA 1 + DGA 1 + DGA 1 + DGA
δνs
δs1
δs2
δs3
≈
+ δL2 ±±±.
DGAD
DG DGA
=(4.6a)(4.6b)(4.6c)

In general, the free running noise of the laser δν is the highest compared to the others, but forthis case, δν is suppressed and becomes δνs ≈ δL2 . Therefore the servo gain G for this loop is not
as demanding as that of the first loop. The main concern for the second servo is to suppress all thenoise associated with the electronics and the VCO (δs2 , δs3 ) below the coating noise level.

4.5

PDH Lock: Residual Amplitude Modulation

Residual amplitude modulation (RAM) usually includes many effects that create amplitude modulation at the frequency of the sideband. RAM causes an offset in the PDH error signal and the57lock becomes susceptible to the other effects (e.g., power fluctuation)1 One of the most importantcomponents that introduce RAM is the EOM used for adding sidebands. The EOM varies the phaseand frequency of the beam by changing the refractive index due to the applied voltage. If the alignment of the EOM axis and the polarization of the incident light do not match, the polarization of
the beam is changed at the same frequency as the sideband, and as the beam propagates throughthe setup and encounters polarizing beam splitters (PBS), the varying polarization is converted tothe amplitude noise. Meanwhile, the axis of the EOM can also be changed due to the stress-induced
birefringence (which occurs due to the temperature change). Thus, to minimize the RAM, the EOM
was temperature-stabilized with insulation and a heater, and then the polarization of the beam wasadjusted by a half wave plate in front of the EOM so that input polarization matches the electricfield axis inside the EOM crystal. To optimize the polarization, the laser frequency is set to be
off resonant from the cavity. The power is reflected off of the cavity on to the RFPD, then theamplitude output is monitored. By slightly changing the polarization input to the EOM, the peakin the output at the sideband frequency can be minimized.

4.6

Noise from Phase Locked Loop Readout

A phase locked loop (PLL) is used to measure the frequency noise in the beat signal. In principle, astable reference signal is controlled so that its phase matches the phase of the target signal, which isthe beat note in this case. By observing how much the control signal has to follow the beat note, thenoise of the beat note can be measured. A schematic diagram of a PLL setup is shown in Fig. 4.5
From the diagram, we can calculate how the noises from RFPD and VCO show up in the final
readout.
δνs = δν ± δn ± δV − δνs DAG=

δν ± δn ± δV
1 + DAGVf b = δνs DG
DG
1 + DAG1≈ (δν ± δn ± δV ) , for G  1
A= (δν ± δn ± δV )(4.7a)(4.7b)(4.7c)(4.7d)(4.7e)

With sufficient gain G, by monitoring Vf b , we can convert it to the beat frequency noise δν by theactuator gain A. (see Fig. 4.7 for an example of the measurement of the calibration factor A).
However, the noise δV from the VCO, and the noise from the RFPD (δn can represent both shotnoise and electronic noise since they combine after the RFPD) show up directly in the beat readout
1 see, e.g., the discussion by Ishibashi et al. [134].

58

RFPD

beat signal @ 160MHz

vco
IFR2023b
160MHz +/- tuning range
RFPD
noise

VCO
freq
noise

G
FFT
SR785

SR560

Figure 4.5: A setup of phase locked loop readout. The RF photodiode used in this work is Newport
1811. The beat note is demodulated by a low noise voltage controlled oscillator (VCO) and theerror signal is fed back to the VCO. This error signal is amplified by the gain servo (G) and becomesa control signal. It tells the VCO output to match the input beat note. With sufficient gain, thiscontrol signal represents the noise in the beat note. For our experiment, the operating setup includes,a level 13 mixer, a low pass filter is at 10 MHz, the servo gain is SR560 with the setup at dc withgain 200. The VCO is IFR2023b with 1 kHz tuning range.

δn
δ νs

Beat δν

D

x
RFPD

δs
δV
A

G

VCO

FFT

Vfb

Figure 4.6: A block diagram of PLL. δν is the frequency noise of the beat note to be measured, δs
is the shot noise of the incident beam, δn is the electronic noise from the beat photodiode, and δV
is the frequency noise of the oscillator.

59(again, the ± sign denotes the incoherent sum from each noise). We have to characterize them tomake sure that their levels are below the expected beat noise.

Frequency Noise from the VCO
The measured frequency noise from the VCO from different settings is shown in Fig. 4.8. The setup
is similar to PLL, but the signal from the beat note is replaced by a signal from another VCO. Bothoscillators have the same settings (i.e., carrier frequency, tuning range), and are assumed to havethe same noise level.

Electronic Noise and Shot Noise
To measure electronic noise in the RFPD, we can observe it at the Vf b (with the input light blocked
off, and the feedback to the VCO removed). Following Fig. 4.6, we can write,
Vf b = δnDG.

(4.8)

But the final noise due to the electronic noise that shows up in the beat signal is δn/A (Eq. 4.7).
Hence, the electronic noise shows up in the beat noise as,
δn
Vf b
Vf b
=
=
.
A
DGA
Open Loop Gain

(4.9)

The open loop gain DGA can be measured from the PLL setup. Similarly, noise due to the shotnoise can be measured with the same method. But instead of blocking the light, we use a white lightsource with the power to match the output level of that from the beat. Then measure the noise dueto the white light.

4.7

Photothermal NoiseAs discussed in section 2.7, fluctuation in laser power changes the effective cavity length via thethermoelastic and thermorefractive coefficients. In the case of a beam whose intensity fluctuationis shot noise limited, the photothermal noise is negligible compared to Brownian thermal noise andthermoelastic noise [76]. However, for a laser with significant intensity noise above the shot-noise
limit, the photothermal noise can be much higher. In the case of the one-laser setup, this excess
photothermal effect appears in both cavities as a common-mode noise. However, this is not thecase for the two-laser setup, and so the photothermal effect has to be carefully characterized andfactored into the noise budget. By using a set of quarter wave plate, EOAM and a PBS, in eachpath to modulate the input power (see Fig. 3.8), we can observe the corresponding modulation in
the beat note frequency using the PLL readout. The results are comparable with the calculations

60

800

data 1linear600

Frequency shift [Hz]

400
200
0
−200
Fit, 712 Hz/V
−400
−600
−800
−1

−0.8

−0.6

−0.4

−0.2

0
Volt

0.2

0.4

0.6

0.8

1

Figure 4.7: An example of a calibration measurement for PLL, the carrier frequency is 160 MHz
with 1 kHz tuning range. The output frequency, which can be measured by a spectrum analyzer
or a frequency meter, shifts from the central frequency due to the input voltage provided by a low
noise voltage calibrator. The fitted slope, 712 Hz/V, is the actuator gain, A, in the block diagram.
This actuator gain is approximately flat in our bandwidth of interest. Use this factor to convert theelectronic control signal to the beat note frequency noise.

61

Frequency Noise from IFR2023B locked to Rb Clock1

10160 MHz /10 kHz
100 MHz /10 kHz
10 MHz /10 kHz
160 MHz /1 kHz
100 MHz /1 kHz
10 MHz /1 kHz
160 MHz /100 Hz
100 MHz /100 Hz
10 MHz /100 Hz
Estimated Coating Noise from 8" cavity0Frequency Noise [Hz/√Hz]

10

−1

10

−2

10

−3

10

−4

10

−1

10

0

10

1

10

2

10
Frequency [Hz]

3

10

4

10

Figure 4.8: Frequency noise from VCO (from IFR2023A and IFR2023B) in PLL setup. The measurements are performed from three carrier frequencies, 10 MHz, 100 MHz, and 160 MHz each withthree different tuning ranges ate 100 Hz, 1 kHz, and 10 kHz. The setup is similar to what is shownin Fig. 4.5, but the RFPD is replaced by another similar VCO with the same setup. Both oscillators are locked to a 10 MHz Rb clock as a frequency reference to suppress their frequency noise. A
transformer is added between the clock and each oscillator to prevent ground loop. The calibration,
A (cf. Fig. 4.6), is measured from each setup. For the chosen carrier frequencies, the calibration
depends only on the frequency tuning range. From the results, we can conclude that the smaller thetuning range, the lower the noise. However, the noise level seems to be comparable around 1 kHz
regardless of the tuning range. We choose 1 kHz tuning frequency range as a default setup because
coating brownian noise is still observable up to almost 1 kHz and the phase locked loop can be locked
sufficiently long enough.

62

coating expansion
substrate expansion
coating refractive
total
measured7[Hz/Watt]

10

6

10

5

10

[Degree]

-1

0

10

10

150
100
50
0
-50 -1
10

10

1

10

0

1

10

[Hz]

2

10

2

10

3

10

3

10

Figure 4.9: A swept sine measurement of beat note frequency fluctuation in response to photothermal
noise. Both the amplitude (top) and phase response (bottom) agree with the calculations from Farsi
et al. For our coatings, the greatest effects are thermal expansion from substrate and coating. The
phase plot shows that the calculation of thermorefractive effect is out of phase with thermoelastic
effect from both the coating and the substrate, as one would expect. The measurement is madepossible with high SNR because of the PLL is not sensitive to amplitude noise from the modulation.
given in Farsi et al. [85] with the assumption of 5 ppm absorption on each mirror, see Fig. 4.9.
Together with the measured RIN in the transmitted cavity beams, the estimated frequency noisedue to photothermal noise can be added to the noise budget.

4.8

Noise from Ambient Temperature

The ambient temperature, i.e., temperature fluctuations in the room or the voltage noise in theheater around the vacuum tank, change the cavities length via thermal expansion coefficient. By
placing the two cavities side by side, the noise becomes common mode between the two. The
frequency of the laser locked to the cavity with length L is
νi =

ni c
,
2Li

(4.10)

63
where subscript i denotes the two cavities, n is the axial mode number. The deviation in frequency
due to thermal expansion of the spacer, to the first order, isδνi ∼ −

ni c
αδT.
2Li

(4.11)

Then, the frequency noise between the two isδν1 − δν2 = −αδT

n1 c
n2 c
−
2L1
2L2



δνbeat = −αδT νbeat .

(4.12a)(4.12b)

Here νbeat is the frequency of the recombined beams. It is around 160 MHz and 120 MHz in the
prototype setup, and in the 2–laser setup, respectively. δT is the ambient temperature fluctuationnear the cavities. It can arise from the ambient room temperature or from the voltage noise on
the heater. The temperature fluctuations transfers through the heat insulation, the vacuum tankto the cavities. Each stage behaves as a low pass filter with a pole around sub-Hz level making
the noise become very small at the bandwidth of interest. To estimate the effect at 1 Hz, δTroom√√is ∼ 10−4 K/ Hz, and δTheater due to the supplying voltage noise is ∼ 10−7 K/ Hz, resulting in√
the beat frequency noise well below 10 µHz/ Hz. However, the real effect can be larger becauseof the temperature gradient due to close by thermal sources from electronic devices or people, or
non–uniformity in the shape of the heat insulation or the heater. These factors lead to higher noiselevel, but it is still not significant compared to other noise sources. Although the temperaturenoise does not dominate in our measurement band, the beat frequency drift due to the temperature
requires a larger tuning range for PLL and this creates a noise floor in the measurement.

64

Chapter 5

AlGaAs Crystalline Coatings
Gallium arsenide, and aluminum gallium arsenide (Alx Ga1−x As with x varied between 0 and 1 toindicate the amount of Al) are semiconductor materials. Because of their similar lattice structure,they can be grown in alternated layer form while preserving their structure. Together with theirsuitable optical properties for light at 1064 nm, coatings fabricated from alternating layer of GaAs
and Alx Ga1−x As can be use as a high reflective coating. Due to their crystalline structure, themechanical loss is expected to be much smaller than that from amorphous materials like SiO2 andT a2 O5 . This chapter will explore AlGaAs properties as high reflective coatings. We will see that
due to its low loss, Brownian thermal noise will not be the limiting noise source anymore. Instead,
substrate thermoelastic noise and coating thermo–optic noise will be the dominant sources, for ourcavity geometry. This situation lets us study more about coating thermo–optic noise which is notobservable in the SiO2 /T a2 O5 coating. Since thermo–optic noise is a combination of thermoelastic
and thermorefractive noise, we can partially cancel it with a careful design of coating structure. A
successful design can set a new fundamental limit of displacement noise in a reference cavity.

