Scuola Normale Superiore di P sa
Corso di Perfezionamento in Fisicai

Tesi di Perfezionamento

Analysis of sensitivity andnoise sources for the Virgogravitational wave
interferometer

AuthorSupervisors

Gabriele VajenteProf. Francesco Fidecaro
Prof. Lorenzo Foá

Acknowledges
Working at the commissioning of the Virgo detector has been a challenging andexciting job. I had the opportunity to work together with many extraordinary
people and researchers, and from everyone I could learn many things. There aremany people I would like to thank here. I will try to remember some of them,even if this might make a quite long list...
To Francesco, for the warm welcome when I rst decided to enter in the
Virgo world, and for all the help I received during my Ph.D. work.
To all the INFN Pisa group, for their support and sympathy.
To Stefano and Matteo, from whom I learnt many things.
To Adalberto, for his great passion and for all the daily questions on theexperiment status.
To Marina, Maddalena and Simona, for all the time spent together sharing
the same oce and for their friendship, which sometimes made the work lighter.
To Paolo, for the hours spent discussing about this strange interferometerand how to improve it.
To my fellow lockers, Enrico and Bas, for all the long night shifts spent
together, eating a quick pizza before going back to work.
To Matt and Lisa, who taught me most of what I know about locking aninterferometer
To Edwige, Julien, Eric, Dominique, Heinrich, Ettore, Giovanni, Irene, Enrico, Francois, Nicolais, Fabien, all the commissioning team, the operators, EGO
and the entire collaboration, for their work and their support.
To Diego, for all the time he spent teaching me control theory and physics,
but most of all for his friendship.
To my friends Luca, Francesca, Ciccio, Ilaria, Angela, Paola, Tiziana, Sandro, . . . for listening to me when I needed somebody to complain to. Ai miei

amici, per avermi ascoltato quando avevo bisogno di qualcuno con cui lamentarmi.
To my parents and my family, without them nothing of this all would havebeen possible. Ai miei genitori e alla mia famiglia, senza di loro niente di tutto

questo sarebbe stato possibile.
To Claudia, and no words are enough. A Claudia, e le parole non bastano.

Introduction

The existence of gravitational radiation, known as gravitational waves, is one of
the most important predictions of the General Theory of Relativity [1, 2]: every
accelerating massive body loses energy by emitting a traveling distortion of thespace-time metric.

Despite this great generality, the gravitational interaction

weakness imposes stringent requirements on the bodies that can emit non negligible amounts of energy by gravitational waves: actually only very massive
astrophysical objects are candidates for being detectable sources of this form ofradiation.
The detection of gravitational waves coming from the universe is a challenging task and even though experimental eorts have been going on since several
decades, so far no direct detection has been achieved.

There are only strong

indirect evidence of the existence of gravitational radiation coming from binary
pulsar systems [3, 4, 5].
Nowadays the most sensitive instruments developed with the aim of detecting
gravitational waves are large scale interferometers [6]: the two LIGO detectors in
the United States [7], the German-British GEO600 in Germany [8], the Japanese
TAMA300 [9] and the French-Italian Virgo [10].

There are also plans for an

Australian interferometer [11].
On May 18th2007 the Virgo detector started a joint data taking run with thest

LIGO Scientic Collaboration until October 1

2007, opening the perspectiveof measuring the sky position of the gravitational wave source. This is also therst Virgo long science run (VSR1: Virgo Science Run 1).
The commissioning of a gravitational wave interferometric detector is a complex work, consisting in a large number of dierent tasks. This thesis is mainly
concerned with the development of strategies for the control of the optical system and with all the eorts directed toward improving the sensitivity of thedetector.

The main topics are:characterization of the longitudinal sensing

and control system; study of the origin of the noise limiting the current detector
sensitivity; development of techniques for the characterization of non-stationary
noises.
The outline of this thesis is the following:vi• The rst two chapters are of an introductory nature.

Chapter one de-scribes very briey the basic theory of gravitational waves, the possible
astrophysical sources and the principles behind their interferometric detection. Chapter two describes in detail the Virgo detector, focusing on all
those aspects that are needed for the discussions of the remaining chapters.

• The following two chapters introduces the principles of the interferometer
sensing and control systems. Chapter 3 describes the longitudinal sensing
and control system, while Chapter 4 discuss the basic principles of theangular one.

• The main part of this thesis, which directly describes the author's work,
starts with Chapter 5. It describes several studies aimed at the characterization of the longitudinal sensing and control system. These activities
have been carried out mainly by the author in collaboration with the lock-ing team, with the main goal of gaining a better understanding of thesystem and to improve its performance in terms of stability, robustness
and noise.

• The second main section of this work contains several chapters dedicatedto the study and characterization of the detector noise and sensitivity.

Chapter 6 introduces the techniques used for the measurement of the sensitivity and some basics tools for noise studies, like the coherence analysis.

Chapter 7 describes the application of these techniques to the Virgo interferometer conguration running during VSR1, being this one of the maincommissioning tasks carried out by the author. Chapter 7 also summarize
all the main sources of noises that couple linearly with the main output
gravitational channel, describing the typical performances obtained during
the science run.

• Chapter 8 describes some tools and techniques developed by the author to
study the long-term variation of the detector output noise. It is based on
the on-line computation of band-limited RMS of the main detector output
signal.

• Finally Chapter 9 describes a fast algorithm, developed within the GEO600
collaboration and implemented in Virgo by the author, to detect fast transient noises in the output signals and its application to the Virgo data.
Several appendices describes some ancillary topics that are relevant for allof the work of this thesis.

Appendix A contains an analytic derivation of the

behavior of elds and signal inside some simple optical systems.

Appendix B

details the lock acquisition procedure used in Virgo to reach science conguration during VSR1.

Appendix C briey describes basic notions about linear

vii
systems and transforms. Appendix D describes the topic of signal analysis, focusing mainly on spectral techniques.

Appendix E describes in some details

the topic of feed-back control systems, widely used in gravitational wave detectors like Virgo. Finally Appendix F describes a software implementation of thedemodulation technique.

Contents

1

Gravitational waves

1

1.1

Linearized Einstein's equation . . . . . . . . . . . . . . . . . . . .

1

1.2

Astrophysical sources . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.1

Neutron stars . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2.2

Coalescing binaries . . . . . . . . . . . . . . . . . . . . . .

5

1.2.3

Supernovae. . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2.4

Stochastic background . . . . . . . . . . . . . . . . . . . .

6Interferometric detection . . . . . . . . . . . . . . . . . . . . . . .

6

1.3.1

DC detection with a Michelson interferometer . . . . . . .

6

1.3.2

Frontal modulation technique . . . . . . . . . . . . . . . .

9

1.3.3

Fabry-Perot cavities. . . . . . . . . . . . . . . . . . . . .

10

1.3.4

1.3

1.4

2

Power recycling . . . . . . . . . . . . . . . . . . . . . . . .

11

Data analysis techniques . . . . . . . . . . . . . . . . . . . . . . .

12

1.4.1

Coalescing binaries . . . . . . . . . . . . . . . . . . . . . .

13

1.4.2

Stochastic background . . . . . . . . . . . . . . . . . . . .

15

1.4.3

Continuous signals . . . . . . . . . . . . . . . . . . . . . .

16

1.4.4

Bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

The Virgo detector

19

2.1

Optical lay-out. . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.2

Injection system. . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.3

Mirrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.4

Suspension and mirror control . . . . . . . . . . . . . . . . . . . .

25

2.4.1

Suspension control . . . . . . . . . . . . . . . . . . . . . .

27

2.5

Detection system . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.6

Data distribution and storage . . . . . . . . . . . . . . . . . . . .

34

2.7

Global control system. . . . . . . . . . . . . . . . . . . . . . . .

34

2.8

Environmental monitoring . . . . . . . . . . . . . . . . . . . . . .

34

2.9

The Virgo design sensitivity . . . . . . . . . . . . . . . . . . . . .

35

x

CONTENTS

I Control systems

39

3

4

5

Longitudinal sensing and control

41

3.1

General control scheme . . . . . . . . . . . . . . . . . . . . . . . .

41

3.2

Laser frequency pre-stabilization. . . . . . . . . . . . . . . . . .

44

3.3

Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.4

Signal extraction

45

3.5

Actuation

3.6

The Variable-Finesse technique

3.7

Lock acquisition strategy. . . . . . . . . . . . . . . . . . . . . . . . . . .

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

48. . . . . . . . . . . . . . . . . . .

49. . . . . . . . . . . . . . . . . . . . . .

50Angular sensing and control

51

4.1

52

Gaussian beam optics. . . . . . . . . . . . . . . . . . . . . . . .

4.1.1

The par-axial diraction equation. . . . . . . . . . . . .

52

4.1.2

Transverse electro-magnetic modes . . . . . . . . . . . . .

53

4.2

Fabry-Perot resonant cavities. . . . . . . . . . . . . . . . . . . .

54

4.3

Cavity response to mis-alignments

4.4

Wave-front sensors

4.5

Coupling of angular noise. . . . . . . . . . . . . . . . . . . . . .

60

4.6

The Anderson technique . . . . . . . . . . . . . . . . . . . . . . .

60

4.7

Virgo angular sensing and control scheme. . . . . . . . . . . . .

61

4.8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64. . . . . . . . . . . . . . . . .

56. . . . . . . . . . . . . . . . . . . . . . . . . .

59

Longitudinal controls characterization

65

5.1

Expected optical response of the system . . . . . . . . . . . . . .

65

5.2

Properties of the control loops . . . . . . . . . . . . . . . . . . . .

71

5.3

Properties of the SSFS loop . . . . . . . . . . . . . . . . . . . . .

71

5.4

Noise subtraction techniques

76

5.5

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

5.4.1

MICH noise subtraction: alpha technique. . . . . . . . .

79

5.4.2

PRCL noise subtraction: beta technique . . . . . . . . . .

81

Reconstruction of the optical matrix. . . . . . . . . . . . . . . .

83

5.5.1

Sensing matrix. . . . . . . . . . . . . . . . . . . . . . . .

84

5.5.2

Filtering matrix. . . . . . . . . . . . . . . . . . . . . . .

84

5.5.3

Driving matrix. . . . . . . . . . . . . . . . . . . . . . . .

84

5.5.4

Measured matrix . . . . . . . . . . . . . . . . . . . . . . .

87

5.5.5

Optical matrix

87. . . . . . . . . . . . . . . . . . . . . . . .

5.6

Performances of the locking system . . . . . . . . . . . . . . . . .

88

5.7

Operating point . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.7.1

Denitions. . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.7.2

Map of locking point . . . . . . . . . . . . . . . . . . . . .

92

5.7.3

Coupling of longitudinal noise to power signals

99

5.8

. . . . . .

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

CONTENTS

xi

II Noise and sensitivity

105

6

107

7

Noise sources and sensitivity
6.1

Sources of noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2

Coherence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.3

Detector sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.4

History of the detector sensitivity . . . . . . . . . . . . . . . . . . 112

Noise budgets
7.1

115

Linear noise projections
7.1.1. . . . . . . . . . . . . . . . . . . . . . . 115

Subtle points and limits of linear noise projections. . . . 117

7.2

Summary of Virgo noise budget during VSR1 . . . . . . . . . . . 119

7.3

Longitudinal control noise . . . . . . . . . . . . . . . . . . . . . . 121

7.4

Frequency noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.5

Angular control noise . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.6

Other sources of control noises

7.3.1

Actuation noise . . . . . . . . . . . . . . . . . . . . . . . . 121. . . . . . . . . . . . . . . . . . . 129

7.6.1

Beam Monitoring System. . . . . . . . . . . . . . . . . . 130

7.6.2

Suspended Detection Bench . . . . . . . . . . . . . . . . . 133

7.7

Laser intensity noise. . . . . . . . . . . . . . . . . . . . . . . . . 135

7.8

Oscillator phase noise. . . . . . . . . . . . . . . . . . . . . . . . 137

7.9

Electronic noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.10 Shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.11 Environmental noise. . . . . . . . . . . . . . . . . . . . . . . . . 139

7.11.1 Electro-magnetic noise . . . . . . . . . . . . . . . . . . . . 139
7.11.2 Acoustic noise. . . . . . . . . . . . . . . . . . . . . . . . 140

7.12 Conclusions and latest development8

Non-stationary noises

155

8.1

Computation of band-limited RMS . . . . . . . . . . . . . . . . . 155

8.2

Analysis of dark fringe non-stationarities . . . . . . . . . . . . . . 158

8.1.1

8.3

9

. . . . . . . . . . . . . . . . 153

Contribution of narrow spectral lines . . . . . . . . . . . . 156

8.2.1

Spectral analysis. . . . . . . . . . . . . . . . . . . . . . . 158

8.2.2

Time domain analysis

8.2.3

Multi-dimensional linear regression . . . . . . . . . . . . . 165

8.2.4

Application of the technique . . . . . . . . . . . . . . . . . 167. . . . . . . . . . . . . . . . . . . . 162

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Fast transient noises

179

9.1

The HACR algorithm

9.2

Analysis of dark fringe clusters

9.3

Analysis of coincidence with auxiliary channels
9.3.1. . . . . . . . . . . . . . . . . . . . . . . . 179. . . . . . . . . . . . . . . . . . . 182. . . . . . . . . . 185

Coincidences with beam monitoring system error signals . 186

xii

CONTENTS
9.3.2
9.4

Coincidences with quadrature signal. . . . . . . . . . . . 187

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

III Conclusions

193

IV Appendices

199

A Fields inside optical systems

201

A.1

Basic conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

A.2

Powers and demodulation

A.3

Simple Michelson interferometer

A.4

Fabry-Perot resonant cavity . . . . . . . . . . . . . . . . . . . . . 208. . . . . . . . . . . . . . . . . . . . . . 203. . . . . . . . . . . . . . . . . . 205

B Lock acquisition technique during VSR1
B.1

213

Locking and pre-alignment of the two cavities . . . . . . . . . . . 213
B.1.1

Locking of the arm cavities and MICH . . . . . . . . . . . 213

B.1.2

Fine pre-alignment . . . . . . . . . . . . . . . . . . . . . . 215

B.1.3

Check of main modulation frequency . . . . . . . . . . . . 215

B.1.4

Force reallocation to top suspension point . . . . . . . . . 217

B.2

Second stage of frequency stabilization . . . . . . . . . . . . . . . 217

B.3

Re-alignment of power recycling mirror . . . . . . . . . . . . . . . 219

B.4

Fine pre-alignment of PR mirror

B.5

Transition to dark fringe . . . . . . . . . . . . . . . . . . . . . . . 222

B.6

The thermal transient. . . . . . . . . . . . . . . . . . 221. . . . . . . . . . . . . . . . . . . . . . . . 222

B.6.1

Description. . . . . . . . . . . . . . . . . . . . . . . . . . 222

B.6.2

Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 224

B.6.3

Locking strategy. . . . . . . . . . . . . . . . . . . . . . . 225

B.7

Locking of output mode-cleaner . . . . . . . . . . . . . . . . . . . 228

B.8

Low noise control lter . . . . . . . . . . . . . . . . . . . . . . . . 229

B.9

Low noise actuators. . . . . . . . . . . . . . . . . . . . . . . . . 231

B.10 Last low noise steps. . . . . . . . . . . . . . . . . . . . . . . . . 232

B.11 Science mode conguration. . . . . . . . . . . . . . . . . . . . . 233

C Linear systems

235

D Signal analysis

239

D.1

Random processes

D.2

Stationarity and ergodicity
D.2.1. . . . . . . . . . . . . . . . . . . . . . . . . . 239. . . . . . . . . . . . . . . . . . . . . 240

Power of a signal . . . . . . . . . . . . . . . . . . . . . . . 241

D.3

Linear transformation of signals . . . . . . . . . . . . . . . . . . . 242

D.4

Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

CONTENTS
D.5

xiii

Digital lters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

E Feed-back control systems

247

E.1

Simple feed-back system . . . . . . . . . . . . . . . . . . . . . . . 247

E.2

Response to variations of the input signal

E.3

Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
E.3.1. . . . . . . . . . . . . 249

Nyquist stability criterion . . . . . . . . . . . . . . . . . . 252

E.4

Bode and Nichols plots . . . . . . . . . . . . . . . . . . . . . . . . 255

E.5

Fabry-Perot cavity

E.6. . . . . . . . . . . . . . . . . . . . . . . . . . 257

E.5.1

Pound-Drever-Hall signal

E.5.2

Frequency response of the cavity. . . . . . . . . . . . . . . . . . 258

E.5.3

Actuators and corrector. . . . . . . . . . . . . . . . . . . 260

Multi-dimensional control systems

F Non-stationary noise couplings. . . . . . . . . . . . . . 259. . . . . . . . . . . . . . . . . 261

265

xiv

CONTENTS

Chapter 1
Gravitational waves
The goal of this chapter is to briey introduce the theory of gravitational radiation, the possible astrophysical sources, the eect on test masses and the basic
principles of interferometric detection.

1.1 Linearized Einstein's equation
Despite that the Einstein's equation for space-time metric is non linear, it ispossible to specialize it to the case of small perturbations of a at Minkowski
metric ηµν :gµν = ηµν + hµν
where ||hµν ||  1. It can be shown (see [1, 2] for a detailed discussion) that there
exists a particular choice of coordinates, corresponding to the Lorentz gauge forthe metric, where the Einstein's equation reduces to∂t2 − ∇2 hµν = 0
with the additional constraint:∂β hβα =1∂α hββ2These two equations are very similar to the ones that describe the propagationof electro-magnetic radiation, and therefore they can be solved by plane waves
with wave-vector kµ :hµν (x) = Aµν eik·x
where the eld amplitude Aµν is transverse:Aµ k µ = 0
The Lorentz gauge is not uniquely xed, and the remaining arbitrariness isenough to impose four further conditions on the Aµν . In particular, given the

2

CHAPTER 1. GRAVITATIONAL WAVES+ POLARIZATIONyxt = T/4

t=0

t = T/2

t = 3T/4yx

X POLARIZATION
Figure 1.1: Eect of plus and cross polarized gravitational waves propagating
along the z axis on a ring of free falling masses.

The strain is shown every

quarter of period.

observer 4-velocity uµ it is possible to perform a further coordinate change suchthat:

Aµν uµ

=

0

Aµµ

=

0

In total there are eight constraints for the ten components of the symmetric amplitude tensor. The two remaining degrees of freedom are indeed the two physical polarizations of the gravitational wave in the transverse-traceless gauge. Fora gravitational wave propagating along the z axis, the metric can be expressed
as



0
 0

hµν = 
 0
0

0
A+
A×
0

0
A×
−A+
0


0
0 
 ik·x e
0 
0The two polarizations are usually known as cross and plus. The importantresult that follows from this solution is that the eect of a gravitational wave on
the space-time metric has an intrinsic dierential nature. This means that the
distance between two nearby test masses in the metric generated by a gravitation
wave is modulated at the wave frequency with opposite sign in the x and y
direction (see g. 1.1).

The relative change in the distance is given by the

amplitude of the gravitational wave h, usually called strain amplitude.

1.2. ASTROPHYSICAL SOURCES

3

For example, assuming a plus-polarized gravitational wave travelling along
the z direction, the eect on the metric is given by

ds2 = c2 dt2 + (1 + h+ ) dx2 + (1 − h+ ) dy 2
It can be shown that the eect on a light ray propagating between two freefalling masses is a change in the optical path and a deviation of the direction of
propagation. This second eect is however negligible as long as the gravitational
wave is weak [1, 2]. Therefore, for a path lying along the x axis, at rst orderin the strain amplitude h:1dx = ±c dt 1 − h+ (t)2The eect has the opposite sign for the y direction.
The propagation of a light ray over a nite distance in the eld of a monochromatic gravitational wave of frequency νg = ωg /2π can be described in terms ofretarded time.

Suppose a light pulse is emitted from a free-falling mass at agiven time, reected by another mass at a distance L, and detected back at therst mass at a certain time t0 . It can be shown [12] that the emission time ofthe pulse depends on the gravitational wave amplitude and frequency and isgiven byL
2L
+ ε h sinctr = t0 −
c
c



ωg L
c


cos (ωg (t − L/c))

where ε = 1 for the x direction and -1 for the y.

Consider two light pulses

emitted on a free falling mass in the origin of the coordinate system toward twoother distant masses at a distance L along the x and y axes, reected back to theorigin and detected there. The two pulses arrive at the origin with a time lag
given by the dierence in the optical paths due to the eect of the gravitational
wave:

δtGW = h

2L
sinc
c



ωg L
c


cos (ωg (t − L/c))(1.1)

1.2 Astrophysical sources
In the far eld and slow motion approximation, the gravitational wave eldgenerated by a source can be expressed as [1, 2]

hµν (r, t) =

2G 1 d2 µνI
c4 r dt2
t−r/c

where r is the distance from the source to the observer and Iµν

is the trace-

reduced quadrupole momentum associated with the energy density ρ(x, t) of thesource:

I

µν

Z
(t) =

3

d x ρ(x, t)1
x x − x2 η µν
3
µ ν



4

CHAPTER 1. GRAVITATIONAL WAVES
The conservation of the stress-energy tensor (which, to rst order, is equivalent
to the conservation of mass, linear and angular momentum) implies that the rstcontribution to the emission of gravitational waves comes from the quadrupoleterm.
To estimate the order of magnitude of the intensity of such radiation thequadrupole momentum of a body of mass M and size R can be approximated2with Q = εR M , where ε is a factor measuring the asymmetry of the mass
distribution.

Expressing the second derivative of R as the typical speed v ofinternal motion of the body one obtainsh ∼ ε2

GM 1  v 2
c2 r cThe multiplicative factor in front of the previous equation is a very small quantityG= 7.4 · 10−28 m/Kg
c2
and since the design sensitivity of interferometric detector like Virgo is aroundh ∼ 10−22 at 100 Hz it is clear that only astrophysical sources can generatestrong enough gravitational radiation to be detectable on earth.
These sources are usually classied depending on the time evolution of thesignal they generate: periodic sources like pulsars and in general spinning neutron stars or other massive bodies; quasi-periodic sources such as coalescing
binary systems of neutron stars or black holes; impulsive sources (bursts) likesupernovae or coalescing systems at the merging point; stochastic background of
cosmological or astrophysical origin.

1.2.1 Neutron stars
Spinning neutron stars can emit gravitational radiation at once and twice therotation frequency f if a mass asymmetry is present:h ≈ 3 · 10−27



10 kpcrf
200 Hz

2 

ε 
10−6

where ε is the asymmetry in the momenta of inertia of the star. Despite theweakness of these signals, their almost periodic nature makes it possible tointegrate over a long period of time to improve the signal-to-noise ratio.
Upper limits on the value of the asymmetry ε can be obtained assuming thatthe entire spin-down rate of the star is due to energy loss through gravitationalradiation.

For the Crab pulsar, spinning at 30 Hz, this gives εcorresponding to an amplitude h

≈ 10

−24

at 60 Hz.

≈ 7 · 10−4 ,This signal is possibly

detectable with one year of integration with the Virgo design sensitivity [13].
Theoretical prediction of ε are aected by large uncertainties and can range
from 10−7−4

up to 10[14].

1.2. ASTROPHYSICAL SOURCES

5

Objects

Range

Expected rate

NS-NS

30 Mpc0.0003-0.3/yr

NS-BH

60 Mpc0.0004-0.5/yr

BH-BH

145 Mpc

0.001-3/yr

Table 1.1: Expected rates for the detection of coalescing binary systems. The
quoted range corresponds to the expected one from rst generation ground based
interferometric detectors.

In our galaxy the number of spinning neutron star is of the order of 108[13] but most of them rotate at a frequency lower than 10 Hz, making them
undetectable with present ground interferometers.

1.2.2 Coalescing binaries
Binary system composed of two neutron stars (NS), two black holes (BH) or aneutron star and a black hole lose energy by means of gravitational radiation.
The frequency of the emitted wave is the double of the orbital one, and it increases the closer the two bodies become, until they merge together. The lifetimeof these objects can be of millions of years, but the gravitational radiation canbe detected only in the last minutes of their life, when the orbital frequency is
above several hertz. When the distance is still large enough (during the inspiralphase ) the orbit can be computed using the post-Newtonian formalism [5] andthe emitted wave-form is known, allowing the use of matched-ltering techniquefor an ecient detection. The expected frequencies can go up to a maximum of
about 1 kHz for NS-NS systems, with a permanence time at a given frequency
scaling like f

8/3.

With the foreseen sensitivity of rst generation gravitational wave detectors
like Virgo, the expected rates of detectable inspiralling events are aected bylarge uncertainties [15, 16] and are listed in table 1.1.
When the two objects become close enough, they merge together. During
this period the gravitational signal is expected to be burst-like, with frequencies
around 1 kHz.

1.2.3 Supernovae
Since the modelling of this kind of events is very dicult and almost unknown,
the expected signal is taken to be burst-like, with a duration of few milliseconds.
The intensity of the gravitational radiation depends strongly on the amount ofenergy ∆E emitted in this form. An estimate can be given by [17]h ≈ 2.7 · 10

−20



∆E
M c2

1/2 

1 kHz
f

1/2 

10 M pcr



6

CHAPTER 1. GRAVITATIONAL WAVES
2

A low value of the emitted energy, expected of the order of ∆E/MSU N c

−6

10

=

, would make the amplitude too small for detecting events in the Virgocluster, where a rate of several supernovae per month is expected [18]. In any
case an event in our galaxy (with rates of few per century) will be detectablewith reasonable signal-to-noise ratio by rst and intermediate gravitational wave
detectors.

1.2.4 Stochastic backgroundA stochastic background can be originated by the incoherent sum of random
gravitational wave signals coming for example from astrophysical galactic sources[19] or it can have a cosmological origin. The expected noise oor from astrophysical sources is however well below the present detector sensitivities [20] andeven below the one of advanced ground detectors.
The estimates for the gravitational background of cosmological origin areaected by large uncertainties. Their are usually expressed in terms of the ratioΩgw between the energy density in the gravitational background and the one
needed to close the universe [21]:Ωgw (f ) =1dρgw
ρcritical d log f

The best sensitivity to this kind of signal is obtained with coincident analysis
between dierent detectors. The network of rst generation ground-based interferometers is expected to be sensitive to Ωgw of the order of 10−5. The standard

inationary model foresees a much smaller value of the order of 10−14, whilesuper-string theories suggest scenarios with a stochastic background of the order
of 10

−9

, strong enough to be detected by advanced ground interferometers [22].

1.3 Interferometric detection
The intrinsic dierential nature of the eect of a gravitational wave on a set oftest masses makes optical interferometers very ecient detection instruments,
since they are extremely sensitive to changes in the relative length of the two
orthogonal arms [6].

1.3.1 DC detection with a Michelson interferometer
The simplest optical conguration for detecting a dierential motion is a Michelson interferometer (see g. 1.2). A laser beam impinges on a half reecting beam
splitter, the two emerging beams are sent toward orthogonal directions, reectedback by two mirrors and they recombine at the beam splitter. The static tuning

α = 2k(la − lb ) of the interferometer is the phase dierence at the beam splitter1.3. INTERFEROMETRIC DETECTION

7

ra

L
rb
Pin
L

BS
Pout

Figure 1.2: Scheme of a simple Michelson interferometer.

between the two reected beams. The power detected at the output port is:

Pout =

Pin ra2 + rb2[1 + C cos (α + ΦGW )]
2
2

(1.2)

where C is the contrast which depends on the amplitude reectivity of the twoarm mirrors ra and rb :C=

2ra rbra2 + rb2

(1.3)

and which for perfectly reecting mirrors is equal to 1. For mirrors like thoseused in Virgo, where r ≈ 1, C is very close to unity as well as the quantity(ra2 + rb2 )/2.
When a gravitational wave passes through the Michelson interferometer, thelight beams coming back from the two arms and recombining at the output port
undergo an additional dierent dephasing given by eq. 1.1:

2πc 2LδΦGW (t) =
h
sinc
λ
c



ωg L
c


cos [ωg (t − L/c)](1.4)

The eect is a variation of the power detected at the output port which can be
expressed at rst order in the gravitational wave amplitude:

δPout (t) =Pin
C sin α · δΦGW (t)
2

(1.5)

The term sinc(ωg L/c) is close to 1 for gravitational waves of frequency below
few thousands of Hz.

8

CHAPTER 1. GRAVITATIONAL WAVES
If there were no sources of noise limiting the ability to detect power uctuations, the best choice for the tuning of the interferometer would be the one thatmaximizes the sine in eq. 1.5, that corresponds to having half of the maximum
power in the output port. This condition is usually called gray fringe, while thetuning that give minimum and maximum powers are called respectively dark

fringe and bright fringe.
In reality there is a fundamental limitation to the accuracy of power measurements, given by the shot noise, which is connected to the quantum nature oflight. This can be intuitively explained as the Poisson distribution related withthe process of counting the photons hitting the detector [23, 24]. The amplitude
spectral density (see app. D) of the shot noise is given by

SP (f ) =p

2 hP ν P

f ≥0

(1.6)

where hP is the Planck constant, ν is the frequency of the light and P is thepower impinging on the photo-detector. This power spectral density is at, i.e.
it does not depend on the frequency.
Therefore the best tuning for a Michelson interferometer is found maximizing
the signal-to-noise ratio :SδP
1
SN R(f ) =
=
SP
2

s

Pin C sin α 2πL√
Sh (f )
hp ν 1 + cos α λ(1.7)

where Sh (f ) is the spectral density of the gravitational wave signal, L is theaverage length of the two arms, and Pin is the input power of the interferometer.
The maximum of the SNR with respect to the static tuning α iscos α =

−1 +√1 − C2C(1.8)

Since the contrast is close to 1, the best tuning of a Michelson interferometeris very close to the dark fringe condition, being cos α ≈ −1 +p2(1 − C). This

technique is usually called DC detection.
The minimum detectable signal is dened as the one having a SNR equal to 1.
It is given by the equivalent spectral density in h of the shot noise. Consideringthat C ≈ 1 the result is

λSh (f ) =
4πL

r

2hP ν
Pin

(1.9)

For a input laser beam of 10W as in Virgo, with a wavelength λ = 1.064 µm theshot-noise limited sensitivity for a simple Michelson interferometer with 3 kmlong arms is

Sh (f ) ≈ 5.4 · 10−21 Hz−1/2(1.10)

1.3. INTERFEROMETRIC DETECTION

9

1.3.2 Frontal modulation technique
The present gravitational wave interferometer uses heterodyne or AC detection

technique. In the frontal modulation scheme the input beam is modulated inphase using an electro-optical modulator driven with a sinusoidal signal at aiωtfrequency Ω. In this way the input electromagnetic eld ψ(t) = A etoi[ωt+m sin(2πΩt)]ψm (t) = A e

=A

+∞
X

is changedJn (m) ei(ω+n·2πΩ)t(1.11)n=−∞where Jn (m) are the Bessel functions and m is the modulation depth. The eect
can be described as the creation of radio-frequency sidebands around the main

carrier frequency, spaced by multiples of the modulation frequency Ω. Moredetails are discussed in appendix A.
The Michelson interferometer is slightly modied introducing a macroscopic
asymmetry ∆L between the two arms, called Schnupp asymmetry. In this way
the resonance condition for the carrier and the sidebands can be dierent. Therefore it is possible to have the carrier tuned at destructive interference, while thesidebands are still present in the output port. Using the same conventions of
the previous section, in this condition the output eld is

Eout ∝

Ein iωt
e
2




2πΩ∆L
J0 (m)(ra − rb ) − 2e2iπΩt e4πiΩL/c J1 (m) sinc

The output signal is extracted by mixing the photo-diode output current witha reference sine or cosine oscillating at Ω and then by applying a low-pass lterwith a corner frequency much lower than the modulation one. In this way thedemodulated signal contains contributions coming from the beating of the carrier
with the two rst order sidebands (see app. A):

(dem)

Pout

 m2 − 41
r2 r3 t21 sin (2Dk)= − e−ikΩ (∆L−2L) −1 + e2i∆LkΩ m
4
2(1.12)

where D is a small deviation from the dark fringe tuning of the interferometer.
In particular this signal is zero in the dark fringe condition.
The eect of a gravitation wave is to introduce an additional dephasing to
the two beams recombining at the dark port. This is for all purposes equivalent
to a dierential deviation from the dark fringe. The response of the demodulated
signal to a gravitational wave is therefore linear.
Using eq. 1.12 to compute the response to a gravitational signal and considering the shot noise level corresponding to the sidebands power at the darkport, it is possible to show that the shot-noise limited sensitivity that can beobtained with the AC detection scheme is the same as in the DC scheme. The
advantages of this dierent method are those already anticipated: the detection
is shifted at the modulation frequency Ω where the laser intensity and frequency
noise are much lower; moreover the control of the interferometer is simpler if

10

CHAPTER 1. GRAVITATIONAL WAVES
the static tuning is chosen in order to have destructive resonance at the dark
port.

1.3.3 Fabry-Perot cavities
The limit on the sensitivity of a Michelson interferometer comes from two mainelements: the arms length and the input laser power. Since it is not technicallyeasy to have arms longer than 3 − 4 km and input power higher than some tensof W, it is important to nd other ways to improve the sensitivity.
One way to increase the eective length of the arms is to substitute themwith resonant cavities. In this way the light is stored for a longer time inside thearms, being reected multiple times back and forth, and therefore enhancing thedephasing due to gravitational waves. For detailed computations of the elds
inside a Fabry-Perot resonant cavity see app. A.
The cavity is on resonance when its length is tuned in order to have constructive interference between the eld transmitted by the input mirror and theone which has done a round trip inside the cavity. In this condition the power
stored inside the cavity can be enhanced by a large factor, which depends onthe nesse of the cavity√
π rI rEF=
1 − rI rE

where rI and rE are the input and end mirror reectivity respectively. With
totally reecting end mirrors, the reection coecient of the cavity aroundresonance is always almost equal to −1.

The phase varies from 0 when thecavity is not resonant, to π at resonance. The response of the cavity to a small
variation of its length around the resonant one is given by

δφ ≈

2F 2π
δL
π λ

For a single mirror at the same distance of the end one, the change in phasewould have been

δφ ≈

2π
δL
λ

The eect of the resonant cavity is therefore to amplify the optical response toa length change by a factor

g=

2F
π

If the two arms of a Michelson interferometer are substituted by two resonantcavities with the same length, the eect of a gravitational wave is enhanced bythis factor. For the Virgo interferometer, the two arm resonant cavities have anesse of about 50, resulting in a gain of a factor 30.
A more precise computation of the dephasing of the reected eld must take
into account a dependence on the frequency of the cavity length perturbation.
In fact one can expect an eect to be visible when the period of the motion

1.3. INTERFEROMETRIC DETECTION

11

is comparable with the storage time of the cavity, which is of the order ofτ = FL/c ≈ 0.5 ms. Indeed the response of the cavity exhibits a pole:
δφ =

2π 2F
r
λ π

1
 2
1 + ffc

at a frequency given by

fc =(1.13)

c4FL(1.14)

which for the Virgo arm cavities is about 500 Hz.
In conclusion if the two arms of a Michelson interferometer are replacedby two resonant Fabry-Perot cavities, the sensitivity that can be obtained isfrequency dependent:

1
Sh (f ) =8FL

r

2chP λ
Pin

s


1+

f
fc

2
(1.15)For Virgo this would give the limit shown in g. 1.3, corresponding to a sensitivity of 2 · 10

−22

Hz−1/2 at 100 Hz. Virgo has been commissioned in this con-guration (recombined interferometer ), but with reduced power, starting fromthe end of 2003, and it has been operated during the commissioning runs C3(April 2004), C4 (June 2004) and C5 (December 2005).

1.3.4 Power recycling
One way to increase the power circulating inside the interferometer, thus reducing the limitation in sensitivity due to shot noise, is the power recycling

technique.

The idea is that when the interferometer is tuned at dark fringe,

since the losses in the arms are very low, all the incoming power is reectedback to the laser. Therefore introducing a mirror (power recycling mirror, see
g. 1.4) between the laser and the beam splitter reecting back this light canstrongly enhance the power circulating inside the interferometer.
The eect of such a mirror can be computed describing the Michelson FabryPerot interferometer with an equivalent mirror of reectivity rM ICH (close to
one in absolute value if the interferometer is tuned at dark fringe). Therefore
the full power recycled interferometer is equivalent to a Fabry-Perot cavity composed by this mirror and the power recycling mirror [12]. The recycling gain isdened as the ratio between the input power and the power circulating inside
the central part of the interferometer:
GP R =tP R
1 − rP R rM ICH

2(1.16)

This increase in the power at the beam splitter mirror translates directly inan improvement of the shot noise limit to the sensitivity. For the nal Virgo

12

CHAPTER 1. GRAVITATIONAL WAVES

Strain sensitivity [Hz-1/2]

10-20

10-21

10

-22

Fabry-Perot Michelson
Michelson

101

102

103

104

Frequency [Hz]Figure 1.3: Spectral density equivalent to shot noise for a Michelson interferometer with 10 W input power, 3 km long arms with Fabry-Perot cavities with
nesse equal to 50, compared to that of a simple Michelson without resonantarms.

conguration the recycling gain is about 50 and therefore the shot-noise-limited
sensitivity is given bys
Sh (f ) = 3 · 10

−23



1+

f
fc

2

Hz−1/2(1.17)

1.4 Data analysis techniques
Discarding all the complexity of a gravitational wave interferometer, from thedata analysis point of view, the detector output is a single time series which
reconstruct as well as possible the gravitational signal:s(t) = h(t) + n(t)where n(t) is the detector noise and h(t) contains all the gravitational information. This is given by the convolution of the directional response of the detector,
the antenna pattern F (Ω̃), and the gravitational signal [25]:h(t) =X Z +∞
A=+,×

−∞

Z
df

dΩ̃ hA (f, Ω̃) e2πif (t−Ω̃·x/c) FA (Ω̃)

S2The antenna pattern is clearly dierent for the two polarizations and depends
on the spatial orientation of the detector [6].

1.4. DATA ANALYSIS TECHNIQUES

13

WE

WI

PR

BS
NE

NI

Figure 1.4: Simplied scheme of a power-recycled, Fabry-Perot interferometer.

The goal of any data analysis technique is to detect the presence of a gravitational wave signal in the detector output, discriminating it from the detector
noise. The expected wave-forms depend strongly on the source, and so do thetechniques used for the detection.

1.4.1 Coalescing binaries
The expected wave-form (chirp ) can be predicted with good accuracy from
Post-Newtonian computations:h(t) = A ν(t)2/3 cos(φ(t) + φ0 )where A is an amplitude depending on the source distance and orientation withrespect to the detector. The time evolution of the frequency can be written, in
the Newtonian approximation, as

8/3

ν(t) = ν0 (1 − t/τ )ν0 is the frequency at the time when the observation starts (t = 0)and τ is the remaining time until coalescence. The phase φ(t) must insteadwhere

be computed in the Post-Newtonian approximation. The typical wave-form isshown in g. 1.5.

14

CHAPTER 1. GRAVITATIONAL WAVES

0.8
0.6
0.4

h [x 10-19]

0.2
0
-0.2
-0.4
-0.6
-0.8
8

8.2

8.4

8.6

8.8
Time [s]

9

9.2

9.4

9.6

Figure 1.5: Example of expected wave-form from a coalescing binary system(chirp).

Being the expected signal known a priori, and assuming the detector noise
to be Gaussian, it is possible to use a matched lter technique to obtain an
optimal detection algorithm.

If the Fourier transform of the expected signal(the template ) and of the detector output are denoted respectively by g̃(f ) ands̃(f ), an inner product can be dened asZ +∞s̃(−f )g̃(f )hs|gi =df
Sn (f )−∞
where Sn is the detector noise power spectrum.