5.1

History of AlGaAs Usage

Doublets of GaAs and Alx Ga1−x As, or collectively called AlGaAs, grown with molecular beam
epitaxial method (MBE), has been used since 1980s as a high reflective device in vertical-cavity
surface-emitting lasers (VCSELs) for wavelengths from 650 nm to 1300 nm [135]. Because of its
application in laser and optic devices, their optical parameters and material parameters, e.g., refractive indices, thermal expansion coefficients, and thermorefractive coefficients, have been studied
extensively [136, 137]. Due to its crystalline structure, AlGaAs is expected to have very low mechanical loss. This makes it a good candidate for a low loss coating material for high reflective
mirrors. Nevertheless, their usage was limited to small devices like VCSELs because the structure
has to be grown on a lattice-matched material, for example, GaAs or germanium which are notsuitable for used as a mirror substrate at 1064nm because of its high optical absorption and cost.

65This problem was overcome when the epitaxial lift off (ELO) technique was developed. This allows
an AlGaAs wafer structure to be grown on a lattice-matched substrate, then removed and applied
on a target substrate. The ELO technique has been widely used, for examples, in solar cell technology [138], LED [139], and lasers [140], where AlGaAs thin films are transferred to silicon, sapphire,
glass substrates. As a result, high reflective mirrors with AlGaAs coatings on fused silica substrates
are made possible. However, the coatings can only be applied to mirrors with radius of curvature of
1.0 m or larger. The smaller radius of curvature will result in excess stress in the coating which cancreate dislocation in the lattice structure, and it cannot be bonded on the substrate. Thus, for thisdissertation, the radius of curvature of the mirrors for AlGaAs coating will be at 1.0 m.
Since a possibility to use AlGaAs as a coating material had shown up, a study to determine its
loss followed. Ting down measurements of thin film AlGaAs had been done. Unlike the ring down
measurements performed on SiO2 /T a2 O5 where the coating has to be deposited on the substrate,
this ring down measurement was performed on a stand-alone thin film AlGaAs. As a result, errors
associated with the substrate loss, the dilution factor (see appendix C.2), can be removed, and theloss in the thin film can be measured with higher accuracy. The wafer samples were created with
gas-phase etching technique [141], with nodal support to minimize structural loss for each bending
mode [142]. The loss angle at room temperature is measured to be 2.5×10−5 , and down to 1.1×10−5
at 20 K) where the coatings still have low optical loss. Together with the bonding technique, thewafer can be grown, transferred, and attached on a fused silica substrate which, is widely used in highquality optics for 1064 nm wavelength. A direct observation of noise on a Fabry–Perot cavity formed
by AlGaAs coatings on fused silica substrates has been measured. The PSD of the displacement
noise is around 10 times lower than that of mirrors with SiO2 /T a2 O5 coatings [143] .
The refractive index of Alx Ga1−x As is a monotonic function of x, cf. Tab. C.1, so it is best to use
a heterostructure of AlAs/GaAs for maximum contrast for reflectivity. But AlAs oxidizes quite fast
in atmosphere, a small amount of Ga has to be added to make the substance more stable. In thiswork, the amount of Ga will be 8 percent by molar, and it is addressed as Al0.92 Ga0.08 As. The end
layers of the coating wafer have to be GaAs on both sides. A table of AlGaAs material parameters
can be found in appendix C

5.2

Thermo–Optic Noise and Coatings Optimization

Because of the high values of thermal expansion and thermal refraction coefficients of AlGaAs,
coating thermo–optic noise, as estimated by Eq. 2.22 become significant for a standard QWL coating
structure. In this section, we will discuss about AlGaAs coating thermo–optic noise and a plan toreduce it.

66

5.2.1

Thermal Noise in AlGaAs Coatings

A noise budget of AlGaAs coatings on a fused silica substrate is shown in Fig. 5.1. Coating Brownian
noise is not the dominating noise source in this situation. Coating thermo–optic noise will be thelimiting noise source around 100 Hz and above.
AlGaAs coatings

0

10

Coating Brownian
Substrate TE
Thermo−Optic
coating TE
coating TR
Total(Br+sub TE + coat TO)−1

Frequency Noise [Hz/√ Hz]

10

−2

10

Spotsize =215µm Cav Length=36.8mm
RoC =1m
T = 134 ppm# of layers = 53 1st cap=0.25 lambda−3

10

Thickness = 4.4e−06 µm−4Finesse = 20398

10

0

10

1

10

2

10
Frequency [Hz]

3

10

4

10

Figure 5.1: Noise budget for AlGaAs coatings with quarter wavelength structure, 53 layers. Dueto the low mechanical loss, thermoelastic from substrate is dominating at low frequency whilethermo–optic noise in coating becomes dominant at frequency above 100 Hz.The total noise is plottedin black dashed line. Other relevant optical properties are shown in the plot.
As discussed in Ref. [74], the effects from thermoelastic and thermorefractive noise in coating
layers cancel each other. By carefully designing the coatings layers thickness, we can reduce thetotal thermo–optic noise while achieving the desired reflectivity of the coatings.
Before we proceed to the optimization discussion, let’s investigate if TO cancellation is possible
with a QWL structure. The contributions from thermoelastic and thermorefractive are proportional
to ᾱc d and β̄λ, respectively, cf. Eq. 2.22. So thermoelastic effect is proportional to the coating
thickness, while thermoelastic effect is roughly independent from the coating thickness (the effect ismostly contributed from the top layers). For the QWL structure with 53 layers, thermoelastic noise
is higher than thermorefractive noise. By reducing the total number of layers, thermoelastic noise
can be varied until it is comparable to thermorefractive noise to achieve thermo–optic cancellation.
This is possible at 33 layers. The noise budget is shown in Fig. 5.2. However, the transmission isaround 3000 ppm, and it results in a rather low cavity finesse (around 1,000).

67
Another method is to add a half–wave cap of AlGaAs on the top layer of a QWL structure
to mitigate thermo–optic noise [143]. This method is intended for a low transmission mirror (81layers, 5 ppm). The cap changes the effective β (cf. Eq. D.7) while maintaining the correct inference
condition. However, AlGaAs is not suitable to be used as an end cap because it can oxidize andchange its optical properties.
AlGaAs coatings

0

10

Coating Brownian
Substrate TE
Thermo−Optic
coating TE
coating TR
Total(Br+sub TE + coat TO)−1

Frequency Noise [Hz/√ Hz]

10

−2

10

Spotsize =215µm Cav Length=36.8mm
RoC =1m
T = 3092 ppm# of layers = 33 1st cap=0.25 lambda−3

10

Thickness = 2.7e−06 µm−4Finesse = 1008.1

10

0

10

1

10

2

10
Frequency [Hz]

3

10

4

10

Figure 5.2: Noise budget of AlGaAs coating QWL structure, 33 layers in total.

5.2.2

Optimization Consideration

The main goal of the optimization is to make thermo optic noise below other noise sources whichcannot be adjusted (e.g., coating Brownian noise, substrate thermoelastic noise), and to demonstratethat such coating is possible. However, as we alter the layer thickness to reduce thermo–optic noise,
other optical properties of the coating is changed too. Here we explain some considerations thathave to be monitored during the optimization.
• Transmission: The desired transmission has to be around 100 ppm to 300 ppm. Smallertransmission (more reflectivity) leads to more power build up in the cavity and higher photothermal noise, cf. sections 2.7 and 4.7. The transmitted power also depends on the mirrors’
transmission when the mirrors have loss (from absorption or scatter). Lower transmission
means that the light will circle in the cavity more often, and lose power each time it reflects

68of the mirrors. The transmitted power from the cavity is given by [144]
Ptrans /Pin =T 2 Re−α,R(1 − Re−α )2(5.1)

Figure 5.3: Transmitted power behind the cavity (left y-axis), and cavity finesse (right y-axis) as afunction of mirror’s transmission. The calculation assumes each mirror has 10 ppm loss.
where T is the power transmission, α is the roundtrip loss. Higher transmission results in a
low finesse cavity which reduces the gain of the frequency discriminator (see sections 4.3, and4.4) and compromises the frequency stabilization performance.
• Phase of the reflected light at the top surface: to avoid surface burning, the reflected
phase at the surface must cancel the incoming phase, the node of the standing wave formed in
the cavity is at the coating surface. In general, for a normal incident wave reflecting from theinterface between medium 1 and 2, the reflected light will have a phase shift of 0 or 180 degrees
depending on the refractive indices of the two medium. For reflection from multiple medium
with various thickness, the back reflection from other interfaces will sum up at the first surface
resulting in a complex value of reflectivity r. Generally, for a coating with QWL structure, thereflection phase will be close to 180 degrees, see different cases of end caps with nH or nL in
appendix D. The optimization will change the path length in coating layers and the effective
phase of the coating reflectivity will not be close to 180 degrees. When that happens, the nodeof the standing wave will move away from the coating surface (see Fig. 5.4) and certain power
can build up on the surface during the light circulation. If there are any impurities on thesurface, they might be burnt from the intra-cavity power. These burnt impurity will create

69
more contamination in the system.
This issue prevents us from optimizing the coating by altering only the cap layer thickness andleave the rest of the layers in QWL structure. Remember that tuning the cap thickness canchange β of the coating effectively [145]. But the reflected phase will be wrong.

Figure 5.4: The power inside a cavity shown in red which is the square of the sum of two electricalfields propagating from left and right directions. The x-axes are the cavity beam line position
normalized by the cavity length such that the coating surfaces are at 0 and 1. In this case, thecavity length is twice the beam wavelength. The y-axes represent the electric field and power inarbitrary units. Left, when the reflected phase of the mirror is close to 180 degrees, the power at
the mirror surfaces (at 0 and 1 on the x-axis) will remain close to zero at all time. Right, when thereflected phase is off by 40 degrees, the power on the mirrors’ surfaces will not always be zero.
In case of reflectivity with phase shift away from 180 degrees, the power build up on the cavity
can be approximated as
Etot = Er + El

(5.2a)≈ Er + rEr = Er (1 + r) for high reflectivity mirror(5.2b)Er and El are electric fields circulating the cavity in right and left directions. The reflectivity
can be written as r = |r|eiπ eiδ , where δ represents the phase shift away from 180 degrees. Ifδ  1 and |r| ≈ 1 the reflectivity can be approximated as r ≈ −eiδ . Then the power on thesurface is

Ptot = |Etot |2 ≈ |Er |2 |(1 − eiδ )|2≈ |Er |2 (δ)2≈Finesse 2 2
FinesseEin δ =
Pin δ 2 .
π
π

(5.3a)(5.3b)(5.3c)

70
For our parameters, the finesse is around 20,000, Pin = 1 mW , and δ ≈ 6 degrees, the power on
the coating will be ≈ 70 mW. With the spot size around 200 µm, the intensity is around 1 MW/cm2(with a safety factor of 2). This intensity level is acceptable for general lab optics (the damage threshold can go up to 10 MW/cm2 for general laser optics1 ), and the risk of burning any contaminationon the surface should be low.

5.2.3

Uncertainties in Material Parameters

Uncertainties in material parameters also lead to deviations from the desired coating properties.
For example, variations in α and β will directly compromise the cancellation in thermo–optic noise.
Refractive indices are also one of the most important factors coming into the calculation. The
amplitude of the reflected phase depends on the refractive indices values. In addition, they alsochange the optical path length in the layers. Since the optimization deals with the optical thickness
in each layer, the associated physical thickness is given byt = dOpt ×

λ
,
n0

(5.4)

Where n0 is the nominal refractive index. Typically, the physical thickness is being controlled during
the crystal growth. As a result, if the actual value of refractive index is n0 6= n0 , the optical thickness
becomesdOpt0 = dOpt ×

n0
,
n0

(5.5)

and the error in optical thickness directly affects the phase of the reflected beam. For the optimized
coating, we have to make sure that the coating properties will still be acceptable even with someuncertainties in the material parameters, see Tab. 5.1 for the approximated uncertainties of AlGaAs
properties.
Parameters

GaAs

Al0.92 Ga0.08 As

Heat capacity, C (×106 ) [J/Km3 ]1.754 ± 5%

1.698 ± 5%

Thermal conductivity, κ [W/mK]

55 ± 5%70 ± 5%Young’s modulus, Y [GPa]100 ± 20%100 ± 20%

Poisson’s ratio, σ0.32 ± 10%0.32 ± 10%

Thermal expansion, α (×10−6 ) [1/K]

5.7 ± 5%5.2 ± 5%

Thermorefractive, β (×10−6 ) [1/K]

366 ± 7179 ± 73.51 ± 0.03

3.0 ± 0.03

Refractive index, n @1064nm

Table 5.1: AlGaAs material parameters and the estimated uncertainties.
1 https://marketplace.idexop.com/store/IdexCustom/PartDetails?pId=145

71

5.2.4

Errors Due to Manufacturing Process

Besides the uncertainties from material parameters, the coating fabrication process introduces errors
in the coating properties in a few following ways:• Thickness Control: Thickness calibration might be off causing the layers thickness to deviate
from the desired value. As a result, the error is not random among each layer. It is rather
constant in each layer type, i.e., all the layers from the same material (nL or nH ) have thesame percentage of error. The errors will have the same sign, that is, if one material is thicker,
another is also thicker, but the magnitudes are uncorrelated. In addition, GaAs has better
thickness control than Al0.92 Ga0.08 As. The standard deviation of GaAs is about 0.5 %, whilethat of Al0.92 Ga0.08 is about 1 % [146].
• Truncated Thickness: The thickness control during the coating process can be controlled
down to 0.5 angstrom [146]. The specification in the design with smaller resolution has to berounded up to the nearest 0.5 angstrom. This is similar to random variations in the thickness
control, see Fig. 5.5. Fortunately, the effect is much smaller than the averaged coating layer(around 800 A), and it will not be a problem for the optimized structure.

0.25
0.2
0.15

Angstrom

0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
−0.25
0

10

20

30
Nth layer

40

50

60

Figure 5.5: Thickness deviation from the design due to the truncation for the optimized coating
structure.

• Error in the Al content: The material properties of Alx Ga1−x As are usually dependent
on x (the amount of Al), see Tab. C.1. According to the coating vendor, the variations in Al
content is x = 92 % ± 0.6 % [146]. For the optimization, one of the most important parameteris the refractive index. It can be extrapolated as n(x) = 3.48 − 0.578x [147, 148]. If x is alteredby 6 %, the refractive index changes by 0.0035. This value is almost a factor of ten smaller

72
than the standard deviation of nL and nH used in the calculation, cf. Tab. 5.1. This is also
true for other parameters . Hence, the effect from the uncertainty in Al amount is negligible.

5.2.5

Optimization Method

The optimization can be done by varying the layers thickness while minimizing a cost function
which is defined by the properties we want to control (i.e., thermo–optic noise level, transmission,and reflected phase). In general, the cancellation will not work at all frequencies due to the finite
thickness of the coating. So, the optimization can be chosen at a single frequency range. For our
calculation, the target frequency f0 of the thermo–optic cancellation was set to 100 Hz, which isabout the central frequency of our detection band. The cost function y is given by

y(S

TO

, T, δ) = STO(f0 ) · w1 +T − T0
T0

2· w2 + δ 2 · w3 ,(5.6)

where w1 , w2 , and w3 are weights assigned to each parameters, T0 is the target transmission set
around 200 ppm, δ is the reflected phase shift from π.
To search for the optimized structure, we used a MATLAB built in function fmincon 2 that isused for finding minimum of a constrained nonlinear multivariable function.
This method will give us the coating structure with the minimized cost function. Note thatthe question about whether the solution will return the absolute minimum is not the main concern
here. Rather, we want to ask if the coating will work even if there are uncertainties in the material
parameters. One of the important parameters for the coating properties are the refractive indices.
To make sure that the designed structure is relatively insensitive to the parameters, we modified thecost function to be
y2 =

XX

y(S T O , T, δ; nH , nL )

(5.7)

nH nL

The new cost function is the sum of cost functions from different values of nH and nL . This
will ensure that the optimized coating’s properties will still not be compromised even though thevalues of refractive indices have some small errors. For our optimization, nH = {3.47, 3.50, 3.53},
and nL = {2.97, 3.00, 3.03}. As a result, the optimized structure with the modified cost function
becomes less sensitive to the variations of refractive indices compared to another optimized structurewith a simple cost function, see Fig. 5.7 and Fig. 5.8. Finally, we checked how the optimized coatingchanged with all uncertainties discussed in section 5.2.3 and section 5.2.4. By running a Monte
Carlo simulation when random errors were added to the parameters. The histograms of coating
properties are shown in Fig. 5.12. Around 70 % of the results, the coating properties will meet therequirements ( 100 < T0 < 300, δ < 7degrees, and ST O < 1.5 × 10−39 m2 /Hz). Details of the final
2 http://www.mathworks.com/help/optim/ug/fmincon.html

73
optimized design are reported in Fig. 5.10 and Tab. 5.2. Coating thermo–optic noise is predicted to
be well below other noise source from DC to a few kHz, see Fig. 5.9.
One way to verify how well the cancellation works it to measure photothermal response of thecoatings. This is similar to the measurement discussed in section 4.7. If the cancellation works, theeffect will mostly be thermal expansion in the substrate. If the cancellation does not work as expect,
the shape of the transfer function should provide a better hint than the direct observation from thecavities displacement noise, see Fig. 5.13.

Figure 5.6: Optical properties of QWL structure with 55 layers as a function of nH and nL . Theseplots serve as baseline values used for comparison with other optimized coatings shown in Fig. 5.7
and Fig. 5.8.

74

Figure 5.7: This optimization was done with a single cost function (cf. Eq. 5.6). Note that thedetail of this structure is not reported here. Although, for nominal values of refractive indices, thethermo–optic cancellation is good, and the transmission is close to the required vale (200ppm), theproperties change rapidly as the refractive indices vary from the central values.

75

Figure 5.8: This optimization is the final design. It was done with the modified cost function(Eq. 5.7). The cancellation for thermo–optic noise is less sensitive to variations in refractive indices,and the transmission will mostly stay in the acceptable range (100–300 ppm).

AlGaAs coatings

0

10

Coating Brownian
Substrate TE
Thermo−Optic
coating TE
coating TR
Total(Br+sub TE + coat TO)−1

Frequency Noise [Hz/√ Hz]

10

−2

10

Spotsize =215µm Cav Length=36.8mm
RoC =1m
T = 141.7 ppm# of layers = 57 1st cap=0.18957 lambda−3

10

Thickness = 4.6e−06 µm−4Finesse = 19427

10

0

10

1

10

2

10
Frequency [Hz]

3

10

4

10

Figure 5.9: Noise budget from the optimized structure with 57 layers in total. Due to the low loss,
substrate thermoelastic noise will be the dominating source from DC to around 100 Hz.

76
Nth

opticalLayernphysicalNthopticalthicknessthickness

Layer

thicknessthickness[wave-[nm][wave-[nm]

lengths]nphysicallengths]1(cap)

0.1896

3.480

57.95

31

0.2528

3.480

77.30

2

0.1121

2.977

40.05

32

0.2457

2.977

87.80

3

0.4995

3.480

152.4

33

0.2539

3.480

77.65

4

0.1

2.977

35.75

34

0.2447

2.977

87.45

5

0.49953.480

140.55350.2549

3.480

77.95

6

0.1695

2.977

60.55

36

0.2439

2.977

87.15

7

0.2760

3.480

84.4

37

0.2556

3.480

78.15

8

0.2145

2.977

76.7

38

0.2433

2.977

86.95

9

0.2510

3.480

76.75

39

0.2562

3.480

78.35

10

0.2388

2.977

85.35

40

0.2427

2.977

86.75

11

0.2403

3.480

73.5

41

0.2566

3.480

78.45

12

0.2508

2.977

89.5

42

0.2423

2.977

86.60

13

0.2368

3.480

72.4

43

0.2571

3.480

78.6

14

0.2553

2.977

91.25

44

0.2420

2.977

86.50

15

0.2375

3.480

72.6

45

0.2574

3.480

78.70

16

0.2571

2.977

91.9

46

0.2417

2.977

86.35

17

0.2391

3.480

73.15

47

0.2577

3.480

78.80

18

0.2564

2.977

91.65

48

0.2414

2.977

86.25

19

0.2414

3.480

73.8

49

0.2579

3.480

78.85

20

0.2550

2.977

91.15

50

0.2412

2.977

86.20

21

0.2437

3.480

74.5

51

0.2581

3.480

78.90

22

0.2533

2.977

90.50

52

0.2409

2.977

86.10

23

0.2459

3.480

75.2

53

0.2585

3.480

79.05

24

0.2515

2.977

89.90

54

0.2405

2.977

85.95

25

0.2480

3.480

75.85

55

0.2587

3.480

79.10

26

0.2498

2.977

89.3

56

0.2401

2.977

85.80

27

0.2498

3.480

76.35

57

0.2556

3.480

78.15

28

0.2482

2.977

88.7

substrate

290.2514

3.480

76.85

30

0.2469

2.977

88.25

Table 5.2: Structure of optimized coating. The physical thickness is truncated to the next 0.05 nm
for the manufacturing process.

77

TO optimized structure
0.5

GaAs(nH)
AlGaAs (nL)

optical thickness [Wavelengths]

0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0

10

20

30
Nth Layer

40

50

160

60

GaAs (nH)
AlGaAs (nL)Physical Thickness [nm]

140
120
100
80
60
40
20
0
0

10

20

30
Nth Layer

40

50

60

Figure 5.10: Above, the optimized structure as shown in optical thickness. Below, the same structure
as shown in physical thickness. The first layer is the air-coating surface, the 57th layer is the coatingsubstate surface. The plots are taken from the data in Tab. 5.2 below. From the plot, it can be seenthat the first few pairs are altered away from quarter wave the most. This is expected as most ofthermorefractive effect is from the very top layers, while all the coating layers contribute equally tothermoelastic effect.

78
0

10

R
T

−1

10

−2

10

−3

R @ 1064nm = 0.9998−4

T @ 1064nm = 150.9 ppm

10

10

−5

10

600

800

1000
1200
wavelength [nm]

1400

1600

Figure 5.11: Reflectivity and transmission of the designed coating.

Figure 5.12: Histograms of the Monte Carlo simulations to determine optical properties of the
optimized structure. Uncertainties from all considerations are added to the calculation.

79

5.2.6

Cavity Parameters
Cavity length / FSR

1.45 inch (3.68 cm) / 4.07 MHz

Finesse20,267

Cavity linewidth (FWHM)/ cavity pole

201 kHz/ 100 kHz

Beam radius on the mirror

w0 = 215 µm

mirror coatings

AlGaAs optimized coatings

mirror transmission

150 ppm

Loss in each mirror

5 ppm

Mirror radius of curvature

1.0 m

Table 5.3: Table of parameters for 1.45 inch cavity and the mirrors with AlGaAs coatings.

Substrate Thermal Expansion
Coating Thermal Expansion
Coating Thermo−Refractive
Total

8

[Hz/Watt]

10

7

10

6

10

5

[Degree]

10 0
10

1

10

2

10

3

10

4

10

100
0
−100
0

10

1

10

2

10

[Hz]

3

10

4

10

Figure 5.13: An estimated result of a swept sine measurement for photothermal effect on AlGaAs
optimized coating.

5.3

Implications for Advanced LIGO

If the limiting noise of the proposed optimized coating structure is measured and the cancellation
does work, we can applied the above method to find optimized structures for Advanced LIGO’s test
masses. The transmissions of an Input Test Mass (ITM) and an End Test Mass (ETM) are 1.4
percent and 5 ppm, respectively. The optimized structure for ITM and ETM are shown below. Here80
the results are only preliminary without any Monte Carlo test. Thorough checks on uncertainties inparameters are required. The complete noise budget for Advanced LIGO with AlGaAs optimized
coating is shown in Fig. 5.16. The calculation is done at room temperature with fused silica substrate.