If the expected signal is notpresent, this gives a zero-mean random variable with variance depending on thetemplate normalization.

Therefore, the following denition of signal-to-noise

ratio is directly proportional to the signal amplitude

hg|si
SN R = phg|gi
For a given choice of the two body masses the expected wave-form is known andits amplitude depends linearly on the inverse of the source distance. Therefore
the previous equation can be inverted to yield the maximum distance of a detectable coalescing event given the detector sensitivity. This gives a useful gureof merit of the detector sensitivity: the maximum distance at which a 1.4 solar
masses NS-NS coalescence is detectable with SNR equal to 8 is usually quoted
as the range or horizon of the detector. The optimal-oriented horizon refers to
the situation of a source with a sky position and orientation such to maximize
the detector eciency. When an average over all sky position and orientationis performed the averaged horizon is obtained.

1.4. DATA ANALYSIS TECHNIQUES

15

During Virgo normal operation the averaged horizon DN S−N S is computed
on-line using a wave-form model based on the stationary phase approximation[26]. This yields the following equation:s
DN S−N S
Mpc= 10−20

Z 2300 Hz
40 Hz

f −7/3df
Sh (f )

where Sh (f ) is the one-sided power spectral density of the detector noise expressed in units of gravitational strain.
The eciency of the matched lter detection algorithm depends on thematching between the template and the real signal parameters.

Since there

is no way to know a priori which are the parameters of the incoming signal,
the search algorithm must be able to detect with good eciency any possible
source.

This is obtained computing and using a suitable template bank: theentire parameters space is covered in tiles and a template is produced for eachof these. The number of tiles is a compromise between computational time anddetection eciency: increasing the number of templates will increase both.

1.4.2 Stochastic background
Being the contribution of a stochastic background indistinguishable from thedetector noise, the only way to detect it is to look for correlations between
at least two dierent detectors.

Indeed the output of each detector can beviewed as the sum of its own instrumental noise plus a stochastic signal fromthe background:si (t) = ni (t) + hi (t)The dierent ni (t) are uncorrelated, while the hi (t) are. They depend on thedetectors position and orientation through the antenna patterns:

hi (t) =XZdΩ̃ FiA (Ω̃)

Z +∞

“
”
x ·Ω̃
2πif t− ic

df eh̃A (f, Ω̃)−∞Awhere h̃A (f, Ω̃) describes the stochastic background distribution in frequency
and angle.
The detection technique is based on the computation of the correlation between the output of two (or more) dierent detectors:

Cij (t, t0 ) = E [si (t)sj (t0 )] = E [hi (t)hj (t0 )]and the second equality follows from the fact that the detector noises are uncorrelated. If the spectrum of the stochastic background is assumed to be broadband, or in other words without narrow spectral features, the correlation canbe written as [21]:

0

Z +∞

Cij (t − t ) =−∞0

df e2πif (t−t ) γ(rij , f ) H(f )16

CHAPTER 1. GRAVITATIONAL WAVES
where γ represent the overlap function of the two detector, which depends ontheir relative orientations and distanceγ(rij ), f ) =

5 X
8π

Z

dΩ̃ e2πifΩ̃·rijc

FiA (Ω̃) FjA (Ω̃)

A

and H(f ) contains the information on the spectral properties of the stochastic
background, here for simplicity assumed isotropic:

H(f ) =

3H02
−3
|f | Ωgw (|f |)16π 2

1.4.3 Continuous signals
In the case of continuous periodical signals, the expected wave-form is wellknown. For example the gravitational wave emitted by a spinning neutron staris given by [27]:
1
1 + cos2 i
h+ (t) = h0 sin α
cos α sin i cos i cos ωτ (t) − sin α
cos2ωτ (t)
2
2


1
cos α sin i sin ωτ (t) − 7 sin α cos i sin 2ωτ (t)h× (t) = h0 sin α
2neglecting the neutron star spin down and Doppler eect related to possible
orbital motion. However it is necessary to take into account the detector andsource relative motion, dominated mainly by the earth rotation and revolution:

τ (t) = t +n̂ · x(t)cwhere x is the receiver position with respect to a xed origin.
If the detector and source were relatively xed, the signal would be monochromatic and the detection would reduce to the analysis of a sine wave in presenceof Gaussian noise. This can be easily carried out by computing the power spectrum of the detector output, integrating for a very long period. The obtained
signal-to-noise ratio would be√TA
SN R = p
2 Snoise (f )where Snoise is the detector noise power spectrum and A the signal amplitude.
However in the real case the Doppler eect due to the earth motion modulates the signal. The most important contribution comes from the change inphase of the received signaln̂ · v(t)
dφ(t)
=ω 1+
dt
c
which, even if small, produces a large eect over long integration periods.

1.4. DATA ANALYSIS TECHNIQUES

17This makes necessary to compensate the Doppler eect to have an ecient
detection. Like in the case of coalescing binaries, if the position in the sky ofthe source is known, the expected wave-form can be computed and a matchedlter can be used. This is equivalent to computing the power spectrum using
non-equispaced samples, considering equal time interval for τ (t) but not for t.
The general method consist in computing the Lomb periodogram [28]. Anotherproposed approach is to directly resample the detector output signal to compensate the Doppler eect and to proceed afterward with a monochromatic signal
search [29]. In both cases the detection algorithm depends on the source position and orientation with respect to the detector. A template search similar tothat used for coalescing binaries must be set up, covering the entire parameter
space with a suitable set of tiles.

1.4.4 Bursts
The characteristics of burst signals are almost completely unknown, and therefore it is not possible to implement matched ltering techniques. A detection
algorithm for bursts must be able to detect short transient signals over a background of stationary Gaussian noise. It usually consist in selecting a suitable listof candidate events based of thresholds on suitable signal statistics. A simple
method, implemented in the HACR algorithm described in chapter 9, computea time-frequency map of the detector output and select triggers based on theexcess of power in each bin with respect to the mean power spectrum statistics.
More sophisticated algorithms used banks of band-pass lters to reconstruct thepower in selected frequency bins (power lters ), or the correlation with a set of
templates of sine-Gaussian signals. More recent algorithm uses a decompositionof the detector output by means of wavelet transforms [30].
In all cases, the expected signal from a gravitational wave burst can bevery similar to transient noises coming from the detector, triggered for example
by environmental events. For this reason a large part of the burst analysis is
devoted to the identication of the time intervals and frequency bands withgood data quality (data conditioning ). Even after this the number of candidate
events can be very large, due to the non-stationary nature of the detector noise.
A large part of these candidates must be discarded by using suitable vetoes :for example it is possible to discard a candidate which is coincident in timewith a noise transient in some channel which cannot contain any gravitational
wave information.

Such channels can come for example from environmental

sensors, like seismometers, microphones or magnetometers. Since the numberof expected gravitational events is very small, such vetoes must have a very low
false alarm rate.
Finally, the best way to exclude candidates coming from noise events is toperform a coincidence analysis between dierent detectors.

Indeed, assuming

18

CHAPTER 1. GRAVITATIONAL WAVES
the noises of the two detectors to be uncorrelated, any coincident candidate is
likely to come from a real burst event.

Chapter 2
The Virgo detector
This chapter describes the Virgo interferometer for gravitational wave detection,in all the main aspects needed for the rest of this thesis.

2.1 Optical lay-out
The Virgo gravitational wave detector is a power recycled interferometer witharms replaced by 3 km long resonant Fabry-Perot cavities. Its schematic opticallay-out is shown in g. 2.1.

The two arm cavities are called north and west.

The two mirrors that compose each cavity are called (north or west) input andend mirror, while the rst mirror encountered by the input beam is the power
recycling mirror. It is customary to abbreviate the name of the mirrors: NI and
NE for north cavity input and end mirrors, WI and WE for the west cavity, BS
for the beam splitter and PR for the power recycling.

The distance between

the beam splitter and the power recycling mirror is approximately 6 m, while
the distance between the beam splitter and the two input mirrors is on average
6 m, with a Schnupp asymmetry of lN − lW = 0.88 m.
The main beams coming out from the interferometer are named as follows:

B1 is the dark fringe beam, the one reected eastward from the beam splitter,
resulting from the recombination of the two beams coming from the arm
cavities.

B2 is the beam reected back by the power recycled mirror.
B5 is the beam coming from the secondary anti-reection-coated surface of thebeam splitter mirror.

This beam therefore sees a small fraction of thebeam reected back from the north cavity, and is used as an indicator ofthe eld inside the recycling cavity.

20

CHAPTER 2. THE VIRGO DETECTOR

Figure 2.1: Schematic optical lay-out of the Virgo detector. Please refer to thetext for more details.

2.2. INJECTION SYSTEM

21
To ITFRFC CAM

RFC quadrants

LASER BENCH

Masterlaser

B2
From IMC
reflectio

From RFC
reflectio

BMS mirror
downstream
BMS
quadrants

Slave laser

IMC quad.
near fiel

B2 photodiodes
IMC quad.
far fielElectro−opticalmodulators

IMC reflection
photodiodeB2 CAM

BMS mirror
upstream

IMC REFB2 quadrants

EXTERNAL INJECTION
BENCH

Figure 2.2: Simplied scheme of the two external optical benches in the laser
laboratory.

B7 is the beam transmitted through the north arm cavity, thus seen past thenorth end mirror.

B8 like the previous one but for the west arm cavity.

2.2 Injection system
The input of the Virgo detector is a 20 W laser beam with a wave-length of1064 nm, generated by a Nd:YVO4 high power slave laser, injection-locked toa high stability solid state Nd:YAG master laser. In this way the higher power
and frequency stability of the master laser is transferred to the slave. The twolasers are hosted in the laser bench, one of two optical benches located inside
the laser laboratory (see g. 2.2). The power produced by the slave laser can becontrolled acting on the current of two pumping diodes. The beam generated by
the slave laser passes through two electro-optical modulators, used to generate
three dierent modulation frequencies at about 6 MHz, 8 MHz and 22 MHz.
The pointing of the beam is controlled at the level of the two laser laboratory
benches by the beam monitoring system (BMS). The beam can be translated
and steered by two mirrors mounted on piezo-electric actuators. The rst mirror(upstream) is placed on the laser bench, while the second mirror (downstream)
is mounted on the external injection bench.

The error signals for the beam

pointing control are obtained from two quadrant-split photo-diodes placed on
the external bench. These are photo-detector with sensitive area divided in four
sectors, similar to those used for the angular sensing and control (see chap. 4).
The dierence between opposite sectors are used to reconstruct signals sensitive
to the spot position. After the BMS system the beam is sent into the vacuum

22

CHAPTER 2. THE VIRGO DETECTOR
to the suspended injection bench.
The external injection bench is also used to collect several beams comingback from the suspended bench: the reection of the input mode cleaner and of
the reference cavity, as well as the beam reected by the interferometer (B2).
These beams are sensed by normal and quadrant photo-detectors and by digital
cameras. The rst kind of detectors are used mainly for the longitudinal sensing
and control system and for the longitudinal control of the injection system.
The quadrants are instead used to control the interferometer angular degrees
of freedom (see chap. 4).

The cameras nally are used mainly for diagnostic

purposes. The laser and external injection benches are contained inside acoustic
isolation enclosures.
Before entering the main interferometer, the beam passes through the input

mode cleaner (IMC), a triangular cavity 144 m long with a nesse of about 1000.
The role of this cavity is to lter the laser beam suppressing all transverse modedierent from the fundamental Gaussian one (see also chap. 4). The IMC cavity
is maintained on resonance for the carrier frequency using a Pound-Drever-Halltechnique based on a 22 MHz demodulation of the IMC reection. The 22 MHz
sidebands are indeed not resonant inside the input mode cleaner, while thecarrier, the 6 MHz and the 8 MHz are.

The length control of the IMC isstrongly linked to the frequency stabilization of the laser. The topology of therst stage of frequency stabilization is explained in deeper details in chap. 3.
The input mode cleaner cavity is made of a highly reecting terminal mirror and of the two input mirrors that composes the dihedron, mounted on thesuspended injection bench (see g. 2.3).

The beam reected by the cavity issent back to the external bench. The transmitted beam passes instead througha Faraday isolator : the beam is completely transmitted from the bench to theinterferometer, while in the other direction it is attenuated by a factor of about
1000. This avoids the light reected by the power recycling mirror to enter the
IMC and interfere with the main one. The reection of the Faraday isolator isused to reconstruct the B2 beam.
The beam transmitted by the Faraday isolator is sent afterward to a telescope
composed of two parabolic mirrors, used to match the beam to the resonantmode of the interferometer and to control its position and pointing. Moreover,a small part of the IMC transmission is sent through a periscope to the lower
part of the bench to the reference cavity RFC, a rigid cavity 30 cm long. Thisis used as part of the laser frequency pre-stabilization system and of the fullinterferometer longitudinal control (see chap. 3).
Part of the beam transmitted by the IMC is sent to an in-vacuum photodetector on the suspended bench, used as error signal for the laser power stabilization. This control system is split in two parts. The high frequency pathuses only the signal above some Hz as error signal for a loop that acts on the2.3. MIRRORS

23

Figure 2.3: Simplied scheme of the suspended injection bench.

slave laser pumping current to stabilize the power transmitted by the IMC witha bandwidth of about 60 kHz. This leaves the mean value of the IMC transmission free to vary.

Another slow servo loop allows its stabilization with abandwidth of about 10 mHz and an adjustable setting point.

2.3 Mirrors
The mirrors used in Virgo are made of high quality fused silica, with a diameterof 35 cm and a mass of about 21 kg for power recycling and cavities mirrors.
The beam splitter is slightly smaller, with a diameter of 23 cm and a mass of

5 kg.

All mirrors are at, except the two end ones which are concave.

Themain parameters for the six main Virgo mirrors are listed in tab. 2.1, 2.2 and2.3: the radius of curvature ROC of the reecting face, the reectivity R andtransmissivity T of the two faces, the losses on the high-reectivity HR andanti-reection coated AR surfaces. In this conguration the nesse of the two
arm Fabry-Perot resonant cavities is about 50 and the recycling gain for carrierand sidebands respectively 40 and 30.

24

CHAPTER 2. THE VIRGO DETECTOR

Figure 2.4: Mechanical scheme of the super-attenuator.

2.4. SUSPENSION AND MIRROR CONTROL
R 1

Mirror
NI
WI

st

nd

T 2

face

132 ± 2 ppm
171 ± 0.6 ppm

face

11.8 ± 0.03%
11.66 ± 0.02%

25

Losses HR

Losses AR

9.25 ppm

0.9 ppm

15.38 ppm

1.7 ppm

Table 2.1: List of input mirrors parameters.

Mirror

ROC

NE

3580 ± 17 m
3601 ± 15 m

WE

nd

T 2

face

Losses HR

42.9 ± 0.02 ppm
38.3 ± 0.7 ppm

4.67 ppm
9.19 ppm

Table 2.2: List of end mirrors parameters.

2.4 Suspension and mirror control
One key feature of the Virgo interferometer is the use of high-performance passive seismic isolation systems, called super-attenuators (see g. 2.4), which exploit the attenuation properties of simple pendulum. Indeed, assuming viscousdamping, any horizontal motion of the suspension point is transmitted to thesuspended mass with an attenuation given by

zM ASSf02=q
2
f 2f 2zT OP[f 2 − f 2 ] + 0
0

(2.1)

Q2

where f0 is the resonant frequency of the pendulum and Q its quality factor.
In particular for frequencies above the resonant one, any motion of the top

2

suspension point is suppressed by a factor (f /f0 ) .
The mirrors are suspended to a chain of seven pendulums, with all longitudinal resonant frequencies below 2.5 Hz [31]. Therefore the seismic isolation
obtained at 10 Hz is of the order of 10

14with respect to the ground motion[32]. Vertical attenuation is obtained replacing the upper ve pendulum terminal masses with seismic lters (see g. 2.5). Two consecutive pendulum wires
are connected by a system of elastic metallic blades. The resonant frequencyof this system would however be too high, given the necessary stiness of themetal to support the weight of the entire suspension. They are therefore combined with two sets of magnets in repelling conguration. They form a magnetic
anti-spring [33] that combined with the blades gives a low resonant frequency,
comparable to that of the horizontal modes.

The overall seismic attenuationof the super-attenuator chain is shown in g. 2.7 for both the horizontal andvertical degrees of freedom. The vertical transfer function is rescaled by a factor
100 to take into account that vertical motions couples much less than horizontal
one with the interferometric signals.
The top stage of the entire suspension chain is another vertical lter (Filter

26

CHAPTER 2. THE VIRGO DETECTOR

to the previous
stage

crossbarm agnets

m agnets

centeringw irescenteringw ires
bell
ow

bell
ow

m ovingelem ent

blade

centeringw irescentralcolum n blade

hinge

to the nextstage

BottomTop

Figure 2.5: Schemes of a vertical isolation lter. The top panel is a schematic
drawing, the bottom ones show the full lter viewed from the bottom and fromthe top.

2.4. SUSPENSION AND MIRROR CONTROL

Mirror
BS
PR

st

R 1

face

50.25 ± 0.18 %1.8 ± 0.2 ppm

R 2

nd

face

nd

T 2

27

face

519 ± 10 ppm

Losses
6.85 ppm (HR)5.13 ± 0.02 %25.54 ppm (HR)
2.5 ppm (AR)

Table 2.3: List of PR and BS mirrors parameters.

0 ), sustained by three legs that form the inverted pendulum [34]. The elasticityof the leg joints, combined with the opposite eect of the gravitational force,
gives a system with a resonance at about 40 mHz. This is used as pre-isolation
stage to reduce the seismic motion of the top suspension point.
The two lower stages of the suspension are quite dierent from the standard lters. Indeed, the mirror is surrounded by a metallic reference mass (seeg. 2.6) which is used as a reaction mass to apply forces to the mirror using
four coil-magnets pairs on the back side of the mirror. Both the mirror and thereference mass are separately suspended to the marionette by four wires each.
In this way it is possible to act on the mirror also steering the marionette. To
this purpose there are four magnets attached to the marionette itself and four
coils attached to pillars coming down from the last vertical isolation lter (Filter

7 ).

2.4.1 Suspension control
The active control of each super-attenuator is split in four parts. The inverted
pendulum is controlled at the level of the top stage of the suspension, the lter
0, by the inertial damping system [35]. The top platform, connected to the inverted pendulum, hosts three accelerometers and three linear voltage dierential

transformers (LVDT) used as absolute position sensor (see g. 2.8). All signals
are acquired by analog-to-digital converters (ADC) and processed by custom
digital-signal-processors (DSP). These signals are combined together, using the
LVDT at frequencies below 50-100 mHz and the accelerometers at higher frequencies, and used to control the top suspension point horizontal motion andthe lter 0 rotation around the vertical axis. The cross-over frequency can bechanged maintaining the control system active, to cope with dierent microseismic conditions.

The correction signals for each suspension are computed

separately by one DSP system, completely devoted to the inertial damping of
the corresponding tower. The computed correction is then applied to the top
stage by coil-magnet actuators.
The length of each suspension stage is also measured by individual LVDT
sensors, summed together and used to control the vertical position of the payload.

28

CHAPTER 2. THE VIRGO DETECTOR

Figure 2.6: Scheme of the bottom part of the super-attenuator, including the
marionette and the payload, composed of a mirror and its reference mass.

2.4. SUSPENSION AND MIRROR CONTROL

29

Figure 2.7: Computed transfer function from seismic motion of the ground to
mirror displacement, for both the vertical and horizontal degrees of freedom.
The vertical transfer function is rescaled by a factor

10−2 , assuming a esti-

mated coupling of vertical to horizontal motion of one hundredth. A lower limit
measurement of this transfer function is described in [32].

The mirror angular and longitudinal (orthogonal to the mirror surface) motions can be controlled both acting at the mirror or at the marionette level (local

control system [36] ). As already anticipated four magnets are glued to the backface of each mirror, in correspondence to four coils attached to the reference
mass.

By controlling the current owing through these coils it is possible toapply longitudinal and torsional forces to the mirror, using another suspended
point as reference. In this way the control is free from any contamination withthe ground micro-seismic motion. Other four magnets are attached to the marionette, in correspondence to coils that are suspended through pillars coming
down from the lter 7. Any current applied to these coils results in a force on
the marionette, which is then transmitted by the last pendulum stage to themirror and the reference mass.
The error signals for the local control systems are obtained using position

sensing devices (PSD): these are optical levers obtained using low power red
laser diodes and quadrant-split photo-detectors (see g. 2.9).

These opticallevers are used to measure the angular and longitudinal position of the mirror
and also the angular one of the marionette, using a small mirror attached to it.
When the motion is too large for the dynamical range of the PSD sensors, the30

CHAPTER 2. THE VIRGO DETECTOR

Accelerometers
Inverted pendulums
Inverted pendulums

Top ring solidal to
the ground
Coil − magnet
Position sensor (LVDT)

Figure 2.8: Scheme of the top stage of the super-attenuator, hosting the sensors
and actuators used for the inertial damping system.

For simplicity only one

LVDT is shown.

Marionette
1.5 mW
RED LASER

Diffusiv
markers

ra

CCD

e
cam

mW
1.5 SERLA
RED

Position Sensing
Device on focal plane

r

inato

Illum

Mirror

Reference
Mass

Position Sensing
Device on image plane

Position Sensing
Device on focal plane

Figure 2.9: Scheme of the local control sensing system, including the marionette
and mirror optical levers.

2.5. DETECTION SYSTEM

31

rough position is reconstructed using a camera which looks at diusive markers
attached to the mirror itself.
All local control signals for each single mirror are acquired by ADC boards
and processed by a dedicated DSP. This system also receives global interferometric signals for the longitudinal and angular sensing and control, transmitted
to the central and terminal buildings by digital optical links.
The current owing into each coil is controlled by custom coil drivers. Indeed, the digital correction computed by the DSPs are converted by digital-

to-analog converters (DAC) to a voltage signal which is suitably amplied andtranslated into a current owing in the coil. These coil drivers have two dierent
control sections: the high power is capable of applying stronger forces to themirror, at the price of a higher noise level, while the low-noise one has a lower
peak-to-peak range and better noise performances.

The rst section is usedduring the lock acquisition procedure (see sec. B) when the free swinging mirror
must be stopped and strong forces are needed. The second one is engaged only
close to the nal detector state.

In order to reduce as much as possible therequired dynamics of the actuators, part of the control force is re-allocated fromthe mirror-reference mass actuators to higher stages: the very low frequency
component is sent to the inertial damping control system through serial links
between the dierent DSPs; the intermediate frequency component is actuatedon the marionette and only the high frequency one is sent to the reference masscoils.

Short suspensions
The two detection and injection suspended optical benches, as well as themode cleaner terminal mirror, are suspended to a short version of the superattenuators. This consists in a shorter inverted pendulum and a suspension with
only two intermediate vertical lters. The isolation performances of this system
are matched to the IMC and suspended benches requirements and clearly lower
than those of the full super-attenuators.

2.5 Detection system
All the relevant beams coming out of the interferometer are sensed using photo-detectors that convert the laser power impinging on them in a electric signal.
The typical power hitting a photo-detector can range from some µW to 100 mW.
The output signal is rst pre-amplied in order to enhance the carrier and therst modulation harmonic with respect to the higher ones. This signals is thensplit in three parts. One is low-passed and used to give a signal measuring thetotal power that hits the detector. The other two are sent to a demodulationelectronic. Here the two signals are mixed with two reference sinusoidal waves

32

CHAPTER 2. THE VIRGO DETECTOR

B1s
B1
Output Mode
Cleaner
B1p

Prism

From theITF

B5

Telescope alignment
quadrants

Figure 2.10: Simplied scheme of the suspended detection bench.

at the modulation frequency, dephased by 90 degrees. The results of the two
mixing are then low-passed to obtain the demodulated in-phase and quadrature
signals. A similar electronics works also for the quadrant-split photo-detectors
used to generate angular error signals.

They generate one power output foreach single sector and demodulated signals for the dierence between opposite
ones.

There are only two exceptions to this scheme:on the B2 beam there

is a photo-detector which is demodulated at three times the main modulation
frequency and a quadrant detector demodulated at 8 MHz.
The secondary face of the beam splitter has a small wedge with respect tothe main one. In this way the beam splitter secondary face pick-o beam B5
and the main dark fringe beam B1 are collimated together toward the suspended

detection bench (see g. 2.10). Here the two beams are focused using a single
telescope.

Its mirrors are mounted on translation stages and can be moved

remotely to optimize the telescope alignment. After the telescope the two beams
are separated by a prism. A small fraction of the B5 beam is sent to in-vacuum
quadrant-split detectors to obtain error signals for the telescope alignment. The
main part of the B5 beam is sent out of the vacuum to the external detection

bench, while the B1 beam is split in two parts: a small fraction of the total
power (about 1%) is sent again to the external bench to the B1p detectors.
The main part of the dark fringe beam is instead sent to a rigid triangularcavity, the output mode cleaner. This cavity can be maintained on resonance bycontrolling its temperature. In this way only the fundamental Gaussian modeis transmitted to the B1 beam and to the external bench, while all other higher
order modes are reected to the B1s beam. The detector contrast is therefore

2.5. DETECTION SYSTEM

33

Figure 2.11: Example of the scanning Fabry-Perot signals. The red curve is theramp applied to the cavity length, while the blue one is the power detected bya photo-diode placed in transmission of the cavity.

enhanced, since all spurious modes generated by mis-matching or mis-alignment
are rejected.
The external detection bench is placed inside an acoustic isolation in thedetection laboratory and hosts all the photo-detectors and quadrant detectors
used to sense the B1p, B1 and B5 beam, as well as several digital cameras used
to monitor the beams shape and position.

Part of the B1p beam is sent toa scanning Fabry-Perot cavity (SFP) [37].

This is a 25 cm long Fabry-Perot

resonant cavity with one of two mirror mounted on a piezo-actuator.

If thelength of the cavity is scanned over one free spectral ranges, the frequency ofthe laser light that resonate changes accordingly. This cavity can therefore beused as an optical spectrum analyzer to separate the power in the carrier and in
the two sidebands. A typical example of the output signal is shown in g. 2.11.
The length of the cavity is scanned at about 10 Hz and the output signal isanalyzed on-line to reconstruct the power in the carrier and sidebands elds.
These values are computed every second and added to the main data stream.
As already explained the B2 beam is sensed by photo-detectors placed on
the external injection bench. The beams transmitted by the two arm cavities
are also used. They are quite large (of the order of 5 cm in diameter) and theirsize is reduced using suitable telescopes, the rst lens being a large doublet.
As well as in the detection bench, the beam is split between cameras, photo-

34

CHAPTER 2. THE VIRGO DETECTOR
detectors and quadrant-split photo-detectors. The two end optical benches arealso surrounded by acoustic enclosures.
All the photo-detectors output are converted to digital signals by ADC
boards and post-processed by the photo-diode read-out processes. These takecare of digitally ltering the signals (for example to compensate for anti-aliasing),of calibrating them in suitable units and to distribute all the results to the dataacquisition system.

2.6 Data distribution and storage
All the digital signals that come from photo-detectors read-out, DSP, environmental probes and so on are collected by the data acquisition system (DAQ)
and redistributed to all the processes that need them. Some of the most criticalsignals are sent independently via optical or serial links.
All the digital signals are distributed on-line with a small latency of a coupleof seconds in frame format [38]. Additional programs that run on local machines
can access through the network any data and process it to produce new signals
and channels that are also available in the main data stream. Example are thenoise monitoring tools like NonStatMoni and VirgoHACR described in chapters
8 and 9 respectively.
The frames are also saved to disk storage farms for o-line analysis andreading.

The actual buer allows a simultaneous storage of all the detector

signals for several months.

2.7 Global control system
The longitudinal and angular control of an interferometer needs to use opticalsignals coming from photo-detectors to compute the force to be applied to themirrors to maintain the correct resonance conditions. The system that take careof collecting suitable error signals, performing all the needed computation andoutput the correction signal is called the global control. It is a piece of software
running on a dedicated real-time Unix machine, directly interfaced with thephoto-diode read-out system and with digital optical links used to send thecomputed correction to the suspensions DSP control systems.

2.8 Environmental monitoring
The environmental conditions inside the laboratories, the buildings and outside
are monitored by several temperature and humidity probes. In particular there
are temperature probes inside each optical bench and inside the external isola-2.9. THE VIRGO DESIGN SENSITIVITY

35

tion oven that surround the suspension towers. Also the external temperature
and wind speed are monitored by a small weather station.
Moreover, acoustic and seismic noise is monitored by microphones, accelerometers and episensors placed in dierent positions: there are sensors attached toeach optical bench, inside the acoustic isolation enclosures, as well as seismometers attached to the ground in each building.

Microphones are also mounted

inside the laboratories and buildings to monitor the environmental acoustic
noise.
Finally several magnetic probes are mounted inside the experimental buildings, to monitor the level of electro-magnetic noise.
All these probes, together with a set of portable ones, have been widelyused to characterize the contamination by environmental noise of the detector
sensitivity (see also sec. 7.11).

2.9 The Virgo design sensitivity
The best possible sensitivity that can be attained by the Virgo detector in thepresent conguration is called the design sensitivity. There are several fundamental sources of noise that limit it, depending on the frequency of interest [39](see g. 2.12).

Seismic noise
The micro-seismic motion of the ground can be approximated by a simple model
given bySx (f ) ≈A0
f2

where the coecient A0 can be estimated to be of the order of 10

(2.2)

−7

−1/2

m Hz

.

Using this estimate and the super-attenuator transfer function, the residual
seismic noise at the level of the mirror can be computed. This is relevant onlyat frequencies below roughly 3 Hz and negligible at higher frequencies, since thesuspension attenuation goes like f

14.

Newtonian noiseThis noise is given by the uctuation of the static gravitational eld, modulated
mainly by ground-air interface motion due to seismic activity. The contribution
to the strain sensitivity can be estimated by [40]:

ShN N (f ) =3 · 10−11x̃seism (f )
f2

(2.3)

This formula is however strongly model dependent [41]. Therefore the contribution of Newtonian noise is shown in g. 2.12, but it is not taken into account

36

CHAPTER 2. THE VIRGO DETECTOR
in the total design noise. The contribution as computed by this model is in any
case negligible inside the observational bandwidth (10-10000 Hz).

Thermal noise
The thermal noise that contributes to the design sensitivity comes from energy
loss processes in the suspension wires and in the mirrors [42, 43].

The maincontributions come from the pendulum thermal noise at frequencies between 3
and 30 Hz. The intermediate region between 30 and 300 Hz is instead limitedby the sum of the contributions coming from the thermal excitation of thesuspension wire violin modes and of the mirror resonant mode.

Shot noise
The shot noise is coming from the intrinsic uctuation of the power detectedby the dark fringe photo-detector [44]. Its contribution to the strain sensitivity,
including the eect of the light modulation [45] is given byrhshot (f ) =

3 1
2 8LF

s

λc
2hP
ηCPin

s



1+

f
fF P

2
(2.4)

where C = 50 is the recycling factor, Pin = 10 W is the input power, F = 50 isthe cavity nesse and fF P their pole:fF P =c∼ 500 Hz
4LF

(2.5)

The shot noise starts to be relevant for frequencies above roughly 300 Hz and
it is the limiting source of noise above 1 kHz.

Total
In conclusion the expected design sensitivity, assuming an input laser power of
10 W, ish

≈ 3 · 10−21 Hz−1/2 at 10 Hzh≈ 7 · 10−23 Hz−1/2 at 100 Hz

2.9. THE VIRGO DESIGN SENSITIVITY

Figure 2.12: Design sensitivity for the Virgo detector, with all the main fundamental sources of noise.

37

Part I
Control systems

Chapter 3
Longitudinal sensing andcontrolThis chapter describes in general the degrees of freedom involved and the control
scheme adopted for the length sensing and control system.

The lock acquisi-tion technique used in Virgo during the science run is described in details inAppendix B.

3.1 General control scheme
In a power-recycled Fabry-Perot interferometer, the length sensing and control
system (called in Virgo locking system ) has the role of reconstructing and controlling the position of six mirrors, in order to maintain the correct resonanceconditions of light elds inside the system:• the carrier eld must be resonant inside the long arm Fabry-Perot cavities,
to enhance as much as possible the optical response to a gravitationalsignal;• the carrier eld must also be resonant in the central part of the interferometer, namely in the compound cavity composed of the power recycling andthe two input mirrors, to prot from the increase of internally circulating
power;• the carrier eld must nally undergo destructive interference at the darkport, to have the most sensitive conguration (dark fringe );• the two sideband elds must be completely reected (anti-resonance condition) from the long arm cavities, to be used as reference for the carrierdephasing;42

CHAPTER 3. LONGITUDINAL SENSING AND CONTROL

Figure 3.1: Denition of lengths in a power-recycled Fabry-Perot interferometer.

• the two sideband elds must also be resonant in the central cavity, toprot again from the increase of circulating power.

In general they are

not interfering destructively at the dark port.
The motion of the main six mirrors (see g. 3.1) in the direction parallelto the beam (longitudinal direction ) are relevant to the above stated resonanceconditions. Only four of these degrees of freedom are in reality important. One
combination of the mirror motions indeed corresponds to a global translationof the entire interferometer and it is clearly irrelevant, since only the relative
distance between mirrors is important. The resonance conditions of the two longarm cavities are independent from the rest of the interferometer and therefore
their lengths can be considered as two of the degrees of freedom to be controlled.
Usually these two lengths are combined together in the common and dierential

modes :

LCARM

=

LDARM

=

LN E + LW E
2
LN E − LW E
2

(3.1)
(3.2)

The lengths of the central part of the interferometer can be combined to form
the following two degrees of freedom:

LP RCLLM ICHLN I + LW I
2= LN I − LW I
= LP R +

(3.3)
(3.4)

The rst gives the length of the power recycling compound cavity, composed of
the PR mirror and the NI and WI ones. The second is instead the dierential3.1. GENERAL CONTROL SCHEME

43

degree of freedom of the central Michelson interferometer. The remaining degreeof freedom can be expressed as

L5 = LP R −LN I + LW I
2

(3.5)

and it has no eect on the dephasing of the light elds inside the central part.
In conclusion four degrees of freedom need to be controlled by the lengths
sensing and control system:• common mode motion of the long cavities (CARM)• dierential mode motion of the long cavities (DARM)• dierential motion of the central Michelson interferometer (MICH)• length change of the power recycling cavity (PRCL)
There is still another degree of freedom which must be taken into account:a change in the laser frequency is equivalent to a global change of scale. Indeed
the dephasing of the light eld when travelling over a distance L is given by

φ = kL = 2πνL/c and therefore the following equivalence equation holds:
δL
δν
=
ν
L

(3.6)

If the propagation happens inside a Fabry-Perot cavity, the dephasing amplication due to the nesse must be considered and the eective length L is larger
than the physical one by the cavity nesse (more details in app. A.4).
Eq. 3.6 implies that also the laser frequency must be controlled, otherwise
it will be seen by the interferometer as a change in lengths. The most sensitiveelements are of course the long Fabry-Perot cavities: to be able to acquire thelock it is necessary that the residual uctuations of the laser frequency aresmall, otherwise the dynamical range of the actuators might not be enough[46] to acquire and maintain the resonance condition. The spectral density of
frequency uctuations of a free running laser like the one employed in Virgo canbe approximated by [47]:δν ≈104 Hz
√
f
Hz

(3.7)

which for a frequency in the infra-red region

ν=

c
c
=
= 2.82 · 1014 Hz
λ1.064 · 10−6 m(3.8)

corresponds to an equivalent length noise for a Fabry-Perot cavity 3000 m long:

δLRM S ≈ 4 · 10−9 m(3.9)

which is of the same order of magnitude of the resonance width of the cavity
itself (see appendix A):

δLW IDT H =λ≈ 5 · 10−9 m
4F(3.10)

44

CHAPTER 3. LONGITUDINAL SENSING AND CONTROL

Figure 3.2:

Scheme of the injection system and the laser frequency pre-stabilization control architecture.

From these consideration it appears clear that in order to be able to acquire thelock of a Fabry-Perot cavity, a pre-stabilization of the laser frequency is needed.

3.2 Laser frequency pre-stabilization
The pre-stabilization is obtained with a Pound-Drever-Hall like technique. The
input beam is phase-modulated at 22 MHz.

This frequency is not resonant

inside the input mode cleaner (IMC) cavity and therefore it can be used forextracting a Pound-Drever-Hall signal in reection. Since at high frequency thecavity is a very good length reference, the PDH signal can be used for locking
the laser frequency to the cavity length. The bandwidth of the corresponding
control loop is about 300 kHz. In this way the laser frequency follows exactly
the lengths variations of the IMC, even at low frequency where the motion of
the suspended injection bench and of the mode cleaner end mirror can be quite
large. To correct this unwanted behavior, the reference cavity (RFC) is used:being a rigid cavity its length variations at low frequencies are small. Therefore
the PDH signal in reection of this cavity is used to lock the length of the IMC,
with a bandwidth of about 100 Hz.
In this conguration (usually called stand-alone injection system ) the frequency of the laser is locked to the reference cavity at low frequencies (below
100 Hz) and to the length of the input mode cleaner at higher frequencies (upto 300 kHz). This is enough to reduce its total RMS uctuation to few Hertz,

3.3. REQUIREMENTS

45

which makes it no more an issue to lock a single Fabry-Perot cavity.
This frequency stabilization is however not enough to reach the design sensitivity for the full interferometer [48] and therefore a second stage of frequency
stabilization (SSFS) is needed. The only possible better length reference are the
3 km long arm cavities.

3.3 Requirements
All lengths must be kept close enough to the correct operating point that realizes the already explained resonance conditions. It is important to understandthe requirements on the accuracy of the control loops, or in other words the
maximum allowed residual RMS motions. In order to keep the optical systemclose to the resonance, all length uctuations should be much smaller than theresonance width. Taking into account the width of the power recycling cavity
resonance and the enhancing eect of the Fabry-Perot cavities, the following
requirements are found [46]:δLP RCL< 2 · 10−10 m (RMS)δLM ICH< 5 · 10−10 m (RMS)δLCARM< 6 · 10−12 m (RMS)δLDARM< 3 · 10−11 m (RMS)

Since the dark fringe signal is the one carrying information about dierential
displacement of the end test masses and in conclusion about gravitational waves,an additional requirement is that the noise in this channel must be as small as
possible.