81

Optical Thickness [Wavelength]

0.4

GaAs (nH)
AlGaAs (nL)

0.35
0.3
0.25
0.2
0.15
0.1
0.05
0

1

3

5

7

9

11 13 15 17 19 21 23 25 27 29
Nth Layer

0

10

R
T

−1

10

−2

10

−3

10

−4

10

R @ 1064nm = 0.986

T @ 1064nm = 1.4 percent

−5

10

600

800

1000
1200
wavelength [nm]Figure 5.14

1400

1600

82

Optical Thickness [Wavelength]

0.7

GaAs (nH)
AlGaAs (nL)

0.6
0.5
0.4
0.3
0.2
0.1
0
0

10

20

30

40
50
Nth layer

60

70

80

0

10

R
T

−1

10

−2

10

−3

R @ 1064nm = 1−4

T @ 1064nm = 5.391 ppm

10

10

−5

10

600

800

1000
1200
wavelength [nm]Figure 5.15

1400

1600

−21

−22

10

−24

10

−23

10

1

10

2

10
Frequency [Hz]3

10

Figure 5.16: The estimated noise budget of Advanced LIGO with TO optimized AlGaAs coatings. The confidence plot as shown in Fig. 1.4 isremoved to display the noise level more clearly. Coating Brownian noise level of SiO2 /T a2 O5 coatings and the corresponding total noise are plottedin dashed brown and grey, respectively. Coating Brownian noise level of AlGaAs coatings and the corresponding total noise are plotted in red andblack, respectively.

Strain [1/√Hz]

10

Quantum noise
Seismic noise
Gravity Gradients
Suspension thermal noise
Coating Brownian noise
Coating Thermo−optic noise
Substrate Brownian noise
Excess Gas
Total noise (AlGaAs)
SiO2/Ta2O5 Coating Brownian noise
SiO2/Ta2O5 Coating Thermo−optic noise
Total noise (SiO2/Ta2O5)

83

84

Chapter 6

Future Upgrade and Application
In this chapter, we will discuss about some possible upgrades for the experiment and its application
as a frequency stabilized light source.

6.1

Future Upgrade

The main goal for the upgrade is to expand the measurement bandwidth. As Mechanical losses inbulk SiO2 and in thin film SiO2 /T a2 O5 from ring down measurements exhibit frequency dependent
behavior [100, 101]. But the frequency dependent component is much smaller than the constant loss
(for example, φSiO2 = 0.4×10−4 +f (2.7)×10−9 [100]) and it is hard to be able to observe this subtle
change in a narrow frequency band. With a wider measurement band, we might be able to study
more about how the frequency dependent loss affects the noise level. To do this, we have to minimize
the relevant technical noise sources as discussed in chapter 4. The measurement from Fig. 3.9 showsthat from DC to around 10 Hz, the current sensitivity is limited by photothermal noise, while, from
1 kHz and above, the sensitivity is limited by shot noise, PLL readout noise and residual frequency
noise. Plans to improve the sensitivity in each regimes are discussed in the following sections.

6.1.1

Improvement in Sensitivity at Low Frequency

To improve the sensitivity at low frequency, we have to• Upgrade ISS: A better intensity stabilization servo with higher gain and output is required
to suppress more intensity noise. The current servo gain uses an SR560 low noise preamplifier
with the output of 10 Vpp1 . This output range limits the dynamic range of the current ISS
to about ±5 %.
• Improve long term beat measurement: To measure the beat note at low frequency,
the PLL has to be locked for a long time. Usually, PLL is out of lock because the beat1 http://www.thinksrs.com/downloads/PDFs/Catalog/SR560c.pdf

85frequency drifts beyond the tuning range. Currently, the drifts around sub Hz level are fromphotothermal noise, which is driven by not only the laser intensity noise, but also the beam
position fluctuations. Temperature variations during the day, can cause the optic mounts, andthe table to warp from their original positions, resulting in the mismatch beam-cavity position.
The power drop due to this effect can be larger than 30 % which is more than the range of thecurrent ISS can compensate. A better controlled air conditioning in the lab should be able toresolve the issue, because the time constant of the setup can be in the order of a few hours.
• Minimize scattered light Since noise from scattered light adds random phase noise to thesignal, its effect usually shows up as a bump in the low frequency region. For this setup,
two places that are most likely to create scattered light are the vacuum window, and thebeat PD. The current 10 inch vacuum window is not high quality optics. The anti-reflective
coating is poor and there are several scratches. Thus, using a high quality optic should reduce
the scattered light. Instead of using a big window, we can install two 1 inch windows on amodified blind flange, see Fig. 6.1, for the two beams entering the chamber. Not only the costfor the window can be reduce, the windows can be easily replaced with commercially available
optics. However, the design should be review, as the usage of o-rings on small windows may
compromise the vacuum pressure.
The second place causing scattered light is the beat PD. The current PD is Newport 1811
(125 MHz bandwidth). Because of its high response time, the detector diameter has to besmall (100 microns2 ). For this tiny PD diameter, we may not be able to precisely focusing thebeam down to 30 microns3 and directing it to the PD without scattering off the nearby casing.
As a result, a photodiode with larger detector is preferred, for example, Newport 2053-FC
with 0.9 mm diameter (10 MHz bandwidth). This means that the beat frequency has to be
below 10 MHz. Temperature stabilization for the two cavities has to be improved in order tokeep the two cavities at the right temperature. One possible way is to use the PLL control
signal as a feedback signal for controlling. With this feedback signal, we can avoid adding two
temperature sensors in the vacuum chamber and have enough pins on the feedthrough for theheaters.
• Redesign Seismic Isolation System: The current double stage seismic isolation that holds
the cavities inside the vacuum chamber have resonant frequencies around 10 Hz. This resonant
frequency is the highest compared to other resonant frequencies from the seismic isolation
stages (air springs and pneumatic table legs). As discussed above, seismic motion introduces
not only displacement noise due to the bending of the cavities, but also scattered lights.
2 http://search.newport.com/?x2=sku&q2=1811-FC
3 The beam’s diameter has to be smaller than the PD’s diameter to prevent any scattered light and noise frombeam pointing fluctuations. A factor of 3 is generally acceptable for a Gaussian beam profile.

86
Unless we can decrease the resonant frequency of the seismic stack, the measurement will becontaminated by scattered light around this frequency. One possible way is to use a suspended
platform to hold the cavity mount. This might push the resonant frequency down to a few Hz
level.

6.1.2

Improvement in Sensitivity at High Frequency

At high frequency, the limiting noise sources are PLL readout noise, shot noise, and residual frequency noise.
• Different Readout Technique: Two main sources that introduce noise in PLL are the
VCO’s frequency noise and the electronic noise from the photodiode (PD). VCO’s frequency
noise can be reduced by using another VCO with lower intrinsic noise or using different readout
technique. Homodyne detection (delay line technique)4 can be used to observe the frequency
noise in the signal. An analog circuit for the homodyne detection can be simple. It includes
a power splitter, cables with different lengths, a mixer, and a low pass filter. However, thismakes the measurement rather sensitive to intensity noise. Sophisticated electronic equipment
has been designed to overcome the problem, for example, see LISA phase meter that is used
as a readout tool in another thermal noise experiment [149].
Another way is to remove the VCO is to measure the beat off-line. The beat signal can be
demodulated using a low noise reference (e.g., Rb clock) to reduce the beat frequency below
the bandwidth of DAQ. The recorded data can be used to extracted the frequency noise of thebeat without any frequency noise introduced by the VCO.
However, electronic noise from the PD will still be a problem. Although electronic noise isusually flat, the noise that shows up in the frequency measurement is proportion to temporalfrequency. This effect is vey undesirable for measuring the signal at high frequency. It is
unavoidable no matter what readout techniques we use. Although the SNR can be improvedby increasing the input power, the saturation level of the PD will limit the maximum input
power.
• FSS with higher UGF In order to measure coating noise at high frequency, the unity gain
frequency (UGF) of the servo has to be increased more. Sideband frequency is one of thefactors that determine the UGF of the servo. The error signal from PDH locking can contains
a term that varies at the same frequency as the sideband. To make sure that it will not be fed
back to the laser, the maximum UGF has to be roughly a factor of 10 smaller than the sideband
frequency. For instance, the maximum UGF of the FSS servo with the sideband frequency
4 For a review on homodyne detection, see Ref. [121].

87
6X
.20
1/4-20 UNC

.50
.50

6X
.20
1/4-20 UNC

.50
.50

2.06
.197
A

.124
2.00°

.196
.106

1.40

1.40

1.484

DETAIL A
SCALE 1 : 1

3.50

UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN INCHESTOLERANCES:
FRACTIONALANGULAR: MACHBEND
TWO PLACE DECIMAL
THREE PLACE DECIMAL
INTERPRET GEOMETRICTOLERANCING PER:PROPRIETARY AND CONFIDENTIAL
THE INFORMATION CONTAINED IN THISDRAWING IS THE SOLE PROPERTY OF<INSERT COMPANY NAME HERE>. ANY
REPRODUCTION IN PART OR AS A WHOLEWITHOUT THE WRITTEN PERMISSION OF<INSERT COMPANY NAME HERE> IS
PROHIBITED.

MATERIAL

USED ON

NEXT ASSYAPPLICATION

DATE

DRAWN

TITLE:CHECKED
ENG APPR.
MFG APPR.
Q.A.
COMMENTS:SIZE DWG. NO.

REV

edit_blank_10_wedge
A

FINISHSHEET 1 OF 1

SCALE: 1:4 WEIGHT:

DO NOT SCALE DRAWING

4

5

NAME

1.88

2

3

1

B
A

C
A

SECTION A-A
SCALE 1 : 2

UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN INCHESTOLERANCES:
FRACTIONALANGULAR: MACHBEND
TWO PLACE DECIMAL
THREE PLACE DECIMAL

2.00°

PROPRIETARY AND CONFIDENTIAL

DETAIL CSCALE 1 : 1

THE INFORMATION CONTAINED IN THISDRAWING IS THE SOLE PROPERTY OF<INSERT COMPANY NAME HERE>. ANY
REPRODUCTION IN PART OR AS A WHOLEWITHOUT THE WRITTEN PERMISSION OF<INSERT COMPANY NAME HERE> IS
PROHIBITED.

5

INTERPRET GEOMETRICTOLERANCING PER:
MATERIAL

USED ON

NEXT ASSYAPPLICATION

4FINISHDO NOT SCALE DRAWING

3

NAME

DATE

DRAWN

TITLE:CHECKED
ENG APPR.
MFG APPR.
Q.A.
COMMENTS:SIZE DWG. NO.

REVwindow_assembly2
A
DETAIL B
2

SCALE: 1:4 WEIGHT:
SCALE
2:1

SHEET 1 OF 1

1

Figure 6.1: Drawings of new vacuum window. Two smaller windows (1 inch diameter, AR coated)
are installed on the blank. Both windows will be tilted to reduce back reflection from the window
surfaces. The grooves on the windows will be filled by O-rings.

88
at 14.75 MHz might be only around 1 MHz. By selecting a higher sideband frequency, forexample at 35.5 MHz, we can increase the measurement band by roughly a factor of 2.

6.2

Application: Frequency Stabilized Light Source

A frequency stabilized light is useful as a stable reference. It can be used for diagnosing noise ina system using interferometric technique. The setup of this work stabilizes laser frequency to thereference cavity’s thermal noise limit and uses only small beam power for the measurement. Thus,
the byproduct of this experiment is a frequency stabilized laser source. Although the short cavity isdesigned for coating thermal noise measurement5 , the laser frequency noise is still much better thanthe NPRO’s free running noise. This frequency stabilized light can be provided as a reference signal
for other experiments. The light can be picked off from the setup and distributed to nearby labs via
optical fiber. Here is a brief discussion on a few experiments at Caltech LIGO labs that can benefit
from having a stable light source.

6.2.1

Laser Gyroscope

Motions on the ground due to shaking and tilting are indistinguishable for a conventional massspring seismometer. If the conventional seismometer is used as a sensor in a feed forward system(e.g., LIGO suspension system), tilt motions, interpreted as seismic motions by the seismometer, will
introduce extra noise to the system. To resolve the issue, tilt meters were developed to monitor only
tilt motion. One example of a tilt sensor is a laser gyroscope (GYRO) which is an interferometerusing Sagnac effect. The setup consists of a single laser, and a ring cavity. The beam from thelaser is split into two beams, each enters the cavity in opposite directions. If the ring rotates, thefrequencies of the two beam will be shifted due to the Sagnac effect. The frequency differences
between the two beam is given by [150]
∆ν =

4 ~
A·ω
~,
λS

(6.1)

~ is the cavity area vector, and ω
where S is the cavity perimeter, A~ is the angular velocity of the~ rotates and causes the frequency shift which
earth rotation. As the surface of the ground tilts, A
can be monitored as a tilt sensor.
The setup for GYRO [151] is conceptually similar to the prototype setup discussed in this work.
The main beam is locked to the ring cavity via PDH lock, the second beam is frequency shifted andlocked to the cavity by an AOM. The frequencies of the two beams are one free spectral range apart(100 MHz).
5 The frequency noise of the laser will be less if it is locked to a longer one, cf. Eq. 3.3

89
Similar to other reference cavities, the ring cavity is subject to displacement noise, for example,
mirror vibrations. Although the noise is common between the two beams and should not affect theread out, the frequency shifts (due to the displacement noise) from different axial modes are not theccsame. The frequencies of the two beams are ν1 = n 2Sand ν2 = (n + 1) 2S. The frequency noise dueto the displacement noise δS for each beam is
δν1
δν2
δS
=
=
,
ν1
ν2
S

(6.2)

so
δνf ake = δν2 − δν1 =

δS
c.
2S 2

(6.3)

This creates a fake GYRO signal δνf ake . If the laser for GYRO is compared to another stablereference signal. The signal due to the cavity displacement noise can be observed and subtracted of
the GYRO signal to achieve better sensitivity.
Primary
Half-wave plate
Faraday isolator
RF photodetector

EOM1064nmNd:YAG
NPRO

VCO 1

VCO 2

AOM

Mode
cleaner

PLL
Sensing
cavitySecondary

Figure 6.2: A schematic diagram of GYRO setup. Picture credit: William Z. Korth.

6.2.2

CrackleMaraging blades that are used for suspending LIGO test masses may be susceptible to crackle
noise, which originates from dislocations in the blades. The noise can be up-converted into theaudio frequency band due to sub-Hz driving motions. To study the phenomenon, a Michelsoninterferometer with both end mirrors attached to the device under test (the blades) can be used.
Each arms are driven simultaneously at low frequency. The output can be observed for any incoherent
crackle noise between the two arms. The frequency noise of the light source masks the output of the90
expected crackle signal ∆L when the lengths of the two arms are mismatched by distance ∆x. The
signal to noise ratio (SNR) at the output port is
SNR =

ν0 ∆L
,
δν ∆x

(6.4)

where ν0 is the laser frequency and δν is the frequency noise. The frequency noise of the main laser
degrades the measurement SNR. By using a frequency stabilized light, the SNR can be improved.
√
For example, at 100 Hz, the free running noise of an NPRO laser is around 100 Hz/ Hz, while the√frequency noise from the laser locked to the 1.45 inch cavity can be below 0.1Hz/ Hz (cf. Fig. 3.9).
This will improve the SNR by 3 orders of magnitude.

Figure 6.3: A schematic diagram of crackle setup. The setup is currently under major upgrade forbetter sensitivity [47].

6.2.3

External Cavity Diode Laser

Conceptually, an external cavity laser diode (ECDL) consists of a diode laser and a diffraction
grating that diffracts a part of the beam back to the diode laser for optical feedback (an example ofan ECDL construction in the Littrow configuration can be found in [152]). It is relatively low costcompared to an NPRO. Thus, the purpose of this project is to design, build, and test an ECDL (withwavelength 1064 nm) for use as a tabletop laser at Caltech LIGO lab [153]. The frequency noise of
the laser diode used in the ECDL is reduced by an optical feedback provided by the diffracted lightfrom a diffraction grating. The noise suppression level and the output wavelength highly depend on
the configurations of the ECDL (e.g., the distance between the diode and the grating, spot size on

91
the grating, the angle of the grating with respect to the incident beam) [154]. A stabilized frequency
light source can be very useful during the setup configuration of the ECDL. By beating the ECDL
output with the reference light and extracting the signal with PLL or delay-line technique, the ECDL
wavelength can be tuned to the desired value, and its frequency noise can be optimized for better
performance.

92

Appendix A

Comparison with Other
Experiments

2

10

Numata2003
TNI 2004
Ludlow 2007
Alnis 2008
Webster 2008
Virgo cavity
Cole 2013
Silicon cav
8−in cavity
1.45−in cavity

1Frequency noise [Hz/√ Hz]

10

0

10

−1

10

−2

10

−3

10 −2
10

−1

10

0

10

1

2

10
10
Frequency [Hz]

3

10

4

10

Figure A.1: Results from several experiments shown in frequency noise.

5

10

93

Numata2003
TNI 2004
Ludlow 2007
Alnis 2008
Webster 2008
Virgo cavity
Cole 2013
Silicon cav
8−in cavity
1.45−in cavity−14

Displacement noise [m/√ Hz]

10

−15

10

−16

10

−17

10

−18

10

−2

10

−1

0

10

10

1

2

3

10
10
Frequency [Hz]

10

4

10

5

10

Figure A.2: Results from several experiments shown in displacement noise.

trace

cavity length

λ [nm]

Ref.

Numata 2003

1 cm

1064

[54]

TNI 2004

1 cm

1064

[55]

Ludlow 2007

7 cm

698

[56]

Alnis 2008

7.75 cm

972[57]

Webster 2008

10 cm

1064

[58]

Virgo cavity

30 cm

1064

[155]

Cole 2013

3.5 cm

1064

[143]

Silicon cavity

21 cm

1550

[156]

Table A.1: Note about each trace on the plots above.
• Numata and TNI: These two experiments were designed to measure coating thermal noise
measurement. The noise plot shows the differential noise between the two cavities.
• Ludlow 2007: The cavity was supported vertically. The substrates and spacer were made of
ULE. The seismic strain coupling is around 5 × 10−11 [1/ms−2 ]• Alnis 2008: Beat between two ECDLs independently stabilized to two FP cavities. The
cavities were supported vertically. The mirror substrates and spacer were made from ULE,
with SiO2 /T a2O5 coatings.

94• Webster2008: Two identical cavities were operated around 10 C for zero thermal expansion
coefficient. The spacers were machined and held horizontally for vibration insensitivity. ULE
spacer and substrates. The strain due to seismic isolation is 3 × 10−12 [1/ms−2 ]• Virgo cavity: A reference cavity for pre-stabilizing the main laser frequency used for VIRGO
detector. The substrates and spacer were made of ULE.
• Cole 2013: The cavity were formed by AlGaAs coatings with fused silica substrate and
Zerodur spacer at room temperature. The laser frequency locked to the cavity was measured
by beating (via frequency comb) with another laser frequency locked by JILA ultrastablecavity.
• Silicon cavity: The cavity noise was measured at 124 K. The cavity spacer and the mirror
substrates were constructed from single crystal silicon. The coatings were SiO2 /T a2 O5 The
cavity was supported vertically. The measurement was reported in modified Allan variance.
The equivalent result in frequency noise shown in the plot was done by David Yeaton-Massey.
The estimated strain ∆L/L is less than 10−11 [1/ms−2 ].
Often, other similar experiments report their results in time domain using the Allan variance
σy2 (τ ), or the modified Allan variance Mod σy2 (τ ) (the square root of these values are called Allan
deviation and Mod Allan deviation). The modified Allan variance is used to improve the estimate
frequency stability of the source when the dominant noise type in the short term is flicker phasemodulation (Sν ∝ f ) of white phase modulation (Sν ∝ f 2 ). The relation between the two is givenby [157]

Mod σy2 (τ ) =24n (πν0 τ0 )2

Z fh
0

ν02 Sν (f )sin6 (πτ f )df,
f 2 sin2 (πτ0 f )(A.1)

where fh is the cutoff frequency of the measured Sν , τ = nτ0 where n is the number of samples
averaged, and ν0 is the carrier frequency. In general, each frequency noise spectrum with different
dependent on f results in Mod σy2 (τ ) with different dependent on τ . The table for converting themodified Allan variance to power spectrum of frequency noise is taken from Ref. [157].
Sν (f )Mod σf2 (f )Random walk frequency

ν02 h−2 f −2

5.42h−2 τ

Flicker frequency

ν02 h−1 f −1

0.936h−1

White frequency

ν02 h0 f 0

h0 /4τ

Flicker Phaseν02 h1 f 1

3.37
1
4π 2 h1 τ

Noise Type

Table A.2: Conversion Factors for Sν (f ) and Mod σf2 (f ). ν0 is the carrier frequency.

95
For example, if the measured Mod σy2 (τ ) can be written as
σy2 (τ ) =C1+ C2 + C3 τ,
τ

(A.2)

where C1 , C2 , and C3 are constants. The corresponding power spectrum of frequency noise, using
the conversion factor in the above table, will beSν (f ) =ν02
ν 2 C2ν 2 C3+ 0+ 0 2.
4C1
0.936f
5.42f

(A.3)

The frequency bandwidth where this conversion is valid will depend on the sampling time and τmaxof the measurement and any cut off frequencies of the low pass filters used in the measurement.

96

Appendix B

Frequency Stabilization Servo• TTFSS setup: Schematic of the table top frequency stabilization servo (TTFSS) used in thiswork can be found in LIGO document control center 1 . The implementation of TTFSS on
the setup with relevant calibrations is shown in Fig. B.1 below. The servo controls the laser
frequency via two actuators, the PZT on the NPRO crystal, and the EOM. The error signal
goes through a common path, then separated to two. The PC path sends the control signal to
the EOM. The Fast path sends the control signal to the NPRO. We pick off the Fast signal,
low pass it, and use it as an error signal for temperature control on the laser crystal (not shownin the diagram) for long term lock.
• Performance of TTFSS: To characterize the servo performance, we measure the open loop
gain transfer function (OLG) and compared it with the calculated OLG from the schematic.
The calculation can be done by computing the transfer function of each stage and multiply
them together. For Fast path only simple low pass stages are used and they are easy to
calculate. For PC path, the schematic becomes more complex. One way to find the transfer
function is by using LISO to simulate the shape of the transfer function, then use a fit function
to determine all poles and zeroes in the servo. For example for the second stage of PC pathin diagram we include the capacitance of the EOM, about 20 pF, as given in the manual. The
transfer function result is shown in Fig. B.2. Finally, the calculated results from the two paths
are combined to get the whole OLG. Then it is compared to the measurement. They agree
well and meet the requirements for reference cavities used in this work. See Fig. B.3. The
requirement is calculated by setting the residual frequency to be 0.3 of the expected signal,then calculate what would be the necessary gain to push the noise down to that level. From ,Eq. 3.4 one gets,required OLG ≈free running noise
residual noise

1 The schematic is on https://dcc.ligo.org/LIGO-D040105.
LIGO-E1000609(B.1)

See the test plan on https://dcc.ligo.org/

97

4.5 MHz/V

15 mRad/V

NPRO

EOM

EOM

C=
20 pF

Frequency Disc
Fast
Path

PC
Path

Adj FG-16dB+24 dB

Fast
Out2

Fast
Out1

Adj CG-16dB+24 dBU3+12 dB
180 degree

CommonOut2

CommonOut1

Figure B.1: A schematic explaining how TTFSS is implemented in the setup. The servo has two
actuators, one is the PZT on the NPRO that changes the refractive index via elastic called Fast Path.
The PZT has flat response from DC up to around 1 kHz, with gain around 4.5 MHz/V. Anotheractuator is the Electro-Optic modulator which changes the phase of the output beam called Pockell
fcell path (PC). The gain is 15 mRad/V or 15 mHz/V [ 1Hz]. Two adjustable gains are for common
path (CG) and for Fast path (FG). The gains are adjustable between -16 dB and +24 dB. The
common gain adjusts the overall magnitude of the servo, while the fast gain alters the shape of theservo. There is a flip sign switch on the fast path to match the phase between the two paths. The
frequency discriminator, Dv, in the setup is around 5.5 V/MHz. This can be measured from theslope of the error signal during the cavity scan.