3.4 Signal extraction
The lock acquisition procedure is the sequence of operations which are needed tobring the interferometer from a completely uncontrolled state to the operating
point with the best possible sensitivity, which is the steady locked condition.
The techniques used for the lock acquisition diers strongly from detector todetector: the LIGO one is described for example in [49], the TAMA techniquein [50] and the GEO600 one in [51].
The error signals for the reconstruction and control of the lengths are obtained with the usual frontal modulation technique. The input beam is phasemodulated at a frequency Ω of about 6.2 MHz. This modulation frequency istransmitted through the input mode cleaner and enters the interferometer. The
output beams are demodulated at Ω to obtain several signals. The combinations

46

CHAPTER 3. LONGITUDINAL SENSING AND CONTROL

Figure 3.3: Simplied scheme of all beams and signals used in the length sensing
and control system.

used for the reconstruction of the error signals change during the lock acquisition procedure [46], since their response to the dierent lengths varies with theevolution of the interferometer state.
The following signals are used, at dierent steps and in dierent combinations (see g. 3.3):• the demodulated in-phase signals obtained from the beams transmitted
by the north and west cavities (their standard names are Pr_B7_ACp and
Pr_B8_ACp);• both in-phase and quadrature demodulated signals from the reectionof the anti-reection-coated face of the beam splitter (Pr_B5_ACp and
Pr_B5_ACq);• the in-phase signal at the dark port, both before and after the output
mode cleaner (Pr_B1p_ACp and Pr_B1_ACp);• the in-phase demodulated signal obtained from the reected beam
(Pr_B2_ACp);• the in-phase signal obtained from the reected beam, demodulated at 3Ω(Pr_B2_3f_ACp);

3.4. SIGNAL EXTRACTION47

The correct demodulation phase for each photo-detector is usually chosen inorder to maximize the contribution from one of the lengths and minimize allothers, and it is experimentally tuned for each step of the lock acquisition sequence.
Moreover, several power signals are used for normalization or for setting up
thresholds:• the powers transmitted by the two arms: Pr_B7_DC and Pr_B8_DC;• the power in the beam reected by the anti-reection coated surface ofthe beam splitter Pr_B5_DC. This is a pick-up of the power reected by
the north arm and if the power recycling mirror is aligned it can be used
as a good estimator of the power circulating inside the recycling cavity;• from the same beam also an estimation of the power in the sidebands canbe obtained, by demodulating at 2Ω. This signal is dominated by the beating of the two sidebands and if their spatial overlap is good it is roughly
equivalent to their mean power. This signal depends of course on its demodulation phase, which is chosen in order to maximize the quadraturecomponent Pr_B5_2f_ACq;
A full discussion of the optical response of each of these signals to the fourdierent longitudinal degrees of freedom can be found in [46] and [52]. In summary the response of each signal depends on which are the elds that beat togenerate it.
The Ω demodulated signal in the B2 beam can be viewed as generated by
the reection of the compound power recycling cavity. It comes from the beating of the carrier eld with the rst order sidebands. All these three elds areresonant inside the compound cavity, but with dierent recycling gains. Moreover the carrier is also resonant inside the long arm cavities and therefore it ismainly sensitive to the CARM degree of freedom.

If this d.o.f.

is controlledwith very high gain with the frequency stabilization servo, its contribution canbe neglected and the demodulated signals from B2 will be sensitive mainly tothe central cavity degrees of freedom MICH and PRCL, in the two dierent
quadratures. The signal obtained still from B2 but demodulated at 3Ω is generated instead mainly by the beating of the rst order with the second order
sidebands. The latter are not resonant inside the recycling cavity: this makes
the situation similar to the standard PDH conguration and therefore this signal will be sensitive mainly to change in the PRCL degree of freedom. In reality
the 3Ω signal has also a small contribution from the beating of the carrier withthe third order sidebands, which again makes a small contamination of CARM
in this signal.
The signals transmitted by the two arm cavities are instead clearly dominated by the variation of the two cavity lengths. If the mean length is controlled

48

CHAPTER 3. LONGITUDINAL SENSING AND CONTROL
with high gain by the frequency stabilization servo, both demodulated transmitted signals are sensitive to the DARM degree of freedom. The same is also
true for the reection of the cavities.

In particular the recombination of thetwo reected elds, that takes place at the dark port, gives the most sensitive
demodulated signal to DARM motion. This is therefore the main gravitational
wave channel.
Finally, the demodulated signals extracted from beam-splitter pick-o beamB5 are sensitive to dephasing of the carrier and sidebands elds inside the central
cavity. Again the carrier dephasing is dominated mainly by the CARM degreeof freedom. This makes one of the two B5 quadratures a very sensitive CARM
error signal. The sideband dephasing depends instead only on the state of thecentral area and in particular of the MICH degree of freedom.
The detailed response of each demodulated signal to every lengths can becomputed analytically or obtained by means of simulation, as it is done insec. 5.1.
Taking into account the full optical path from the interferometer to thephoto-diodes it is possible to convert the power actually impinging on the detector with the one inside the interferometer. For B7 and B8 beams the transmission of the end mirror must be taken into account ( 42.9 ± 0.2 ppm for NE
mirror, 38.3 ± 0.7 ppm for WE, see [53]) together with the properties of all mirrors and lenses on the external benches. The conversion factors from the power
impinging on the photo-diode and the one inside the two cavities are 5.2 · 10

4north cavity and 5.8 · 10

4

forfor west cavity. In a similar way the transmission to

the B5 beam must take into account the reectivity of the secondary face of the3

beam splitter mirror (519 ± 10 ppm [53]) and the conversion factor is 2.5 · 10 .

3.5 Actuation
The outcome of the longitudinal sensing and control system is a vector of corrections to be applied to the mirrors.

The signals expressed in terms of thephysical degrees of freedom (DARM, CARM, MICH, PRCL) must be converted
to the actual corrections to be sent to every single mirror.

This is a simple

geometrical computation. DARM and CARM are summed and subtracted togive the NE and WE corrections:

CN E= CCARM + CDARM

CW E= CCARM − CDARM

The corrections for MICH and PRCL are mixed together since a motion of the
BS mirror changes also the power recycling cavity length. A displacement of BS,
perpendicular to its surface in the PR and WI mirrors direction has the eectof making the PR-BS distance shorter, the BS-NI one longer and of leaving3.6. THE VARIABLE-FINESSE TECHNIQUE
unchanged the distance BS-WI. A displacement of the PR mirror instead onlyaects the distance between PR and BS. In summary:

∆LN I

√
∆P R − 2∆BS
√
=
2∆BS

∆LW I

=

∆LP R

=

0

By inverting these equations it is possible to nd the correct combination of
MICH and PRCL corrections to be applied to each mirror:

CBS
CP R

1
√ CM ICH
2
1= CP RCL + CM ICH
2=and therefore the PRCL correction is applied only on the PR mirror while the
MICH one is split between the two.
The actual coecients in the driving matrix are dierent from those reported
here since they must take into account also the dierent gain of coils-magnets
actuators for dierent mirrors.

3.6 The Variable-Finesse technique
The lock acquisition procedure used in Virgo is based on the variable-nesse

technique [46]: the nal state of the interferometer is reached passing throughseveral intermediate stable states.
If the power recycling mirror is removed (or misaligned in order to avoid its
reections interfering with other beams inside the interferometer) the two arm
cavities are completely independent and they can be controlled using the PDH
signals extracted from the transmitted beams. If the two cavities are locked inthis way, their reection is stable and the entire interferometer can be considered
as a simple Michelson with the two locked cavities in place of two simple mirrors.
Therefore the beam splitter can be controlled independently from the arms.
The second observation is that the full interferometer can be viewed as aresonant cavity composed by the power recycling mirror and an equivalent one
which models the total reectivity and transmissivity of the optical system composed by the beam splitter and the two arm cavities. If these are maintainedat resonance, the reectivity of this compound mirror is determined by the interference condition at the dark port:

the more is the power which exit theinterferometer through the dark port, the lower is the reectivity of the equivalent mirror. In conclusion it is possible to vary the nesse of the equivalentcavity by tuning the operating point of the beam splitter mirror. In a low nesseconguration, the mixing of dierent degrees of freedom in the optical signals is

49

50

CHAPTER 3. LONGITUDINAL SENSING AND CONTROL
small. This makes the control of the interferometer easier. For a more detailed
discussion of the variable nesse technique refer to [46].

3.7 Lock acquisition strategy
The lock acquisition strategy used during the Science Run is explained in details in Appendix B. The state of longitudinal sensing and control in the nalconguration, at the end of the lock acquisition procedure, can be summarized
as follows:• the DARM degree of freedom is controlled using the dark fringe itself as
error signal (Pr_B1_ACp) with unity gain frequency at 100 Hz;• the PRCL degree of freedom uses the Pr_B2_3f_ACp signal below 5 Hz
and Pr_B2_ACp above. The unity gain frequency of this loop is at 40 Hz.
• the MICH degree of freedom is controlled using Pr_B5_ACq. Its control
loop has a unity gain frequency at 15 Hz and a strong roll-o above 50 Hz;• the gains of these three longitudinal loops are stabilized by slow servo
using as error signal the demodulation of calibration lines;• the laser frequency is stabilized on the mean length of the arm cavities,
using Pr_B5_ACp as error signal and with unity gain frequency at about
25 kHz; at the same time the common mode of the arm cavities is lockedon the demodulated reection of the reference cavity, with a bandwidthof about 1 Hz;• all longitudinal actuators are in low-noise conguration: the very low frequency part of the correction is applied also at the top stage of the suspension, while for the two cavity end mirrors the correction is re-allocatedat intermediate frequencies to the marionette;• the re-injection of MICH and PRCL noise in the dark fringe signal isreduced by using noise subtraction techniques.

Chapter 4
Angular sensing and control
The discussion of resonant conditions inside optical systems, as carried out sofar, implicitly considered good alignment conditions of the mirrors with respect
to the optical axes. If left free in their angular degrees of freedom, the mirrors
will be subjected to motions due to the residual seismic noise at low frequency.
When the interferometer is not locked, every suspension works in stand-alone
condition and the angular position of the payload is controlled by local sensors
which measure the mirror pointing with respect to the ground.

This control

system is however not sucient to obtain good performances of the detectorin science mode state, for two main reasons.

First of all the typical residualuctuation of the angular degrees of freedom of a mirror is of the order of
1 µrad, usually too large to obtain the required sensitivity.

Second the local

control reconstruction of the mirror angular position is subjected to slow drifts(with time scale of the order of hours), driven mainly by temperature eects,
and therefore it is not possible to maintain for long times a correct alignmentof the optics.

Moreover, the local system clearly can not guarantee a correct

global alignment of all the mirrors with respect to the input beam, since it hasaccess only to local data.
For all these reasons a global automatic alignment system is needed. Thissystem must be able to measure the angular position of the mirrors with respect
to the beam and to act in order to correct possible mis-alignments. This chapter
describes the basic principles of functioning of the angular sensing and control
system, usually called alignment system.

For a more detailed description see[54].
The following convention is adopted to refer to the three angular degrees of
freedom of a mirror (see g. 4.1):

ϑx is the rotation around the horizontal axis in the mirror plane (pitch)ϑy is the rotation around the vertical axis in the mirror plane (yaw)52

CHAPTER 4.

ANGULAR SENSING AND CONTROL

ϑy

ϑz
ϑx

Figure 4.1: Denition of the mirror three angular degrees of freedom.

ϑz is the rotation around the axis perpendicular to the mirror (roll) whichcorresponds usually with the beam axis

4.1 Gaussian beam optics
4.1.1 The par-axial diraction equation
In the general case the propagation of light inside an optical system is describedby the Maxwell equations and by the corresponding wave equation

∆ + k 2 Ek (x, y, z) = 0(4.1)

In the simpler case of a laser beam inside an interferometer, the propagation iswell described in the par-axial approximation [12, 55]. The light eld is supposed
to propagate mainly along one direction called z and all its transverse variations
to be small. If the eld is written as

E(x, y, z) = eikz E(x, y, z)

(4.2)

∂E
∂z

 kE , the wave equation(2ik∂z + ∆trans ) E = 0(4.3)

and the variation along z of the envelope is small
simplies to the par-axial diraction equation :which can be used to describe the propagation of an almost monochromatic
beam concentrated along the z direction.

Under the assumption of cylindri-

cal symmetry the fundamental Gaussian solution of the par-axial diraction
equation can be foundΨ(r, z) = q

2

1
1+

2

− xw2+y
(z)

e
z2
b2

x2 +y 2zeik 2R(z) e−i arctan b eikz

(4.4)

4.1. GAUSSIAN BEAM OPTICS

53

1

r [m]

0.5
0
-0.5
-1
-3000

-2000

-1000

0

1000

2000

3000

z [m]

Figure 4.2: Intensity map of the fundamental solution of the par-axial diraction
equation. The color scale is logarithmic.

where

r

z2
= w0 1 + 2bb2
R(z) = z 1 + 2
z
2
kw0
b =
2
w(z)

The transverse intensity shape is a Gaussian function for every z . Its widthis minimum when z = 0, at the waist of the beam, and increases for larger z
according to the function w(z) (see g. 4.2). The constant phase surfaces of thisbeam are parabolic, with a decreasing curvature given at r = 0 by R(z). When
propagating along z the eld undergoes an addition dephasing with respect tothe one predicted in the plane-wave approximation. This is the Gouy phase

φG = − arctan

z
b

(4.5)

4.1.2 Transverse electro-magnetic modes
Starting from the fundamental solution of eq. 4.4 it is possible to relax the requirement of cylindrical symmetry and search for solutions with dierent transverse shape.

The most useful set of solution is given by the Hermite-Gauss

functions, also called electro-magnetic transverse modes (TEM). They depends
on two integer indexes n and m:√

T EMmn (x, y, z)ikz= Nmn (z)e

Hm

2x
w(z)

√

!

2y
w(z)

Hn
x2 +y 22!

2

− xw2+y(z)e−i(n+m+1) arctan(z/b) eik 2R(z) e

(4.6)

54

CHAPTER 4.

ANGULAR SENSING AND CONTROL

Order

Polynomial

012x4x2 − 28x3 − 12x1
2
3

Table 4.1: Hermite polynomials of lowest order.

where the normalization coecient Nmn (z) is chosen in order to make this set
orthonormal for any xed z :s2πw(z)2 2n+m m!n!

Nmn (z) =(4.7)

and the Hn (t) are the Hermite polynomials that can be dened for example as
[56]:

t2



d
−
dt

n

2

e−t

(4.8)

Given a value of the beam size at the waist

w0 , these solutions form aHn (t) = e

complete, orthonormal set, meaning that every beam can be expressed as alinear combination of these functions at xed z .
The rst Hermite polynomials are listed in table 4.1 and the transverse
intensity pattern of the rst TEM modes are shown in g. 4.3.

4.2 Fabry-Perot resonant cavities
The eect of the reection of a beam on a mirror can be described with the usual
plane-wave contribution plus a variable dephasing which takes into account theshape of the mirror:R(x, y) = ire2ikf (x,y)(4.9)

where f (x, y) describes the position of the reecting surface as a displacementfrom a plane with constant z taken as reference and r is the usual ampitude
reection coecient.

In general if a TEM mode with given waist size andposition is reected by an arbitrary mirror, its shape will change and in general
it will not be possible to describe it as a single TEM mode with the same waist
size and position. Therefore only a fraction of the reected beam power will bein the same mode of the incident beam, thus reducing the eective reectivityof the mirror for the mode.
However, if the shape of the mirror follows exactly one constant phase surfaceof the incident beam, the reected one will have exactly the same mode. Thismeans that the optimal shape for the mirrors in a resonant cavity is curved.

4.2. FABRY-PEROT RESONANT CAVITIES55

TEM0,0TEM1,0

TEM0,1TEM1,1

TEM2,0TEM2,1Figure 4.3: Transverse intensity maps of the rst TEM modes.

56

CHAPTER 4.

ANGULAR SENSING AND CONTROL

If the cavity is long enough, the parabolic constant phase surfaces are almost
indistinguishable from spherical one.
The typical example of a resonant cavity is given by a at input mirror anda curved terminal one.

The input mirror must be placed at the waist of theincoming beam, which is the only position where the constant phase surface isplane. The curvature of the constant phase surface at the position of the end
mirror is given byb2
R(L) = L 1 + 2
L(4.10)To have an optimally matched cavity this must be equal to the radius of curvature of the end mirror.

This can be viewed in a dierent way, saying thatthe length L of the cavity and the radius of curvature RC of the end mirror
unambiguously determines the waist size of the modes resonant in the cavity:

w02 =

2p
L(RC − L)k(4.11)

In particular a resonant mode exists only if the radius of curvature of the end
mirror is larger than the length of the cavity.
The resonance condition for the dierent TEM modes can be obtained as
usual imposing the total phase lag over a round trip inside the cavity to be amultiple of 2π . From eq. 4.6:

∆φ = 2kL + π − 2(m + n + 1) arctanL= 2π Nb(4.12)

In particular this equation shows that if the T EM0,0 mode is resonant, all theother modes are not, except in the case of degeneracy.

The dierence of theresonant frequency of a T EMm,n mode with respect to the fundamental one isgiven by∆νm,n =

c
2L


P+

m+n
L
arctanπb
(4.13)

being P a suitable integer number.

4.3 Cavity response to mis-alignments
The optical axis of a resonant cavity can be easily dened in a at-curved
or curved-curved cavity, to be the only line perpendicular to both the mirror
surfaces at the points where it intersects them. The resonant modes describedin the previous section are always referred to this optical axis. A tilt of one oftwo mirrors has the eect of changing the cavity axis, see g. 4.4.
Simple geometrical considerations show that a tilt of the input mirror of anangle α has the eect of rotating the cavity axis by the same amount and oftranslating it by

aIN = (RC − L) sin α cos α(4.14)

4.3. CAVITY RESPONSE TO MIS-ALIGNMENTS

57

α
β

δL
a

cavityaxis
cavity axis

cavity axis

acavity axis(a)Figure 4.4:

(b)

δL

Scheme of mis-alignments of input and end mirrors in a resonantcavity.

where L is the cavity length when aligned and RC the radius of curvature of
the end mirror. Moreover, the length of the cavity changes by

δLIN = (RC − L)(1 − cos α)(4.15)

Since only small deviation of the mirrors from their aligned position are relevant
here, the previous equations can be approximated at the rst order in the angle.
The eect of an input mirror misalignment is to tilt and translate the cavity
axis, since the length changes only at second order:aIN∼ (RC − L)α(4.16)

θIN= α(4.17)In a similar way, the eect of a tilt of the end mirror by an angle β produces
only a translation of the axis and a change in the cavity length:aEN D= RC sin β(4.18)

δLEN D= RC (1 − cos β)(4.19)

and as before the change in the cavity length is second order in the angle, while
the axis translation can be approximatedaEN D ∼ RC β(4.20)The input beam is assumed to be matched to the aligned cavity resonant
mode: this means that the waist of the beam is in the input mirror plane andits size correspond to the one of the cavity resonant mode.
The eect of the mis-alignments is derived exploiting the property that theset of cavity TEM modes is complete: therefore it is always possible to describe

58

CHAPTER 4.

ANGULAR SENSING AND CONTROL

the mode of the input beam in terms of the cavity eigen-modes. This is most
simply done by considering two coordinate systems: the rst one S corresponding to the beam axis and the second one S

0

to the cavity axis. The resonant

0

modes of the cavity are easily expressed in the S system. Considering rst only
a translation of the cavity axis, assumed to be along x, the link between thetwo coordinate systems is given by

x0

= x−a

0

= y

z0

= z

y

The expansion of the input beam in cavity modes can be performed at any
longitudinal coordinate z .

The simplest choice is of course the waist z= 0.

The input beam is described by eq. 4.6 in the S coordinate system. By simple
substitution and considering only terms of rst order in a the result, in the0cavity coordinate system S , is found to be:IN
T EM0,0(0, x, y)



ax0
CAV
= T EM0,0(0, x0 , y 0 ) 1 + 2 2
w0
a
CAV
CAV
0 0
T EM1,0= T EM0,0 (0, x , y ) +(0, x0 , y 0 )w0

In the case of a rotation, the link between the two coordinate systems isgiven byx = x0 cos θ − z 0 sin θ
y

= y0

z

= x0 sin θ + z 0 cos θ

The magnitude of the mode can be expressed as
02

2 x 2 sin2 θINCAVT EM0,0= T EM0,0e w0(4.21)and the dierence is irrelevant being second order in the angle. However, thebeam is also subjected to a x-dependent phase dierence since the waist plane
is dierent in the two coordinate systems. This phase is given by

φ(x0 ) = kx0 sin θ ∼ kx0 θ(4.22)and thereforeIN
T EM0,0(0, x, y)CAV= T EM0,0(0, x0 , y 0 ) (1 + 2ikθx0 )
k
CAV
CAV
= T EM0,0(0, x0 , y 0 ) + i
T EM1,0(0, x0 , y 0 )2w0

In conclusion, the eect of input or end mirror misalignment can be described, in the cavity optical axis reference system, as the creation of additional

4.4. WAVE-FRONT SENSORS

59transverse modes proportional to the tilts:

IN
T EM0,0CAV= T EM0,0+αRC − Lk
CAV+i
T EM1,0
w0
2w0

RC
CAV
T EM1,0(4.23)
w0
In the general case only the T EM0,0 mode is resonant inside the cavity and+ βtherefore all other components are reected.

4.4 Wave-front sensors
To build a system capable of controlling the alignment of a resonant cavity
it is necessary to extract error signals that measure the position of the mirrors
with respect to the beam axis. The most common technique is called wave-front

sensing [57, 58]. It uses quadrant-split photo-detectors (quadrants in brief ): thesensitive area of these photo-diodes being divided into four sectors. The light
power impinging on each of them can be measured separately. These sensors
can be used in + or  × conguration. In the rst case the dierence between
the two opposite halves of the sensors can be obtained and used to extract botha power or a demodulated signal.

In the second case the dierence is taken

between opposite quadrants.
These sensors are able to detect the superposition of dierent TEM modes.
Suppose that the beam hitting a + sensor can be described asE(x, y) = α T EM0,0 + β T EM1,0(4.24)

Assuming that the size of the beam is much smaller than the sensor radius, theright and left halves of the sensor will detect a power given by

Z

PL

PR

Z
2
2
|E(x, y)| dxdy = |α|2
|T EM0,0 | dxdy
x<0
x<0
Z
Z
2
∗
+|β|2
|T EM1,0 | dxdy + 2Re α∗ β
T EM0,0
T EM1,0 dxdy
x<0
x<0
Z
Z
2
2
=
|E(x, y)| dxdy = |α|2
|T EM0,0 | dxdy
x>0
x>0
Z
Z
2
2
∗
∗
|T EM1,0 | dxdy − 2Re α β
T EM0,0+|β|
T EM1,0 dxdy
=

x>0

x>0

Since the magnitude of all TEM modes is even with respect to both x and y , inthe dierence between the two signals only the interference term survives:

D = 4Re α∗ β

Z∗
T EM0,0
T EM1,0 dxdy

(4.25)

x>0
In conclusion the dierence signals from quadrant-split photo-detectors are sensitive only to the interference between even and odd TEM modes, and therefore
makes a perfect candidate to obtain an error signal for an alignment control
system.

60

CHAPTER 4.

ANGULAR SENSING AND CONTROL
a

d

(a)

(b)

Figure 4.5: Eect of beam mis-centering with respect to the rotation axis of themirror.

4.5 Coupling of angular noise
From the discussion of the previous two sections it follows that angular motionsof the mirrors are seen by normal photo-detector only at the second order inthe tilt angle, since they detects only the total power impinging on them. Thisis however true only if the beam axis passes through the mirror rotation center,
which is essentially dened by the way the angular actuation is performed.
Referring to g. 4.5, if the beam axis is translated of d from the mirror center,a rotation of an angle α will produce also an eective longitudinal displacement
given bya = d tan α ∼ dα

(4.26)

This displacement is at any eect indistinguishable from a real longitudinalmotion of the mirror and it acts as an external source of noise for the longitudinal
control system.
It is therefore important to ensure that the rotation center of the mirror
coincides as well as possible with the beam axis, in order to reduce the linear
term in the coupling of angular noise into the detector main output signals.

4.6 The Anderson technique
The technique used in Virgo to extract complete information on the mis-alignmentof the cavity mirror has been rstly developed by Anderson [59] and diers fromthe one used in other interferometric detectors like LIGO [60] since it implies
an additional constraint on the main modulation frequency.
When a cavity is locked, the T EM0,0 mode is resonant for the carrier andanti-resonant for the two sidebands. In this condition the T EM1,0 and T EM0,1
modes of the carrier are not resonant in the cavity. Any mis-alignment of the4.7. VIRGO ANGULAR SENSING AND CONTROL SCHEME
cavity mirror will generate one of these modes but they will not be present ina signicant fraction in transmission of the cavity since they are not resonant.
On the contrary, the two sidebands T EM0,0 modes will be anti-resonant andalmost completely reected by the cavity. Since the resonant frequency of the

T EM1,0 mode is dierent from that of the fundamental mode, it is however
possible to choose the modulation frequency to be an integer multiple of thisfrequency dierence.

In this way the T EM1,0 and T EM0,1 modes of one of

the two sidebands, the upper for example, will be resonant inside the cavity.
Any misalignment will generate one of these mode that, being resonant, will
be present in the transmission port. Here a wave-front sensor will detect the
beating of any T EM0,0 and T EM1,0 modes: the only possible combination isthe carrier for the rst and the upper sideband for the second. This component
can be extracted using a demodulation technique on the dierence signals of
the quadrant sensor and this can be used as an error signal for the alignment
system.
From eq. 4.23 it follows that tilts of the input or end mirrors will result in asuperimposition of the two modes with dierent phase. This implies that using
both the in-phase and quadrature signal from the wave-front sensor allows a fullreconstruction of the two mirrors position.
Moreover, as already pointed out when discussing eq. 4.6, dierent transverse modes acquire dierent dephasing when propagating, due to the Gouy
phase term. It is therefore possible to place a pair of wave-front sensors on thesame beam, at dierent distances to prot of the dierent Gouy phase that thefundamental and rst transverse modes will acquire in the propagation. However, since the distance needed for a signicant change in the phase is of theorder of hundreds of meters, normally telescopes are used for this goal.

4.7 Virgo angular sensing and control scheme
The automatic alignment system in a complex interferometer like Virgo deals
with the control of two degrees of freedom for each of the six main mirrors.
Moreover a control of the input beam pointing must be implemented to avoid
slow drifts that might move the beams out of the mirror centers, leading toworse performances of the detector.
The error signals are obtained from a set of quadrant sensors placed on all themain beams. Referring to g. 4.6, there are two couples of sensors in the beams
transmitted by the two cavities (named Q71, Q72 and Q81, Q82 respectively);
two other quadrants are placed on the beam reected by the interferometer(named Q21 and Q22); one quadrant is placed on the dark fringe beam, before
the output mode cleaner (Q1p) and nally one on the beam splitter pick-o
beam (Q51).

Each quadrant produces several output signals: the power im-

61

62

CHAPTER 4.

Figure 4.6:

ANGULAR SENSING AND CONTROL

Position of the various quadrant sensors used for the automaticalignment system of the Virgo interferometer.

4.7. VIRGO ANGULAR SENSING AND CONTROL SCHEME
pinging on each one of the four sectors and the demodulated dierence between
opposite sectors. The normalized dierence of the powers detected by opposite
sectors can be computed to obtain signals sensitive to the beam position on thequadrant.
In total there are 8 quadrants giving a total of 16 demodulated and 16 power
asymmetry signals:

therefore a total of 32 to control 14 degrees of freedom.

These are organized and controlled in the following way:

• The tilts of the two end mirrors, both in ϑx and ϑy are controlled in common and dierential combinations.

These degrees of freedom are calleddierential and common end modes. The dierential mode is controlledusing the signal coming from the dark fringe quadrant (Q1p) while thecommon mode error signal is obtained from the quadrants in reectionof the interferometer (Q21 and Q22), both demodulated and asymmetry
signals.

• The power recycling mirror is controlled using error signals obtained fromthe quadrant on the beam splitter pick-o beam (Q51).

• The two input mirrors are controlled using as error signals the spot position in transmission of the two cavities, measured using cameras on thesebeams (B7 and B8).

• The beam splitter mirror control uses two signals coming from one of thequadrants placed on the beam transmitted by the west arm cavity (Q81).

• The input beam is nally steered using error signals coming from one of
the quadrants placed in reection of the interferometer (Q21).
There are two dierent congurations for the control of the various angular degrees of freedom. In the automatic alignment conguration the angulardegrees of freedom are controlled using only the global error signals obtained
from the quadrant sensors. The bandwidth of the control in this conguration
is of the order of a couple of Hz. The angular correction is normally appliedat the level of the marionette. In this conguration the local control system iscompletely switched o. A typical example of the open loop transfer function
for these controls is shown in g. 4.7.
On the other hand, in the drift control conguration the local control of themirror is still active, and the global signals are used, after passing through a low
pass lter, only to adjust the set point. In this way the high frequency motionof the mirror is still controlled by the robust local system, with a bandwidthof some Hz, while slow drifts are compensated using the global signals.

This

control path has a bandwidth of the order of tens of mHz.
During the science run the automatic alignment conguration was used forthe end mirrors common and dierential modes, for the power recycling mirror

63

64

CHAPTER 4.

ANGULAR SENSING AND CONTROL

104

tx
ty

Magnitude

102
0

10

-2

10

10-4
10-6
4
3
Phase [deg.]

2
1
0
-1
-2
-3
-4
10-2

Figure 4.7:

10-1

100
Frequency [Hz]

101

102

Theoretical open loop transfer functions for the angular control ofthe end mirror dierential mode in ϑx and ϑy .

and for the ϑx d.o.f. of the beam splitter. All other degrees of freedom were
working in drift control conguration.
The noise performances are described in sec. 7.5. Several actions has been
performed to reduce as much as possible the contribution of angular control noise
to the sensitivity: mainly the implementation of better control lters to reduce
the re-introduction of noise above the unitary gain frequency and a systematic
campaign of beam centering on the mirrors to reduce as much as possible thecoupling of angular noise to longitudinal signals.

4.8 Conclusions
The angular sensing and control system has proved to have very good performances, both from the point of view of long term reliability and stability andof noise performances (see sec. 7.5 for the noise performances and [54] for moredetails on the alignment system). The study of the relation between the angularand longitudinal degrees of freedom has been started. In particular a reductionof angular to longitudinal coupling has been obtained checking and maintaining the centering of the beam on mirrors. An important topic is the relation
between angular and longitudinal working points, which will be studied in thenear future.

Chapter 5
Longitudinal controls
characterization
The task of characterizing the longitudinal sensing and control system deals
with several aspects, which are described in the following sections:• study of the loop properties, stability and performances;• measurement of the system transfer functions and reconstruction of theproperties of the optical part.

• identication of the operating point and its dependence on parameters;
The performances of the locking system in terms of control noise are discussed
in more details in Sec. 7.3.

5.1 Expected optical response of the system
One of the most important topic is the characterization of the interferometer
optical transfer functions.

These describe the response of each photo-diodedemodulated signal to motions of the mirrors or better to the physical degrees
of freedom.
The theoretical transfer functions for an ideal interferometer can be obtainedby analytical computations [46, 52] or by means of simulations.

Using thissecond approach, it is easier to use frequency domain simulations, since they
can directly give results in terms of transfer function.

Here the Finesse [61]program has been used, with a conguration which uses nominal parameters for
the Virgo interferometer, listed in table 5.1. In this simulation all demodulation
phases have been tuned using a procedure as close as possible to the one used
experimentally, mainly by maximizing the in-phase component with respect to

66

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION

Parameter

Value

Input power

10 W

Length of cavities

2999.9 m

Distance PR-BS

6.08 mDistance BS-NI

6.20 mDistance BS-WI

5.41 mModulation freq.

6.264150 MHz

PR re. and trans.

0.95, 0.05

NI re. and trans.

0.8819, 0.118

WI re. and trans.

0.8833, 0.1166

NE re. and trans.

0.99985, 42.9 · 10−6

WE re. and trans.

0.99985, 38.3 · 10

−6

BS sec. face re.

520 · 10−6

Table 5.1: List of main parameters used in the Finesse simulation.

the quadrature one in correspondence of suitable calibration lines.

However,these results should be intended mainly as a rough estimation of the generalfeatures that characterize the optical transfer function.
The results are shown in g. 5.1 and 5.2 for all four demodulated signals
used to control the longitudinal degrees of freedom and frequency noise. It isclear that most of the signals are dominated by the contribution of the CARM
loop, which corresponds to the laser frequency noise. This is another point infavour of having a very high gain frequency stabilization servo. The response of
the SSFS error signal (Pr_B5_ACp) shows a single pole at low frequency, which
is typical of a system composed of two coupled cavities [46, 52], in this casethe power recycling and the two arm cavities. The transfer functions related tothe dark fringe signal (Pr_B1_ACp) are characterized by the presence of the arm
cavity pole at about 500 Hz (g. 5.1).
Since the frequency stabilization is so peculiar, being engaged since the beginning of the lock acquisition procedure, and having such a high gain at thefrequency of interest of the other loops, it is customary to decouple it from theother controls. In other words, from the point of view of locking characterization, the SSFS servo can be considered almost as a part of the optical system. It
has however two main eects: it reduces almost to zero the contribution of frequency noise and it changes the frequency response of the various photo-diodes
to longitudinal motions.
Referring to g. 5.3, the SSFS loop can be modeled by its optical transfer
function Ff 2 from frequency noise f to the error signal s2 and its corrector lter

G2 . A generic longitudinal loop is instead composed of its own optical transfer
function Fz1 and the corresponding corrector lter G1 . The o-diagonal matrix

5.1. EXPECTED OPTICAL RESPONSE OF THE SYSTEM

Pr_B1_ACp
10-5

Magnitude [a.u.]

10-6
10-7

PRCL
MICH
DARM
CARM

10-8
10-9
10-10
10-11
200
150

Phase [deg]

100
50
0
-50
-100
-150
-200
10-2

10-1

100

101
Frequency [Hz]

102

103

104

2

10

3

10

Pr_B2_3f_ACp-610

Magnitude [a.u.]

10-7
10-8

PRCL
MICH
DARM
CARM

10-9
10-10
-11

10

10-12
200
150
Phase [deg]

100
50
0
-50
-100
-150
-200
-2
10

Figure 5.1:

-1

10

0

10

1

10
Frequency [Hz]10

4

Optical transfer function from longitudinal motions to the photo-

diodes used for longitudinal sensing and control. These results are computed
using frequency domain simulations.
signal Pr_B1_ACp.

Bottom:Top: dark fringe in-phase demodulated

in-phase demodulated signal (at 3Ω) from theinterferometer reected beam Pr_B2_3f_ACp.

67

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION

Pr_B5_ACp
10-2
PRCL
MICH
DARM
CARM

Magnitude [a.u.]

10-3
10-4
10-5
10-6
10-7
10-8
10-9
200
150
Phase [deg]

100
50
0
-50
-100
-150
-200
10-2

10-1

100

101
Frequency [Hz]

102

103

104

Pr_B5_ACq

Magnitude [a.u.]

10-4

PRCL
MICH
DARM
CARM

-5

10

-6

10

10-7
10-8
10-9
200
150
100

Phase [deg]

68

50
0
-50
-100
-150
-200
-2
10

Figure 5.2:

-1

10

0

10

1

10
Frequency [Hz]

2

10

3

10

4

10

Optical transfer function from longitudinal motions to the photo-

diodes used for longitudinal sensing and control. These results are computed
using frequency domain simulations.

Top: in-phase demodulated signal fromthe beam splitter pick-o Pr_B5_ACp. Bottom: quadrature demodulated signal
from the beam splitter pick-o Pr_B5_ACp.

5.1. EXPECTED OPTICAL RESPONSE OF THE SYSTEM

69

G1
z1

−

z

s1

Fz1
Fz2
Ff1

f

s2

Ff2

−

G2
Figure 5.3:

Scheme of a double control loop, used in the text to compute theeect of the high gain SSFS loop on the optical transfer functions.

Here

z

denotes a generic longitudinal degree of freedom, while f is the laser frequency.
Optical transfer functions are denoted with the capital letter F . The corrector
lter are denoted with the capital letter G.

elements which links the two loops are denoted by Fz2 and Ff 1 . The two error
signals are linked together by the following linear system of equations:(s1 = Fz1 (z − G1 s1 ) + Ff 1 (f − G2 s2 )s2 = Ff 2 (f − G2 s2 ) + Fz2 (z − G1 s1 )(5.1)

The part one is interested in is the eect of the loop 2 on the response of therst error signal to z .

Therefore terms containing f can be ignored and it ispossible to solve for s1

F

F

G

f 1 z2 2
Fz1 − 1+F
f 2 G2
z
s1 =Ff 1 Fz2 G2
1 + G1 Fz1 − 1+Ff 2 G2

(5.2)

Similarly the expression giving the residual longitudinal motion z1 can be obtained:

z1 =

1


F

F

G

f 1 z2 2
1 + G1 Fz1 − 1+F
f 2 G2z(5.3)

and nally the optical transfer function, dened as the response in function of
frequency of the error signal to a longitudinal motion, is given by the ratio of
eq.5.2 and 5.3:

0
Fz1
= Fz1 −

Ff 1 Fz2 G2
1 + Ff 2 G 2(5.4)

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION

Pr_B1_ACp
10-5

Magnitude [a.u.]

10-6

PRCL
MICH
DARM

10-7
10-8
10-9
10-10
10-11
200
150

Phase [deg]

100
50
0
-50
-100
-150
-200
10-2

10-1

100

101
Frequency [Hz]

102

103

104

102

103

104

Pr_B2_3f_ACp
10-6-7

Magnitude [a.u.]

10

PRCL
MICH
DARM

10-8
-9

10

-10

10

10-11
10-12
200
150
Phase [deg]

100
50
0
-50
-100
-150
-200
10-2

10-1

100

101
Frequency [Hz]Pr_B5_ACq-4

Magnitude [a.u.]

10

PRCL
MICH
DARM

10-5
10-6
10-7
10-8
10-9
200
150
100

Phase [deg]

70

50
0
-50
-100
-150
-200
10-2

10-1

100

101
Frequency [Hz]

102

103

104

Figure 5.4: Optical transfer function from longitudinal motions to photo-diodes,
corrected by the action of the SSFS high gain loop (thick lines) compared withthe free one (thin lines). The attening eect is clearly visible.

5.2. PROPERTIES OF THE CONTROL LOOPS

71

If the open loop gain of the SSFS loop, given by Ff 2 G2 is much larger than one
at the frequencies of interest, this expression can be very well approximated by0
Fz1≈ Ff 1



Fz1
Fz2
−
Ff 1
Ff 2


(5.5)

which is independent from the shape of the frequency servo open loop transfer
function. Therefore eq. 5.4 can be used assuming a corrector lter for the SSFS
which gives a simple integrator with unity gain frequency at 25 kHz as open
loop transfer function. The equivalent optical transfer functions are shown ing. 5.4.

The main eect of the SSFS loop is to atten all transfer functions

up to the cavity pole at 500 Hz for the dark fringe signal and up to several
thousands of Hz for the other ones.
In conclusion, the optical matrix, composed of all these optical transfer functions, is expected to be constant in all the frequency range of interest for thelongitudinal sensing and control system.

5.2 Properties of the control loops
As explained in appendix E, adding a suitable amount of noise inside a control
loop, for example to the error signal, it is possible to measure its open loop gain.
This measurement is indeed very useful to understand the stability margins of
the control systems.
The result is shown in g. 5.5 for the DARM degree of freedom. The gain
margins are ×1.5 and ×1/2.7, meaning that the overall gain of the loop can be
decreased of a factor 2.7 or increased of a factor 1.5 before reaching an instability.
The typical instability frequencies are 40 Hz for too low gain and 150 Hz for too
high gain. The phase margin of the loop is 15 degrees. The unity gain frequency
is at 100 Hz.
Fig. 5.6 shows the results for the PRCL degree of freedom. The gain margins
are ×3.7 and ×1/4.8 and the typical instability frequencies are around 6 Hz for
too low gain and 100 Hz for too high gain. The phase margin of the loop is 60
degrees.
Finally g. 5.7 shows the results for MICH. The gain margins are ×1.7 and×1/2.1 with instability points at about 6 Hz and 30 Hz. The phase margin is
30 degrees.
In conclusions all loops behave as expected and with reasonable stability
margins.

5.3 Properties of the SSFS loop
Because of its analog nature and its high unity gain frequency, the properties of
the SSFS loop can not be measured using the standard data acquisition system,

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION
2

10

1

10
Magnitude

0

10

-1

10

-2

10

10-3
-4

10

150
Phase [deg]

100
50
0
-50
-100
-150
1

Coherence

0.8
0.6
0.4
0.2
0
0
10

1

10

2

10
Frequency [Hz]

10

3

2

10

1

10

100
Magnitude

72

-1

10

10-2

-3

10

-480 -440 -400 -360 -320 -280 -240 -200 -160 -120
Phase [deg]Figure 5.5:

-80

-40

Measurement of the DARM open loop gain.