98
The requirement for a longer cavity is higher because of the smaller signal. It requires higher
servo gain to suppress the free running noise down to the cavity’s displacement noise.

Figure B.2: Transfer function of PC path as obtained from LISO model, the blue plot is the result
from the simulation. The red plot is the calculation from poles and zeros obtained from the fit.

0

100

200

−50 1
10

0

50

100

2

10

2

10

10

10

3

3

4

10
Frequency [Hz]410

RCAV Open Loop Gain TF

5

10

5

10

10

10

6

6

7

10

7

10

FAST
PC
RCAV OLG − Measurement
RCAV OLG − Computed
requirement for 1.45" cavity

Figure B.3: Open Loop Gain transfer function of TTFSS, 1mW input power. The requirements assume that the residual noise is within 5 percentof the suppressed frequency noise. The boost, which is an integrator stage with a zero at 4.6 kHz is turned off during the measurement. From theplot, the servo is good enough up to 1 kHz for the setup with 1.45 inch cavities. The inconsistency between the measurement and the calculated
TF might come from the PC path. We could not measure the TF of this path individually due to the limited accessibility of the test points. The
calculated TF might be different from the real situation due to incorrect model and non-ideal behavior of the op amps. Also, the TF of the RFPD
for PDH lock is not taken into account. This will result in the deviation of the measured phase at frequency around 10 MHz an above due to theresonant circuit in the RFPD.

−200 1
10−100

Magnitude [dB]

Phase [Degree]

150

99

100

Appendix C

Material parameters for
SiO2/Ta2O5 and AlGaAs
C.1

Table of Parameters

Parameters

SiO2

T a2 O5GaAs

Al0.92 Ga0.08 As

Al1−x Gax As

Heat capacity, C (×106 ) [J/Km3 ]

1.64

2.1

1.76

1.69

1.76 + 0.114x − 0.19x2

Thermal conductivity, κ [W/mK]

1.38

33

55

70

55 − 212x + 248x2

Young’s modulus, Y [GPa]

72

140

100

100

-

0.17

0.23

0.32

0.32

-

) [1/K]

0.51

3.6

5.7

5.2

5.73 − 0.53x

Thermorefractive, β (×10−6 ) [1/K]

8

14

366

179

366 − 203x

Refractive index, n @1064nm

1.45

2.06

3.48

2.98

3.48 − 0.578x

Mechanical loss angle, φ (×10−4 )

1

4

0.25

0.25

0.25

Poisson’s ratio, σ−6Thermal expansion, α (×10

Table C.1: Material parameters used in this work.

C.2

Uncertainties in Parameters

Errors in the material parameters will cause error in thermal noise calculation. For a high reflective
dielectric mirror where Brownian thermal noise in coating dominates, uncertainties in loss angles willbe the most significant parameters to determine the noise level accurately. Loss angles are usuallyobtained from a ring down measurements. The coating is applied on a blade or a disc with low loss.
The system is excited and the ring down is measured. The coating loss and the whole system loss
is related as
Utot φtot ≈ Us φs + Uc φc ,

(C.1)

101
where U is the elastic energy in each ring down mode. Note that if φbulk and φshear are different,
Eq. C.1 has to be separated in bulk and shear modes accordingly. The dilution factor Us /Uc can becalculated analytically, or by FEA. For example, the energy loss from a disc with a thin coating canbe written as:
φtot = φs +

Uc
φc .
Us

(C.2)

φs is the loss of the substrate that can be measured before applying the coating, φtot is the total
loss of the coating-substrate system. For each mode, the dilution factor will be different. For the
first drumhead mode the dilution factor is given by [158]
Yc d
Uc
=3
,
Us
Ys t

(C.3)

where d is the coating thickness, and t is the substrate thickness, Yc is the averaged Young’s modulusof the coating. For a wafer structure of the coating, the bending mode is mostly stretching andcompressing along the surface layer. Yc can be calculated asYc =Y1 d1 + Y2 d2,
d1 + d2(C.4)

where d1 and d2 are the total thickness of material 1 and 2. This calculation is similar to a system ofconnected springs. The dilution factor obtained from this analytical result agrees with the calculationfrom FEA (as reported in Ref. [99]) within 5 %. Hence, uncertainties in Young’s moduli lead to
uncertainty in the dilution factor which is used for extracting the coating loss angle from ring down
measurements. These errors in Young’s moduli propagate through the extraction of φ (Eq. C.2) andshow up in the errors of the loss angle. The result can be much larger than the error in the coating
calculation due to uncertainties in Young’s moduli alone (as done in Ref. [53]).
Development in nano-indentation technique allows us to measure Young’s modulus in thin film
form with better accuracy although the samples used are still much thicker, from 500nm to a few
microns, than a single layer of the dielectric coating (around 200 nm or less). Young’s moduli of
SiO2 and Ta2O5 are measured to be 70 GPa and 140 GPa respectively [159].

C.3

Note for AlGaAs

The elastic parameters of AlGaAs are taken from Ref. [160, 148]. However, due to the crystalline
structure of AlGaAs, these parameters are directional dependent. The values here are only averaged
values intended to be used in the amorphous coating calculation as an approximation.

102

Appendix D

High Reflective Coating Structure
and its Average Properties
D.1

Why Half Wave Cap?

For quarter wave layer stack (QWL) SiO2/Ta2O5 coatings, SiO2 and Ta2O5 are materials with low(nL ) and high refractive indices (nH ), respectively. Due to the stronger structure of SiO2, a cap of
SiO2 as a protective layer is usually on top. However, if the SiO2 cap is 1/4 thick, the first reflectedbeam at the air–cap interface layer will destructively interfere with the rest of the reflected beams,
causing the reflectivity to go down, see Fig. D.1(A).
If the cap has thickness of 1/2 wave length, the reflected beam from the interface between thecap and all the following layers will be in phase with the first reflected beam at the air–coating
surface, see Fig. D.1(B).
For a cap with high index material, it can be 1/4 wave length thickness, and the phase fromevery reflected beams still interferes constructively, see Fig. D.1(C).
λ/2 λ/4 λ/4 λ/4nL nH nL nH

λ/4 λ/4 λ/4 λ/4nL nH nL nH

λ/4 λ/4 λ/4 λ/4nH nL nH nL
0

0

0
∏

∏

∏

2∏

∏

∏

2∏

∏

∏

2∏

∏

∏

(A)

(B)

(C)

Figure D.1: Reflection phase from a few top layers of different coating structures. (A) QWL with
low index material (light blue) on top. (B) QWL with a half wave cap of low index material. (C)QWL with high index material (blue) on top

103
For Fig. D.1(B) and Fig. D.1(C), the incoming beam and the reflected beam are 180 degrees outof phase. It means that the electric field at the coatings surface will always be zero. This prevents
the burning on the surface of the coating. With this, the standing wave in a Fabry–Perot cavity
formed by two such mirrors will always have zero electric field at the surfaces.
For AlGaAs coatings, the situation is different from SiO2 /T a2 O5 coatings. The cap has to be
GaAs (nH) because Al0.92 Ga0.08 As will oxidize and change its material properties. With the highindex cap, the thickness has to be 1/4 wavelength.
If the substrate’s refractive index is smaller than both nH and nL, the last layer at coatingsubstrate interface has to be the high index material to have constructively interfere reflected beam.
However, the effect is smaller as the numbers of the layer increase.

D.2

Averaged Material Parameters for Coatings

With the length scales of ω0 , and rT are large compared to the layer thickness, we can use averagedmaterial properties to represent the coating. The equations here are taken from Ref. [78, 74]. We
include them here for completeness. The thermal expansion coefficient for a given layer k in thecoating is
1 + σsᾱk = αk1 − σk

1 + σkYk+ (1 − 2σs )1 + σsYs(D.1)

and the volume average coefficient for a coating with N layers each of thickness dk is

ᾱc =

N
X

dk
d

ᾱk

k=1

(D.2)

where d is the total coating thickness
d=

N
X

dk

(D.3)

k=1

The heat capacity calculated by volume averaging,

Cc =

N
X
k=1

Ck

dk
,
d

(D.4)

the average thermal conductivity,
κc =

N
X
1 dk
k=1

κk d

,

(D.5)

the effective β used here follows the notation in [74] where β is chosen to be positive and the negative
sign is written explicitly. An effective TF coefficient, β̄ is defined such that
∂ZT R
= −β̄λ.
∂T

(D.6)

104
For β̄ made of 1/4-wave doublets (SiO2 /T a2 O5 ) with a 1/2-wave SiO2 cap layer (or any coating
materials with nL as a cap), β̄ can be approximated byβ̄ 'BH + BL (2(nH /nL )2 − 1).
4(n2H − n2L )(D.7)

Where BX is the fractional change in optical path length with respect to temperature in materialX,
BX = βX + ᾱ0 X nX ,

(D.8)

where
ᾱ0 X = αX

1 + σX
1 − σX(D.9)

is the correction for the fractional path length change due to thermal expansion inside a layer [161].
For the cap with arbitrary thickness, β̄ can be found in Ref. [145]. The numerical calculation for β̄
is given in Ref. [74] which is used for the TO optimized coating in this thesis.

105

Appendix E

1.45 Inch Reference Cavity: Design
and Assembly
E.1

Choosing Cavity Length

To raise the coating Brownian thermal noise level, we need to decrease the beam size. This canbe done by shortening the cavity length while keeping the same mirrors. However, there are otherconsiderations to choose the optimal length of the cavity. Mainly, we have to• Be able to tune the cavity length by changing the temperature. The shorter the cavity, thehigher the temperature required to change the length through thermal expansion, see Fig. E.1.
• Keep the cavity stable.
The condition where the beam can propagate repeatedly without running off from the cavity
is given by [124]

0<

1−

L
R1



L
1−
= g1 g2 < 1,
R2

(E.1)

Where L is the total length, R1 and R2 are the radius of curvature of the mirrors. The valueof g1 g2 vs cavity length is shown in Fig. E.2.
• Make sure no higher order modes will be in resonance simultaneously with the carrier. The
allowed resonant frequencies for the cavity is given by Eq. 3.1. However, the incident beam
does not consist of only a single frequency, it also has two sidebands. Additionally, the othermodes besides TEM00 (from the carrier and the sidebands) can be in resonance with thecavity’s other axial modes due to the Gouy’s phase shift is [124]Φmn (z) = (2m + n + 1)arctan(λz/πw02 ),(E.2)

where m, n are integers denoting the mode number in Hermite–Gaussian modes. The phasevelocity increases with higher mode number and leads to differences in the resonant frequencies.

106
The one–way Gouy’s phase shift written as a function of g–parameters is√δφmn = (2m + n + 1)arccos( g1 g2 ).

(E.3)

This phase shift is related to the frequency shift asδφmn =π
2πδfL=δfmn .
c
FSR

(E.4)

If any of these modes are in resonance simultaneously with the carrier, the PDH error signal
will be disturbed and becomes unstable. This problem can be avoided by choosing the rightcavity length or the right sideband frequency that does not allow other higher order mode
frequencies to be close to the cavity’s allowed frequencies, see Fig. E.3.
The sensitivity gain due to the shorter cavity is shown in Fig. E.4. With all the considerations
discussed above, the length of the cavity is chosen to be 1.45 inch (36.83 mm). The outer diameterand the inner diameter are 1.5 inch (38.1 mm), and 0.375 inch (9.525 mm), respectively, see Fig. E.5.
The outer radius is chosen based on how the cavity will be mounted, see section E.4.

E.2

Assembling Reference Cavity with Optical Contact

Optical contact method is used for joining mirrors and a spacer to form a cavity. The jointed objects
becomes monolithic through covalent bonds, and the associated noise should be minimum comparedto other available bonding techniques. Furthermore, the method is relatively easy and can be donein a lab with a flow bench. Here is an instruction for optical contacting process.
• Clean the flow bench area.
• Prepare isopropanol, methanol and acetone (should be spectrophotometric/reagent grade).
The solvents should be stored in glass containers. Use glass syringes with needle metal todispense the solvents on cleaning wipe. Do not use rubber droppers.
• Clean the target surfaces thoroughly by drag wipe method. For cleaning optical surface ingeneral, wipe the surface with acetone first, follow by methanol. Finally, use isopropanol tofinish off any residue. The reason for this solvent order is that acetone is good to remove
organic compound, but it dries up quickly and redeposits the contaminants. Thus, methanol
is used to dissolve the pre-cleaned surface. Isopropanol can be used to rinse methanol andacetone and to remove any particles on the surface because of its relatively slow evaporation.
Make sure not to use excessive amounts of isopropanol. Otherwise it can leave drying markson the optic. If the stain on the surface still persists, use a pair of forceps to hold a folded

107

Figure E.1: Changing temperature is required to tune the allowed frequency by one FSR. Generally,
shorter cavities require more heating power to change the allowed frequency via thermal expansion.
We would like to avoid heating the cavity up more than 20 K (for 1/2 FSR tuning). These conditions
set a lower bound for the cavity length to 1 inch. (thermal tuning coefficient for a fused silica
nm1(α = 5.5e−7 K)).
≈ 14 K∗in108

Figure E.2: Cavity g-factor vs cavity length for a 2-mirror FP-cavity with identical mirrors for
ROC=0.5m and ROC=1m. There will be no stability issues for the mirrors.

Figure E.3: Higher order modes up to n+m=40 for cavities with a length of 1 in to 2.5 in (mirrors
ROC=0.5 m). The sideband frequency is14.75 MHz. Yellow shaded areas are preferred as they are
free of higher order modes for any sideband frequency up to 20 MHz. The three widest areas (andso less sensitive to tolerances in spacer fabrication and changes in mirror curvature) are 1.45 ± 0.02
in, 1.88 in and 2.02 in. Legend: red-upper sideband; green-lower sideband; blue-carrier

109

Figure E.4: Coating thermal noise level ratio between the shorter cavity, and the existing LIGO
reference cavities (length=8 in, mirrors ROC=0.5 m, concave-concave configuration). The plot shows
the results from two different ROC (0.5 m and 1 m) . The lower limit in length is given by the thermal
tuning requirement (see Fig. E.1). At 1.45 inches, the total increase in thermal noise is a factor of
8.7

8

5

6

4

2

3

1

3

D

0.125

-A-

4x

0.0

1

2

D

X4

5°

7

1.50

1

0.050 A
0.375

0.020 B

C

C

0.005 B

0.725

0.020 A

1.45
-B-

B

B

1

end faces: polished for optical contacting

2

outer cylinder surface: standard polish, no
special requirements

3

venting hole: diameter can be different,
but should not exceed 3/16"

UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN INCHESTOLERANCES:
FRACTIONAL 1/16
ANGULAR: MACH 0.5
TWO PLACE DECIMAL.01
THREE PLACE DECIMAL.005

A

INTERPRET GEOMETRICTOLERANCING PER:PROPRIETARY AND CONFIDENTIAL
THE INFORMATION CONTAINED IN THISDRAWING IS THE SOLE PROPERTY OF<INSERT COMPANY NAME HERE>. ANY
REPRODUCTION IN PART OR AS A WHOLEWITHOUT THE WRITTEN PERMISSION OF<INSERT COMPANY NAME HERE> IS
PROHIBITED.

8

7

6

5

MATERIAL

USED ON

NEXT ASSYAPPLICATION

4FINISH

DRAWN

NAME

DATE

Frank

03/23/12

TITLE:CHECKED
ENG APPR.

cavity_spacer_1.45inx1.5in

MFG APPR.
Q.A.
COMMENTS:SIZE DWG. NO.

Bcavity_spacer_1.45inx1.5in

fused silica
polished

3

Figure E.5: A drawing of 1.45 inch spacer

SHEET 1 OF 1

SCALE: 1:1 WEIGHT:

DO NOT SCALE DRAWING

2

REV

1

A

110
cleaning wipe to apply slight pressure while cleaning the surface with the solvents. Then
repeat the process with the drag wipe method. For more details on how to clean optics, see
Ref. [162, 163, 164].
• Place the two pieces together, for our case, a cavity and a mirror. A temporary fixture to
guide the two pieces in contact should be used, as the surface is very slippery. We use a pieceof Delrin (Polyoxymethylene plastic) with diameter a few in thousand inch larger than the
mirror and the spacer to hold them together coaxially, see Fig. E.6 a.
• Apply slight force to keep the bond going. If the two surfaces are sufficiently clean and smooth,the bond should be formed instantly. If there is a dust particle on the surface, fringes on thebonding area will appear, see Fig. E.6 b. Remove the parts and clean the surfaces again.

Figure E.6: a) A pair of fixtures made from Delrin keep the mirrors and the spacer in place during
the bonding process. b) Unsuccessful bonding attempt can be seen from the fringes on the outeredge of the mirror.
• Leave the assembly for several hours, as the bond becomes stronger as time passes [165, 87].
It is possible that the pieces fall apart from one another due to thermal change, vibration after
the bonding process. Make sure to check that the pieces are well contacted before install themin a setup.
• To remove the bonded pieces, use a razor blade and try to wedge in between the contacted
surfaces. Apply a lot of isopropanol when the blade presses on the contact area.

111

E.3

Eigenmode study

Since all the thermal noise calculations from all parts of the cavity are done under the quasi-static
assumption. This means that the thermal noise in the frequency of interest has to be lower than thefirst body mode of the reference cavity. It is important to calculate the eigenfrequency of the lowest
body mode to make sure that the measured bandwidth is below the first eigenfrequency. COMSOL
can calculate eigenmodes and associated eigenfrequencies of a body. First we try a cylindrical shape
to compare the results between COMSOL and an analytical result given in situation 7b, table 16.1
of Ref. [166] as
1.57
f1 =
2π

s

Y
.
ρl2

(E.5)

Where Y is the Young’s modulus of the material (72 GPa for fused silica). ρ is the mass density(2200 kg/m3 ). We use l as half of the cavity length because our case has both ends of the cavity are
free while the formula is for a bar with one end fixed, another end free. The first longitudinal mode
frequency obtained from both methods are 74 kHz and 77 kHz, respectively, see Fig. E.7.

Figure E.7: COMSOL simulation for the first longitudinal mode of a cylindrical object with thesame dimension as the spacer. The frequency is at 74 kHz. The color on the surface represents therms displacement from all directions.
Once we verify that the result from COMSOL is reliable, we simulate the complete cavity with
COMSOL. The lowest body mode is a radial mode at 46 kHz, and the first longitudinal mode is
at 60,kHz, see Fig. E.8. The frequencies are high enough that the thermal noise calculation withquasi-static assumption is still valid.

112

Figure E.8: COMSOL simulation for eigenfrequencies of the cavity. Left, the lowest body mode,
which expands and contracts radially along the beam line, at 46.7,kHz. Right, the first longitudinalmode at 60.1,kHz. The result from the first longitudinal mode is included here because this mode
directly couples into the displacement noise of the cavity and can limit the bandwidth of the frequency
stabilization servo.

E.4

Searching for Optimum Support Point with COMSOL

To find the optimum support point, we ran simulations on COMSOL to compute the bending effectfrom different support points on the surface of the cylindrical spacer. Since the first body mode of
the spacer is up to 46 kHz, cf. section E.3, we can use quasi static study for the FEA to calculate
the strain at DC response and apply the result to the noise budget at frequency lower than thefirst body mode. The FEA results show that there is no way to completely cancel the displacement
along the beam-line axis. However, tilting of the mirror can be canceled, see Fig. E.10. Our support
position is chosen to be 1.17 inches apart at 30 degrees measured down from the horizontal plane.

E.5

Mount Design and Assembly

The main components of the cavity mount assembly are the following:• Holder: The cavity holders keep the cavity above the stainless steel mount. They are madefrom Polyether ether ketone (PEEK). It is chosen for its thermal insulation and vacuum compatible [131]. They are cut from a cylinder rod and placed orthogonally to the spacer to achieve
point contact support, see Fig. E.11.

113

Figure E.9: COMSOL simulation for the cavity on the support bending under its weight. The
simulation can be done on 1/4 of the cavity due to its symmetries. The support position is shownin the picture. The color represents the displacement along the beamline axis measured from its
original position. Numbers without unit are in meter.

114

Figure E.10: Results from COMSOL simulations. The top plot shows the displacement along thebeam line, the bottom plot shows how much the mirror tilt due to vertical acceleration. The plots
have results from four different angle positions, 15, 30, 45, and 60 degrees. From the plots, it appears
that with smaller angle (the support position is closer to the horizontal plane), the cavity will be lesssusceptible to the applied acceleration. However, there are a few major drawbacks. It is not practical
for the design of the mount, the force between the mount and the cavity is increased and it mightlead to extra loss on the spacer surface, and the tilt cannot be cancelled anymore. Hence, the chosen
support position is at 30 degrees, 1.17 inch apart. The strain sensitivity to vertical acceleration is2
about 6 × 10−12 sm

115• Thermal shield: A copper tube with a heater wire wrapped around the outer surface is sued
as a temperature control on the cavity. The tube has four slots to let the peek supports touch
the cavity.
• Mount base: The main structure that holds the PEEK holders, thermal shields and thecavities together. It is screwed down to the seismic isolation stack. The geometry of themount is based on the optimum support design. The drawing is shown in Fig. E.12.

.13 THRU ALL
.22
.11

.25
.188

.38

.6250

.25
.125

R.25

UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN INCHESTOLERANCES:
FRACTIONALANGULAR: MACHBEND
TWO PLACE DECIMAL
THREE PLACE DECIMAL
INTERPRET GEOMETRICTOLERANCING PER:PROPRIETARY AND CONFIDENTIAL
THE INFORMATION CONTAINED IN THISDRAWING IS THE SOLE PROPERTY OF<INSERT COMPANY NAME HERE>. ANY
REPRODUCTION IN PART OR AS A WHOLEWITHOUT THE WRITTEN PERMISSION OF<INSERT COMPANY NAME HERE> IS
PROHIBITED.

5

MATERIAL

USED ON

NEXT ASSYAPPLICATION

4

NAME

DATE

DRAWN

TITLE:CHECKED
ENG APPR.
MFG APPR.
Q.A.
COMMENTS:SIZE DWG. NO.

A peek_v2

FINISH

3

SHEET 1 OF 1

SCALE: 2:1 WEIGHT:

DO NOT SCALE DRAWING

2

REV

1

Figure E.11: A drawing of PEEK support. A piece can be cut from a commercially available PEEK
rod.

116

.09
4X
4-40 UNC
.50

.35
.22
6.10

1.50

.14
6X
8-32 UNC

.49
.33

.02

A

.91
3.00 2.00

1.17

1.170

DETAIL A
SCALE 1 : 1.92
4X.18 THRU ALL.31.16#4-40 Tapped Hole
.409

3.05
.50

.40

.326

.50

120.00°

.60

1.20

1.50

1.217
6.60
UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN INCHESTOLERANCES:
FRACTIONALANGULAR: MACHBEND
TWO PLACE DECIMAL
THREE PLACE DECIMAL
INTERPRET GEOMETRICTOLERANCING PER:PROPRIETARY AND CONFIDENTIAL
THE INFORMATION CONTAINED IN THISDRAWING IS THE SOLE PROPERTY OF<INSERT COMPANY NAME HERE>. ANY
REPRODUCTION IN PART OR AS A WHOLEWITHOUT THE WRITTEN PERMISSION OF<INSERT COMPANY NAME HERE> IS
PROHIBITED.

5

MATERIAL

USED ON

NEXT ASSYAPPLICATION

4

NAME

DATE

DRAWN

TITLE:CHECKED
ENG APPR.
MFG APPR.
Q.A.
COMMENTS:SIZE DWG. NO.

REV

dual_cavity_mount_base_v11
A

FINISH

3

SHEET 1 OF 1

SCALE: 1:2 WEIGHT:

DO NOT SCALE DRAWING

2

Figure E.12: A drawing of dual cavity mount base.