05.3. PROPERTIES OF THE SSFS LOOP

73

1

10

0

10
Magnitude

-1

10

-2

10

-3

10

10-4
-5

10

150
Phase [deg]

100
50
0
-50
-100
-150
1

Coherence

0.8
0.6
0.4
0.2
0
0
10

1

2

10

10
Frequency [Hz]

10

3

1

10

0

Magnitude

10

-1

10

-2

10

-400

-360

Figure 5.6:

-320

-280

-240
-200
Phase [deg]

-160

-120

-80

Measurement of the PRCL open loop gain.

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION
1

10

0

Magnitude

10

-1

10

-2

10

-3

10

-4

10

150
Phase [deg]

100
50
0
-50
-100
-150
1

Coherence

0.8
0.6
0.4
0.2
0
0
10

1

2

10
Frequency [Hz]

10

1

10

100

Magnitude

74

-1

10

-2

10

-400

Figure 5.7:

-360

-320

-280

-240
Phase [deg]

-200

-160

-120

Measurement of the MICH open loop transfer function.

5.3. PROPERTIES OF THE SSFS LOOP

75

100

Magnitude [a.u.]

10-1

10-2

10-3

10-4
300
200

Phase [deg]

100
0
-100
-200
-300
-400
-500
-600
103

104
Frequency [Hz]105

Figure 5.8: Optical transfer function for the SSFS loop: it is measured between
the correction and the error signals of the SSFS loop, while adding noise to theerror point.

which is limited to a sampling frequency of 20 kHz. All the following measurement have therefore been performed using a spectrum analyzer, by adding sweep
sine excitations to the error signal and measuring both the open loop transfer
function and the optical transfer function of the interferometer.
The optical transfer function (g. 5.8) is measured form the correction to
the error signal of the SSFS loop. It clearly shows a deep notch at exactly halfline-width of the Fabry-Perot cavities (about 25 kHz) and at its multiples. The
corrector lter (g. 5.9) has been developed in order to compensate as well as
possible the feature at 25 kHz, otherwise the phase rotation at this frequency
would made impossible to have high UGF.
The open loop gain of the SSFS, when the interferometer is in its nal state,is shown in g. 5.10.

The unity gain frequency is at about 23 kHz, the gain

margin is 2 in both directions and the phase margin is of about 25 degrees. The
gain at 1 kHz is about 500 and it is increasing approximately like 1/f for lower
frequencies, since the optical transfer function is at below the cavity pole at
500 Hz and the corrector lter behave like a simple integrator in this region.

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION

Magnitude [a.u.]

103

102

101

100
60
40
20
Phase [deg]

76

0
-20
-40
-60
-80
-100
-120
103

104
Frequency [Hz]105

Figure 5.9: Transfer function of the analog corrector lter for the SSFS loop,
from the error to the correction signals.

5.4 Noise subtraction techniques
In order to reduce the contribution of MICH and PRCL control noises to thedark fringe signal, a noise subtraction technique is used [62].

The idea is tocancel the contribution of the control noise a longitudinal loop introduce in thedark fringe signal by adding a suitable correction to the end mirror dierential
mode. The sum of the two paths, one passing through the longitudinal noise
coupling and the other through the added correction, can be made to vanish.
Referring to g. 5.11, the DARM loop is labeled by the number 1 and its error
signal, denoted as b, is the dark fringe. The other control loop, which is here
assumed to be MICH, is denoted by M and its error signal by e. The couplingfrom MICH motion to the dark fringe signal is modeled by the (unknown)transfer function FM 1 .
The coupling of MICH noise can be seen as a bad diagonalization of the
DARM error signal. To improve it the dark fringe can be mixed with the signal
used for MICH. However, this approach is not the one used, since it is preferred
to use the dark fringe signal as it is to reconstruct the gravitational information.
The subtraction is therefore performed at the level of the correction signals:
the one sent to the MICH degree of freedom is also added, after passing through a5.4. NOISE SUBTRACTION TECHNIQUES

77

3

10

2

Magnitude

10

1

10

0

10

10-1
10-2
200
150

Phase [deg]

100
50
0
-50
-100
-150
-200
3
10

4

5

10
Frequency [Hz]

10

3

10

2

10

Magnitude

101

0

10

-1

10

-2

10

-300

-280

-260

-240

-220

-200

-180

-160

-140

-120

-100

-80

Phase [deg.]

Figure 5.10: Open loop gain for the SSFS measured when the interferometer islocked in its nal state.

78

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION

Figure 5.11:

Scheme of the control loops involved in the measurement of thenoise subtraction parameter, here α.

lter (denoted by α) to the dierential mode one. This can be seen as a change
in the way the dierential and Michelson degree of freedom are driven.

Oneadvantage of this method is that the noise in the dark fringe signal is actually
reduced. Moreover, if the noise which limits the MICH error signal comes fromsensing noise instead of actuation noise, as it is the case, the correction signal
subtraction is able to cancel it.
Therefore, assuming a noise subtraction at the level of the corrections, theeect on the dark fringe can be expressed as:

b = F1 [d0 − G1 b − αGM (n + e)] + FM 1 [m0 − GM (n + e)](5.6)

The contribution from the auxiliary loop can be written in terms of the corresponding correction, given by

mz = GM (e + n)

(5.7)

In this way, assuming that the noise coming from the auxiliary loop is dominant,the dark fringe signal isb=−1(αF1 + FM 1 ) mz
1 + F1 G 1(5.8)

This shows rst of all that by using a suitable α it is indeed possible to cancel thenoise contribution from the auxiliary loop. Moreover, if an external perturbationis added to this loop (to the error signal for example, as shown in g. 5.11), thetransfer function T from the correction signal to dark fringe can be measured.
This also allows reconstructing the coupling FM 1 between the two loops:

FM 1 = −αF1 − T (1 + F1 G1 )(5.9)

5.4. NOISE SUBTRACTION TECHNIQUES

79

10-4
Off
On

Dark fringe spectrum [W Hz-1/2]

10-5

10-6

10-7

10-8

10-9

10-10 110

Figure 5.12:

102
Frequency [Hz]Performances of the MICH and PRCL noise subtraction (α and βtechniques). The two curves compares the dark fringe noise with and without
noise subtraction. The noise comes mainly from MICH below 80 Hz and from
PRCL above the same frequency.

and gives a way to compute the correct shape of the α lter, as a function of
the previous value of the lter itself and of the measured transfer function:

αnew = α + T

1 + F1 G 1
F1(5.10)The only additional ingredient needed for this computation is the open looptransfer function of the DARM loop, which can be easily measured by injecting
noise into the DARM loop itself, or modeled.

5.4.1 MICH noise subtraction: alpha technique
A typical example of the result of this kind of measurement for the MICH noise
coupling is shown in g. 5.13 together with the best α computed using eq. 5.10,
compared to the noise subtraction lter used during the measurement. As it isvisible there is a small mismatch between the running lter and the measured
one, that can be easily solved by implementing a better lter.
The performances of MICH noise subtraction, usually called alpha technique,are very good.

Indeed a reduction of the contribution of MICH noise to thedark fringe signal by a factor of roughly 50-100 can be normally achieved in afrequency range between 10 and 200 Hz (see g. 5.12).
The long term stability of this noise subtraction has been studied in detail (see g. 5.14). It turned out that the shape of the lter does not change

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION

10-1

Magnitude [a.u.]

10-2
10-3
10-4
10-5
10-6
10-7
150

Phase [deg]

100
50
0
-50
-100
-150
1
Coherence

0.8
0.6
0.4
0.2
0
101

102
Frequency [Hz]

103

0.3

Magnitude [a.u.]

0.25

0.2

0.15

0.1

40
20
Phase [deg]

80

0
-20
-40
100

101

102

103

Frequency [Hz]Figure 5.13:

Example of the measurement of MICH noise coupling to the dark

fringe signal. The top plot shows the transfer function from MICH correction
to dark fringe, during the noise injection. The bottom plot shows instead thecomputed α transfer function (red) together with the one which was running at
the moment of the measurement (blue).

5.4. NOISE SUBTRACTION TECHNIQUES

81

0.14

2
Alpha
Beta x10-3

1.5
1

0.136

0.5

0.134

0
-0.5

Beta coefficient

Alpha coefficient

0.138

0.132
-1
0.13
0

Figure 5.14:

5

10

15

20
25
Time [days]

30

35

40

-1.5
45

Trend of variation of the α and β coecients over two months

during the science run.

The slow uctuations are mainly driven by the arm

cavity nesses.

signicantly. This can be explained by the fact it should depend only on thedierence in the BS and end mirrors actuators response.

The overall gain ofthe lter changes slowly with time, following variations in the mean nesse of
the two cavities, caused by temperature-driven Etalon eect in the two input
mirrors [63, 64]. This change is compensated easily by a slow servo system.

5.4.2 PRCL noise subtraction: beta technique
The same subtraction technique has been applied to longitudinal control noise
coming from PRCL. By injecting noise into the PRCL loop it is possible tomeasure the transfer function T between the correction and dark fringe signals.
Eq. 5.10 can be used for computing a new value of beta knowing the one runningat the moment of the measurement. Typical results are shown in g. 5.15. The
measured coherence between PRCL correction and dark fringe is not as good
as for the MICH case. This is related to the fact that the coupling of PRCL
noise is much lower than in the MICH case. This also implies that a very goodcancellation of PRCL noise is not needed. For this reason so far only frequency
independent noise subtraction has been used. Indeed the results shown in thegure agrees quite well with a constant beta within some percents, so there is
no real need of a better subtraction.
This subtraction parameter also changes slowly with time and it is constantly
adjusted by a servo system (see g. 5.14).

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION
10-2

Magnitude [a.u.]

10-3
10-4
10-5
10-6
10-7
10-8
150

Phase [deg]

100
50
0
-50
-100
-150
1
Coherence

0.8
0.6
0.4
0.2
0
101

102
Frequency [Hz]

103

10
9

Magnitude [x10-4]

8
7
6
5
4
3
2
1
200
Measured
Running190

Phase [deg]

82

180

170

160

150

102

103
Frequency [Hz]Figure 5.15:

Example of the result of measurement of the PRCL noise coupling

to the dark fringe signal. The top plot shows the transfer function from PRCL
correction to dark fringe, during the noise injection in the PRCL loop.

Thebottom plot shows the measured β transfer function together with the one which
was running at the moment of the measurement.

5.5. RECONSTRUCTION OF THE OPTICAL MATRIX

83

G is theunknown optical matrix of the interferometer, S is the sensing matrix, F collects
the corrector lter, D is the driving matrix. Refer to the text for more details.
Figure 5.16:

Scheme of the longitudinal sensing and control loops:

5.5 Reconstruction of the optical matrix
Performing suitable noise injection in the longitudinal control loops, it is possible
to derive an estimation of the interferometer optical transfer functions [65],which can be very useful for a better understanding and modeling of the system.
In this analysis the two common mode loops has not been considered. The eectof the SSFS loop is indeed well decoupled, as already explained, because of itsvery high gain. Moreover, the error signal used for the CARM loop (the one
acting on the common mode of the end mirror with low UGF, see Sec. B.2) is
not sensitive to any other degree of freedom, since it senses only variations of thelaser frequency, and therefore it is also decoupled from the other longitudinal
degrees of freedom.
Referring to g. 5.16, the optical matrix

G of the system is the set of alltransfer functions from the three physical d.o.f. (DARM, PRCL and MICH) to
the photo-diodes used to reconstruct the error signals (Pr_B1_ACp, the mix of

Pr_B2_3f_ACp and Pr_B2_ACp, Pr_B5_ACq). The measurement of this matrix
is the goal of this procedure.
The longitudinal sensing and control system is modelled as follows.
photo-diode signals are mixed together using a sensing matrix

The

S which recon-

structs the three error signals. To these an external noise source can be added.
The result is then passed through the three correction lters F . The correction
obtained for each d.o.f.

is sent to the dierent mirror actuators.

This is de-scribed by the driving matrix D . All these matrices, except for the optical one,
are known or can be measured independently from the locking loops.

84

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION

5.5.1 Sensing matrix
This matrix simply takes the three photo-diodes signals and reconstruct threeerror signals multiplying by suitable gains. These are all set inside the digital
global control system and therefore known.

The signals are rst normalizedusing suitable functions of the carrier and sidebands powers inside the recycling
cavities.

Afterward the DARM error signal is reconstructed from B1 alone

multiplying it by the loop gain.

Similarly the PRCL error signal is obtained

from B2 alone, after normalization and multiplication by the loop gain.

The

MICH error signal comes instead from a mixing of B2 and B5, controlled bya cross-term gain. Since most of the locking parameters are controlled by thealready described slow servo systems, the value of this sensing matrix changes
slightly with time. A typical example is:



 
EDARM
−9.6 · 10−7 0
 EP RCL  = 
EM ICH00
7.8 · 102−8.7 · 104 
0
B1_ACp 
0   B2_ACp 
1.5 · 102
B5_ACq5.5.2 Filtering matrix
This matrix is in reality diagonal, since every degree of freedom has its own
control lter and there is no mixing between dierent degrees of freedom at thisstage. Therefore there are only three dierent lters, shown in g. 5.17, together
with the CARM one for reference.

5.5.3 Driving matrix
The driving matrix considered here is the one that converts the DARM, PRCL
and MICH correction computed by the control system to real motions of these
degrees of freedom. This is composed by three contributions.
The rst one is a numerical matrix used by the global control system toconvert from the computed d.o.f.

corrections to the ones sent to the mirror

actuators. This matrix is given by:

 
1
CN E
 C
  −1.1
 WE  = CP R   0
0
CBS


β
−1.1 · β
−1
0




α
CDARM

−1.1 · α    CP RCL −2
CM ICH1(5.11)√
2

Here α and β are the coecients that control the noise subtractions.
ready explained, β is a constant value (typically about −4.8 · 10−5As al-) and α isinstead frequency dependent, with the shape shown in g. 5.13 and a gain at
low frequency of the order of 0.17. The global control part of the driving matrix includes also two notches for BS suspension violin modes at 167.05 Hz and167.35 Hz.

5.5. RECONSTRUCTION OF THE OPTICAL MATRIX

CARM
108

13

10

7

10

1012

106

Magnitude [a.u.]

1014

1011
1010
9

10

108

105
104
3

10

102

107

101

10

6

10

150

150

100

100

50

50

0

Phase [deg]Phase [deg]Magnitude [a.u.]

DARM

0
-50
-100

0
-50
-100

-150
10-1

85

-150
100

101
Frequency [Hz]

102

103

10-1

100

PRCL

101
Frequency [Hz]

102

103

102

103

MICH

104

101
0

10
Magnitude [a.u.]

Magnitude [a.u.]

3

10

102

1

10

10-1
10-2
-3

10

10-4

10

10-6

150

150

100

100

50

50

Phase [deg]Phase [deg]

10-5
0

0
-50
-100

-100

-150
10-1

0
-50

-150
100

101
Frequency [Hz]

102

103

10-1

100

101
Frequency [Hz]Figure 5.17: Bode plots of the lters used for the four longitudinal control loops.

86

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION

Table 5.2:

ActuatorGain (µm/V )

NE

12.1

WE

10.9

BS

78

Actuators gains for longitudinal correction. All measurements havean uncertainty of about 5%.

The second contribution to the total driving matrix is given by the frequency
response of the mirror actuators, from the applied correction to the resulting
displacement. This response must be modeled dierently for the end mirrors,
where part of the locking force is re-allocated to the marionette, and for the
BS and PR mirrors. For the last two the response can be described in rst approximation as that of a simple pendulum, with resonance frequency at about
600 mHz.

For NE and WE the transfer function must take into account themarionette mechanical transfer function and the compensator lters. The overall behavior is quite close to a simple pendulum, with some signicant dierence
around the cross-over frequency. The low frequency gain of the actuators has
been measured for NE, WE and BS [66, 67] and are listed in table 5.2.

Thegain for the PR actuators has not been measured yet, so the nominal value of

13 µm/V is used here.

An additional frequency dependence of NE and WE

actuators has been measured.

This is modeled using a pole at 100 Hz and azero at 125 Hz.
Finally, the whole control chain must include a constant delay of about
600 µs coming from the data acquisition and control systems.
The third and last contribution to the total driving is given by a geometricalmatrix that converts motions of the mirrors to the physical degrees of freedom.
This has already been discussed in sec. 3.1 and can be summarized by







1
DARM

 
=
P
RCL

  0
M ICH
0

−1
0
0


zN E
0
0


  zW E 
1 − √12   zP R 
√
0
2
zBS




(5.12)The total driving matrix is obtained by multiplication of these three parts.
For a perfect driving the result should be diagonal.

This is not the case forthe present conguration of the Virgo locking system: the MICH and PRCL
degrees of freedom are not completely decoupled. This might be improved in
the future by measuring the PR actuator gain and computing the corresponding
global control driving matrix.

5.5. RECONSTRUCTION OF THE OPTICAL MATRIX

87

5.5.4 Measured matrix
Referring to g. 5.16, to perform the measurement necessary for reconstructing
the optical matrix, noise is added to the error signals of the three locking loops
at dierent times. For each noise injection, the transfer functions between the0

injected noise n and all three error signals e , taken after the addition of thenoise, are measured, as well as the corresponding coherences. This gives a 3 × 3
matrix A of transfer functions. For each one of these, only frequency bins where
the coherence is better than 0.5 are considered. All other bins are set to 0.
The measurement of the coupling from DARM noise to MICH and PRCL
error signals has not been possible, since the coherence between the added noise
and the error signals was always close to zero. The possible explanation is thatthe couplings factors are small and the intensity of the noise added to the DARM
loop was not enough to generate an eect higher than the normal noise present
in the error signals. For this reason in the following analysis the DARM noise
coupling to other error signals has been assumed to be zero.

5.5.5 Optical matrix
0

The transfer function between the error signals e and the noise n can be computed solving the feed-back system as in appendix E with additional care inmaintaining the correct order of operands in matrix computations.

Referring0

again to g. 5.16, the error signals e can be expressed by

e0 = n + SGDF e0(5.13)and this equation can be inverted to obtain an analytical expression of themeasured matrix A:−1e0 = A n = (1 − SGDF )n(5.14)The unknown optical matrix G can be easily extracted since all other matrices
are measured or known:

G = S −1 1 − A−1 F −1 D−1(5.15)The results of this computation are shown in gures 5.18 and 5.19.

Therst gure shows the transfer functions to the dark fringe demodulated signal

Pr_B1_ACp. As expected they are almost at except for the arm cavities poleat about 500 Hz. The results compensated for the pole are shown in bottom
panel of g. 5.18. The residual frequency dependence can be well modeled witha simple pole at 3 kHz, which correspond to an anti-aliasing lter used inside
the photo-diodes read-out system, and with a double zero at 24 Hz with Q
equal to one and two poles with the same Q at 28 Hz.

This low frequency

structure is related to a frequency dependence on the actuator response. The
transfer function from the MICH d.o.f. is as expected the second most important

88

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION
one. Its magnitude is about a factor 30 lower that that of the DARM transfer
function. This ratio is of the order of magnitude of the expected one (see g. 5.4),
given by the resonant enhancement factor of the response to displacement in the
Fabry-Perot cavities, given by

T FDARM →DF
2F∼
T FM ICH→DFπ
The transfer function of PRCL motion is instead higher than expected (seeagain g. 5.4).

The reason is that the driving is not perfectly diagonal andtherefore the noise added to the PRCL error signal couples also to the MICH
correction. Therefore the path followed by PRCL noise to reach the dark fringe
signal passes through the MICH correction.
The plots of g. 5.19 show the optical transfer functions to the demodulated
signal from the beam in reection of the interferometer Pr_B2_ACp (top) and tothe one from the beam splitter pick-up beam Pr_B5_ACq (bottom). For thesetwo photo-diode signals the measurements are not very good above roughly
100 Hz, mainly for the MICH transfer functions.

This happens because thestrong roll-o of the MICH control lter makes almost impossible to inject asignicant amount of noise above that frequency.
The reconstruction of the optical matrix has been performed for several
dierent noise injections, during dierent locks. The results of all these measurements are compatible within 10%, meaning that the optical congurationof the interferometer is reasonably constant in time.

Since the transfer func-

tions, after compensating for the cavity pole, are very close to be at, the entire
optical matrix G can be described with frequency independent coecient. The
mean values and uncertainties, given in W/µm are: 
1.91 ± 2%
B1

 
0
 B2  = B502.2 · 10−3 ± 10%
1.0 · 10−3 ± 8%0.61 ± 8%



DARM
7.86 · 10−2 ± 2%1.90 · 10−4 ± 4%   P RCL 
7.5 · 10−2 ± 10%
M ICH(5.16)

5.6 Performances of the locking system
Starting from the measurement of the optical matrix, several other propertiesof the longitudinal sensing and control system can be obtained. First of all, themultiplication of the optical matrix with the sensing matrix gives the responseof each longitudinal error signal to the motion of the dierent degrees of freedom(in arbitrary units):

−1.8 · 10−6

S·G=
0
0

−2.3 · 10−9
1.0
2.1


−7.7 · 10−8

0.18

−4.8

(5.17)

5.6. PERFORMANCES OF THE LOCKING SYSTEM

DARM
PRCL
MICH

1

Magnitude [W/µm]

10

100
-1

10

10-2
10-3
-4

10

150

Phase [deg]

100
50
0
-50
-100
-150
102

103
Frequency [Hz]

DARM
PRCL
MICH

1

Magnitude [W/µm]

10

100
10-1
-2

10

-3

10

10-4
150

Phase [deg]

100
50
0
-50
-100
-150
102

103
Frequency [Hz]Figure 5.18:

Plots of reconstructed optical matrix.

The top plot shows thetransfer functions from DARM, MICH and PRCL motion to the dark fringe
signal Pr_B1_ACp. The bottom one shows the same transfer functions, corrected
for the eect of the cavity pole at 500 Hz.

89

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION

10-2

Magnitude [W/µm]

PRCL
MICH

-3

10

10-4

150

Phase [deg]

100
50
0
-50
-100
-150
101

102
Frequency [Hz]

103

3

10

MICH
PRCL

2

Magnitude [W/µm]

10

101
100
10-1
10-2
10-3
10-4
150
100

Phase [deg]

90

50
0
-50
-100
-150
102

103
Frequency [Hz]Figure 5.19:

Plots of reconstructed optical matrix.

The top plot shows thetransfer functions to the reected beam signal Pr_B2_ACp and the bottom one
to the beam splitter pick-up signal Pr_B5_ACq.

5.6. PERFORMANCES OF THE LOCKING SYSTEM

91

This matrix also gives an estimate of the decoupling of dierent degrees of
freedom in the error signals. Each row determines how much the corresponding
error signal see an equal motion of the DARM, PRCL and MICH. Therefore
if each row is normalized in order to have unity elements on the diagonal, theo-diagonal ones give estimates of the coupling to other motions:



1




1/800
1
1/2


1/25

1/5 
1

(5.18)

It appears clear that the DARM error signal, which by the way coincides withthe dark fringe signal, has a good decoupling with respect to motions of MICH
and PRCL, while the other two error signals has a much poorer decoupling. In
particular the MICH error signal has only a decoupling of a factor of 2 withrespect of PRCL motion.
From the matrix of eq. 5.17 a calibration of the error signals in µm can beobtained, by simple inversion: −5.4 · 105
zDARM

 
0
 zP RCL  = 
0zM ICH−1.8 · 10−2
0.93
0.40



eDARM
8.1 · 10−3
3.5 · 10−2   eP RCL −0.20
eM ICHThis calibration matrix is clearly valid only at low frequencies, below 100 Hz,
but this is the region where the residual motion of the mirrors are relevant. Using
this matrix it is possible to reconstruct the real motion of the various degrees of
freedom. The locking accuracy is usually dened as their residual RMS motions:

RM SDARM=

3.3 · 10−12 m

RM SP RCL=

2.8 · 10−11 m

RM SM ICH=

8.4 · 10−11

(5.19)

m

The actual motion of the DARM degree of freedom is much higher than what one
should expect by considering the detector sensitivity to dierential displacement(or gravitational waves).

The reason is that the noise subtraction techniques

operate in order to maintain the noise seen by the dark fringe signal as small
as possible, at the price of introducing a larger dierential motion of the end
mirrors.

This motion is exactly the one needed to cancel the contribution of

PRCL and MICH from the dark fringe signal.

In this way, even if the real

motion of the end mirrors is much higher than what is expected, the sensitivityof the dark fringe signal to dierential displacement is still given by the (1, 1)
element alone of the matrix in eq. 5.16. Therefore once all contribution fromother degrees of freedom are reduced as much as possible, the dark fringe signal
can be considered as the limiting noise in sensing dierential displacement.

92

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION

5.7 Operating point
5.7.1 Denitions
The locking point can be dened as the point where all longitudinal control loops
keep the interferometer locked, which correspond to the relative position of all
mirrors that makes all the error signal simultaneously zero. One can distinguishthe operating point which is the point where the best resonance conditions are
obtained. These can be translated in terms of power signals:• the carrier must be resonant inside the recycling cavity, meaning that
Pr_B5_DC must be at its maximum;• the sidebands must also be resonant in the recycling cavity, or in otherwords Pr_B5_2f_ACq must be maximum;• the dark port must be tuned at destructive interference for the carrier:
Pr_B1_DC must be at its minimum;• nally, as a consequence of all the other conditions, the sidebands should
be well balanced everywhere inside the interferometer, and in particularat the dark port.
For an ideal interferometer the locking point coincides with the operating point,since all error signals are zero exactly when the interferometer is at the bestresonance condition. However, in a real interferometer there are many causesthat can spoil this equivalence. For example the presence of higher-order Gaussian modes (for example due to mis-matching of the input beam with respect
to the interferometer mode, or due to even small mis-alignments) can introduce
relative osets between error signals. In Virgo it is known that one of the maineects of the thermal transient is to change the curvature of the input mirror,
therefore changing the resonance condition and mode of the sidebands in therecycling cavity. This is the main source of high-order modes and longitudinalosets.

5.7.2 Map of locking point
This rst technique to characterize the locking point was applied shortly after
the beginning of the WSR program and gave very important information, that
nally resulted in the implementation of the previously described oset servo
for the balancing of the sidebands.
Since all longitudinal control loops are realized in the digital domain, it isquite easy to add constant osets to their error signals. They must be maintained small to keep the locking point still inside the linear region of the error

5.7. OPERATING POINT

93

Figure 5.20: The two dimensional space of Pr_B2_3f_ACp and Pr_B5_ACq osets
is covered with a square spiral travelled at constant speed. In this way every
region of the plan is sampled for a similar time.

signals and because too big osets might move the locking point quite far fromthe optimal one, resulting in saturation of photo-diode signals.
For technical reasons the osets were added not directly to the error signals,
but to the demodulated output of the photo-diodes, immediately after the ADC
boards. For the DARM and PRCL loops this makes no dierence, since theirerror signals are reconstructed using only one photo-diode signal each.

The

MICH error signal instead is reconstructed from a combination of two signals:

Pr_B5_ACq and Pr_B2_3f_ACp which is also used for PRCL. Therefore the addition of an osets to this last signal results in a simultaneous motion of both
MICH and PRCL.
The rst observations showed that any oset added to the DARM or SSFS
loops produced always a worsening of the interferometer performances, for example an increase of the power at the dark port and a decrease of the power inthe recycling cavity. The eect was moreover symmetrical with respect to thezero oset. Therefore it is possible to conclude that the locking point of these
two loops are coincident with the best operating point.
A similar experiment performed adding separately osets to Pr_B5_ACq or
Pr_B2_3f_ACp showed powers and noise levels changing in a non symmetricalway.

This was a clear indication that the locking point was not correct, andthat the two signals must be considered together in order to fully understandthe position of the best operating point.
For this reason a scan of the two-dimensional osets space has been performed. Technically, this has been obtained by starting from a given reference
point and changing the two osets separately with slow ramps in order to cover

94

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION
the bi-dimensional plane with a square spiral (see g. 5.20), with the osets point
moving at constant speed. This kind of measurements requires long times, typically of the order of hours, since the osets must be moved slowly enough to
avoid triggering instabilities of the interferometer.
After the measurement itself, a post-processing of the collected data is necessary to produce intelligible maps of interesting signals versus the value of thetwo osets. The procedure is simple: the entire region covered by the two osets are divided in square tiles to obtain a two-dimension histogram. Then every
second the mean value of the osets and of the analyzed signals are computed.
The latter is stored in the tile which includes both values of the osets. When
all the measurement time has been processed, the values stored for each tile are
averaged, to obtain an estimation of the mean value of the signal for that binof the two-dimensional histogram.
Some results of this mapping of the operating point are shown in gures
5.21, 5.22 and 5.23. These maps are not centered around the zero oset position,
since this gave by far not the best locking point. Actually the (0, 0) position isclose to the top left corner of the maps.
In g. 5.21 the changes in some interferometer powers are shown.

It isevident that moving away from the (0, 0) position to the center of the plot, thecarrier power (Pr_B5_DC) slightly change by about 5%, meaning that its overall
recycling gain is decreasing. However at the same time also the power in thedark fringe is decreasing, and it has indeed a minimum near the center-rightof the plot. If the ratio between the power inside the recycling cavity and thepower at dark fringe is computed (see the lower right plot) it becomes evident
that even if the recycling gain is decreased, the contrast of the interferometer
becomes better.

Finally, more or less in correspondence with the minimumof the dark fringe power, the recycling gain of the sidebands is maximum, as
visible from the plot of Pr_B5_2f_ACq signal (bottom left). From these maps
the conclusion can be drawn that the photo-diodes signals zero-crossing point
does not give a good operating point:a better conguration is obtained byadding suitable osets.
Moreover, the same conclusion can be drawn from the analysis of the maps
shown in g. 5.22, which plots the behavior of the carrier and sidebands elds atthe dark port. These signals are obtained as already explained from a scanning
Fabry-Perot cavity, and here they have been calibrated in watts so that the sumof carrier and sidebands powers is equal to the power read by the B1p photodiode. The top row maps show the expected behavior of the upper and lower
sidebands, which should have their maximum gain at two dierent positions(this is an eect of the Schnupp asymmetry). From these values the sidebands
imbalance can be computed, as

I=

PU SB − PLSBPU SB + PLSB

(5.20)

5.7. OPERATING POINT

95(a) B5 DC power [mW](b) B1 DC power [mW]

0

45
44

-5

43
-10
42
-15

41

-20

30
B5 d2 ACq offset [x 1000 ADC counts]

B5 d2 ACq offset [x 1000 ADC counts]

46

28

0

26
24

-5

22
-10

20
18

-15

16
14

-20

40
0
2
4
6
8 10 12 14 16 18
B2 3f d1 ACp offset [x 100 ADC counts]

02
4
6
8 10 12 14 16 18
B2 3f d1 ACp offset [x 100 ADC counts](c) B5 sidebands power [µW](d) B1 DC / B5 DC

0

54
52

-5
50
-10

48
46

-15

44
-20

0.65
B5 d2 ACq offset [x 1000 ADC counts]

B5 d2 ACq offset [x 1000 ADC counts]

56

0.6

0

0.55
-5
0.5
0.45

-10

0.4
-15
0.35
-20

0
2
4
6
8 10 12 14 16 18
B2 3f d1 ACp offset [x 100 ADC counts]

Figure 5.21:

0.3
0
2
4
6
8 10 12 14 16 18
B2 3f d1 ACp offset [x 100 ADC counts]

Eect of changing longitudinal osets on interferometer powers.

The axis represents the value of electronic osets added to the two photo-diodes
used to reconstruct MICH and PRCL error signals. Top left: Power inside therecycling cavity.
cleaner).

Top right: Power at the dark port (after the output mode

Bottom left: Sidebands power inside the recycling cavity. Bottom

right: Ratio of the power at the dark port and inside the recycling cavity. This
gives an estimate of the darkness of the fringe.

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION(b) Lower sideband power [µW]
120

0
100
-5
80
-10

60
40

-15

20

B5 d2 ACq offset [x 1000 ADC counts]

B5 d2 ACq offset [x 1000 ADC counts](a) Upper sideband power [µW]

-20

180

0

160
-5

140
120

-10

100
-15
80
60

-20
0
2
4
6
8 10 12 14 16 18
B2 3f d1 ACp offset [x 100 ADC counts]

02
4
6
8 10 12 14 16 18
B2 3f d1 ACp offset [x 100 ADC counts](c) Carrier power [µW](d) Sidebands balancing (USB - LSB)/(USB + LSB)

0

100

-5

90

-10

80

-15

70

60

-20

1
B5 d2 ACq offset [x 1000 ADC counts]

110
B5 d2 ACq offset [x 1000 ADC counts]

96

0
0.5
-5
0
-10

-20

0
2
4
6
8 10 12 14 16 18
B2 3f d1 ACp offset [x 100 ADC counts]

Figure 5.22:

-0.5

-15

-1
0
2
4
6
8 10 12 14 16 18
B2 3f d1 ACp offset [x 100 ADC counts]

Eect of changing longitudinal osets on carrier and sidebands

powers at the dark port.

The axis represent the value of electronic osets

added to the two photo-diodes used for reconstructing MICH and PRCL error
signals.

Top left:

power in the upper sideband.

Top right:

power in thelower sideband. Bottom left: power in the carrier. Bottom right: sidebands
imbalance, computed as the dierence of the tow sidebands divided by theirsum.

5.7. OPERATING POINT

97

The region where this value is close to zero corresponds roughly to the previously
identied one, which gives the best interferometer contrast.

The sidebands

should be balanced as better as possible, both for stability issues and to reducethe coupling of several noise sources [68].

Finally, the eect of changing the locking point on the detector sensitivity
is shown in g. 5.23. The top left plot shows the general improvement in thesensitivity, as measured by the NS-NS optimal oriented horizon.

There is asignicant improvement when the locking point is in the good region. A moredetailed picture of the detector sensitivity can be obtained by looking at thenoise level in dierent frequency bands, measured by mean of band-limited
RMS (see sec. 8.1 for more details). The change in the BRMS of some bands isshown in the remaining plots of g. 5.23.

In conclusion, this study allowed to prove that the locking point obtained
without any careful tuning of the photo-diode osets is not the best one, and that
it is possible to greatly increase both the performances and the sensitivity of thedetector by moving to a better point. The main outcome of this work has been
the understanding of the necessity of a control of the locking osets. Finally,
after several experimental trials, the sidebands imbalance has been chosen as
the error signal for the slow servo that changes only one of the two osets. The
use of a servo system has been necessary since the correct osets to add to thephoto-diode signals are strongly sensitive to the interferometer state: rst, they
can change quite a lot because of mirrors heating, even for about ten hours after
the lock acquisition; second, they also depend strongly on the overall alignment
condition of the interferometer; nally, as expected, they change accordingly tothe demodulation phase of the photo-diodes. The oset long term evolution andaccuracy are shown in g. 5.24.

It is also interesting to convert the osets so far expressed in terms of ADC
counts to the equivalent displacement for the MICH and PRCL degrees of freedom. There are two steps in this conversion. The rst one is to compute thecorrect combinations of

Pr_B2_3f_ACp and Pr_B5_ACq that gives the MICH

and PRCL error signals. The second one is the translation of error signals osets to real displacement. Both these computations can be easily carried out.
Some examples of the results are visible in g. 5.25, where some of the maps
already shown in gures 5.21, 5.22 and 5.23 are plotted against d.o.f. displacements. From this analysis it follows quite clearly that the most relevant eects
are obtained by changing the MICH locking point, while the PRCL one plays
a signicant role only in changing the sidebands recycling gain, as visible fromthe 2f signal inside the recycling cavity.

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION(b) BRMS 200 300 Hz [x 10-9 W Hz-1/2]
4.4

0

B5 d2 ACq offset [x 1000 ADC counts]

B5 d2 ACq offset [x 1000 ADC counts](a) Horizon NS-NS Optimal [Mpc]

4.2
4

-5

3.8
3.6

-10

3.4
3.2

-15

3
-20

3.5
0
3
-5

2.5

-10

2
1.5

-15

1
-20

0
2
4
6
8 10 12 14 16 18
B2 3f d1 ACp offset [x 100 ADC counts]

02
4
6
8 10 12 14 16 18
B2 3f d1 ACp offset [x 100 ADC counts](c) BRMS 500 1000 Hz [x 10-9 W Hz-1/2](d) BRMS 1 - 2 kHz [x 10-12 W Hz-1/2]
500
B5 d2 ACq offset [x 1000 ADC counts]

10
B5 d2 ACq offset [x 1000 ADC counts]

98

9

0

8
7
-5
6
5
-10

4
3

-15

2
1

-20

450

0

400
-5

350
300

-10

250
-15

200
150

-20
0
2
4
6
8 10 12 14 16 18
B2 3f d1 ACp offset [x 100 ADC counts]

Figure 5.23:
and noise.

02
4
6
8 10 12 14 16 18
B2 3f d1 ACp offset [x 100 ADC counts]

Eect of changing longitudinal osets on interferometer sensitivity

Top left:

Maximum range for optimal oriented 1.4 solar masses

coalescing neutron stars.

Top right:

signal between 200 and 300 Hz.

Band-limited RMS of the dark fringe

Bottom left: Band-limited RMS of the dark

fringe signal between 500 and 1000 Hz.

Bottom right: Band-limited RMS of

the dark fringe signal between 1 and 2 kHz.

99

10

8000

5

7000
6000

0
5000
-5

CountsPhoto-diode offset [x103 ADC counts]

5.7. OPERATING POINT

-10

4000
3000

-15
2000
-20

1000

-25
0

5

10

15

20

25

30

35

40

45

Time [days]

Figure 5.24:

0
-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Sidebands im-balance

Performances of the sidebands balancing servo.

The left plotshows the evolution of the added oset during one and a half months of thescience run. Like the alpha and beta coecients of g. 5.14, the oset changes
accordingly to variation of the cavity nesses. The right plot shows the residual
im-balance of the sidebands, during a single lock of 90 hours.

The accuracy

in the balancing is of the order of 10% and is mainly limited by the error inmeasuring the sideband amplitudes.

5.7.3 Coupling of longitudinal noise to power signals
Another possible way to understand if the locking point gives the correct resonance conditions is to search for coupling of longitudinal noise into the signalsthat measure the power in dierent places of the interferometer. Powers should
be maximum or minimum in correspondence of the operating point, and therefore any residual motion along the longitudinal degrees of freedom should notcouple to power signals at rst order, but only at the second and higher ones
(see g. 5.26). If instead the locking point is displaced from the maximum or
minimum of the power, any motion of one d.o.f. can couple linearly to powersignals.
As already explained calibration lines are permanently added to the error
point of the longitudinal loops: 24 Hz to MICH, 62 Hz to PRCL, 379 Hz to
DARM and 1111 Hz to the laser frequency.

It is therefore possible to search

for these lines in the spectra of all power signals. The best way to check if aline is visible, meaning that the corresponding longitudinal excitation couples
directly into the power signal, is to check the coherence between the latter andthe added excitation. If the coherence is not negligible, then a linear coupling
between the longitudinal motion induced by the calibration line and the powersignal is present, meaning that the locking point does not correspond to the topof the resonance peak.
In g. 5.27 the results of this analysis, for the interferometer conguration of

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION(a) B5 DC power [mW](b) B1 DC power [mW]
46

30

1.6

1.6
45

1.2

44

1
43

0.8
0.6

42

28

1.4
PRCL offset [nm]

PRCL offset [nm]

1.4

0.4

26

1.2

24

1
22
0.8
20

0.6

18

0.4
41

0.2
0

0.2

16

0

14

40
-1.2

-1

-0.8 -0.6 -0.4 -0.2
MICH offset [nm]

0

-1.2

-1

(c) B5 sidebands power [µW]-0.8 -0.6 -0.4 -0.2
MICH offset [nm]0(d) B1 DC / B5 DC
56

0.65

1.6

1.6
54
52

1.2
1

50

0.8
48

0.6
0.4

46

0.2
44

0

0.6

1.4
PRCL offset [nm]

PRCL offset [nm]

1.4

1.2

0.55

1

0.5

0.8

0.45

0.6
0.4

0.4

0.2

0.35

0
0.3

-1.2

-1

-0.8 -0.6 -0.4 -0.2
MICH offset [nm]

0

-1.2

(e) Upper sideband power [µW]-1-0.8 -0.6 -0.4 -0.2
MICH offset [nm]0(f) Lower sideband power [µW]
120

1.6
1.4

1.6

100

1.2
80

1
0.8

60

0.6
0.4

40

0.2

180

1.4
160
PRCL offset [nm]

PRCL offset [nm]

100

1.2
1

140

0.8

120

0.6
100

0.4

80

0.2
20

0
-1.2

-1

Figure 5.25:-0.8 -0.6 -0.4 -0.2
MICH offset [nm]

0

0

60
-1.2

-1

-0.8 -0.6 -0.4 -0.2
MICH offset [nm]0Map of operating point, re-expressed in terms of MICH and PRCL

displacement in nanometers.

In the rst row the power inside the recycling

cavity and at the dark port are shown; in the second row the sidebands power
inside the recycling cavity and the fringe darkness; in the last row the two
sidebands behavior.

5.8. CONCLUSIONS

Figure 5.26:

101

Scheme of the mechanism that couples longitudinal noise into

power signals if the interferometer is not locked at the correct resonance condition.

the rst part of the science run are shown. The plots in the rst two rows referto the MICH and PRCL degrees of freedom. The coherence of all powers withthe longitudinal excitations are very close to 1, meaning that the linear relationis quite high and stable in time. This is also conrmed by the clear visibilityof the calibration line in the spectra of powers. The plots in the last two rows
instead does not show any signicant coherence and the lines are not visible.
In conclusion this analysis shows that for what concern the dierential modeand the frequency stabilization, the working point of the two control loops isindeed the good one, while MICH and PRCL are not yet optimal.

This canin principle lead to a bigger coupling of longitudinal noise into the dark fringe
signal.
These results suggest a possible procedure to nd the good locking point andto maintain it. The power signals can be demodulated at the frequency of thecalibration lines, in a way similar to what is done to reconstruct the unity gain
frequency of the longitudinal loop, obtaining signals that can give information on
the value of MICH and PRCL osets. Since the coupling coecient changes signwhen passing the resonance, they give also information on the sign of the osets,
and therefore they can be used as error signals for a slow servo that changes
MICH and PRCL longitudinal osets to minimize the coupling of control noise
into power signals. Using a servo instead of a xed value will allow compensating
for slow variations of the osets.

5.8 Conclusions
In summary, a technique has been developed to reconstruct the optical responseof the Virgo interferometer to longitudinal motion, using measurements basedon (almost) white noise injections. The results conrm in the general behavior

CHAPTER 5. LONGITUDINAL CONTROLS CHARACTERIZATION

10

-5

1
B1
B2
B5
B5_2f
B7
B8

-6

B1
B2
B5
B5_2f
B7
B8

0.8

0.6

0.4

Coherence

Spectrum [W Hz-1/2]

10

10-7
0.2

10-8

10

0
20

21

22

23
24
25
Frequency [Hz]

26

27

28

20

21

22

23
24
25
Frequency [Hz]

26

27

28

-6

1
B1
B2
B5
B5_2f
B7
B8

-7

B1
B2
B5
B5_2f
B7
B8

0.8

0.6

0.4
10

Coherence

Spectrum [W Hz-1/2]

10

-8

0.2

10

-9

0
58

59

60

61
62
63
Frequency [Hz]

64

65

66

58

59

60

61
62
63
Frequency [Hz]

64

65

66

10

-7

B1
B2
B5
B5_2f
B7
B8

B1
B2
B5
B5_2f
B7
B8

0.8

0.6

10

0.4

-8

Coherence

Spectrum [W Hz-1/2]

1

0.2

10

-9

10

-6

375

10-7

376

377

378
379
380
Frequency [Hz]

381

382

383

375

376

377

378
379
380
Frequency [Hz]

381

382

0
383
1

B1
B2
B5
B5_2f
B7
B8

B1
B2
B5
B5_2f
B7
B8

0.8

0.6

10-8

0.4

Coherence

Spectrum [W Hz-1/2]

102

0.2
10-9
1107

1108

Figure 5.27:

1109

1110 1111 1112
Frequency [Hz]

1113

1114

1115

1107

1108

1109

1110 1111 1112
Frequency [Hz]

1113

1114

0
1115

Coupling of longitudinal calibration lines into power signals. The

left plots show the spectra of various power signals, while the right ones thecoherence with the added excitation.

From the top:

MICH (24 Hz), PRCL

(62 Hz), DARM (379 Hz), frequency stabilization (1111 Hz).

5.8. CONCLUSIONS

103

the expected transfer functions obtained from simulations. They moreover havebeen very useful to gain a better understanding of the detector and to develop
noise projection, as is explained in chap. 7. There are several points that needa further development. More precise measurements would be useful to obtainbetter estimates of the optical matrix, mainly extending the results already obtained at lower and higher frequencies. A possible new measurement techniquemight replace the white noise with a sweep sine : a single line can be injected,
with frequency slowing varying over a suitable frequency band.

This method

allows having a much higher amplitude of the excitation than with white noise,
without perturbing too much the interferometer control and without saturating
the dynamical range of the actuators.

This will give a better signal-to-noise

ratio in the measurement and will allow extending its validity range to lower
frequencies. This method has not been used yet because of technical limitationsof the present control architecture.
More eorts must also be devoted toward a better characterization of thelongitudinal working point, to understand which is the best one and how tomaintain it. The actual conguration is somehow defective, since the slow longitudinal oset servo uses only one error signal, given by the sidebands balancing, to control both MICH and PRCL locking osets. It would be preferable
to have two independent error signals to control these two osets. A possible
technique could imply the use of signals generated by demodulation of powers
at the frequency corresponding to longitudinal calibration lines. Some work inthis direction has already been started during the last period of the science run.
Finally, the task of improving the robustness and performances of the longitudinal control system is a priority one, since it is crucial to reduce the noise
re-introduced at low frequency (see chap. 7). Some work has been done during
the science run, mainly developing a better control lter for the CARM loop andstudying the possibility of doing the same for MICH and PRCL. Moreover, thenoise of the error signals is not yet completely understood and it is much higher
than the expected sensor noise. Therefore the study of this noise might prove
to be mandatory to reduce the longitudinal control noise at low frequencies.

Part II
Noise and sensitivity

Chapter 6
Noise sources and sensitivity

6.1 Sources of noise
The noise sources that limits a detector sensitivity or in general one interesting signal can be classied with respect to several dierent criteria.

Given atime scale of interest, that can be minutes, hours or days, a noise source canbe called stationary if its statistical properties do not vary signicantly overthe chosen time period.

The most clear implication of this statement is thatthe contribution of the noise source to the signal power spectrum is constantin time. A noise source that does not full this requirement is called instead

non-stationary.

There are many examples: noise sources that couple with acoecient modulated by slow angular motion of the mirrors or by environmental conditions; micro-seismic residual motion of the suspensions that changes
accordingly to the weather conditions, etc.
Another way to classify noises relies on their origin:Fundamental noises. These are sources of noise which come from intrinsic
limit of the detector. Examples can be seismic noise not completely lteredby the super-attenuators or shot-noise that limits the photo-diodes output
depending on the level of power hitting them.

Control noises.

These are those noises reintroduced in the system or am-plied by the control loops used to maintain the correct operating point.
They can be due for example to not optimal control lters or to noisy
error signals or actuators.

Technical noise. More in general these are those noises coming not from fundamental limits but rather from the actual implementation of the detector.
Examples are laser frequency and power noises.

Environmental noises, like electro-magnetic elds coupling to the detector

108

CHAPTER 6. NOISE SOURCES AND SENSITIVITY
output or to the coils used for the actuation, or acoustic and seismic
vibration outside the vacuum system that can re-enter the interferometerby means of diused and scattered light.
Several dierent analysis methods can be used to discover and characterize
noise sources:

Coherence analysis. A rst method to have some indications of the probableorigin of noise is to look at the coherence of the dark fringe with othersignals. If there is some kind of linear relation between them this will beshown as a non zero coherence. This method however can not give any
indication on the coupling mechanism. This will be discussed in the next
section

6.2.

Stationary noises analysis. All noises coming from control system, technicallimitations and similar are essentially constant over quite long periods oftime. The main method used to study them is based on linear noise projec-

tions. These are active techniques, in the sense that the characterizationof the noise source passes through a measurement that implies a perturbation of the system.

First the coupling transfer function from a noise

source to the dark fringe is measured, by adding an external perturbation
to the system, and then the result is used to compute the contribution ofthis source to the total noise. Noise budgets are presented in chap. 7.

Slow non-stationary noises analysis. If part of the sensitivity is modulatedwith time scales going from seconds to several minutes and hours, theevolution in time of the noise power can be tracked and correlated toother monitoring channels coming from the interferometer. This method
is passive, since no perturbation is done to the system. It often allows theidentication of the sources of the noise power uctuations. This analysis
is the subject of chap. 8.

Fast transient analysis.

This kind of analysis looks for transient events,

which are short periods of time (fractions of second) when the noise power
is sensibly higher than the usual value. These can be real transient eventsof fake events due to glitches in the data (wrong or missing samples forexample). This is discussed in chap. 9.

6.2 Coherence analysis
If two signals are connected by a linear relation, the coherence

1 between themis not zero. Therefore a non-negligible coherence between for example the dark

1 The denition of coherence is given in appendix D.

6.3. DETECTOR SENSITIVITY

109

fringe and another detector channel in a given frequency band is a hint of a
possible noise source limiting the main signal. This is however not conclusive,
since coherence can not discriminate the direction of the coupling mechanism:supposing that signals A and B are coherent, it is not possible to tell if the noise
in A is coupled to B or if it is the other way wrong, unless some other information
are at disposal allowing to exclude one of the paths (for example if the dark
fringe signal is coherent with an environmental channel like a microphone, it isreasonable to assume the coupling to be from acoustic to dark fringe). Moreover,

A and B can be coherent because both are contaminated by a third source ofnoise C .
Even the lack of coherence is not a guarantee of uncoupled signals. Indeed, if
the noise measured by a channel B couples to the main signal A, but the transfer
function is non-stationary, meaning that it varies strongly with time, both inmagnitude and phase, the averaging process involved in the coherence estimation
can strongly under-estimate the linear relation between the two signal, resulting
in a very small coherence even if the noise is dominant.
It is also important to consider that usually there is not only one source ofnoise that contributes to a given signal. Suppose that the coherence between
a signal A and a possible noise source B is needed, but the rst signal is also
contaminated by another noise source C . The coherence that can be measured
is given by

2

CA+C,B =2|SA+C,B ||SA,B + SC,B |=SA+C,A+C SB,B
(SA,A + SC,A + SA,C + SC,C )SB,B

and supposing that A and C are uncorrelated, as well as B and C , this simpliesto

CA+C,B = CA,B

SAA
SAA + SCC

and therefore if the noise coming from C is dominating A, the measured coherence can be decreased by a large factor, even if A and B are strongly related.
In conclusion, coherence analysis is for sure a valuable tool, since it can give
hints on the origin of the dominant noise, but it is in general not conclusive
and it usually can not give information about noises that are sub-dominant. An
extension of coherence analysis to multiple channels, which can to some extent
provide information on non-dominant noises, is obtained with the multicoherencetechniques [69], which is not discussed here.

6.3 Detector sensitivity
Even if for commissioning purposes it is usually enough to analyze the spectrumof the dark fringe demodulated signal, for data analysis purposes and to gaina better knowledge of the detector sensitivity, it is necessary to convert it to110

CHAPTER 6. NOISE SOURCES AND SENSITIVITY

Figure 6.1: Scheme of the transfer functions involved in the measurement of thedark fringe signal calibration.

Here M is the mechanical transfer function of

the actuators, G the optical transfer function from dierential motion to dark
fringe and F the corrector lter of the DARM loop.

a calibrated gravitational wave signal.

In this way the sensitivity is directly

expressed in terms of dierential strain.
The calibration factor that perform this conversion is frequency dependent:a rst dependence comes from the optical response of the dark fringe signal todierential motion of the end mirrors, which is described by a simple pole atabout 500 Hz; the second dependence comes from the longitudinal DARM control loop, which corrects dierential displacements inside its bandwidth (roughly
100 Hz as discussed in section 5.2).
The standard way to recover the calibration transfer function is to mimic theaction of a gravitational wave, adding a known source of white noise to one of theend mirrors longitudinal correction. This noise will produce both a dierential
and common mode motion of the end mirrors, but the latter is compensated by
the action of the SSFS loop. Knowing the actuator gain, namely the transfer
function from the correction signal to real displacement of the mirrors, it ispossible to extract the needed calibration.

The response of the dark fringe

signal to the injected noise and to the strain given by a gravitational wave are
given by (refer to g. 6.1)

bn

=

bh

=

MG
n
1 + M GF
GLh
1 + M GF

Therefore the calibration can be recovered from the measured transfer function
between the injected noise and the dark fringe signal if the frequency responseof the actuators M is known:
TFh→df = L

1
TFn→dfM

6.3. DETECTOR SENSITIVITY

111

Transfer function [W-1]

10-10

10-11

10-12

10-13 1
10

102

103

104

Frequency [Hz]Figure 6.2: Measured calibration transfer function. The dark fringe spectrum,
multiplied by this transfer function, gives the detector sensitivity.

This calibration includes both the frequency response of the optics and the eect
induced by the DARM control loop. The typical result is shown in g. 6.2. The
calibration is measured below 700 Hz and tted to a model at higher frequencies.
This model includes the cavity pole and the eect of anti-aliasing lters. The
structures below 400 Hz are related to the closed loop suppression of the DARM
control.
The frequency response of the actuators is close to that of a simple pendulum,
given by the last stage of the suspension system (see chap. 2). There are small
deviations from this behavior that can be measured separately, for example
by performing experiments in a free swinging Michelson conguration.

The

origin of these deviation are mainly ascribed to the frequency dependence of theelectronic response.
The entire process of measuring the detector sensitivity takes about 10 minutes and it is used periodically to obtain reference calibrations.
The method discussed so far to measure Virgo sensitivity is based on frequency domain techniques. It is however important, mainly for data analysis
purposes, to obtain a time domain calibrated version of the detector output,
namely a signal that reconstruct as well as possible the dierential strain. Thisis obtained using an adaptive algorithm that compensate on-line for the actionof the longitudinal control system, tracking the amplitude of several calibration

112

CHAPTER 6. NOISE SOURCES AND SENSITIVITY
lines to recover the variation of optical gains. The h reconstruction system is
described in [70].

6.4 History of the detector sensitivity
The sensitivity of the Virgo detector has improved greatly over the past years.
This section briey outlines the story of the Virgo sensitivity, explaining themain actions that resulted in the improvements.

The evolution is shown ing. 6.3, starting from the rst commissioning run in recycling conguration(C5) performed in December 2004. Between C5 and C6 the activity was mainly
focused on reducing the longitudinal control noise, improving the frequency stabilization servo and implementing the automatic alignment system.

The runC6 was the rst one with the angular degrees of freedom controlled using slow
servos in the drift control conguration. This helped a lot in improving the interferometer contrast and long term stability. The automatic alignment system
has then been upgraded to a full bandwidth one, which operated during the C7
run. Other improvements between C6 and C7 were related to the reduction of
actuator noise.
After the C7 run a long shutdown took place, with the two main goals of
replacing the power recycling mirror and the suspended injection bench. The
latter modication was necessary to be able to inject full power into the interferometer without suering from back reected light.

After roughly one yearof commissioning, the rst week-end science run took place, with an improved
high frequency sensitivity due to the increased input power.
Between WSR1 and WSR9 lots of commissioning activities has been performed: acoustic and seismic noise investigations to reduce the impact of scattered and diused light; actuation noise reduction; implementation of the frequency dependent alpha technique to improve the low frequency sensitivity;
modication of the second stage of frequency stabilization; improvements in theautomatic alignment system. All these activities, together with many others,
led to the detector sensitivity at the beginning of the rst Virgo Science Run.
During the run some smaller commissioning activities were performed, obtaining
further sensitivity improvements at low frequency.
After the end of the run many other signicant improvements have been
attained, mainly at low frequencies.

These are the results of improvementsin the longitudinal and angular control systems and of the identication andmitigation of diused light problems.

Figure 6.3:

years, since the rst commissioning run in recycled conguration.

101

10-22

10-21

10-20

10-19

10-18

-17

10

10

-16

10

-15

10-14

102
Frequency [Hz]103

Virgo Design
C5 (Dec. 2004)
C6 (Aug. 2005)
C7 (Sep. 2005)
WSR1 (Sep. 2006)
WSR9 (Feb. 2007)
VSR1 (Sep. 2007)
Last (Dec. 2007)

104

6.4. HISTORY OF THE DETECTOR SENSITIVITY
113

Evolution of the Virgo detector sensitivity during the last three

Strain sensitivity [Hz-1/2]

Chapter 7
Noise budgets

7.1 Linear noise projections
Once the lock is acquired, linear controllers are used to maintain it. Therefore
control noises can be simply modeled with linear relations between the error
signal of the loops and the dark fringe. This makes easy to identify this kind ofnoise, if it is the dominant one, using a simple coherence analysis. It is however
useful to have a method to precisely measure the contribution of a loop noise
to the dark fringe, even in the case of sub-dominant sources.
The linear noise projection technique [71] can be used to characterize thenoise coming from a control loop.

This method is made of two steps.

First,

an external perturbation is added inside the control loop to the error or tothe correction signals.

The noise level and spectral shape must be chosen inorder to dominate the normal noise in the control loop at all the interesting
frequencies.

Using this noise injection it is possible to measure the transfer

function between the error or correction signal and the desired main one (thedark fringe for example).
Indeed, referring to g. 7.1, the eect of noise injections at the level of theerror signal or of the correction can be easily computed:

s =
e =

C
(z0 − nc − F ne )1 + FGG(z0 − nc − F ne )1 + FG

Considering only the case of noise added to the error signal (the other one
being similar), at the frequencies where the added perturbation is dominant the

116

CHAPTER 7.

NOISE BUDGETS

Figure 7.1: Scheme of one control loop and its noise contribution to anothersignal, used in the text for explaining the linear noise projection technique.

previous formulas simplies toCFne
1 + FGGF
e = −ne
1 + FGs = −The error signal after the addition point of the noise is given instead by

e0 =

1
ne
1 + FG

Therefore the transfer functions from the error signal before and after the noise
addition to the dark fringe are given by:

s
e
s
e0

=

C
G

= CF

In a quiet period, when no noise is added, the dark fringe and error signals are
given byG
z0
1 + FGC
squiet =z0
1 + FG
equiet =The eect of the loop is two-fold. It reduces the motion of the controlled degreeof freedom, but it can also introduce additional noise, coming from example
from the sensing part. The residual motion of the d.o.f. can be estimated by

z=

e
G

7.1. LINEAR NOISE PROJECTIONS

117

and its contribution to the dark fringe signals is given by

s=

C
e
G

This is indeed the transfer function measured between the error signal before thenoise addition and the dark fringe. It gives an estimate of the contribution of thed.o.f. residual motion to the main signal. Indeed, inside the active bandwidthof the loop the correction applied is really needed to maintain the working
position of the controlled degree of freedom.

Therefore only the contributionof the residual motion to the dark fringe signal is interesting.

This can be

estimated multiplying the spectrum of the error signal (in quiet condition) withthe absolute value of the transfer function between the point before the noise
addition and dark fringe T Fe→s (f ). This gives the noise projection of the control
loop:

Nsresidual (f ) = |T Fe→s (f )|with noise · |ASDe (f )|without noise(7.1)

where ASD is the amplitude spectral density of the signal.
On the other hand the loop can increase the motion of the d.o.f. throughthe correction it applies:s = −C c = −CF eand the link is given this time by the transfer function measured between theerror signal after the noise addition and the dark fringe.

Outside the active

bandwidth of the loop, the applied correction has no eect in reducing theresidual motion, but it can introduce additional noise in the dark fringe signal.
This can be estimated in a way similar to eq.7.1, using this time the transfer
function between the point after noise addition and the dark fringe:

Nsloop (f ) = |T Fe0 →s (f )|with noise · |ASDe (f )|without noise(7.2)

Which one of the two noise projections should be used depends mainly on
whether the frequencies one is interested in are inside or outside the active
bandwidth of the loop.
Finally, it is worthwhile saying that even if in principle it would be muchbetter to measure the noise transfer function with the feed-back loop open, toavoid suppression of the added noise, it is seldom possible to carry out sucha measurement in complex systems like Virgo, since if one of the controls is
not active, the interferometer is usually not stable at the operating point, andtherefore the measurement, even if possible, is not performed in the correctstate.

7.1.1 Subtle points and limits of linear noise projections
Even if this method for tracking the contribution of a control loop works wellin most of the cases, there are some subtle points and some situation in which
it might give wrong results. This section describes the most common issues.

118

CHAPTER 7.

NOISE BUDGETS

In principle there should be no dierence between computing a noise projection using the error or the correction signal, since in between the two there
is only the control lter. However since both signal are somehow acquired andconverted to digital format by some electronic board, they might be dierentlyaected by unwanted noise. For example the error signal might be limited bysome sensing noise ne (for example shot noise, electronic noise, ADC noise) andtherefore in some bands it might not describe correctly the residual motion of
the system. If this happens outside the control bandwidth of the loop, this noise
is usually re-injected into the system and if the corrector lter is not cleverly
designed, it might increase the noise in the dark fringe.
A similar case is that of a loop with very high gain: inside its bandwidth theerror signal is kept close to zero with great accuracy. The residual uctuations
might be smaller than the sensing noises ne (typically ADC noise). In this casethe error signal is no more a good estimate of the residual motion of the system,
giving only an upper estimate. A solution to this limitation is to have anotherindependent signal, sensitive to the same degree of freedom but not used for the
feed-back. This avoids the problem of large loop suppression, and gives a goodprojection if the signal is not limited by sensing noise. This kind of solution isseldom applied mainly because it is not usual to have multiple signals than canbe used for monitoring the same degree of freedom. Moreover, if the out-of-loop
signal to be used for the projection has a better sensitivity than the one used
for the feed-back, the role of the two will be soon switched: the most sensitive
signal will be used as error signal, while the other one will be left out-of-loop,
but likely blinded by sensing noise.
In summary whenever the error signal is dominated by some kind of sensing
noise ne , the corresponding noise projection must be considered with care.
Another dierence between projections made with error or correction signals
becomes clear when actuation noise nc is considered: the correction sent to theactuators might be dominated at some frequencies by actuator noise. A typicalexample is DAC noise: since most of the control loops are digital, the output of
the correction lter must be converted to an analog signal by a DAC board; if
the amplitude of the correction is too small it can fall below the level of noise
introduced by the conversion. In this case the projection of the correction signal
gives a lower estimate of the real noise.

For this reason it is always a betteridea to use, if possible, the error signal of the loop for the projection, since ittakes automatically into account the possible excess of noise re-injection due toactuation noise.
Moreover, while the error signal, if well reconstructed, gives a real estimateof the residual motion, the correction, inside the control bandwidth at least,
contain also information on other noise sources that enters after the signal readout and that are corrected by the loop: this means that the correction might be7.2. SUMMARY OF VIRGO NOISE BUDGET DURING VSR1
much higher than the real motion of the system. A simple example is the locking
of a single cavity: the error signal is taken from the transmission photo-diodeand the actuation is done by applying a force to the end mirror. If the input
mirror is strongly excited, the eective length of the cavity would have large
variations. Since the loop has no way to damp the input mirror, it simply actswith a large correction applied to the end one, which is moved in order to keep
the distance between the two mirrors constant. Therefore even if the correction
is large the residual motion and the re-injected noise are small.
The last point to discuss is the eect of multi-dimensional control systems,
which is the case of interferometric detectors. If the loops that control dierent
degrees of freedom are not perfectly decoupled (which is usually the case), thesame noise can be counted multiple times in dierent loops. For example if theerror signal used for controlling A is contaminated by a large amount of noise
coming from residual motion of B , when projecting its noise also a contribution
from B will be considered.
when projecting B itself.

But this contribution is also taken into accountIn conclusion, the total sum of all noise projection

can give an under or over estimate of the total amount of noise, depending on
the relative coupling of noises in the various loops.

7.2 Summary of Virgo noise budget during VSR1
The typical sensitivity and the total noise budget during the rst two months of
the Science Run are shown in g. 7.2. The various contributions are explained
in details in the following sections.
At low frequencies, below roughly 50 Hz, the limiting source of noise comesfrom longitudinal controls (sec. 7.3), with a contribution from angular noise(sec. 7.5) between 30 and 40 Hz.

Between roughly 50 and 100 Hz the noise

comes from dierent sources, mainly actuation (sec. 7.3.1) and noise comingfrom eddy currents in the metallic reference masses. Between 100 and 1000 Hz,
the noise oor is well described by shot noise (sec. 7.10) and still actuator andeddy currents noises, but there are many structures not completely understood.
Part of these has been identied and cured during the last month of the sciencerun. They were originated by longitudinal noise in the IMC cavity caused bynoisy piezo-electric actuators mounted on the suspended injection bench. The
most likely origin of the remaining structures is environmental noise (sec. 7.11)
that can couple through diused light or input beam jitter (sec. 7.6.1). Above
1 kHz the main sensitivity limitation comes from the dark fringe shot-noise andin a small frequency band around 6 kHz from frequency noise (sec. 7.4). Thisbump was also caused by the noisy piezo-electric actuators and it disappeared
after the intervention.

119

CHAPTER 7.

NOISE BUDGETS

10-23 1
10

10-22

10-21

10-20

10-19

10-18

102

Frequency [Hz]103

Sensitivity
Total
Longitudinal
AngularFrequency
Shot
Phase
Electronic
Actuation
Eddy currents
Design104

120

Sensitivity [Hz-1/2]
Figure 7.2: Summary of the detector noise budget for the conguration runningduring the rst two months of VSR1. Refer to the text for more details.

7.3. LONGITUDINAL CONTROL NOISE

121

7.3 Longitudinal control noise
Since the noise of DARM is the gravitational channel itself, and the eect of
the DARM loop is taken into account during the calibration of the dark fringe
signal, the contributions of longitudinal control noise that are relevant are thoseof MICH, PRCL and CARM loops.
An automatic procedure [72] has been implemented and periodically usedduring all the run to measure the longitudinal noise budget.

This procedure

performs the required noise injections, computes the transfer functions from theerror signals after the noise addition to the dark fringe and uses them to obtainthe control noises projection. Since the error signals are used, the projections
are valid only if they are not limited by sensing noise.

This is true for theerror signals used for the longitudinal sensing and control system.

Moreover,the computed projections include the eects of actuation noise.
A typical result of the longitudinal noise budget measurement is shown ing. 7.3.

Both the coherence and the transfer function from the longitudinalloop error signals (after the adding point of the noise) to the dark fringe arecomputed. The transfer functions are used, at frequencies where the coherenceis good, to project the normal level of noise.

The typical results for the rst

part of the science run are shown in g. 7.4. The noise coming from the PRCL
loop is about a factor 10 below the actual sensitivity, but it is strongly variablewith time. The CARM control noise is one of the two concurrent sources that
limits the sensitivity below 20-30 Hz. The MICH control noise is very close to
the sensitivity below 20 Hz and limiting between 20 and 40 Hz roughly. The
steep roll-o of the control lter above about 50 Hz ensures that this loop does
not aect the sensitivity above that frequency.
In order to improve the low frequencies sensitivity, the control noise comingfrom CARM and MICH had to be reduced. For what concerns the rst degreeof freedom, this has been obtained in two steps. First, any mis-balancing of theactuation at the end mirrors results in a leaking of the CARM correction to
the dierential mode and this couples directly with the dark fringe signal. By
simply tuning the balancing of the CARM actuation of the end mirror it has
been possible to reduce the coupling of this control noise by roughly a factor
ten. The second action has been the implementation of a better control lterwith a steeper cut-o at about 9 Hz (shown in g. 5.17).

The total eect ofthese two modication brought the CARM control noise well below the actual
sensitivity, and even below the design one above 30 Hz.

7.3.1 Actuation noise
An important contribution to the noise in the dark fringe signal comes from
actuation noise. For what concerns the beam splitter and power recycling mirror

122

CHAPTER 7.

NOISE BUDGETS

1
0.9
0.8
0.7
Coherence

0.6
0.5
0.4
0.3
0.2
MICH
PRCL
CARM

0.1
0
1
10

2

3

10
Frequency [Hz]

10

1012
MICH
PRCL
CARM
DARM

10

10

108
6

Transfer function

10

4

10

102
100
10-2
10-4
10-6
10-8
101

102
Frequency [Hz]103

Figure 7.3: Example of the measurements performed to compute the longitudinal noise projections. The top plot shows the coherence between the error signals
and the dark fringe during the corresponding noise injection. The bottom plotshows the obtained transfer functions.

7.3. LONGITUDINAL CONTROL NOISE

123

10-18

Strain sensitivity [h/rHz]

10-19

10-20

10-21

10-22

10-23

10-24 1
10

MICH
PRCL
CARM
Sensitivity
Design102
Frequency [Hz]103

Figure 7.4: Projection of longitudinal noise in the dark fringe signal, calibrated
in units of strain sensitivity, measured during VSR1. The color of the lines isgrey when the measurement of the corresponding transfer function showed low
value of coherence, meaning that the projection is not meaningful. The thickblack line is the measured sensitivity of the detector while the thin black one isthe design one.

this is already taken into account by the longitudinal budget, but it is stillinteresting to understand what fraction of the noise introduced by the MICH
and PRCL loop can be ascribed to actuation noise. Instead the contribution of
the end mirrors actuators must be considered separately.
A scheme of the coil driver low-noise section, responsible of converting thedigital correction to the actual current owing in the coil actuators, is shownin g. 7.5. The main source of actuation noise comes from the digital-to-analog
converter. The boards used in Virgo have a peak-to-peak range of 10 V and aoor noise that can change accordingly to the input signal. For null or sinusoidal√

input it was measured to be of about 300 nV /Hz . If a signal that mimic thecorrection applied during normal operations is used, the noise introduced has a√f −1/2 shape and a value of about 1.5µV / Hz at 100 Hz.

In the science mode low noise conguration the maximum value of the force
that it is necessary to actuate using the reference mass coils is small.

This

allows the use of two tricks to reduce the impact of the DAC noise. The correction is rst digitally amplied (by a factor 600 for the end mirrors) before the124

CHAPTER 7.

NOISE BUDGETS

Figure 7.5: Scheme of one of the coil drivers used to convert the longitudinalcorrection to the current to be applied to the coil actuators.

DAC, thus enhancing the signal-to-noise ratio.

The amplication is compen-sated analogically adding a resistor in series with the coil. Moreover, the highfrequency part of the correction is amplied by emphasis lters, compensatedafter the DAC by analog counterparts. In this way a further improvement of
the signal-to-noise ratio of the correction is obtained at high frequencies. For
the end mirrors these lters are made by a couple of real poles at 3 and 20 Hz
and a couple of real zeros at 80 and 100 Hz. A similar conguration is also used
for the BS, while the PR has no emphasis/de-emphasis lters.
The contribution of the DAC noise to the dark fringe signal can be estimatedas follows. First, the DAC noise can be expressed in terms of equivalent noiseat the level of the longitudinal correction dividing it by the emphasis lter andthe gain:nzCorr =nDAC
g Femph(7.3)

The corresponding mirror displacement can be computed using the known frequency response of the mechanics:

nz = FmechnDAC
g Femph(7.4)

This noise estimate refers to one single coil. Therefore for each mirror the total
displacement noise is given by the incoherent sum of the one coming from eachcoil (there are two of them for end mirrors and power recycling, and four forthe beam splitter). Assuming the same level of noise for each coil and similarlters, this gives an additional factor√

2 for end mirrors and power recycling,

and a factor 2 for the beam splitter.
For the two end mirrors, the displacement noise translates directly in a
DARM motion and it can be expressed in units of sensitivity dividing by the7.4. FREQUENCY NOISE

125

Magnitude

100

10-1

-2

10

10-1

100

101
Frequency [Hz]

102

103

Figure 7.6: Typical shape of the analog de-emphasis lters used to reduce theactuation noise.

This allows a reduction of more than a factor 100 at higher

frequencies.

arms mean length and by the cavity response:p
hend =

n2zN E + n2zW E
LFcav

(7.5)

The displacement noise from the beam splitter and the power recycling mirror
can be recombined in terms of MICH and PRCL noise, using the geometricaldriving matrix described in sec. 3.5.

The PRCL displacement noise can bepropagated to the gravitational channel using the measured optical matrix (seesec. 5.5), since the eect of the noise subtraction technique can be neglected.
The same is not true for the MICH noise, since α can give cancellation factors up
to 100 in the relevant frequency band. The predicted actuation noise projectionuses a theoretical factor to convert from MICH motion to dark fringe signal[73, 74]. The real transfer function can be obtained taking into account the eectof the noise subtraction or more easily with a direct measurement injecting noise
directly at the level of the BS actuators. The two projections are compatible.
The level of actuation noise in the Science Run conguration is shown ing. 7.7. The total is about a factor 2 below the detector sensitivity between 10
and 80 Hz. This plot also shows an estimation of the noise due to eddy currents
induced in the metallic reference mass [75]. This is an intrinsic limitation forthe current payload conguration and it is indeed close to the actual sensitivity.
The long term plans for Virgo+ include the replacement of the metallic reference
mass with dielectric ones.

126

CHAPTER 7.

NOISE BUDGETS

-18

10

Sensitivity
End mirrors
BS mirror
PR mirror
Total
DesignEddy currents-19

Sensitivity [Hz

-1/2

]

10

-20

10

-21

10

-22

10

-23

10

1

2

10

10
Frequency [Hz]Figure 7.7: Projection of actuation noise in the gravitational wave signal. The
total noise is about a factor 2 below the actual sensitivity up to roughly 80 Hz.
An upper limit for the noise coming from reference mass eddy currents is also
shown.

7.4 Frequency noise
Since the unitary gain frequency of the second stage of frequency stabilization
is higher than the observational bandwidth, a noise budget for frequency noisemust take into account the coupling of residual frequency noise to the dark fringe
signal. The measure is performed by adding noise to the error signal of the SSFS
and measuring the transfer function from the signal before the addition to thedark fringe.

The result is shown in g. 7.8.

This transfer function does notgive directly the optical coupling between the SSFS error signal and the dark
fringe, since it is necessary to correct for the eect of the DARM loop which
reduces the eect of frequency noise like any external disturbance. If a constanttransfer function is considered as the real coupling of frequency noise between
the two signals, the eect of the DARM loop can be easily computed using
the open loop transfer function described in sec. 5.2.

The result is shown ing. 7.8. The agreement with the measurement is very good above about 30 Hz.
Below this frequency the measured transfer function shows some un-expected
sharp features, which are likely the eect of cross-coupling with some otherlongitudinal control loops.
In any case, using the modeled transfer function, the residual noise in theerror signal can be projected in the dark fringe. The result is shown in g. 7.9.
The contribution of frequency noise to the detector noise is negligible for almost

7.5. ANGULAR CONTROL NOISE

127

10-6
MeasuredModel-7

Magnitude [a.u.]

10

10-8
10-9
10-10
10-11
150

Phase [deg]

100
50
0
-50
-100
-150
10

100

1000

10000

Frequency [Hz]Figure 7.8: Measured transfer function for frequency noise, from the error signalof the second stage of frequency stabilization to the dark fringe demodulated
signal. The measurement agrees very well with the expected model (refer to thetext for details).

all frequencies, excluding a narrow region between 5800 and 5950 Hz, where itis the limiting source of noise.
The level of frequency noise has been strongly decreased by the modicationin the control of the suspended injection bench piezo-electric actuators.

In

particular, the noise bump at about 6 kHz has completely disappeared.

7.5 Angular control noise
As explained in chapter 4 all angular degrees of freedom of the interferometerare controlled by using global servo systems, most of them with a bandwidthof the order of 3 Hz. The control noise coming from these global loops can be
estimated using an automatic procedure similar to that implemented for thelongitudinal degrees of freedom. The angular control of the two input mirrors
and of the θy degree of freedom of the beam splitter are controlled using only
local controls.

Since the error signals for these loops are local, a coherence

128

CHAPTER 7.

NOISE BUDGETS

10-17
Sensitivity
Frequency noise
Design-18

10

Sensitivity [Hz-1/2]

10-19
10-20
10-21
10-22
10-23
10-24
10-25 1
10

102

103

104

Frequency [Hz]10-20
Sensitivity
Frequency noise

Sensitivity [Hz-1/2]

-21

10

10-22

-23

10

Figure 7.9:

5500

5600

5700

5800
5900
Frequency [Hz]

6000

6100

6200

Projection of the residual frequency noise into the gravitational

channel signal, during VSR1.

The bottom plot is a zoom around a narrow

region where this noise is actually limiting the sensitivity.

7.6. OTHER SOURCES OF CONTROL NOISES

Figure 7.10: Projection of angular noise in the dark fringe signal, calibrated
in units of strain sensitivity, measured during VSR1. The color of the lines isgrey when the measurement of the corresponding transfer function showed low
value of coherence, meaning that the projection is not meaningful. The thickblack line is the measured sensitivity of the detector while the thin black one
is the design. This angular noise budget includes only those degrees of freedom
controlled in automatic alignment conguration.

analysis is enough to see if they are contributing to the dark fringe noise. This
is not the case. It might be anyhow interesting to add them to the noise budget,
but this has not been done yet.
The noise budget for the global angular control system, measured in the
VSR1 conguration, is shown in g. 7.10.

The end mirror common mode θxcontrol is one of the two dominant sources of noise in a small region between 30
and 40 Hz. Excluding this d.o.f., all other noise contributions are even below
the design sensitivity above 30-40 Hz. At lower frequencies they are one of thesources that concur to give the measured noise, even if not the dominant one.

7.6 Other sources of control noises
There are many other control loops working during the normal detector operation of the Virgo interferometer, and most of them do not aect the sensitivity
with control noise.

The most critical ones are perhaps those linked with the

129

130

CHAPTER 7.

NOISE BUDGETS

10-5

Transfer function [V-1]

10-6
10-7
10-8
10-9
10-10
MDh
MDv
MUh
MUv

10-11
10-12 110102
Frequency [Hz]Figure 7.11:

Transfer functions from the four BMS correction signals to thedark fringe. Several resonant structures are visible above 100 Hz.

injection system, mainly with the angular and length control of the input modecleaner. These are mainly relevant if the modulation frequency is not well tuned
with respect to the IMC length, as explained in sec. B. However, even by articially increasing the noise in the loops more then a factor 10 no eect was
visible in the dark fringe signal. Therefore it is safe to conclude that these loops
are not relevant.
There are at least other two sources of control noise that has proved to be
relevant.

The rst is the Beam Monitoring System (see sec. 2.2) which cangenerate input beam jitter than couples with the dark fringe signal [76]. The
second source is linked with the local controls of the suspended detection bench.