1

117

Figure E.13: Top, the cavity mount during assembly. Note: the shields are not shown here. Below,
the final assembly during the final installation. The walls are for preventing thermal coupling
between the two cavities.

118

Appendix F

Pre-Mode Cleaner Cavity
The pre-mode cleaner is used for filtering the beam profile, intensity filtering. It also acts as a stablereference for beam position. The design parameters of the PMC is shown in the table below [167].

Round trip length / FSR

0.42 m / 357 MHz

Finesse50 (designed for P-polarization)

Cavity linewidth (FWHM)

14 MHzHigher order mode suppression (power)

TEM01 (vertical):200, TEM10 (horizontal):780

Actuator range

PZT:2.7 FSR for 0–375V

Actuator bandwidthFirst resonant of mounted PZT 10 kHz

Beam waist radius

w0 = 370 µm
r 
 2
2
√
dx
dα
+
=
< (3 × 10−5 /f ) Hz at f > 10Hz
w0
ΘD

Pointing relative to optical table

Table F.1: Design parameters of the first prototype PMC w0 is the beam radius, ΘD is the half
angle beam divergence, dx is the displacement fluctuations, dα is the angular fluctuations.

F.1

Eigenmode Study

The eigenmode study is done on the PMC in order to determine its body mode frequency on thefirst longitudinal mode. Since the PZT actuator that keeps the cavity length moves in this direction,
the highest unity gain frequency (UGF) is affected by the resonant frequencies of the PMC bodyand the PZT. In this case, the first resonant frequency of PZT is around 10 kHz, and 17 kHz for the
PMC body. This limits the UGF to around a few kHz.

119

Figure F.1: First longitudinal mode of the PMC. The eigenfrequency is around 16,600 kHz.

120

Appendix G

Mathematical Note
G.1

Note About Fabry–Perot Cavity

Ein

Ea

Er

Eb
t,r

Eout

t,r

Figure G.1: E field inside a cavity formed by two similar mirrors.
δ = 2πλ L is the phase change during a single trip inside the cavity. We can write
Eout = Ea te−iδ ,(G.1a)

Ea = tEin + rEb ,(G.1b)

Eb = Ea re−2iδ .

(G.1c)

Eoutt2 e−iδ=.
Ein
1 − r2 e−2iδ(G.2)

Solving the above equations, we getThis equation explains why only certain optical frequencies are allowed to pass through a Fabry–Perotcavity (Eq. 3.1).
Next, Let’s calculate the power build up in the cavity. We have to take Ea and Eb into account.
For a high finesse cavity Ea ≈ Eb , and they form a standing wave pattern. The total maximum
electric field is Ea + Eb ≈ 2Eb . Thus the amplification factor is the square of that ratio between

121
in-cavity electric field and the input field averaged over time (divided by a factor of 2) because of
the standing wave pattern inside the cavity,
1 2Eb 2||2 Ein
2t2 r2=(1 − r2 )2
2F
≈
,
π

|Pcav /Pin | ≈(G.3a)(G.3b)(G.3c)

r
where F ≈ π 1−r
2 is defined as the cavity finesse. The power build up as a function of frequency

due to the cavity response is [77],|Pcav /Pin (f )| =2F/π,
1 + (f /fcav )2(G.4)

where fcav is the pole of the cavity.

G.2

Temperature Fluctuations for TO Calculation

The temperature fluctuation sensed by a Gaussian beam for TR calculation is given in Ref. [84] as(in a very simplified form where all constants in front of the integral are absorbed into one constant),

TO
S∆T= const

Z ∞
k⊥ e

2
−k⊥
w02 /4Z ∞dk⊥0

k2 + k2
dkz 2 z 2 2⊥ 2
(kz + k⊥ ) + Ω−∞



1
1 + kz2 l2


.

(G.5)

Since the penetration depth l is typically much smaller than the beam radius, l  lth  ω0 , theintegral is solved while assuming the last term as unity, and the denominator as constant while
integrating over k⊥ .
For small spot size or low frequency where the assumption is not held, the integral can be solved
analytically while ignoring the last term. The integral for kz can be converted to a complex contour
integral as

Z ∞
dkz
−∞

2
kz2 + k⊥→2
2
2
(kz + k⊥ ) + Ω2Idzz 2 + k2
(z 2 + k 2 )2 + Ω2

(G.6a)

z 2 + k2
(G.6b)(z 2 + k 2 + iΩ)(z 2 + k 2 − iΩ)Iz 2 + k2
√
√
√
√
= dz
(z + k 2 + iΩ)(z − i k 2 + iΩ)(z + k 2 − iΩ)(z − i k 2 − iΩ)
I

=

dz

(G.6c)
2

I
=

dz

2
+ k⊥

z
.
(z − z1 )(z − z2 )(z − z3 )(z − z4 )(G.6d)

122
where,pz1 = −i k 2 − iΩpz2 = i k 2 + iΩpz3 = i k 2 − iΩpz4 = −i k 2 + iΩ

(G.7a)(G.7b)(G.7c)(G.7d)

Next is to find the positions of all the poles on the complex plane. k 2 + iΩ will be in the first√√
quadrant. Its two roots, k 2 + iΩ, − k 2 + iΩ will be in the first and third quadrant, respectively.
iπ

Finally, after being multiplied by i = e 2 that rotates the complex vector counter clockwise by π/2√√
radian, i k 2 + iΩ and −i k 2 + iΩ are in the second and fourth quadrants. By the same token, one√√
can show that i k 2 − iΩ and −i k 2 − iΩ are in the third and first quadrants. Then,we can takethe contour integral to cover the real axis and on the top half of the complex plane. So only z1and z2 have to be considered during the contour integral. Using Cauchy’s residue theorem, we canreduce the integral toIZf (z)dz = 0 =Zf (z)dz +

s1

Z

Z

f (z)dz +s2

f (z)dz +
s3

(G.8)

S1

Im z

S4

f (z)dz,
s4

S3
z2

z1

S2
Re z

z3

z4

Figure G.2: The poles and their positions in the complex plane are shown in the figure. The contour
line for the integrals can be separated into four line integrals. The integral along S2 is the original
integral to be solved.
where s1 is the upper half of the circle, s2 is the line along the real axis, s3 is the circle around123
z1 , and s4 is the circle around z2 , see Fig. G.2. The first integral goes to zero, the second integral
is the original integral we want to solve, the third and fourth integrals have to be evaluated,z12 + k 2
(z1 − z2 )(z1 − z3 )(z1 − z4 )
π
,
= √
2
2 k + iΩZ

f (z)dz = 2πis3(G.9a)(G.9b)

andZ

z22 + k 2
(z2 − z2 )(z2 − z3 )(z2 − z4 )−π
= √.
2 k 2 − iΩ

f (z)dz = 2πis4(G.10a)(G.10b)Thus,
Z ∞k2 + k2
dkz 2 z 2 2⊥ 2 =(kz + k⊥ ) + Ω
−∞

Z

Z
f (z)dz = −s2Zf (z)dz +f (z)dz!
√√πk 2 + iΩ − k 2 − iΩ√=2k 4 + Ω2√
Re[ k 2 + iΩ].
=π √k 4 + Ω2
s3

(G.11a)s4(G.11b)(G.11c)By changing the variable k to u and rearranging all the constants, we show that the double integralsof kz and k⊥ in Eq. G.5 can be reduced to a single integral as seen in Eq. 2.25

G.3

Error Analysis of the Measurement

As we use Eq. 2.16 to fit for the effective loss angle φc , uncertainties in various parameters can cause
the error in φc . The summary of the effects from each parameters are given in Tab. G.1. An error
in each parameter is propagated forward by thermal noise equation using Eq. 2.20. The root squaresum of the error is 7.34 percent in [Hz 2 /Hz]. This yields the error in the effective loss angle to beδφc = ±0.3.

124Parameter

Value

Contribution @100 Hz in % PSD

YSiO2

72 ± 6 GPa

0.3

YT a2 O5140 ± 20 GPa

7.2

Coating thickness

4.54 ± 0.02 µm

0.4

Cavity length1.45 ± 0.01 inch

0.6

Mirror ROC

500 ± 3 mm

(causes spot size error)

Spot size

182 ± 0.5 µm

0.55

308 ± 2,K

0.65

710 ± 5 Hz/V

0.7

Temperature
PLL calibration
errors in the noise budget

0.4

root square sum

7.34

Table G.1: Errors contributed from various consideration. Values and errors of the material parameters are taken from [101, 159, 168]

G.4

Bayesian Analysis

We use Bayesian probability to determine the mechanical loss angles of SiO2 and T a2 O5 because it
should provide us with more reliable answers. Due to the rapid improvement of coating technology
over the past decade, we only have a single coating measurement where the sample was manufacturedby the similar vendor with similar equipment to those of our samples. We will use the result from [51]as the prior for our Bayesian analysis.
Given φc , a knowledge of the parameters of our coatings, and prior observations of coating loss
angles, we can make a Bayesian estimate of φL and φH . To do this, we first write down a formula
relating φc , φL and φH :Mφc = ΞL NL dL φL + ΞH NH dH φH .

(G.12)

Here M = (1 + σs )(1 − 2σs )d/Es , NL = 15, NH = 14, dL = λ/4nL , and dH = λ/4nH . The coefficients
ΞL and ΞH are found by combining Table 1, and Eqs. 94 and 96 from Ref. [53], assuming zero light
penetration into the coating. These coefficients depend only on the coating parameters. Next wewrite down Bayes’s theorem [169]:
p(φL , φH |φ̂c ) =1
L(φL , φH |φ̂c ) p(φL , φH ),Z(G.13)

where Z is a normalization. As a prior, we use data from the ring down measurements in Ref. [51],since these measurements were performed on coatings from the same manufacturer as in our experiment, and were made during a similar time period. Harry’s quoted quantity φk is related to the125material loss angles φL and φH via(EL dL + EH dH )φk = EL dL φL + EH dH φH .

(G.14)We use φ̂k ± σφ̂k = (5.2 ± 0.3) × 10−4 as the value measured by Harry et al. 1 .
We then construct the prior(φ̂k − φk )21
1
,p(φL , φH ) =exp −
Z02 σφ̂2 + σφ2 k(G.15)kwhere Z0 is a normalization, φk is related to φL and φH via eq. G.14, and σφk is found by propagating
forward the uncertainties on the material parameters as given in appendix C.
As a likelihood we take"1 (φ̂c − φc )2
L(φL , φH |φ̂c ) = exp −2 σφ̂2 + σφ2 c#(G.16)

cwith φ̂c given by our measurement, and φc given by Equation G.12.
The prior, the likelihood, and the resulting posterior are shown in Figures G.3–G.5. By marginal−4
izing the posterior for each loss angle, we find that the median estimates are φL = 1.6+1.3−1.0 × 10+1.4
and φH = 8.0−1.4
× 10−4 , where the error bars indicate 68 % credible intervals.

1 Harry originally determined φ̂

−4 using a coating thickness that was 5 times the actual
k ± σφ̂k = (1.0 ± 0.3) × 10

value. Taking into account the correction given in Penn et al. [99], we arrive at φ̂k ±σφ̂ = (5.2±0.3)×10−4 , assumingkthat the uncertainty is dominated by systematics that do not scale with the nominal value of the loss angle.

126

×10−3

1.000
0.875

1.2

0.750

ϕH

1.0

0.625

0.8

0.500
0.375

0.6

0.250

0.4
0.2

0.125
0.0

0.2

0.4

0.6

0.8

ϕL

1.0
×10−3

0.000

Figure G.3: Prior PDF, after Harry et al. [51]. Picture credit: Evan Hall.

×10−3

1.000
0.875

1.2

0.750

ϕH

1.0

0.625

0.8

0.500
0.375

0.6

0.250

0.4
0.2

0.125
0.0

0.2

0.4

0.6
ϕL

0.8

1.0
×10−3

Figure G.4: Likelihood. Picture credit: Evan Hall.

0.000

127

×10−3
1.2

ϕH

1.0
0.8
0.6
0.4
0.2

0.0

0.2

0.4

0.6
ϕL

0.8

1.0
×10−3

Figure G.5: Posterior PDF. Picture credit: Evan Hall.

×107
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0

128

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