7.6.1 Beam Monitoring System
The BMS system (see sec. 2.2) uses two quadrant photo-diodes placed on theexternal injection bench to extract error signals for stabilizing the position of
the beam before the IMC. The four error signals (two quadrants, both verticaland horizontal signals) are combined together with a reconstruction matrix tocontrol two mirrors mounted on piezo-electric actuators.
The projection of BMS control noise into the dark fringe signal has been
performed adding noise between roughly 20 and 300 Hz to each of the four

7.6. OTHER SOURCES OF CONTROL NOISES

Figure 7.12: Projection of the BMS correction signals in the detector sensitivity.
The dotted lines represent the projection of the DAC noise.

correction signals and measuring the transfer functions to the dark fringe. The
results of this measurements are shown in g. 7.11. Above 100 Hz all transfer
functions show several structures, which are related to internal resonances ofthe mirror mounts and of the actuators: they are mainly at 145 Hz, 170 Hz,
190 Hz, 230 Hz and 245 Hz.
Using these transfer functions it is possible to compute the projection of thefour correction signals in the detector sensitivity (see g. 7.12). However it isnecessary to take into account DAC noise, which turns out to be much higher
than the requested correction signal above 100 Hz (the DAC noise is dottedin g. 7.12). However even considering the DAC noise, it might seem that the
BMS contribution is completely negligible.
However, projecting the correction signal it is not possible to correctly recover the contribution of any source of actuation noise that comes after theread-out of the correction. For the BMS system this means not counting any
additional motion of the beam steering mirrors: for example noise introducedby the piezo-actuators themselves or seismic noise that can excite directly theresonant modes of the mirror mounts. The only way to take into account thecontribution from these sources of noise is to compute the projections using theerror signals as starting point. The transfer functions from error signals to dark

131

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NOISE BUDGETS

10-16
Sensitivity
MUh
MUv
MDh
MDv

10-17

Sensitivity [Hz-1/2]

10-18
10-19
10-20
10-21
10-22
10-23
10-24 110102
Frequency [Hz]10-19
Sensitivity
MUh
MUv
MDh
MDv

Sensitivity [Hz-1/2]

10-20

10-21

10-22

10-23

10-24
100

Figure 7.13:

150

200
Frequency [Hz]

250

300

Projection of the BMS error signals in the detector sensitivity.

The lines are grey where the signals are dominated by sensing noise. At thosefrequencies the projections are not meaningful. The bottom plot is a zoom athigher frequencies.

7.6. OTHER SOURCES OF CONTROL NOISES
fringe can be easily computed knowing the one already measured and the matrix
used to reconstruct the corrections. These projections are however valid only at
frequencies where the error signals are not dominated by sensing noise. Its levelcan be easily measured when there is no light impinging on the photo-detectors
and assuming that the level of this dark noise does not change with the amountof light hitting the detector. For the quadrant photo-detectors used for the BMS
system unfortunately the dark noise is almost dominant above few tens of Hz.
Only some small frequency regions, corresponding to the mounts resonances,
show a signal signicantly higher than the dark noise.
The result of projecting the BMS error signals is shown in g. 7.13. Where
the lines are grey the error signals are dominated by sensing noise, and therefore
the projections are not meaningful. However there are some frequency regions,around the BMS mount resonances, that are meaningful: in these regions thebeam jitter introduced by the BMS system is very close to the sensitivity. Thisis consistent with a coherence analysis: the BMS error signals are coherent withthe dark fringe around the resonant frequencies.

Moreover they are coherent

with seismic and acoustic sensors on the optical benches, proving that the originof this noise is likely seismic motions of the mirror mounts. A further proof thatbeam jitter noise coming from the BMS system is limiting the sensitivity comesfrom the analysis of fast non-stationarities or glitches (see chap. 9): around theresonance frequencies of the mounts, several glitches are found in coincidencewith ones in the BMS error signals.
In conclusion, even if a complete noise projection for these degrees of freedom
is not possible because the error signals are dominated by sensing noise, there arestrong indications that beam jitter into the input mode cleaner is contributing
to the dark fringe noise in the region between 100 and 300 Hz.

7.6.2 Suspended Detection Bench
The position of all suspended benches must be controlled, at least at low frequency, to maintain their correct orientation.

The local control loops imple-

mented for the suspended detection bench have a unitary gain frequency of
about 1 Hz, therefore any correction sent to the coil actuators above this frequency can introduce spurious motions of the bench. These couple to the dark
fringe signal in a complex and not completely understood way: the main candidate is some form of clipping of the main beam.
The control noise coming from the output bench was indeed limiting thesensitivity in a quite wide region between 20 and 40 Hz, even if the coherence
between the dark fringe signal and bench correction was quite poor, since as
expected the coupling of such noise can be strongly modulated in amplitude
and phase.
After the suspicion of noise coming from the bench controls arose, improve-

133

134

CHAPTER 7.

NOISE BUDGETS

10-2
Old
New

Magnitude [a.u]

10-3
10-4
10-5
10-6
10-7
10-8
10-9
200

Phase [deg.]

150
100
50
0
-50
-100
-150
-200
10-1

100

101
Frequency [Hz]

102

103

10-2
10-3

Spectum [V Hz-1/2]

10-4
-5

10

-6

10

10-7
10-8
10-9
Old txNew tx
DAC noise

-10

10

-11

10

-1

10

0

1

10

10

2

10

Frequency [Hz]Figure 7.14: Improvements in the suspended detection bench controls. The top
plot shows a comparison between the two corrector lters, while the bottom one
shows the corresponding reduction in the correction signal. These plots refers
to the θx degree of freedom. The results for the others are similar. The pinkline shows the level of DAC noise.

7.7. LASER INTENSITY NOISE

135

10-18

Sensitivity [Hz-1/2]

10-19

10-20

10-21

Before
After
10

20

30
40
Frequency [Hz]50

60

Figure 7.15: Improvements in the detector sensitivity after the implementationof the improved control lters for the suspended detection bench.

ments were introduced implementing more ecient control lters, capable ofmaintaining the same accuracy in the bench control while reducing the correction above 10 Hz by at least a factor 10 (see g. 7.14). The computed correction
actually falls below DAC noise above about 30 Hz.
The eect on the detector sensitivity is quite impressive and it is shown ing. 7.15. The most striking structure that has disappeared is a quite big line at
28 Hz with several sidebands around it. This is a further proof that the noise
coupling was strongly modulated.

From the result it is clear that the benchcontrol was a quite strong limiting source of noise in the entire region between
20 and 50 Hz.

7.7 Laser intensity noise
Fluctuations of the input laser intensity have a component at the modulation
frequency which is not stabilized by any control system, but the intrinsic relative
intensity noise of the free laser at 6 MHz is of the order of

δP6 M Hz< 9 · 10−9 Hz−1/2
P

(7.6)

and this does not give a relevant contribution to the dark fringe noise [73].
The component at low frequency of the laser intensity noise, inside the detector observational bandwidth, is much higher for the free laser and therefore
it is stabilized using the power stabilization loop described in sec. 2.2.

This

136

CHAPTER 7.

NOISE BUDGETS

105
Power Stab. OFF
Power Stab. ON
Dark noise

Spectrum [V Hz-1/2]

104

103

102

101

100

10-1 1
10

102

103

104

Frequency [Hz]Figure 7.16: Plot of the power stabilization error signal in three dierent conditions: when the loop is on, when it is o, and when no light is reaching thephoto-detector. This last curve shows the dark noise of the detector.

intensity noise can couple to static or RMS uctuations of the DARM degree of
freedom around the operating point. The simplest possible model gives [73]:Sh 'RM SDARM SδP
L
P

(7.7)

where L is the mean arm length, P the dark fringe power, RM SDARM gives
the RMS uctuations of the dark fringe signal expressed in terms of equivalent
DARM noise and SδP is the laser intensity noise spectrum. The contribution
coming from the input laser power uctuations can be estimated substituting
in the above equation the dark fringe power with the input one (g. 7.17).
So far no complete measurement of the real transfer function from power
noise to the dark fringe has been performed, mainly due to the lack of a goodout of loop sensor measuring the residual intensity noise when the high frequency
power stabilization loop is engaged. Indeed, as shown in g. 7.16, the very highgain of the loop reduces the noise in the error signal below the level of darknoise. For this reason it can not be used reliably to perform a noise projection.
However, a simple upper limit to the contribution of power noise can be obtainedby comparing the dark fringe spectrum when the stabilization is on and o. The
increase of noise in the sensitivity is almost null: only between 300 Hz and some
kHz there is an increase of the order of 10%. Therefore an upper limit can be
set by lowering the dark fringe spectrum by an amount given by the ratio of
the power stabilization error signal when the loop is on and o. The results isshown in g. 7.17, when both the dark noise and the spectrum of the signal in7.8. OSCILLATOR PHASE NOISE

137

10-17
Sensitivity
Upper limit (signal)
Upper limit (dark noise)
Design10-18

Sensitivity [Hz-1/2]

10-19
10-20
10-21
10-22
10-23
10-24
10-25
10-26 1
10

102

103

104

Frequency [Hz]Figure 7.17: Upper-limit to the contribution of laser intensity noise to the sensitivity. The two curves uses the spectrum of the power stabilization error signal
with the loop closed and its dark noise to obtain the estimate.

normal condition are used. This shows that the residual laser intensity noise isalmost everywhere a safe factor below the actual and design sensitivities.

7.8 Oscillator phase noise
If the reference oscillator that generate the 6 MHz signal used for the demodulation process has a small amount of phase jitter, the in-phase demodulated
signal of the dark fringe can be contaminated by a small contribution from thequadrature one. This can be estimated as [73]:Ss ' δϕ · ACq · TFdf →h(7.8)

δϕ is the spectral density of phase noise, ACq is the RMS power inthe quadrature signal and T Fdf →h is the calibration transfer function, that
where

converts the dark fringe signal to the gravitational wave strain.

The level ofphase noise of the signal generator used presently in Virgo has been estimated√to be about 0.15µrad/

Hz , using data from a period when big uctuations of

the quadrature signal were present. In this way a linear relation between thelevel of high frequency noise and the value of the dark-fringe quadrature signal
has been found.
This gives the estimation of contribution of phase noise to the sensitivity
shown in g. 7.18. The projection is safely below the actual sensitivity and in

138

CHAPTER 7.

NOISE BUDGETS

10-17
Sensitivity
DesignShot noise
Phase noiseElectronic noise

Sensitivity [Hz-1/2]

10-18

10-19

10-20

10-21

10-22

10-23 1
10

102

103

104

Frequency [Hz]Figure 7.18: Summary of noises relevant at high frequencies: shot noise, oscillator phase noise and dark fringe photo-detector electronic noise.

any case no indication of phase noise contamination of the dark fringe signal
has been found so far. For this reason there has not been strong eorts in the
direction of a full measurement of the coupling of phase noise.

7.9 Electronic noise
The name electronic noise usually indicates all the noises introduced by thephoto-diode itself and by the entire read-out chain.

Therefore it can include

dark noise of the photo-diode, additional noise introduced by the read-out anddemodulation board and nally also ADC noise. The sum of all these noises areeasily estimated by measuring the output signal during a period when there isno light impinging on the detector. The implicit assumption is that this noise
does not change signicantly when there is light.
The most relevant electronic noise is that coming from the dark fringe
diode itself.

It was measured, with no light impinging on the diode, to be

4 · 10−11 W Hz −1/2 .

The corresponding contribution to the strain sensitivity,

shown in g. 7.18, is well below the actual detector sensitivity.

Contribution

from other photo-diodes can enter through longitudinal and alignment loops.
They are therefore already taken into account by the corresponding noise budgets and in any case they are negligible.

7.10. SHOT NOISE

139

10-20

1
All onWI off
WI BS off
WI BS NI off

All onWI off
WI BS off
WI BS NI off

Coherence

Sensitivity [Hz

-1/2

]

0.8

0.6

0.4

10-21

0.2

0
80

85

90

95

100
105
Frequency [Hz]

110

115

120

80

85

90

95

100
105
Frequency [Hz]

110

115

120

Figure 7.19: Example of noise of electro-magnetic origin. The eld was generated by simple power supplies used for the local control illuminators. The twoplots show the eect of switching them o on the sensitivity (left) and on thecoherence between dark fringe signal and a magnetic probe (right).

7.10 Shot noise
The actual limitation at high frequency is shot noise. Assuming that the power
reaching the dark fringe is dominated by the contribution of the sidebands, theshot noise level can be estimated as a function of the power measured on theB1 diode [73, 74, 77]:r
Sh = TFdf →h ·3√ p
2 2hP νPB1
2

!
(7.9)

where T Fdf →h is the calibration transfer function and PB1 is the power impinging on the dark fringe photo-diode.

This gives the result shown in g. 7.18,

which explains well the oor noise level above some kHz.

7.11 Environmental noise
7.11.1 Electro-magnetic noise
The electromagnetic elds generated by electrical and electronic components
can couple to the main detector output through some electro-magnetic pickup in the read-out or actuation electronic, or through direct coupling to themagnets used for the mirror control. This kind of noise is very dicult to ndand debug, since the electro-magnetic eld can change signicantly in space andit is usually very dicult to nd any coherence between the dark fringe signal
and magnetic probes.

Even if a coherence is found, therefore indicating thatthe source of the noise is electro-magnetic, it is usually dicult to identify theorigin of the disturbance and to cure it.

140

CHAPTER 7.

NOISE BUDGETS

An example of noise coming from elds irradiated by electronic devices isshown in g. 7.19. The source was tracked down to power supply boxes used
for the local control illuminators. These boxes were placed close to each tower,
just outside the vacuum tanks. A rst experiment was performed by switchingo the boxes one by one. As clearly visible in g. 7.19 the largest noises camefrom the power supplies placed inside west input and beam splitter ovens.
This noise coupled directly with the magnets used for the mirror control.
This high coupling is explained by the fact that for the two input mirrors andfor the beam splitter these coils have been attached with the wrong relative
polarity: therefore the total magnetic dipole momentum is signicantly dierent
from zero.
This source of electro-magnetic noise was nally strongly reduced by displacing of several meters the power supplies from the towers.

7.11.2 Acoustic noise
Even if all the mirrors and the main optics of the detector are suspended to seismic attenuators and placed in vacuum, all injection and detection components
are mounted on benches in air and sitting on the ground. Seismic motions of
the benches and of the components mounted on them can introduce noise into
the dark fringe. To distinguish from the seismic motion of the mirrors, whichis completely ltered by the super-attenuators (see sec. 2.4), this noise is usually called acoustic noise, since it is important mainly in the acoustic range of
frequencies (between some Hz and some thousands Hz) and because one of themain origin is real acoustic noise close to the benches.
A mirror or a lens mounted on an optical bench can vibrate and introduce
a phase modulation on the beam reected or transmitted. For a moving mirror
the eect can be described in terms of the reected electric eld:x(t)ER (t) = e2ikx(t) = e4πi λ(7.10)

where x(t) is the mirror displacement with respect to a xed reference, as afunction of time. Its spectral component are usually concentrated in the acoustic
region.

The depth of the phase modulation depends on the amplitude of themirror motion. If the amplitude of x(t) is much smaller than the laser wavelength, the exponent in eq. 7.10 is small and the eect is roughly linear. The
noise introduced in the detector output will resemble the spectral structuresof the mirror motion which usually is larger in correspondence of mechanicalresonances of the mount or of the optical bench.

On the other hand, if theamplitude of the motion is larger than one wave-length, the non-linear natureof eq. 7.10 is dominant and the detector noise spectrum can be very dierent
from the one of the optics motion.

7.11. ENVIRONMENTAL NOISE

141

The modulation depends also on the laser frequency. This means that carrierand sidebands will be aected in a dierent way:
ω

REF
ECAR

= e2i c t ECAR

REF
ESB

= e2i c t e±2i c t ECARωΩ

The common term between the three elds is not relevant for the main interferometric beams: it cancels out in any power or demodulated signal measured by aphoto-detector, since they depends on the product of the eld with its complex
conjugate (see for example eq. A.2). The additional dephasing of the sidebandsaects instead all demodulated signals. Assuming a mirror motion of the orderof one wave-length for reference, the dephasing has an amplitude of the order
of 10

−8

radians. The eect on the demodulated signal can be estimated fromeq. A.3 for example:


ACP (t) ∼ cos2Ω
x(t) ACp0 (t) ∼ 1 − O(10−16 ) ACp0 (t)c0where ACP is the signal that would be there without the eect of the moving
optics. This contribution is usually neglegible.
The common de-phasing becomes important if the reected beam is a spurious one which might enter again the interferometer with a superposition ofGaussian modes that may have large low order components.

These can res-onate and be transmitted to the dark port with some small amplitude and aphase carrying information about the mirror motion:

EDF = E0 (t) + εeiφ(t) E1 (t)being E0 the main eld amplitude and E1 the one of the original beam before
scattering. This additional component will add spurious noise to the detector
output.

A typical example of this situation comes from the arm cavity end

benches: if the transmitted beam hits a lens perpendicular to it, part of the beamis scattered back inside the cavity, with exactly the geometry of the resonantmode.
Of course the amount of noise reintroduced by these mechanisms depends
directly on how much the optical elements move. The origin of these motions
can be tracked down to essentially two categories: seismic motion of the ground,both of natural and human origin or acoustic noise that directly excites lenses
and mirrors.

In both cases it is possible to reduce the motion having better

seismic and acoustic isolation of the benches or making the mounts more rigid.
In particular it is important to avoid that an optical element has a resonancebelow 1-2 kHz since this might enhance the motion at that frequency.

142

CHAPTER 7.

NOISE BUDGETS

Analysis techniques
Noise from diused and scattered light is very dicult to identify by meansof coherence analysis with acoustic and seismic sensors.

The main reason is

the highly non-stationary and non-linear coupling. Moreover, the real motionof the scattering element might be very dierent from what is recorded byseismometers and microphones, unless they are placed exactly on top of it. In
the worse case one can imagine for example to have the seismic sensor placedon a node and the optical element on a maximum of the resonance pattern ofa bench: therefore the element can move with a large amplitude and the sensor
will not be able to detect it.
The simplest, and often the only, way to study the coupling of acoustic noise
is to increase as much as possible the environmental disturbances and look at
the eect on the dark fringe signal. One possibility is to use loudspeakers toinject acoustic noise at dierent places. This method has usually been veryeective in exciting almost all optical components. However it is intrinsicallynot selective: it is not possible to clearly identify which is the particular object
that is coupling the noise to dark fringe. Another similar method consists inusing a shaker attached to the benches:in this way it is possible to excite

seismically without coupling to acoustic noise. Like loudspeakers, this method
is not selective.

The only way to better identify the culprit between all the

optical components in a bench is to perform tapping tests hitting directly single
objects.

Even if this is not a precise quantitative method, it can give useful

information on which element is the most critical one.
All these techniques have been widely used in Virgo to characterize the contribution of acoustic noise and to identify the sources. Acoustic and seismic noise
injections have been performed around all optical benches in air: the laser andexternal injection benches in the laser laboratory, the external detection benchin the detection laboratory and the two terminal benches in the end stations.
Moreover there are at least two in-vacuum optical elements that can play a signicant role in re-introducing seismic noise. Indeed, the injection and detectionsuspended benches are separated from the remaining of the vacuum system bymeans of two Brewster windows, namely anti-reection coated windows placedat the Brewster angle with respect to the beam. The reason of these two separations is that the vacuum inside the detection and injection towers is not as
good as the one in the main part of the interferometer. The optical elements
mounted on the two suspended benches can pollute the vacuum and therefore
they must be separated from the main area to avoid any contamination of themirrors. Since these windows are attached to the vacuum tube, they can movebecause of the ground seismic motion. These two Brewster windows have also
been excited using the shaker to measure their contribution to the dark fringe
noise.

7.11. ENVIRONMENTAL NOISE

143

10-3

Transfer function [W/V]

10-4

10-5

10-6

10-7 1
10

102

103

104

Frequency [Hz]Figure 7.20: Transfer function from seismic excitation of the detection Brewster
window to the dark fringe signal, measured using single lines.

In all stations acoustic and seismic lines at dierent frequencies have been
injected, with the main goal of understanding the non-linear properties of thecouplings:• The coupling of acoustic and seismic noise depends strongly on the line
frequency (see g. 7.20). The coupling coecient of a line can change oforders of magnitude with relatively small (tens of Hz) shifts in frequency.
This is expected since the coupling passes mainly through the excitationof mirrors mounts and optical components resonances, which can be quite
narrow.

• Even using a very strong line, only the linear coupling term is important:eects in the dark fringe signal are visible only around the line frequency
and not at multiples of it.

• The coupling is strongly non-stationary on quite fast time-scales, up totens of Hz. This is clearly seen in the dark fringe signal: the eect of asingle line is not limited to the line frequency, but instead it is spread ona bump, composed of strong sidebands around the central frequency (seeg. 7.21).

The width of the noise bump is strongly varying dependingon where the acoustic or seismic excitation is applied and also on its
frequency. This non-stationarity is not surprising when dealing with noise
coming from diused and scattered light, since the noise is modulated by
the power in the beam which is diused.

144

CHAPTER 7.

-6

NOISE BUDGETS

-1

10

10

-2

]
-1/2

10

Spectrum [V Hz

Spectrum [W Hz-1/2]

10
-7

10-8

10-3

-4

10

-5

10

10-9

65

70
Frequency [Hz]

75

10-6

80

]
-1/2

10-8

-9

10

385

390
395
Frequency [Hz]70
Frequency [Hz]

75

80

10-4

-5

10

10-6

400

101

10-6

100
]

10-5

-1/2

10-7

Spectrum [V Hz

Spectrum [W Hz-1/2]

10-10

-8

10

10-9
10-10
10-11

65

10-3

Spectrum [V Hz

Spectrum [W Hz-1/2]

10-7

385

390
395
Frequency [Hz]

400

10-1
-2

10

10-3
10-4

910

Figure 7.21:

915

920
Frequency [Hz]

925

10-5

910

915

920
Frequency [Hz]925

Injection of seismic lines on the detection Brewster window. The

plots in the left column show the eect on the dark fringe signal, while the rightcolumn ones show the lines as detected by a seismometer. The red curves show
the ambient noise with no injections.

7.11. ENVIRONMENTAL NOISE

145

This last point explains the diculties in characterizing acoustic and seismic
noise. The strong non-stationarity spoils the coherence between dark fringe andseismometers or microphones, even if the noise is dominant.

Moreover, theenvironmental sensors can not usually be placed exactly on top of the sourceof the scattering, since it is usually not known a priori. This means that what
is measured by the sensor can be quite dierent from the real motion of thescattering element, because of the intrinsic non-localized nature of acoustic andseismic waves. For these two reasons it turned out to be quite dicult to produce
accurate projections of the contribution of this kind of noise to the gravitationalsignal.
Projections of the environmental noise in the two end stations and in thecentral area laboratories have been measured using white noise acoustic andseismic injections. Since the coherence between any environmental sensor anddark fringe is in any case poor, the transfer function have been estimated using
the ratio between the power spectra of dark fringe and sensor signals. Clearly the
projection should be considered meaningful only when there is a visible increaseof the dark fringe noise during the measurement.

It happens sometimes thatthe projection obtained in this way gives an estimate of the noise higher than
the real one. This might indicate that the microphone or seismometer is moresensitive than the scattering element to the injected noise.
The results of this analysis for the Virgo detector are shown in plots from
g. 7.22 to 7.27. These projections refer to the situation in the period just before
the start of the science run. Several interventions had already been performed:
acoustic isolation enclosures were placed around the two laser laboratory opticalbenches, the two end ones and the external detection bench; several opticalmounts were replaced with more rigid one.
In gures

7.22 and7.23 the contribution of the acoustic noise in the twoterminal optical benches is shown.

After the installation of the two acoustic

enclosures around the benches, the level of noise that could be injected using
loudspeakers has proved to be barely sucient to introduce visible eects in thedark fringe signal. Therefore for most frequencies only an upper limit can becomputed: assuming that no eect is visible in the dark fringe when noise isinjected, it is given by the actual sensitivity reduced by a factor equal to theincrease of noise in the environmental sensor.

For both west and north end

stations, the estimated contribution from the acoustic noise is well below theactual sensitivity.
Similar noise projections for the optical benches in the detection and laser
laboratories are shown in gures 7.24 and 7.25.

At some frequencies, corre-

sponding to resonances of optical mounts on the benches, there are some visibleeects on the dark fringe signal. At these frequencies it is possible to computea real noise projection, while at other ones only an upper limit can be obtained.

146

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NOISE BUDGETS

The contribution of acoustic noise in the laser laboratory is negligible, mainly
because of the good performances of the acoustic enclosure installed around theoptical benches. For what concern the detection laboratory the noise projection
can be computed only for some small frequency regions, where it turns out tobe quite close to the actual sensitivity: mainly around 110 Hz, between 400 and
500 Hz, around 600 Hz and 1600 Hz.

For all other regions only upper limits

could be computed. Not all of them are signicant, because the noise injected
could not dominate by a large factor the normal environmental noise.
Finally, the results of the seismic noise injections on the two Brewster windows are shown in gures 7.26 and 7.27. The contribution from the injection
Brewster is well below the actual sensitivity. On the contrary, the projection of
the detection Brewster seismic noise is close to the actual sensitivity in several
bands between 100 and 1000 Hz:

between 240 and 260 Hz, around 300 Hz,

390 Hz, 470 Hz, 500 Hz, between 580 and 780 Hz, between 1480 and 1780 Hz
and between 1930 and 2200 Hz. In some of these bands the projection is higher
than the actual noise. This might be caused by some of the eects already discussed before: the high non-stationarity and non-linearity of the noise coupling
and the fact that the environmental sensors might not read exactly the motionof the diusing element.
In conclusion, it is very dicult to state accurately which elements are themost relevant ones for introducing acoustic noise through diused light. However
it seems possible to conclude that at least some of the structures visible in thesensitivity between 100 and 1000 Hz are likely to be generated by diused andscattered light, coming from the detection laboratory and neighbouring opticalelements. The most likely candidates are however the Brewster window between
the main vacuum and the detection tower and the tower itself.

7.11. ENVIRONMENTAL NOISE

147

Dark fringe [W Hz-1/2]

10-6
Quiet
Noise

10-7
10-8
10-9
10-10
10-11

Microphone [Pa Hz-1/2]

103
102
101
100
10-1
10-2
10-19
Sensitivity
Upper limit
DesignSensitivity [Hz-1/2]

10-20

10-21

10-22

10-23

10-24 1
10

102

103

104

Frequency [Hz]Figure 7.22: Projection of acoustic noise in the north terminal station. The two
top plots compare the spectra of the dark fringe signal and of a microphone
inside the north terminal optical bench in quiet condition and during the noise
injections. The bottom plot shows the computed upper limit on the contributionof acoustic noise to the sensitivity.

148

CHAPTER 7.

NOISE BUDGETS

Dark fringe [W Hz-1/2]

10-6
Quiet
Noise

10-7
10-8
10-9
10-10
10-11

Microphone [Pa Hz-1/2]

103
102
101
100
10-1
10-2
10-19
Sensitivity
Projection
Upper limit
DesignSensitivity [Hz-1/2]

10-20

10-21

10-22

10-23

10-24 1
10

102

103

104

Frequency [Hz]Figure 7.23: Projection of acoustic noise in the west terminal station. The two
top plots compare the spectra of the dark fringe signal and of a microphone
inside the west terminal optical bench in quiet condition and during the noise
injections. The bottom plot shows the computed projection and upper limit on
the contribution of acoustic noise to the sensitivity.

7.11. ENVIRONMENTAL NOISE

149

Dark fringe [W Hz-1/2]

10-6
Quiet
Noise

10-7
10-8
10-9
10-10
10-11

Microphone [Pa Hz-1/2]

102

101

100

10-1
10-19
Sensitivity
Projection
Upper limit
DesignSensitivity [Hz-1/2]

10-20

10-21

10-22

10-23

10-24 1
10

102

103

104

Frequency [Hz]Figure 7.24: Projection of acoustic noise in the detection laboratory. The two
top plots compare the spectra of the dark fringe signal and of a microphone
inside the external detection bench in quiet condition and during the noise
injections. The bottom plot shows the computed projection and upper limit on
the contribution of acoustic noise to the sensitivity.

150

CHAPTER 7.

NOISE BUDGETS

Dark fringe [W Hz-1/2]

10-5
Quiet
Noise

10-6
10-7
10-8
10-9
10-10
10-11

Microphone [Pa Hz-1/2]

102
101
100
10-1
10-2
10-3
10-19
Sensitivity
Projection
Upper limit
DesignSensitivity [Hz-1/2]

10-20

10-21

10-22

10-23

10-24 1
10

102

103

104

Frequency [Hz]Figure 7.25: Projection of acoustic noise in the laser laboratory. The two top
plots compare the spectra of the dark fringe signal and of a microphone inside theexternal injection bench in quiet condition and during the noise injections. The
bottom plot shows the computed projection and upper limit on the contributionof acoustic noise to the sensitivity.

7.11. ENVIRONMENTAL NOISE

151

Dark fringe [W Hz-1/2]

10-6
Quiet
Noise

10-7
10-8
10-9
10-10

Seismometer [m s-2 Hz-1/2]

10-11
102
101
100
10-1
10-2
10-3

10-19
Sensitivity
Projection
DesignSensitivity [Hz-1/2]

10-20

10-21

10-22

10-23

10-24 1
10

102

103

104

Frequency [Hz]Figure 7.26: Projection of seismic noise of the detection Brewster window. The
two top plots compare the spectra of the dark fringe signal and of a seismometerattached to the window in quiet condition and during the noise injections. The
bottom plot shows the computed projection and upper limit on the contributionof seismic noise to the sensitivity.

152

CHAPTER 7.

NOISE BUDGETS

Dark fringe [W Hz-1/2]

10-5
Quiet
Noise

10-6
10-7
10-8
10-9
10-10
10-11

Seimometer [m s-2 Hz-1/2]

102
101
100
10-1
10-2
10-3
10-19
Sensitivity
Projection
Upper limit
DesignSensitivity [Hz-1/2]

10-20

10-21

10-22

10-23

10-24 1
10

102

103

104

Frequency [Hz]Figure 7.27: Projection of seismic noise of the injection Brewster window. The
two top plots compare the spectra of the dark fringe signal and of a seismometerattached to the window in quiet condition and during the noise injections. The
bottom plot shows the computed projection and upper limit on the contributionof seismic noise to the sensitivity.

7.12. CONCLUSIONS AND LATEST DEVELOPMENT

7.12 Conclusions and latest development
The linear noise projection technique has proved to be very important to gaina full understanding of the noise sources limiting the detector, even if it has
been deeply applied only to longitudinal and angular degrees of freedom and tolaser frequency noise. There are still many sources of control noise that mightprot from a similar analysis, even if they are known to be non dominant: forexample suspended benches local controls, input beam jitter, laser power noise,
modulation amplitude and phase noise. The automation of noise budget measurements is very important, since it provides a very powerful and simple tool to
obtain quantitative indications of the improvements brought by commissioning
activities.
The analysis of noise coming from diused and scattered light has been avery active item in the months immediately before the beginning of the run, and
it is likely to maintain its importance even afterward. The technique developed
for the characterization of this noise and the estimation of projection on thedetector sensitivity have been and will be useful, even if they are intrinsicallynot very precise. In any case these methods have been widely used to compare
the situation before and after actions devoted to diused light mitigation.
After the end of the scientic run the commissioning activities brought many
improvements to the detector sensitivity, mainly at low frequency (g. 7.28):• the longitudinal sensing and control system has been strongly optimized,
improving the performances of the noise subtraction techniques and improving the error signal noise.

In this new conguration control noise

coming from these loops is a factor 3 to 5 lower than the sensitivity;• the angular controls have been improved also, mainly developing better
corrector lters and centering better the beam on all mirrors. Also angularcontrol noise is no more limiting the sensitivity;• an extensive campaign of diused light investigation and mitigation helped
removing many structures up to some hundred Hz.

153

CHAPTER 7.

NOISE BUDGETS

10

10-22

10-21

-20

10

10-19

1

10

2

Frequency [Hz]103

Virgo Design
VSR1 End (Oct. 2007)
Post VSR1 (Dec. 2007)

10

4

154

Strain sensitivity [Hz-1/2]
Figure 7.28:

Improvements in the detector sensitivity obtained in the twomonths following the end of VSR1.

Chapter 8
Non-stationary noises
The level of noise in the dark fringe spectrum is not constant in time: there arenormally variations over dierent time scales, from seconds to hours and days.
The study of these variations is useful to characterize the long term behavior ofthe detector and to understand their origin, with the goal of reducing them asmuch as possible.

8.1 Computation of band-limited RMS
To easily perform quantitative analysis on the dark fringe noise variations overtime, it is useful to compute the band-limited RMS, which is the total powerof the noise in a limited frequency band.

If this computation is performed

periodically it is possible to track the time evolution of the noise.
There are at least two methods that can be used to compute band-limited
RMS. First, given the edge frequencies of the desired band [f1 ; f2 ], the signal canbe processed using a tuned band-pass lter. If the lter roll-o is steep enough(usually a second order one is sucient), the total RMS of the output, averaged
over a suitable time window, gives the total power in the band.

The second

method consist in estimating the power spectrum of the signal and summing
the power of all bins inside the frequency band. The two methods are clearly
equivalent, since summing only bins into a band in frequency domain amounts
to multiplying the spectrum by a square window, which is equivalent to time
domain band-pass ltering. The rst method requires less computational time
for a single band. The by-product of the second method is an estimate of thefull power spectrum. If the same signal has to be monitored in multiple bands,
which is usually the case, the second method is better.
In summary, whatever method is used for estimating the BRMS, given thedark fringe signal and a list of frequency bands

 min max 
f1 , f1, . . . , fnmin , fnmax

156

CHAPTER 8. NON-STATIONARY NOISESSpectrum
Noise floor
Lines SNR>3

Power spectral Density [W Hz-1/2]

10-6

10-7

10-8

10-9

10-10
102

103
Frequency [Hz]104

Figure 8.1: Result of the line identication algorithm applied to the dark fringe
signal. The red curve is the full spectrum while the green one is the reconstructed
noise oor. The blue circles are the lines found with SNR greater than 3.

and chosen a suitable time window T for the averaging or the computation of
the power spectrum, the results are n new signals, sampled at F = 1/T , whichgive the noise power in each band as a function of time. It is possible to obtainthese signals at a larger sampling rate using overlapping time intervals for thecomputations: for example it is possible to use 10 s of data for each estimation,with an overlapping of 90%, meaning that every second the last 10 s of data areused for the computations. This is useful to have a better frequency resolution.
Indeed, considering the case of power spectrum estimation, if a time window of

T seconds is used for the estimation, using nav averages with 50% overlap, thelength of each single Fourier transform is given by

∆T =

2T
nav + 1

(8.1)

and therefore the frequency resolution of the Fourier transform isδf =nav + 1
2T

(8.2)

Therefore, the use of longer overlapped intervals allows gaining a better frequency resolution and moving the monitored bands to lower frequencies.

8.1.1 Contribution of narrow spectral lines
If a signal contains a strong sinusoidal contribution at a given frequency (calledin brief spectral line or simply line ), the BRMS of a band containing it might8.1. COMPUTATION OF BAND-LIMITED RMS

157

be completely dominated by the power in the line itself. Every uctuation of
the noise oor is hidden by the line amplitude. This situation is quite common
when analyzing signals coming from interferometric detectors, because stronglines are usual and often added for calibration and monitoring purposes.

Analgorithm has been developed to separate the contribution to the BRMS comingfrom the noise oor and from lines [78].
The algorithm works in two main steps: rst it estimates the noise oor and1

afterward it searches for lines and estimates their parameters :1. The starting point is an estimate of the power spectrum Sk . Its logarithm
is computed Lk = log Sk . This logarithmic transform is applied to reducethe contribution of strong lines in the computation of mean values andstandard deviations.
2. The full frequency range, between the minimum frequency fmin and the
maximum fmax , is divided into N segments (typically 200 between 1 and
10 kHz). In each of these segments, the minimum value of Lk is found andassociated with the central frequency of the segment.

All the N points

obtained in this way are linearly interpolated to all frequencies between

fmin and fmax , obtaining a rst lower estimate LFk of the logarithm of
the noise oor.
3. The dierence between the real logarithm of the spectrum and the latter
estimate is computed Dk = Lk − LFk . This has the eect of attening
the noise oor over the entire frequency range.
4. The mean value and standard deviation of the dierence Dk is computed
over all the frequency range. All points which are more than three time
the standard deviation far from the mean value are discarded. Then themean value and standard deviation are computed again and the process
iterated a couple of times.
5. The lower estimate is corrected adding the computed mean value to obtainthe nal estimate of the noise oor: Fk = exp

LFk + Dk

6. All local maxima of the spectrum are found and their signal-to-noise ratio(SNR) computed dividing their peak value by the value of the noise oorat the corresponding frequency. Only those with SNR higher than a given
threshold SN Rmin are considered.
7. For each of these maxima, side bins are inspected to nd the frequencies where the spectrum stop to decrease. These are dened as the limit
frequencies of the peak

1 A dierent algorithm for extracting lines is described in [79]

158

CHAPTER 8. NON-STATIONARY NOISES
8. Adjacent peaks are merged together if the ratio of their amplitudes are
below the SNR threshold.

This is done to avoid counting a peak withsmall sidebands multiple times. The line frequency is dened as the oneof the strongest peak
9. For each of the surviving lines the frequency and peak amplitude arecomputed, as well as the SNR, dened as before as the peak value overthe noise oor value.
This algorithm can be used to separate the contribution of lines to the BRMS
from the noise oor, therefore allowing to track separately the evolution in timeof the power in lines and in the bulk of the noise spectrum. It is also used foron-line high resolution search of lines in the dark fringe spectrum.

8.2 Analysis of dark fringe non-stationarities
During all the science run the dark fringe signal has been monitored by anon-line process, called NonStatMoni [80], which performed the computation of
band-limited RMS every second, with dierent time windows, and saved theresults in the main data stream. The list of all the monitored bands is reported
in table 8.1.
Two dierent strategies has been adopted for the analysis of noise nonstationarities.

The typical duration of a continuous segment of science mode

data is of the order of several tens of hours.

This time window is enough toperform a Fourier analysis of the BRMS values. This allows the identication of
periodicities in the noise variations, that can afterward be studied to understandtheir origin. The typical frequencies are between 1 and 500 mHz.
The noise level in dierent bands however shows also slower and non-periodic
uctuations, that are more easily studied in the time domain, by looking forcorrelation with other auxiliary channels.

8.2.1 Spectral analysis
Using the data from a single continuous science mode data segment, the main
periodical non-stationarities can be studied computing the power spectrum of
the time series given by the value of the BMRSs for each second. In this way the
maximum detectable frequency of the uctuations is given by half the inverseof the time shift between two consecutive windows:

fM AX =

1
2S

(8.3)

which in the case of Virgo science run data is 500 mHz. The minimum frequency,
which correspond also to the spectrum resolution, is given by a formula similar8.2. ANALYSIS OF DARK FRINGE NON-STATIONARITIESMinimum frequency

Maximum frequency

Time window

0 Hz

10 Hz

10 s

10 Hz

30 Hz

10 s

30 Hz

50 Hz

10 s

50 Hz

100 Hz

10 s

100 Hz200 Hz

5 s

200 Hz300 Hz

5 s

300 Hz400 Hz

5 s

400 Hz500 Hz

5 s

500 Hz600 Hz5 s

600 Hz700 Hz

5 s

700 Hz800 Hz

5 s

800 Hz900 Hz

5 s

900 Hz1000 Hz

5 s

500 Hz

1 kHz

1 s

1 kHz

2 kHz

1 s

2 kHz

3 kHz

1 s

3 kHz

4 kHz

1 s

4 kHz

5 kHz

1 s

5 kHz

6 kHz

1 s

6 kHz

7 kHz

1 s

7 kHz

8 kHz

1 s

8 kHz

9 kHz

1 s

9 kHz

10 kHz

1 s

Table 8.1: List of all the bands monitored during the Science Run by the NonStatMoni process. The output sampling rate is 1 Hz.

159

160

CHAPTER 8. NON-STATIONARY NOISES
10-1
5 s over 1 s
10 s over 1 s

Transfer function

10-2

10-3

10-4

Figure 8.2:

0

0.1

0.2
0.3
Frequency [Hz]

0.4

0.5

Example of the ratio between spectra of BRMS estimated withdierent time interval.

This is the measured ratio between values computedwith 10 s and 1 s and between 5 s and 1 s.

to eq. 8.2, where now T is the duration of the data segment and nav is thenumber of averages: for example a single stretch of data 50 hours long, taking

20 averages, the frequency resolution is about 70 µHz.
There is one subtle point in this estimation of the uctuation spectrum, if
the overlap of the time intervals is not zero. In this case two consecutive samples
for BRMSs are not completely independent, and it might seem not possible torecover information on frequencies higher than 1/2T . However using overlappedintervals is not really equivalent to over-sampling the time series. Indeed, theoperation of estimating the BRMS can be seen as an average of the instantaneous power in the band over the time interval used for the computation of thespectrum. This is equivalent to a time convolution with a rectangular windowfunction: therefore the eect on the spectrum is simply the multiplication by
the Fourier transform of this window, which is the well-known sinc function.
In conclusion, given the BRMS values of a signal, computed with two dierent
windows T1 and T2 > T1 , the ratio of the spectra of the two time series is givenby (see g. 8.2):

T F (f ) =sincπf T2
sincπf T1(8.4)

Once the frequencies of the main fast non-stationarities have been identied, it is interesting to correlate the variation of the BRMS in dierent bands
with global variables, like environmental conditions (temperatures, seismic activity, etc.), laser power levels at dierent output ports, alignment conditions,8.2. ANALYSIS OF DARK FRINGE NON-STATIONARITIES
etc. One possible way to do this is to compute the coherence between the BRMS
values and slow signals monitoring the detector: a high value of the coherenceindicates that the dominating sources of noise in that particular band is modulated like the uctuations of the considered monitoring signal. If the BRMS
taken into account is computed using overlapped segments, the coherence is notaected by the transfer function of eq. 8.4, since a linear transformation does
not change the coherence between two signals.
These frequency-domain analysis are performed automatically by an o-line
analysis tool, which reads the output data from NonStatMoni, computes spectra
and coherences with a large set of auxiliary interferometer channels. The results
are organized as web pages [81].

Analysis of the science run data
An example of the results obtained for a long stretch of data during the sciencerun is shown in gures 8.3, 8.4 and 8.5. These spectra are almost all very similar
and does not show many relevant structures. The main dierence concerns theband 5-6 kHz where the BRMS is dominated by strong lines corresponding tomirror internal resonant modes. The other spectra are, with some exception,quite smooth, as expected for noise without strong non-stationarities.
The coherence analysis of these BRMSs shows some important relations indierent frequency regions:• referring to g. 8.3,8.4 and 8.5, the very low frequency non-stationarities(below 100 µHz) are coherent with variation in the seismic activity, as
measured by the RMS of seismometers in dierent bands, and also withthe mode cleaner terminal mirror angular motions.

The two things arelikely related: high seismic motion excites more the short MC suspension;• referring to g. 8.3,8.4 and 8.5, the uctuation at frequencies between
1 mHz and 10 mHz are coherent with angular motions of the beam splitter
and west arm mirrors. This has also an eect visible on the dark fringe
power and in the sidebands amplitudes at the dark port, before the output
mode cleaner;• referring to g. 8.3, the narrow line at about 20 mHz is due to angularmotion of the PR and BS mirrors in the ϑy degree of freedom;• referring to g. 8.3 and 8.4, the structure between 25 and 30 mHz is also
related to beam splitter and west input mirror motions, but in the ϑxdegree of freedom;• referring to g. 8.4, the structures visible between 100 and 200 mHz are
coherent with the angular motion of PR and BS in the ϑx degree of freedom

161

162

CHAPTER 8. NON-STATIONARY NOISESNormalized band-limited RMS [W Hz-1/2 Hz-1/2]

10-5
500-1000 Hz
1000-2000 Hz
2000-3000 Hz
3000-4000 Hz
4000-5000 Hz
5000-6000 Hz
6000-7000 Hz
7000-8000 Hz
8000-9000 Hz
9000-10000 Hz

10-6

10-7

10-8

10-9

10-10
10-4

10-3

10-2
Frequency [Hz]10-1

Figure 8.3: Spectrum of dark fringe band-limited RMS variation for high frequency bands.

In summary, on the timescale of one hour or less, the detector noise is quite
stable and the main sources of semi-periodic non-stationarities are linked toresidual angular motions.

8.2.2 Time domain analysis
The non-stationarities on long time scales (typically days) are more easily analyzed in time-domain. For the rst three months of the run, all the science mode
segments of data have been analyzed, computing the root mean square value ofthe BRMS over time windows of 100 s. In this way only the slower uctuations
are taken into account. Moreover, the nal 100 seconds of each science segment
has been discarded, to avoid possible perturbation of the BRMS values by preunlock transient. Also the rst 3000 seconds of each lock has been discarded
since several frequency bands are dominated by damping resonances, like violin
and mirror internal modes, that are excited during the lock acquisition.
An example of the BRMS variations over the long period is shown in g. 8.6.
It is important to correlate these uctuations with changes in the detector con-

8.2. ANALYSIS OF DARK FRINGE NON-STATIONARITIESNormalized band-limited RMS [W Hz-1/2 Hz-1/2]

10-6
100-200 Hz
200-300 Hz
300-400 Hz
400-500 Hz
500-600 Hz
600-700 Hz
700-800 Hz
800-900 Hz
900-1000 Hz

10-7

10-8

10-9

10-10

10-11

10-12 -4
10

10-3

10-2
Frequency [Hz]10-1

Figure 8.4: Spectrum of dark fringe band-limited RMS variation for intermediate frequency bands.

ditions. For this reason the mean value of several auxiliary channels has been
computed over the same 100 s periods used for the BRMS:• signals concerning the length sensing and control system:

the DARM,

MICH and PRCL loops electronic gains, the value of the sensing crossterm between MICH and PRCL, the alpha and beta coecients, the oset
added to balance the sidebands (see chap. 3);• the powers detected by all photo-diodes: B1, B1p, B5, B7 and B8 as wellas the signal used to estimate the sidebands power inside the recycling
cavity B5_2f;• the power transmitted by the input mode cleaner;• the amplitude of carrier and sidebands at the dark port, as detected by
the scanning Fabry-Perot system (see sec. 2.5 and B.6.3);• the total RMS of angular corrections for all six main mirrors and for the
injection and detection suspended benches;

163

164

CHAPTER 8. NON-STATIONARY NOISESNormalized band-limited RMS [W Hz-1/2 Hz-1/2]

10-4
0-10 Hz
10-15 Hz
15-20 Hz20-30 Hz30-50 Hz80-90 Hz50-100 Hz

10-5

10-6

10-7

10-8

10-9

10-10
10-4

10-3

10-2
Frequency [Hz]10-1

Figure 8.5: Spectrum of dark fringe band-limited RMS variation for low frequency bands.

• the seismic activity in dierent frequency bands between 30 and 4000 mHz;
• the seismic noise measured on top of the detection Brewster window (seesec. 7.11.2);• the motion of the top stage of the mirror suspensions;• the vertical position of all main mirrors and of the two suspended benches;• the external temperature as well as those measured inside the main buildings, the optical benches and the towers;• the centering signals of all alignment quadrants (see chap. 4);• the beam position measured with respect to the output mode cleaner by
quadrants mounted on the suspended detection bench, as well as the OMC
locking signals;• the input beam position measured by the BMS system as well as thecorresponding RMS motions;8.2. ANALYSIS OF DARK FRINGE NON-STATIONARITIESFigure 8.6: Typical example of the variation over long period of the band-limited
RMS, in this case computed between 500 and 1000 Hz.

• the optical gains measured on-line by the calibration process, which trackthe dark fringe response to mirror motions at specic frequencies.

This makes a total of about 100 auxiliary channels to search for correlation withthe dark fringe noise.

8.2.3 Multi-dimensional linear regression
The task of nding the possible correlations between the time evolution of the
BRMS in a given band and a set of auxiliary channels can be tackled withstatistical methods, in particular applying a multi-dimensional linear regression[82]. The values at n dierent times yi of the BRMS in a given band can beviewed as the noisy version of a linear combination of the k auxiliary channels
measured at the same times x1...k,1...n :yi = β0 + β1 x1,i + · · · + βk xk,i + εi
Here

(8.5)

βj are the unknown coecients to be estimated and εi is a source ofadditional noise, that can be considered as coming from the measurement of theyi or also from the auxiliary channels. In other words, the analysis starts fromthe hypothesis that the input x signals are deterministic and that the output isa random variable given by an unknown linear combination of the x plus somerandom noise. This noise is assumed to be normally distributed, with zero mean,
uncorrelated for dierent times and with constant unknown variance σ . These
hypothesis are reasonably satised in the case analyzed. Under these assumption
the Gauss-Markov theorem [83] states that the best un-biased estimate of theβ coecients can be obtained with a least square method.
It is easier to carry out the computation in matrix form. With the following

165

166

CHAPTER 8. NON-STATIONARY NOISESdenitions:

Y



y1




= 


.
.
.





yn

X

β

ε

1

x1,1

 .
.
.
.
= 
.
 .
1 xn,1


β0
 . 
. 
= 
 . 
βk


ε1
 . 
. 
= 
 . 
εn

...
...

x1,k



.
.
.





xn,k

equation 8.5 can be rewritten in a more compact form:

Y = Xβ + ε
and the sum of squared errors is

S

n
X

=

2

(yi − β0 − β1 x1,i − · · · − βk xk,i ) = (Y − Xβ)T (Y − Xβ)

i=1T= Y Y − 2β T (X T Y ) + β T (X T X)βThe best estimate of the β coecients is obtained by minimization of this quantity, or in other words by solving the equation:

0=∂S= −2X T Y − 2(X T X)β∂β T

If all the auxiliary signals are linearly independent, the X matrix has maximum
rank and X

T

X is invertible. Therefore the solution of the previous equation isβ̂ = (X T X)−1 X T Y(8.6)

and the predicted output signal isŶ = X β̂
The estimated parameters are linear combination of the measured Y which
is a realization of a normal random variable. It follows that also β is normally
distributed, and its covariance matrix is given by


 
T
C(β̂) = (X T X)−1 X T σ 2 (X T X)−1 X T = σ 2 (X T X)−1

8.2. ANALYSIS OF DARK FRINGE NON-STATIONARITIESTherefore the variance of the parameters β̂ can be computed if the value of

σ is known.

An un-biased estimator can be found analyzing the probability

distribution of the sum of squared residuals, dened as[ = (Y − Ŷ )T (Y − Ŷ ) = (Y − X β̂)T (Y − X β̂)
SSR

(8.7)

Some algebraic computations yieldY − X β̂ = 1 − X(X T X)2 X T Y
From the geometrical point of view the operator inside brackets is an orthogonal
projection into the subspace complementary to the one spanned by the columnsof X . Therefore its dimension is exactly n − k − 1. Eq. 8.7 represent therefore
a sum of n − k − 1 squared normal random variables, each one with variancegiven by σ . In other words the following variable[
SSR(Y − X β̂)T (Y − X β̂)
=
σ2
σ2

(8.8)

2

is distributed like a (n − k − 1)-dimensional χ . This nally gives the needed
un-biased estimator of σ asσ̂ 2 =(Y − X β̂)T (Y − X β̂)
n−k−1

(8.9)

Using this the variance of the β̂ parameters can be computed:ŝ(β̂j ) = σ̂q
(X T X)−1
j,j

Finally the variableT =β̂j − βjŝ(β̂j )follows a Student's t distribution with n − k − 1 degrees of freedom and it allows
the determination of condence intervals for the estimated β̂j as

β̂j − tα/2 ŝ(β̂j ) < βj < β̂j + tα/2 ŝ(β̂j )(8.10)

where the condence level is 1 − α.

8.2.4 Application of the techniqueSince the linear regression technique is quite sensitive to out-layers, all channels
have been scanned for periods of bad behavior, and these have been excluded
from the analysis. At the end about 50000 data points remained, covering therst three month of the science run. All auxiliary channels have been normalizedusing their mean, standard deviation and the number of points:

xN
j = √

1 xj − µjσjNP(8.11)

167

168

CHAPTER 8. NON-STATIONARY NOISESIn this way all the signals have similar order of magnitude. Moreover, in thisway numerical problems due to the high number of multiplications of signals in
the computation of the X

T

X matrix are reduced.

The result of the analysis is a set of about 100 coecients for the BRMS
in each frequency band. The condence intervals of these estimations can beobtained as explained before. However, given the very high number of degrees of
freedom involved in this linear regression, the Student's distribution can be very
well approximated by a normal one. The condence intervals have therefore been
computed at the 3σ level, corresponding to a condence of 99.7%. Afterward
all coecients compatible with zero were discarded. The number of survivingβ̂j is still high, of the order of 60.

Therefore a method to identify the lesssignicative ones has been constructed as follows: the standard deviation of agiven channel contribution to the predicted BRMS is divided by the standard
deviation of the residual.

This gives an estimation of the importance of thatsignal for the prediction. All signals with this ratio lower than a given thresholdwere discarded. The best value has been found esperimentally to be 1/10. This
procedure left a number of coecients of the order of 20-30 for each band-limited
RMS.
Finally, the linear regression analysis has been performed again for each bandusing only the relevant channels identied as explained above. In this way theprediction of the BRMS values improved.
The results are shown in gures from 8.7 to 8.11 for all the analyzed bands.
The agreement between the real variations of the BRMS and the predictedvalues is good for all bands, except for the 5-6 kHz. This is expected since thisBRMS is dominated by very large mirror internal resonance peaks, which arequite stable with time.
A further check of the obtained models can be performed by plotting thehistogram of the residuals. Supposing that the model could completely describe
the non-stationarities, the residuals should be distributed like the normal noiseof eq. 8.5. Some of the histograms are shown in g. 8.12. Their distribution isusually not too far from a normal one, with frequent tails on the high values side.
This is an indication that not all the non-stationary behavior can be explainedon the simple basis of the chosen set of auxiliary channels.
Although this technique is very useful to understand which amount of thetotal noise non-stationarity can actually be predicted from a given set of auxiliary channels, it is quite dicult to extract a small number of channels thatcan give some useful information for the detector characterization and commissioning. The main reason is that the auxiliary channels are indeed independent,
otherwise the analysis would fail, but they are not orthogonal, in the sense that
there are close relationship between them.

For example the seismic activity

at very low frequency can aect the interferometer sensitivity by spoiling the8.2. ANALYSIS OF DARK FRINGE NON-STATIONARITIES

169

9.0 10-6
0 10 Hz
Prediction

8.0 10-6

BRMS [W Hz-1/2]

7.0 10-6
6.0 10-6
5.0 10-6
4.0 10-6
3.0 10-6
2.0 10-6
1.0 10-6
0.0 100

0

10

20

30

40

50

60

70

80

90

Time [days]
1.2 10-7
10 30 Hz
Prediction

1.0 10-7

BRMS [W Hz-1/2]

8.0 10-8
6.0 10-8
4.0 10-8
2.0 10-8
0.0 100-2.0 10-8

0

10

20

30

40

50

60

70

80

90

Time [days]
4.0 10-830 50 Hz
Prediction

3.5 10-8

BRMS [W Hz-1/2]

3.0 10-8
2.5 10-8
2.0 10-8
1.5 10-8
1.0 10-8
5.0 10-9
0.0 100-5.0 10-9

0

10

20

30

40

50

60

70

80

90

Time [days]
1.8 10-7
50 100 Hz
Prediction

1.7 10-7

BRMS [W Hz-1/2]

1.6 10-7
1.5 10-7
1.4 10-7
1.3 10-7
1.2 10-7
1.1 10-7
1.0 10-7

0

10

20

30

40

50

60

70

80

90

Time [days]
9.0 10-9
100 200 Hz
Prediction

8.5 10-9

BRMS [W Hz-1/2]

8.0 10-9
7.5 10-9
7.0 10-9
6.5 10-9
6.0 10-9
5.5 10-9
5.0 10-9

0

10

20

30

40

50

60

70

80

90

Time [days]

Figure 8.7: Variation of the BRMS in low frequency bands and predicted values
from the linear regression analysis.

170

CHAPTER 8. NON-STATIONARY NOISES

2.0 10-9
200 300 Hz
Prediction

1.8 10-9

BRMS [W Hz

-1/2

]

1.6 10-9
1.4 10-9
1.2 10-9
1.0 10-9
8.0 10-10
6.0 10-10
4.0 10-10
2.0 10-10

0

10

20

30

40

50

60

70

80

90

Time [days]
6.5 10-9
300 400 Hz
Prediction

6.0 10-9
5.5 10-9
BRMS [W Hz-1/2]

5.0 10-9
4.5 10-9
4.0 10-9
3.5 10-9
3.0 10-9
2.5 10-9
2.0 10-9
1.5 10-9

0

10

20

30

40

50

60

70

80

90

Time [days]
4.5 10-10
400 500 Hz
Prediction

BRMS [W Hz

-1/2

]

4.0 10-10
3.5 10-10
3.0 10-10
2.5 10-10
2.0 10-10
1.5 10-10

0

10

20

30

40

50

60

70

80

90

Time [days]
2.6 10-10
500 600 Hz
Prediction

2.4 10-10

BRMS [W Hz

-1/2

]

2.2 10-10
2.0 10-10
1.8 10-10
1.6 10-10
1.4 10-10
1.2 10-10

0

10

20

30

40

50

60

70

80

90

Time [days]
9.0 10-10
600 700 Hz
Prediction

BRMS [W Hz

-1/2

]

8.0 10-10
7.0 10-10
6.0 10-10
5.0 10-10
4.0 10-10
3.0 10-10

0

10

20

30

40

50

60

70

80

90

Time [days]

Figure 8.8: Variation of the BRMS in intermediate frequency bands and predicted values from the linear regression analysis.

8.2. ANALYSIS OF DARK FRINGE NON-STATIONARITIES

171

8.5 10-10
700 800 Hz
Prediction

8.0 10-10

BRMS [W Hz-1/2]

7.5 10-10
7.0 10-10
6.5 10-10
6.0 10-10
5.5 10-10
5.0 10-10
4.5 10-10
4.0 10-10

0

10

20

30

40

50

60

70

80

90

Time [days]
5.5 10-10
800 900 Hz
Prediction

5.0 10-10

BRMS [W Hz-1/2]

4.5 10-10
4.0 10-10
3.5 10-10
3.0 10-10
2.5 10-10
2.0 10-10
1.5 10-10
1.0 10-10

0

10

20

30

40

50

60

70

80

90

Time [days]
2.6 10-10
900 1000 Hz
Prediction

2.4 10-10

BRMS [W Hz-1/2]

2.2 10-10
2.0 10-10
1.8 10-10
1.6 10-10
1.4 10-10
1.2 10-10

0

10

20

30

40

50

60

70

80

90

Time [days]
7.5 10-91000 2000 Hz
Prediction

7.0 10-9

BRMS [W Hz-1/2]

6.5 10-9
6.0 10-9
5.5 10-9
5.0 10-9
4.5 10-9
4.0 10-9

0

10

20

30

40

50

60

70

80

90

Time [days]
6.5 10-92000 3000 Hz
Prediction

BRMS [W Hz-1/2]

6.0 10-9
5.5 10-9
5.0 10-9
4.5 10-9
4.0 10-9
3.5 10-9

0

10

20

30

40

50

60

70

80

90

Time [days]

Figure 8.9: Variation of the BRMS in intermediate frequency bands and predicted values from the linear regression analysis.

172

CHAPTER 8. NON-STATIONARY NOISES

6.5 10-92000 3000 Hz
Prediction

BRMS [W Hz-1/2]

6.0 10-9
5.5 10-9
5.0 10-9
4.5 10-9
4.0 10-9
3.5 10-9

0

10

20

30

40

50

60

70

80

90

Time [days]
5.4 10-93000 4000 Hz
Prediction

5.2 10-9

BRMS [W Hz-1/2]

5.0 10-9
4.8 10-9
4.6 10-9
4.4 10-9
4.2 10-9
4.0 10-9
3.8 10-9
3.6 10-9

0

10

20

30

40

50

60

70

80

90

Time [days]
4.9 10-94000 5000 Hz
Prediction

4.8 10-9
4.7 10-9

BRMS [W Hz-1/2]

4.6 10-9
4.5 10-9
4.4 10-9
4.3 10-9
4.2 10-9
4.1 10-9
4.0 10-9
3.9 10-9

0

10

20

30

40

50

60

70

80

90

Time [days]
1.6 10-7
5000 6000 Hz
Prediction

BRMS [W Hz-1/2]

1.4 10-7
1.2 10-7
1.0 10-7
8.0 10-8
6.0 10-8
4.0 10-8

0

10

20

30

40

50

60

70

80

90

Time [days]
4.6 10-96000 7000 Hz
Prediction

BRMS [W Hz-1/2]

4.4 10-9
4.2 10-9
4.0 10-9
3.8 10-9
3.6 10-9
3.4 10-9

0

10

20

30

40

50

60

70

80

90

Time [days]

Figure 8.10:

Variation of the BRMS in high frequency bands and predictedvalues from the linear regression analysis.

8.2. ANALYSIS OF DARK FRINGE NON-STATIONARITIES

173

5.6 10-9
5.4 107000 8000 Hz
Prediction-9

5.2 10-9

BRMS [W Hz-1/2]

5.0 10-9
4.8 10-9
4.6 10-9
4.4 10-9
4.2 10-9
4.0 10-9
3.8 10-9
3.6 10-9
3.4 10-9

0

10

20

30

40

50

60

70

80

90

Time [days]
9.0 10-98000 9000 Hz
Prediction

8.0 10-9

BRMS [W Hz-1/2]

7.0 10-9
6.0 10-9
5.0 10-9
4.0 10-9
3.0 10-9
2.0 10-9

0

10

20

30

40

50

60

70

80

90

Time [days]
4.4 10-99000 10000 Hz
Prediction

4.2 10-9
4.0 10-9

BRMS [W Hz-1/2]

3.8 10-9
3.6 10-9
3.4 10-9
3.2 10-9
3.0 10-9
2.8 10-9
2.6 10-9
2.4 10-9

0

10

20

30

40

50

60

70

80

90

Time [days]

Figure 8.11:

Variation of the BRMS in high frequency bands and predictedvalues from the linear regression analysis.

174

CHAPTER 8. NON-STATIONARY NOISES

104

104
0 10 Hz
Gaussian fit

10 30 Hz
Gaussian fit

103

CountsCounts

103

102

101

102

101

100
-4000 -3000 -2000 -1000

0

1000

2000

3000

4000

5000

6000

100
-40

7000

-20

0

Residual [x10-9 W Hz-1/2]

20

40

60

80

100

Residual [x10-9 W Hz-1/2]

104

103
50 100 Hz
Gaussian fit

100 200 Hz
Gaussian fit

103

CountsCounts

102

102

101
101

100
-40

-30

-20

-10

0

10

20

100
-1.5

30

-1

-0.5

Residual [x10-9 W Hz-1/2]

0

0.5

1

1.5

2

Residual [x10-9 W Hz-1/2]

103

103
200 300 Hz
Gaussian fit

800 900 Hz
Gaussian fit

Counts102

Counts

102

101

100
-0.4

101

-0.2

0

0.2

0.4

0.6

0.8

1

100
-0.1

1.2

-0.05

0

Residual [x10-9 W Hz-1/2]

0.05

0.1

0.15

0.2

0.25

Residual [x10-9 W Hz-1/2]

104

103
2000 3000 Hz
Gaussian fit

6000 7000 Hz
Gaussian fit

103

CountsCounts

102

102

101
101

100
-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Residual [x10-9 W Hz-1/2]

1

1.2

1.4

1.6

100
-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Residual [x10-9 W Hz-1/2]

Figure 8.12: Some example of histograms of the residuals in some frequency
bands.

0.3

8.2. ANALYSIS OF DARK FRINGE NON-STATIONARITIES
accuracy of the alignment loops.

175

Therefore both the seismic sensors and theRMS of angular corrections are good signals and even if the original cause is
the micro-seism, the multi-dimensional regression might end up with using only
the alignment signal.
To overcome all these diculties, a dierent approach has been developed to
be able to sort the auxiliary channels in order of relevance for the predictionof the BRMS in a given band. The rst step is a one-dimensional linear regression analysis of the BRMS time series with each auxiliary channel separately.
The one that minimize the residual standard deviation is then chosen and its
predicted contribution is subtracted from the BRMS values. Starting from thissubtracted time series the analysis is re-iterated, taking every time into accountonly the auxiliary channel that minimize the residual standard deviation.

Inthis way the nal prediction is not the optimal one and not as good as the one
obtained with the full linear regression, but at least it is possible to extract a
reduced number of channels (usually 5) that can reconstruct most of the slowest trends of the BRMS. This analysis usually fails to reproduce the fastest
variations or the shortest events.
With this technique it has been possible to determine, for each frequency
band, those auxiliary channels that are most relevant to explain the non-stationarities. Some of these channels can be pointed out as the real origin of the noise
variation, like for example an increase of some correction RMS which can betracked down to a real increase of the residual motion of one degree of freedom.
Some other channels are not likely to be the direct culprit of the noise increase,but simply an indicator of something changing inside the detector. A typicalexample are the powers detected by photo-diodes.
In summary, the non-stationarities are linearly depending on several dierent
channels. The noise at high frequencies, above 1 kHz, is changing accordingly
to• the power in the dark fringe, as detected before the OMC. This is an indication of changes in the interferometer contrast due to higher order modes,
usually due to small alignment drifts o the operating point.

Moreoverthere is a linear dependence on the dark fringe quadrant centering signals,
meaning that the beam might be moving in front of the quadrants and of
the OMC. This is further conrmed by the dependence on the signals coming from the suspended detection bench quadrants, used for the alignmentof the bench telescope;• the RMS of angular corrections of the suspended injection and detection
benches. The rst couple to the main interferometer as beam jitter;• some temperature inside the central and terminal buildings, that mightcause drifts in the quadrant centering that worsen the overall alignment

176

CHAPTER 8. NON-STATIONARY NOISES
of the interferometer.
The noise at intermediate frequencies, between roughly 100 and 1000 Hz is
instead changing like• the RMS of suspended injection bench angular controls, even more thanin the high frequency bands;• the vertical position of several main mirrors, mainly the arm cavity ones;• the RMS of the angular corrections of the end mirrors. This is directly
related to micro-seismic conditions: when the seismic noise at very low
frequency is higher, the accuracy of the angular controls is reduced;• the dark fringe power before the OMC;
• the variation of some length sensing and control signals, like the optical
gains and the sensing cross term. It is unlikely that there is a direct relation between the values of these signals and the noise in the 100-1000 Hz
band. It is more likely that these signals are indicators of something changing inside the interferometer, like for example the arm cavity nesses, due
to changes in the input mirror temperature.
Finally the noise at low frequency, roughly below 100 Hz, known to come mainly
from controls (see chap. 7), changes accordingly to• mainly the micro-seismic conditions at the lowest frequencies;• the vertical position of several payloads, both mirrors and the suspended
detection bench;• the values of the noise subtraction coecients alpha and beta (see chap.
5.4), which directly aect the noise in this region;• the power transmitted by the arm cavities, which is likely to indicate adependence on the nesse variations;• the RMS of angular corrections of both the injection suspended bench andthe input mode cleaner terminal mirror.

8.3 Conclusions
The study of noise non-stationarities gives a dierent perspective on the detector
sensitivity than other kind of analysis. The computation of band-limited RMS
from on-line data has been running since a couple of years even before the run,and has been widely used when a quantitative measurement of the noise level8.3. CONCLUSIONS

177

was needed and a simple comparison of spectra at two dierent time was notenough.
The study of semi-periodic non-stationarity allowed to identify several sourcesof noise uctuation, mainly linked to residual angular motions of the mirrors,
and it gave the necessary indications to trigger improvements. Indeed the stationarity of the noise during the run was very good and there were few periodical
modulation still present, and all quite small.
The time domain analysis of the noise level during long periods has been rstcarried out during the science run, since this was the right occasion to collect
long stretches of data in controlled and stable conditions.

The linear regres-sion analysis is only the starting point of a deeper investigation, and showedvery interesting results.

Indeed most of the noise level variations can be ex-plained with simple linear relations with several interferometer monitoring andauxiliary channels.

The foreseen and natural evolution of this analysis is not

the blind attempt to add more and more channels to reconstruct every noise
non-stationarity, but instead a deeper investigation of the results already obtained.

The goal is to understand the relations between a given BRMS andsome relevant auxiliary channel, to be able to kill the noise source or to control
the suitable interferometer degree of freedom in order to minimize the coupling.
Moreover, this linear regression analysis should be extended to include quadratic
coupling with auxiliary channels, since these might be the dominant one in the
hypothesis that the detector working point is the optimal one.
Finally, the understanding of a large number of sources of non-stationarities
is the rst step toward the subtraction of their contribution and the study of
the remaining noise level variation. The most interesting analysis is for sure thesearch for noise modulation with a period of one sidereal day, which might indicate a non-terrestrial origin, for example from a gravitational wave astrophysicalgalactic background [19].

Chapter 9
Fast transient noises
Another kind of non-stationary noises that can appear in interferometric detectors consists in fast transients: brief periods of time, lasting usually less than
one second, of increased noise in the full frequency band or in a restricted one.
The origin of this kind of events are many: brief instabilities in some control
loops, real glitches in the signals, dust falling in front of photo-detectors, fastvarying noise couplings, environmental noise, etc. As a matter of nomenclature,in the following these transient events will be called clusters.

9.1 The HACR algorithm
There are lots of dierent algorithms that can be used to detect such events,
most of them developed in the context of impulsive gravitational wave sources(bursts) analysis. The algorithm implemented as part of this work and describedhere is based on HACR (hierarchical algorithm for curves and ridges ) [84] developed inside the GEO600 collaboration. This algorithm is quite fast, allowing
a simultaneous monitoring of tens of channels. This characteristic makes it very
useful to perform coincidence analysis.
The idea behind this algorithm is to to search for times and frequencies
where there is an excess of power with respect to the mean statistics. Given an
input signal x(t), the algorithm rst compute short windowed Fourier transforms(using for example 50 ms of data) without averages. In this way a time-frequency
map F (t, f ) of the signal is created using the squared values of these transforms,
with good time and quite low frequency resolutions (see g. 9.1a). The second
step is the computation of a mean value of the spectrum, by running a decay
average for each frequency bin: the result is a map of the mean values of thesignal power spectrum over the past seconds (see g. 9.1b) for each time:

M (t, f ) = F (t − ∆t < τ < t, f )

(9.1)

180

CHAPTER 9. FAST TRANSIENT NOISES

(a)

(b)
0

4600

-2

10
4300

10-3

4200

2.1

2.15

2.2

2.25
Time [s]

2.3

2.35

10-1

4400
-2

10
4300

10-3

4200

10-4

4100

10

4500

10-1

4400

Frequency [Hz]Frequency [Hz]

4500

0

4600

10

10-4

4100

2.4

2.1

2.15

2.2

(c)

10-1
4300
10-2

2.3

2.35

2.4

4400

4300

4200

10-3

4200

4100

10-4

4100

2.25
Time [s]2.44500Frequency [Hz]Frequency [Hz]

100
4400

2.2

2.35

4600

101

4500

2.15

2.3

(d)
102

4600

2.1

2.25
Time [s]

2.3

2.35

2.4

2.1

2.15

2.2

2.25
Time [s]Figure 9.1: Example of how the HACR algorithm works, using simulated data
made of white noise plus a sine-Gaussian burst. Top left (a) is a time-frequency
map of single short Fourier transforms of the signal.

Top right (b) is themean value of the power spectrum obtained by time averages from the previous map. Bottom left (c) is the signicance map computed from the previous= 20= 3: black bins are those with signicance above THIGH , grey one
with signicance above TLOW .

ones.

Bottom right (d) is the corresponding triggers map, with THIGH

and TLOW

At the same time also the variance of each frequency bin is computed in a similarway:2

σ 2 (t, f ) = (F (t, f ) − M (t, f ))(9.2)

The mean value is recomputed excluding bins with deviations greater than 3σ .
At this point the signicance of each time-frequency bin is computed as thedeviation from the mean value in units of σ :s(t, f ) =|F (t, f ) − M (t, f )|σ(t, f )(9.3)

This signicance map (see g. 9.1c) is the starting point for nding regionswith excess of power: the algorithm triggers every time the signicance of abin exceeds a given high threshold THIGH . All these triggers are then analyzed
one by one. Given one of them, all neighbouring bins are considered: if theirsignicance exceed a lower threshold TLOW they are clustered together with the9.1. THE HACR ALGORITHM

181

triggering bin. This procedure is iterated for all the bins in this newly formedcluster in order to recover all adjacent bins with signicance above the lower
threshold. The reason for this operation is that a transient event usually spreads
its power over more than one bin and therefore a clustering is necessary to beable both to recover the full event power and to avoid multiple counting of thesame event.
For each of these clusters several quantities are computed. The mean timeand frequency are obtained with weighted averages based on bin powers:

t

=

f

=

P
ti F (ti , fi )Pi∈C
F (ti , fi )P i∈Cfi F (ti , fi )Pi∈Ci∈C F (ti , fi )

(9.4)

(9.5)

Similarly the total power is computed summing over all bins in the cluster:PT OT = αX

F (ti , fi )

(9.6)

i∈C
where α is a normalization parameter depending on the window function used
and on the overlapping of subsequent segments used in computing the spectrum
map.
The maximum and mean signicances are also computed. An estimation of
the signal-to-noise ratio (SNR) is given dividing the power in the cluster withthe one in the mean spectrum (which is what would have been measured without
any transient):P
F (ti , fi )SN R = P i∈Ci∈C M (ti , fi )(9.7)

Other interesting parameters are the number of points in the cluster and its
time and frequency widths:

sP
∆t

2
i∈C (ti − t) F (ti , fi )=Pi∈C F (ti , fi )

(9.8)

sP
∆f

=

2
i∈C (fi − f ) F (ti , fi )Pi∈C F (ti , fi )(9.9)

All these results are saved in lists of events than can be used for various
analysis:• The number of clusters per minute can give a quantitative indication of
the stability of the detector and on the data quality.

• Looking at the time and frequency distributions of the clusters it is often possible to identify periodicities and particular frequency bands thatgenerate classes of transients.

182

CHAPTER 9. FAST TRANSIENT NOISES• The use of this algorithm for commissioning purposes can help in identifying the origin of not understood noises. For example frequent clusters canbe dicult to discover looking only at spectra, since they can be easily
mistaken by a stationary incoherent noise.

• From the data analysis point of view, list of triggers of this kind can be
useful to set up statistical veto procedures, making coincidence analysis
between dierent channels [85].

9.2 Analysis of dark fringe clusters
During the science run the HACR algorithm has been running on-line [86], looking for clusters in several channels: the dark fringe in-phase and quadrature, thelongitudinal corrections and errors, the laser frequency and power stabilization
errors, the beam monitoring system (BMS) errors.

All the results have been

stored in a centralized database. This was periodically mined by an automatic
script to produce short reports about some of the analyzed channels.

Thesereports are available via a web page [87].
The analysis of HACR results starts naturally from the clusters detectedin the dark fringe in-phase demodulated signal. In the following the analysis is1

described for two dierent periods of science data : the main dierence between
the two is the disappearance of clusters generated by the beam monitoring
system, due to the replacement of a broken piezo-electric actuator.
Time-frequency maps are shown in g. 9.2 for the dark fringe signal. These
give a global overview of the number of clusters in the analyzed channel, allowing
the identication of variations over time and of the most interesting frequencies.
The distribution in SNR, shown in g. 9.3 allows to identify two classes ofclusters:

the main part of the distribution has SNR greater than 9, while adierent population has lower SNR, even below 1. One should remember thatthe HACR algorithm does not consider any cut in the clusters SNR, but only
on their maximum signicance: therefore these low SNR clusters must in any
case have high signicance, meaning that they are actually regions of the timefrequency power map where there is a large uctuation of the noise. A deeper
investigation showed that this class of clusters are related to strong lines in thedark fringe spectrum: calibration lines or thermal modes of wires and mirrors.
Therefore in the following only clusters with SNR greater than 9 are considered.
In g. 9.4 the frequency distribution of the cluster is shown. It is possible
to distinguish several classes of clusters depending on their frequency (see table9.1). The clusters between 7.8 and 9.3 kHz are a relevant fraction of the total
and correspond to a structure well visible in the dark fringe spectrum, which is

1 The rst one spans from GPS time 867827080 to 868066580,

869885490 to 869932490.

the second one from

9.2. ANALYSIS OF DARK FRINGE CLUSTERS

183

104

103

SNR

Frequency [Hz]

102

103

101

102

100
0

2

4

6
Time [h]

8

10

12

104

103

SNR

Frequency [Hz]

102

103

101

102

100
0

2

4

6
Time [h]

8

10

12

Figure 9.2: Time-frequency map of clusters in the dark fringe in-phase demodulated signal during 12 hours of two dierent science mode periods. The colorof the dots represent the reconstructed SNR of the clusters. The bottom plot
refers to the rst period, before repairing the BMS system, the top one to thesecond period.

184

CHAPTER 9. FAST TRANSIENT NOISES
10-1

Counts (normal.)

10-2

10-3

10-4

10-5

0

20

40

60

80

100

60

80

100

SNR

Counts (normal.)

10-2

10-3

10-4

10-5

0

20

40
SNR

Figure 9.3: SNR distribution of the clusters detected in the dark fringe in-phase
demodulated signal. The top plot refers to the rst period and the bottom tothe second. Note that the time length of the two periods is not the same.

believed to be electro-magnetic noise coming from the main power supplies. The
clusters between 5.8 and 5.96 kHz corresponds to the frequency noise bump (seesec. 7.4). All other cluster classes correspond to structures in the dark fringe
whose origin is not well identied.

The classes at lower frequencies, namely

below 300 Hz were mainly related to the BMS malfunctioning: the number ofclusters detected in this region is indeed strongly reduced after the replacementof the broken actuator.
It is interesting to investigate if the detected clusters are distributed randomly in time or if there is some periodicity in their presence. This possibility
can be studied computing an auto-correlogram : for each cluster the time difference with all later ones is computed and organized in an histogram.

The

results for all the clusters and for those in some selected frequency bands areshown in g. 9.5 for the second period of science data. While the glitches at
low frequencies does not show any signicant periodicity, those in the rst class(between 7.8 and 9.3 kHz) shows a clear repetitive structure with a period of
about 2 s. This means that the source of noise coupling responsible for these9.3. ANALYSIS OF COINCIDENCE WITH AUXILIARY CHANNELS
103

Counts

102

101

100

0

1000

2000

3000

4000
5000
6000
Frequency [Hz]

7000

8000

9000

10000

0

1000

2000

3000

4000
5000
6000
Frequency [Hz]

7000

8000

9000

10000

Counts

102

101

100

Figure 9.4: Frequency distribution of the clusters detected in the dark fringe
in-phase demodulated signal.

The top plot refers to the rst period and thebottom to the second. Note that the time length of the two periods is not thesame.

glitches is modulated with a frequency of about 500 mHz.

9.3 Analysis of coincidence with auxiliary channels
A powerful method to investigate the origin of the dark fringe clusters is to correlate them with those detected in other auxiliary channels. In general, given
two lists of clusters for two channels, a rst analysis can look for time coincidences. Given a time window δt (usually of the order of 100 ms), two clusters are
considered coincident if their times dier less than δt. However, if the numberof clusters is large enough, even with small time window, some of them might
be considered accidentally coincident. To discriminate this possibility, the coincidence analysis is repeated several times with dierent time shifts added to
one of the two series of clusters. If the coincidences have a physical meaning,

185

186

CHAPTER 9. FAST TRANSIENT NOISESClass no.

Min. freq. [Hz]Max. freq. [Hz]Description

1

7800

9300

Power noise

2

5800

5960

Frequency noise

3

2990

3050

4

2520

2550

5

2200

2230

6

2050

2070

7

1700

1760

8

1280

1320

9

1020

1040

10

800

860

11

600

710

12

480

510

13

410

440

14

200

300

BMS

15

120

140

BMS

Table 9.1: List of cluster classied depending on their frequency.

the time-shift histogram, built counting the number of coincidences found foreach time shift, should have a large peak around zero. Moreover the number ofcoincidences for large time shift gives an estimate of the accidental rate. Finally,
the width of the peak around zero gives a estimate of the best time window to be
used for catching the largest number of real coincident clusters and the smallest
number of accidental ones.
Since the clusters detected by HACR have both a mean time and frequency
information, it is possible to study the simultaneous coincidence both in timeand frequency, with dierent window sizes (100 ms for time and 200 Hz for
frequency).

This usually further restricts the number of coincidences found,removing all those pairs of clusters which are not centered around the same
frequency. This is normally a desired feature if the coupling mechanism of thenoise is known to be linear. Otherwise a more general analysis can be performedby adding simultaneous time and frequency shifts to one of the two series ofclusters and building a double-shift histogram.

9.3.1 Coincidences with beam monitoring system error
signals
As already anticipated, during the rst period of science mode most of theclusters detected at frequencies below 600 Hz were identied to be caused by
the bad behavior of the beam monitoring system. For this reason it is interesting9.3. ANALYSIS OF COINCIDENCE WITH AUXILIARY CHANNELS
105

CountsAll glitches
104

103

102
105
7800-9300 Hz

Counts

4

10

103

102
103

Counts< 800 Hz
102

101

100

Figure 9.5:

0

10

20

30
Time [s]

40

50

60

Auto-correlogram of the dark fringe in-phase signal clusters, forsome selected frequency bands.

to carry out a coincidence analysis between the dark fringe clusters and those
detected in the four BMS error signals. The result of the time-shift analysis isshown in g. 9.6. The coincidences found at zero time-shift are physical, even
if their number is not large. The coincidences both in time (δt = 100 ms) andfrequency (δf = 500 Hz ) are concentrated at frequencies below 250 Hz, mainly
in correspondence of the BMS mount resonances (see sec. 7.6.1).

However,comparing the frequency distribution of dark fringe clusters before and after the
BMS repairing (see g. 9.7), it appears that also clusters at higher frequencies,
up to 800 Hz, are less frequent, suggesting a non-linear propagation of BMS
noise to the dark fringe.

9.3.2 Coincidences with quadrature signal
Since the quadrature demodulated signal of the dark fringe beam is almost
completely not sensitive to gravitational waves, it is interesting to analyze the

187

188

CHAPTER 9. FAST TRANSIENT NOISESBMS FFv
300

250

250

200

200
CountsCountsBMS FFh
300

150

150

100

100

50

50

0

0
-1

-0.5

0
Time shift [s]

0.5

1

-1

-0.5

BMS NFv

0
Time shift [s]

0.5

1

0.5

1

BMS NFh

200

200

150

150
Counts250

Counts

250

100

100

50

50

0

0
-1

-0.5

0
Time shift [s]

0.5

1

-1

-0.5

0
Time shift [s]Figure 9.6: Time shift plots for coincidences between the dark fringe signal andthe four BMS error signal.

coincidence between clusters in the main in-phase signal and in the quadratureone. The result of the time-shift analysis is shown in g. 9.8. The distribution
is indeed strongly peaked, showing that the coincidences found have physicalmeaning, but the maximum is not found for zero time shift, and the distribution
appears signicantly asymmetric. However, if also frequency coincidence is required, this asymmetry disappears. A double time and frequency shift analysis
gave the result shown in g. 9.9. The distribution shows at least three peaks:• the expected one at zero shifts;• a big and wide peak centered around ∆t = 70 ms and ∆f = 8100 Hz;• a smaller one centered around ∆t = 60 ms and ∆f = 2500 Hz;
The last two peaks mean that low frequency clusters in the quadrature demodulated signal are coincident with high frequency ones in the phase signal (seeg. 9.10). In particular almost all clusters detected in the 7.5-9.5 kHz region
in the in-phase signal are coincident with quadrature clusters below 900 Hz.
Similarly the in-phase signal clusters around 2 and 3 kHz are coincident withsome in the quadrature between 400 and 900 Hz.
In conclusion a relevant part of the clusters detected in the dark fringe
signal in the low frequency region (below 1 kHz) are coincident in both time

9.4. CONCLUSIONS

189

0.006
First lockSecond lockCounts (normal.)

0.005
0.004
0.003
0.002
0.001
0
100

200

300

400

500
600
Frequency [Hz]

700

800

900

1000

Figure 9.7: Comparison of the normalized frequency distribution of dark fringe
clusters before and after the repairing of the BMS actuator.

and frequency with similar clusters in the quadrature signal.

Instead, thosearound 2 and 3 kHz are coincident in time with clusters in the quadrature signal,
but the frequency must be shifted by 2.5 kHz to have double coincidence: thequadrature clusters below 1 kHz are coincident with the in-phase one at higher
frequency. The same is true for almost all the in-phase clusters in the 8-9 kHz
region: they are coincident with quadrature clusters in the same low frequency
region.
The mechanism that creates this non-linear relation between in-phase andquadrature clusters is still under investigation.

9.4 Conclusions
The strength of the HACR algorithm is its speed: it is possible to simultaneously
analyze tens of channels without the need of large computing power.

This

makes it very useful for commissioning purposes, since it can give quick andquantitative indications of the signals glitchiness.

Moreover the results canbe used to perform coincident analysis. This gives another perspective on thedetector noise characterization. It is not unlikely for example that a noise which
is normally not limiting the sensitivity, and therefore is not considered importantby a linear projection analysis, can couple in a highly non-stationary way andgenerate clusters in the main signal as well as in other ones.
During the science run, the periodic report web pages gave a valuable data
quality tool, since it allowed to easily compare the number and distribution ofcluster during dierent locks and periods, pointing out possible problems in thedetector.

190

CHAPTER 9. FAST TRANSIENT NOISES

2500

2000

Counts

1500

1000

500

0
-1

-0.5

0
Time shift [s]0.5

1Figure 9.8: Time shift plot for coincidences between the dark fringe in-phaseand quadrature signals.

5000

102
CountsFrequency shift [Hz]

10000

0
101
-5000

100

-10000
-0.4

-0.2

0
Time shift [s]

0.2

0.4

Figure 9.9: Time and frequency shifts histogram for double coincidences between
the dark fringe in-phase and quadrature signals.

9.4. CONCLUSIONS

191

In-phase signal

Counts (normal.)

103

102

101

100
Quadrature signal
103

Counts (normal.)All(0 ms, 0 Hz)(70 ms, 8100 Hz)(60 ms, 2500 Hz)
102

101

100

0

1000

2000

3000

4000
5000
6000
Frequency [Hz]

7000

8000

9000

10000

Figure 9.10: Frequency distribution of the in-phase and quadrature signal clusters found to be coincident with dierent time and frequency shifts, corresponding to the peaks in g. 9.9.

Part III
Conclusions

Conclusions
The work done for the commissioning of the Virgo interferometer in the last
three years, since the rst lock of the recycled conguration, has brought thedetector to a remarkable sensitivity, comparable with those of the LIGO scientic collaboration (see g. 9.11). The duty cycle of science mode data duringthe rst Virgo science run has been higher than 80%, with stable sensitivity.
During the time spent working in the Virgo project I have been involved
mainly in commissioning work, under several dierent aspects.

Angular sensing and control system. I worked for one year with the alignment group to the rst implementation of the angular control of the mirrors in the full recycled conguration of the interferometer, before the C6
an C7 commissioning runs (see chap. 4).

Length sensing and control system. The main part of my commissioning
work was carried out together with the locking group, for the re-implementation of the control system after the increase of the input power.
After the rst re-locks of the detectors I have been strongly involved in
the development of better control strategies to improve the stability andperformances of the locking: the slow servo systems to stabilize on thelong term the unitary gain frequencies of the loops and other locking
parameters; control lters improvements to reduce the noise re-injectedin the dark fringe signal (see chap. 3).

Moreover I worked on locking

characterization: to the reconstruction of the longitudinal optical matrix(see sec. 5.5); to the identication of the correct operating point which leadto the implementation of the sidebands balancing servo system, acting on
the longitudinal locking point (see sec. 5.7). These studies made possible
to further improve the performance of the longitudinal control system,
bringing it to the level of being no more a limiting source of noise at any
frequency.

Control noise studies. I started a systematic campaign of characterization of
the contribution to the dark fringe noise coming from control systems. The
main outcomes of this work have been: the implementation of an auto-

196

10-17
LIGO LHO 4km S5
LIGO LLO 4km S5
LIGO LHO 2km S5
GEO S5
Virgo VSR1 (Sept. 2007)
Virgo Design

Strain sensitivity [Hz-1/2]

10-18

10-19

10-20

10-21

10-22

10-23 1
10

102

103

104

Frequency [Hz]Figure 9.11: Comparison of the typical Virgo sensitivity during the rst sciencerun with that of the LSC detectors during the S5 run.

matic procedure for the measurement of longitudinal and angular control
noise budgets (see sec. 7.3 and 7.5), now routinely used during the sciencerun; the estimation of the contribution of frequency noise to the sensitivity
and the measurement of the transfer function between frequency noise anddark fringe signal (see sec. 7.4); the implementation of improved control
lters to reduce the noise coming from some servos (see chap. 7).

Noise studies. I focused mainly on the implementation of tools and analysis
techniques for the characterization of non-stationary noises (see chap. 8).
To this goal I developed several noise analysis tools that are providing
on-line data during the run: NonStatMoni to monitor the variation withtime of band-limited RMS of the dark fringe and environmental signals,
together with an o-line analysis tool that can periodically generate web
reports; LineMonitor (see sec. 8.1.1), which computes the spectrum of asignal and identify the main narrow lines, writing the results in the main
data stream; VirgoHACR to detect short transient noises on many dierent channels (see chap. 9). I also developed several techniques to analyze
the results of these tools, like the non-stationarity studies explained insec. 8.2.2.

Environmental noise. I collaborated with the environmental studies group

197
to characterize the coupling of acoustic and seismic noise through diusedand scattered light, implementing analysis techniques to better understandthe non-linear behavior of these noise couplings (see sec: 7.11).
In terms of the maximum detection range for coalescing neutron stars, inthe last three years the detector sensitivity has improved enormously, passing
from 80 kPc during the C6 commissioning run (September 2005) to more than
5 MPc in the months after the end of VSR1 (December 2007).

There is still

a lot of work to be done to improve the detector sensitivity: the main sourceof noise below some hundreds Hz is understood to be of environmental origin,coupling through diused light. Moreover the sensitivity is close to be limitedby actuation noise at the lowest frequencies (10-30 Hz).
In the near future major upgrades of the detector is planned, moving toward
the Virgo+ project [88]. The laser power will be increased from 20 to 50 W.
This implies that thermal eects in the input mirrors will increase accordingly.
For this reason a thermal compensation system in under development, and it
will be installed and commissioned before the increase of input power. Other
upgrades are foreseen: the entire digital data acquisition and control system will
be replaced; the two arm cavity mirrors will also be replaced with the goal ofincreasing the nesse from 50 to 150. Finally it is foreseen to install monolithic
suspensions to reduce the thermal noise at low frequencies. This should increase
the design coalescing binary horizon up to about 50 MPc [88, 89].

Part IV
AppendicesAppendix A
Fields inside optical systems
This appendix collects several computations of elds and signals in dierent
kind of optical systems. All the computations are carried out in the plane wave
approximation, without considering the eect of higher order modes.

Moregeneral computations can be found also in [90].

A.1 Basic conventions
In the plane wave approximation, the beam entering in an optical system canbe described by its (complex) amplitude and its frequency:

E(t) = E0 eiωt

(A.1)

In general the frequency ω/2π can be that of the main laser (which will be
indicated simply by ω ) or of one of the radio-frequency sidebands generated by
frontal modulation, namely ω ± Ω.
The eect of the reection on a mirror can be described as follows (seeg. A.1). If the mirror is still, the amplitude of the transmitted and reectedbeams can be written asET= t E0ER= ir E0

where t and r are real amplitude transmission and reection coecients and theconventions used here are those of [12].
Any motion of the mirror can be described as a displacement x(t) from axed reference point. If this displacement is constant, the eect on the reectedbeam would be simply a dephasing given by

ER = ir e2ikx E0202

APPENDIX A. FIELDS INSIDE OPTICAL SYSTEMS

Figure A.1:

Conventions for transmitted and reected beams by a semi-reecting mirror.

where k =ωc . This results is still true if the variation of the displacement is small

in the time the light needs to cover it, which is equivalent to the requirementthat the speed of the mirror must be much less than the speed of light. This isof course true with very good accuracy. Therefore the eect of a moving mirror
is a time varying dephasing:ER (t) = ir e2ikx(t) E0
If the displacement of the mirror is smaller than the wave-length of the laser
beam, namely|x(t)|
 1, then the previous expression can be approximated toλrst order:

ER (t) ≈ ir [1 + 2ikx(t)] E0
To compute transfer function from mirror motions to optical signals, it is useful
to consider monochromatic motions given for example byx(t) = x0 cos(2πωx t) =


x0 2iπωx te+ e−2iπωx t2

In conclusion to completely characterize an optical system (at the linear order) it is sucient to consider an input eld with two components: one oscillating at the main frequency ω and one at ω + ωx and to compute the transmissioncoecients of both components to the output port, taking into account the effect of the perturbation added inside the system. If the elds are described bybi-dimensional vectors, this can be expressed in matrix notation:

E0
Ex

!
=M
out

E0
Ex

!
in

where M is a two by two matrix. For example the eect of a moving mirror isA.2. POWERS AND DEMODULATION

203

1given by"
M = ir 1 +

0
2ikx0

0
0

#!

and the propagation over a distance L by"
P(L) =
where kx =

eikL
0

0

#

ei(k+kx )Lωxc .

A.2 Powers and demodulation
The input eld can be decomposed in a carrier and two RF sidebands. For small
modulation index miωt

EIN (t) = E0 (t)e




m2
m iΩt
m −iΩt1−+i e +i e
422where the amplitude E0 (t) can vary slowly (with respect to ω and Ω) to take
into account a possible perturbation added to the input, like amplitude, phaseor frequency noise. In general the output eld can be written in similar way
EOU T (t) = eiωt A0 (t) + A+ (t)eiΩt + A− (t)e−iΩtwhere the A coecients can be computed given the system matrix dened above

Mij (k, kx ) as a function of the eld and perturbation frequencies:m2
E0 (t) M11 (k, 0) + M12 (k, kx )eiωx t
A0 (t) =1−
4
+ M12 (k, −kx )e−iωx tm
A+ (t) = i E0 (t) M11 (k + kΩ , 0) + M12 (k + kΩ , kx )eiωx t
2
+ M12 (k + kΩ , −kx )e−iωx tm
A− (t) = i E0 (t) M11 (k + kΩ , 0) + M12 (k − kΩ , kx )eiωx t
2
+ M12 (k − kΩ , −kx )e−iωx t
A photo-diode placed at the system output port detects a total power given
by:

POU T (t)2

= |EOU T (t)|222

= |A0 | + |A+ | + |A− |+ eiΩt A∗0 A+ + A0 A∗− + e−iΩt A∗0 A− + A0 A∗++ e2iΩt A∗− A+ + e−2iΩt A∗+ A−
1 This is not the most general form of this matrix. Indeed, the (1, 2) o-diagonal matrix
element could be dierent from zero, introducing a down-conversion eect of the eld at ω.
However, this is neglected here since, assuming the eld at ω to be small, it would give a second
order eect. Moreover, this form of M is enough to compute all optical transfer functions
from mirror displacement to any eld.

204

APPENDIX A. FIELDS INSIDE OPTICAL SYSTEMSAs usual the mean power, or DC signal, can be extracted, neglecting all component with oscillatory behavior:

2

2

2

DC(t) = |A0 | + |A+ | + |A− |(A.2)

and the in-phase and quadrature demodulated signals can be obtained by mixing
with a sine or cosine wave at the modulation frequency Ω and taking only the
low frequency part of the result:


1 
∗
∗
Re A0 A− + A0 A+
21 = POU T (t) · sin Ωt = Im A0 A∗− + A∗0 A+
2= POU T (t) · cos Ωt =ACp (t)ACq (t)

(A.3)
(A.4)

More in general the demodulation can be done with a cosine wave shifted by anarbitrary phase φ and the result will be

ACφ (t) =
1  −iφRe e
A0 A∗− + A∗0 A+ ≡ Re e−iφ AC
2

(A.5)

2

where the complex demodulated signal has been dened .
In a similar way the frequency response of the output demodulated signal
to a perturbation added inside the system can be computed by specialization of
the above formulas:AC(ωx )



im
m2 h
∗
= −
1−
− M12
(k, kx )M11 (k − kΩ , 0)
4
4∗
−M11 (k, 0)M12(k − kΩ , −kx )∗+M12(k, −kx )M11 (k + kΩ , 0)i∗
+M11(k, 0)M12 (k + kΩ , kx )(A.6)

If the perturbation is added to the input beam instead of inside the system,
this result is slightly dierent. The input eld can be given in general by

i
h

m
m
EIN (t) = 1 + αeiωt t + βe−iωx t E0 (t)eiωt 1 + i eiΩt + i e−iΩt22and the frequency response of the demodulated signals is given byAC(ωx )



im
m2 h
∗
1−
− αM22(k, kx )M11 (k − kΩ , 0)
4
4∗
−βM11 (k, 0)M22(k − kΩ , −kx )

= −

∗
+βM22
(k, −kx )M11 (k + kΩ , 0)i∗
+αM11 (k, 0)M22 (k + kΩ , kx )

A.3. SIMPLE MICHELSON INTERFEROMETER

Figure A.2: Scheme of a simple Michelson interferometer, with the names conventions used and explained in the text.

A.3 Simple Michelson interferometer
Referring to g. A.2, the beam splitter is the mirror number 1 and its transmission and reection coecients are denoted by t1 and r1 (they are usually both√

equal to 1/

2). Similarly the two arm mirrors are denoted by 2 and 3 and theirdistance from the beam splitter by L2 and L3 .

All three mirrors are allowed

to move at the signal frequency ωx , with amplitudes given by xi . The naming
convention for elds are shown in g. A.2.
The two interesting elds are at the interferometer reection E10 and at thedark port E9 . Let the propagation matrix along the two arms be P2 and P3 andthe reection matrices for the three mirrors be

where the factor

√

R̂1



√
= ir1 1 ± 2R1

R̂2= ir2 (1 + R2 )R̂3

= ir3 (1 + R3 )2 for the beam splitter is necessary to take into account its

45 degrees orientation and the sign depends on the direction the incoming beamis hitting the mirror from. The Ri matrices are simply given by"#
0 0
Ri = 2ikxi
1 0
2 Note that in this expression the eld amplitudes are used. In the case of a single cavity,

the carrier can be treated as a real eld, while the two sidebands are pure imaginary numbers.
The complex conjugation then introduces a sign dierence between the two terms in eq. A.5.

205

206

APPENDIX A. FIELDS INSIDE OPTICAL SYSTEMS
1
Dark port
Reflection
Total

In-phase
Quadrature

0.004

Demodulated signal [W]Power [W]

0.8

0.6

0.4

0.002

0

-0.0020.2-0.004
0
-1

-0.5

0
Differential offset [µm]

0.5

1

-1

-0.5

0
Differential offset [µm]0.51Figure A.3: Example of output signals from a simple Michelson interferometer.

√
r1 = t1 = 1/ 2, r2 = 0.8, r3 =0.95, L = 3000 m, ∆L = 1.2 m, λ = 1 µm, m = 0.1, Ω = 2π · 6.2 · 106 rad/s.
The input power is 1 W. On the left the powers reected and transmitted to the
The parameters have been chosen as follows:

dark port are shown. On the right the in-phase and quadrature demodulated
signals at the dark port are plotted.

With these conventions, the elds reected and transmitted by the interferometer are given by

TM ICH =RM ICH =

E9
E0

E10
E0

= −r1 r2 t1 (1 −√

2R1 )P2 (1 + R2 )P2√−r1 r3 t1 P3 (1 + R3 )P3 (1 + 2R1 )√√= −ir12 r3 (1 + 2R1 )P3 (1 + R3 )P3 (1 + 2R1 )+it21 r2 P2 (1 + R2 )P2Since all mirror motions are considered small and only linear eect are to beconsidered, the above expressions can be simplied by keeping only terms containing at most one R operator:

TM ICHRM ICH√= −r1 r2 t1 P22 + P2 R2 P2 − 2R1 P22

√−r1 r3 t1 P32 + P3 R3 P3 + 2P32 R1
= −ir12 r2 P22 + P2 R2 P2
√−ir12 r3 P32 + P3 R3 P3 + 2R1 P32

(A.7)

(A.8)

From these equations the full expression of the two operators can be easily
computed. It is customary to dene common mode and dierential mode lengths

A.3. SIMPLE MICHELSON INTERFEROMETER

207

and motions by

L2
L3
x2
x3

∆L
2
∆L
= L−
2
xd
= xc +
2xd= xc −
2= L+

Equations A.7 and A.8 have to be specialized in dierent ways for carrierand sidebands, since the frequency and the interference conditions are dierent.
The carrier is usually kept near destructive interference at the dark port. Itstransmission coecient can be derived form the (1, 1) element of eq. A.7:

−e−ik(∆L−2L) r1 e2i∆Lk r2 + r3 t1
2i∆L0 kThe destructive interference is obtained when e= −1.

Therefore thedierential mode length can be expressed as a displacement from this value∆L = ∆L0 + D.

In a similar way the value of the common mode length

2iL0 kcan be expressed as the displacement C from the point dened by e

= 1.

These displacements are often called osets. Afterward, some approximations
can be considered. First of all kx  k can be used everywhere except inside
the exponents, where some care is necessary.

kx D ≈ 0, kx C ≈ 0.

Indeed it is possible to assume

For the sidebands, the same conditions must be used,with the approximation kΩ  k . The nal expressions for the transmission andreection matrices are quite long and are not reported explicitly here.
Using the approximated version of eq. A.7 and the expressions found in theprevious section for DC and demodulated signals (eq. A.2 and A.3) it is possible
to compute the power reaching the dark port and the one reected as a functionof the common and dierential mode osets. From eq. A.2 the transmitted andreected powers can be computed:

DCREF

DCT RA= PIN r14 r32 + r22 t41 + r12 r2 r3 t21 cos(2Dk)
2 − m2 + m2 cos(2∆LkΩ )
= PIN r12 t21 r22 + r32 + r2 r3 cos(2Dk)
m2 − 2 − m2 cos(2∆LkΩ )(A.9)(A.10)The power at the dark port is indeed minimum when the dierential modeoset is zero (see g. A.3).
maximum.

In this condition the reected power is of course

The residual power at dark fringe has both a carrier component

determined by the asymmetry in the arm end mirror reections and a sidebands
component, determined mainly by the Schnupp asymmetry ∆L0 . Both reected
and transmitted powers are independent of the common mode oset C .

208

APPENDIX A. FIELDS INSIDE OPTICAL SYSTEMS

Figure A.4: Scheme of a Fabry-Perot resonant cavity, with the names conventions used and explained in the text.

Similarly, from eq. A.3 and eq. A.5 the demodulated signals at the dark port
can be obtained:1
AC = − e−ikΩ (∆L−2L) −1 + e2i∆LkΩ m(m2 − 4)r12 r2 r3 t21 sin (2Dk)
8This signal is zero both when the ITF is tuned at dark or bright fringe (destructive or constructive interference at the dark port), but the crossings havedierent slopes, see g. A.3. This signal has a constant phase, therefore there
is one demodulation phase for which the in-phase component is zero and thequadrature contains all the information.
Using eq. A.6 the transfer function from a dierential mode motion to thedark fringe signal can be computed:

TF

=


1 − i(∆L−2L)(ckΩ +f π)c−1 + e2i∆LkΩ
e
8

1+e

2i∆Lf πc

km(m2 − 4)r12 r2 r3 t21

This transfer function is almost at in amplitude up to a frequency which depends only on the asymmetry in the arm lengths (for an asymmetry of 1.2 m
this frequency is about 40 MHz).

A.4 Fabry-Perot resonant cavity
Referring to g. A.4, let 1 and 2 denote the input and end mirrors, and ri , ti thecorresponding reection and transmission coecients. The length of the cavity
is denoted by L. It is sucient to consider only motions of one of the mirrors,for example the end one.

First of all the eld inside the cavity E1 must be

computed. It can be expressed in terms of the following matrix equation:

E1 = t1 E0 − r1 r2 PL (1 + R2 )PL E1
which can be solved, with some care in the order of the operands, obtaining thefollowing result:−1

E1 = t1 [1 + r1 r2 PL (1 + R2 )PL ]E0(A.11)

A.4. FABRY-PEROT RESONANT CAVITY

209

1.00000.9995

Total
Carrier
Sidebands

10-2

0.99900.9985
Power [W]Power [W]10-3
0.99800.99750.997010-4

0.996510-50.99600.99550.9950-1-0.5

Figure A.5:

0
Cavity length offset [µm]

0.5

1

10-6

-1

-0.5

0
Cavity length offset [µm]0.51Power reected (left) and transmitted (right) by a Fabry-Perotcavity with parameters similar to those of Virgo arm cavities:

L = 3000 m,

r1 = 0.93883, r2 = 0.99992, m = 0.3.
From this the transmitted and reected elds can be easily derived:

ET
ER

−1

= t1 t2 PL [1 + r1 r2 PL (1 + R2 )PL ] E0
= i r1 + r2 t21 PL (1 + R)PLi
−1
[1 + r1 r2 PL (1 + R2 )PL ]E0

The carrier and sidebands elds have dierent resonance conditions inside
the cavity. Typically the cavity length is tuned in order to have the carrier atresonance. The modulation frequency is chosen in order to have the sidebands
anti-resonant. The (1, 1) component of the matrix in eq. A.11 is:

t1
1 + r1 r2 e2ikL
Therefore the correct resonance conditions are obtained choosing for the carrier

e2ikL0 = −1 and for the sidebands e2i(k±kΩ )L0 = 1. The previous equations cantherefore be simplied and the cavity length can be expressed as an oset fromthe resonance.

Approximations similar to the one adopted for the Michelsoninterferometer can be applied also here. The nal expression for the transmission
an reection matrices are quite long and are not reported here.
The transmitted and reected powers can be computed, using eq. A.2, as afunction of the oset from resonance:

"
PT RA

= PIN


#
t22 t21 1 + r22 r12 − 2 −1 + m2 r2 r1 cos (2δLk)
2 + 2r24 r14 − 4r22 r12 cos (4δLk)

"

PREF


t21 −1 + r22 (r12 + t21 )1 + r24 r12 − 2r1 r2 cos(2δLk)
#

2 2
2
2
2 2r1 r2 t1 −1 + r2 (r1 + r2 ) cos(2δLk)−m
1 + r24 r14 − 2r12 r22 cos(4δLk)= PIN r12 + t21 +210

APPENDIX A. FIELDS INSIDE OPTICAL SYSTEMS

0.2

In-phase
Quadrature

In-phase
Quadrature

2

Demodulated signal [x 10-5 W]

Demodulated signal [W]

0.15
0.1
0.05
0
-0.05
-0.1

1

0

-1

-0.15
-2
-0.2
-0.4

-0.2

0
Cavity length offset [µm]

0.2

0.4

-0.4

-0.2

0
Cavity length offset [µm]

0.2

0.4

Figure A.6: Demodulated signals in reection (left) and transmission (right) ofa Fabry-Perot cavity with the same parameters of g. A.5.

The transmitted power is maximum when the cavity is resonant for the carrier,see g. A.5. A second smaller peak is visible when the sidebands are resonant.
The distance between two dierent resonant peaks is called free spectral range(FSR) and it can be expressed both in terms of a change in the cavity length
or in the laser frequency:

δL =
δν

=

λ
2
c
2L

It is customary to dene the nesse of the cavity as√
π r1 r2F=
1 − r1 r2With these denition the width of the resonance peak (at half height) is givenbyδνF W HM =δνFFor the Virgo arm cavities the nesse is about 50.
The demodulated signals in reection and transmission can be computed
using eq. A.3 and eq. A.5:

ACREF
ACT RA
2 2
2m 2
2 1 + r2 (r1 + t1 ) sin(2δLk)=(m − 4)r1 r2 t1
4
1 + r14 r24 − 2r12 r22 cos(4δLk)m
r1 r2 t21 t22 sin(2δLk)= i (m2 − 4)
4
1 + r14 r24 − 2r12 r22 cos(4δLk)(A.12)(A.13)

These are the well known Pound-Drever-Hall signals that can be used to control
the cavity length (see g. A.6 and chapter E).
Finally, using eq. A.6 the transfer function from a motion of the end mirror
to the transmitted signal can be computed:

T FT RA = −im(m2 − 4) 4πiLf
r1 r2 t21 t22
e c
8πiLf
2
2
2 (r1 r2 − 1)
1 − e c r12 r22(A.14)

A.4. FABRY-PEROT RESONANT CAVITY

211

Transfer function [W/m]

104

103

Exact
Approximated
102 0
10

101

102
Frequency [Hz]

103

104

Figure A.7: Transfer function from longitudinal motion of the cavity end mirror
to the transmitted demodulated signals, for a Fabry-Perot cavity with the same
parameters of g. A.5. The red curve is the exact expression, while the blue oneis the usual one-pole approximation.

For small enough frequencies, the exponential can be approximated at rst orderand the transfer function is given by a simple pole (see g. A.7):

T FT RA (f ) = i

km(m2 − 4)r1 r2 t21 t22
1
2(r12 r22 − 1)2
1 − i ffp

The pole frequency is given byfp =1 c 1 − r12 r22c≈
2π 2L 1 + r12 r22
4LF

For a Fabry-Perot cavity like Virgo arms, this pole is at about 500 Hz.
The main feature of a Fabry-Perot cavity, that can be deduced from eq. A.12
or A.14, is that the optical response of the reected eld to a cavity length change
is amplied, with respect to a simple mirror, by a factor

G=

2F
π

This is the reason why resonant cavities are used as optical amplier for gravitational wave signals. For the Virgo arm cavities, this amplication factor isabout 32.

Appendix B
Lock acquisition techniqueduring VSR1

This appendix describes in details the lock acquisition technique used in Virgoduring all the science run.

This is the result of the work of the entire lock-ing, alignment and commissioning teams during the last years. After the endof VSR1 it has further evolved, gaining in speed, robustness and noise performances. This is not discussed here, see for example [91].
The full locking acquisition sequence in Virgo is composed of 12 steps, eachleaving the interferometer in a stable state. The following sections describe all
the actions performed during the procedure. Please refer to [92] and [93] for amore technical description.

B.1 Locking and pre-alignment of the two cavities
B.1.1 Locking of the arm cavities and MICH
The standard state of the uncontrolled Virgo interferometer is with all mirrors
roughly aligned except the power recycling one, which is intentionally misalignedby 150 µrad in order to prevent its reected beam to interfere inside the optical system.

In the rst step of the locking acquisition procedure, due to thismisalignment, the two cavities are locked independently using the in-phase demodulated signals in transmission (Pr_B7_ACp and Pr_B8_ACp, see g. B.1) witha standard Pound-Drever-Hall technique. In order to increase the linear rangeof the error signals, they are normalized using the power transmitted by the214

APPENDIX B. LOCK ACQUISITION TECHNIQUE DURING VSR1

Figure B.1: Scheme of the control strategy for the rst step of the lock acquisition procedure. See the text for more explanations

cavities themselves [46]:

EN

=

EW

=

Pr_B7_ACp
Pr_B7_DC
Pr_B8_ACp
Pr_B8_DC

The resonance condition at the dark port is chosen on order to be at half

fringe : the power transmitted there is half of the maximum possible.

This

condition is well outside the linear range of the demodulated signal, even withnormalization. For this reason a DC locking is used: the fringe is locked directly
to the amount of power which is transmitted to the dark port, normalized by
the maximum possible power which is estimated using the power reected by
the north arm and by the secondary face of the beam splitter. In other words,
the error signal is given by

EM ICH = α

Pr_B1p_DC−φ
Pr_B5_DC

where the dark fringe beam used is the one which does not pass through theoutput mode-cleaner, which is not on resonance yet. In the previous formulas α
is a normalization constant which depends essentially on the reectivity of theB.1. LOCKING AND PRE-ALIGNMENT OF THE TWO CAVITIES
secondary face of the BS mirror and on the fraction of the beam that hits thephoto-detectors and φ is the fringe oset which is set to 0.5 in this step andafterward it is reduced down close to 0 which corresponds to the dark fringe
condition. The response of this signal to MICH motions becomes smaller andsmaller as the fringe oset decreases, since the dark port power has a typicalcosine dependence on the MICH oset.
The PR mirrors does not need to be controlled yet, since it is misaligned.

B.1.2 Fine pre-alignment
After the two cavities and MICH are locked, a ne pre-alignment procedure
is run with several goals: to align the input beam in order to center it in thenorth arm cavity; to align the beam splitter to center the reected beam in thewest arm cavity; to align the input and end mirror of both cavity in order tosuperimpose the beam and cavity axes. To do this, a global automatic alignment
system is engaged for the two cavities separately. This system, which will be
explained more in details in chapter 4, uses the demodulated signals obtained
from wave-front sensors placed in transmission of the cavities to reconstruct
suitable error signals for the angular position of the input and end mirrors. The
misalignment of the PR mirror ensures that the two cavities are independent.
These error signals are passed through pure integrator lters to remove theirhigh frequency component and then used as set-points for the local angularcontrol systems of the mirrors. The bandwidth of this drift control system isof the order of few tens of mHz. Its main advantage is that the high frequency
motion of the mirrors are still controlled by the local systems, easily providing
enough gain to damp the suspension angular resonances, while ensuring thatthe cavity is well aligned with respect to the laser beam.
While this system keeps the cavities aligned, the position of the transmitted
beams are measured by cameras and the pointing of the input beam and the
BS position are changed accordingly to meet xed reference position.
All this pre-alignment procedure takes in normal condition few minutes andafterward the drift control system is dis-engaged, since it cannot work withthe same error signals during all the lock acquisition sequence. Therefore themirrors are controlled by the local systems alone keeping them close the correctlyaligned position.

B.1.3 Check of main modulation frequency
The main modulation frequency Ω must be precisely tuned to match the resonant peak of the input mode cleaner. The carrier is always kept on resonanceby the locking servos of the IMC. If the modulation frequency is chosen in order to match exactly an integer multiple of the IMC free spectral range (about

215

216

APPENDIX B. LOCK ACQUISITION TECHNIQUE DURING VSR1

Figure B.2: Eect of a mis-tuning of the main modulation frequency with respect
to the IMC length. (a) the frequency is tuned to make the two sidebands exactly
resonant (b) the sidebands are not perfectly resonant in the IMC.

1 MHz), then any length variation of the cavity does not couple linearly to achange in amplitude of the sidebands transmitted into the main interferometer(see g. B.2a). If instead the matching is not perfect (see g. B.2b), any change
in the length of the IMC can couple linearly and dierentially to the sideband
amplitudes. In other words if the modulation frequency is not well matched to
the length of the IMC, any length noise of this cavity can couple directly to thedark fringe signal.
For this reason, during the pre-alignment procedure, a check of the relative matching of the IMC length and modulation frequency is performed.

In

normal condition a calibration line at 1111 Hz is always added as an external
perturbation to the laser frequency. For the IMC cavity this line is equivalent
to length noise and it aects the sidebands only if the modulation frequency is
not exactly tuned. A fraction of the beam transmitted by the IMC is sent toa photo-detector and demodulated rst at Ω and then at 1111 Hz. This signal
gives the amplitude of the calibration line as seen by the beating between the
carrier and the rst order sideband elds. It is therefore directly related to themis-match between Ω and the IMC length and it can be used as an error signal
to tune the modulation frequency.
If the mismatch is small, it is corrected simply changing the modulation
frequency. Otherwise the length of the IMC is changed, by unlocking it, moving
the top suspension point of the end mirror of the needed amount and re-locking
it. After this operation is performed, the lock acquisition can be started again.
The needed accuracy for this tuning, at the present Virgo sensitivity, is a bit
less than one Hz, and it can be easily obtained by only checking the matchingat the beginning of each lock.

B.2. SECOND STAGE OF FREQUENCY STABILIZATION

Figure B.3:

Control scheme for the laser frequency and the common mode of

the interferometer when the second stage of frequency stabilization (SSFS) is
engagedB.1.4 Force reallocation to top suspension point
Another action performed during this rst step of lock acquisition is the reallocation of the very low frequency part of the longitudinal corrections to the top
stage of the suspensions. This tidal control, engaged at the rst step for the BS
mirror, consist in using the longitudinal correction as set point for the top stage
local control after passing through an integrator lter with unity gain around
1 mHz. In this way it is possible to reduce strongly the force applied directly
to the mirror and the necessary dynamical range of the lower stage actuators.
For the NE and WE mirror the longitudinal correction is not used for adjusting the set point, but it completely replaces the local control error signals
coming from position sensors:

the longitudinal correction is therefore mixed

with the accelerometer signal, with cross-over frequency at 70 mHz. In this way
the same eect of the tidal control is obtained, plus the advantage of avoiding
the use of noisy local signal for the suspension control. This particular system
is called global inverted pendulum control (GIPC).

B.2 Second stage of frequency stabilization
In the second step of the lock acquisition procedure the control strategy for thetwo arm cavities is modied. As already stated, a change in the laser frequency
is equivalent to a change in the length scale.

The most sensitive degrees of

freedom are those related to the long arm cavities: in particular a change in

217

218

APPENDIX B. LOCK ACQUISITION TECHNIQUE DURING VSR1
their mean length (CARM) is equivalent to a change in laser frequency (eq. 3.6).
Moreover the residual frequency noise, with only a stabilization on the IMC and
RFC active, is still too high for reaching a good sensitivity. A further stage ofstabilization is needed and the only better possible reference available is themean length of the arms itself.
Therefore during the second step of lock acquisition the second stage of fre-quency stabilization (SSFS) is engaged (see g. B.3). It uses the error signal
coming from the reection of the north arm (as seen by the Pr_B5_ACp signal).
The correction is obtained by a suitable control lter to give a unity gain frequency (UGF) which varies between 100 Hz and 25 kHz depending on the stateof the interferometer. This is then added to the error signal of the rst stage of
frequency stabilization (coming from the reection of the IMC). It also replaces
the reference cavity signal for the lock of the input mode cleaner. Indeed, due
to the very high gain of the rst stage of stabilization, the sum of the IMC
reection and of the SSFS correction signal can be assumed to be zero in theactive band of the second stage of frequency stabilization. Therefore the IMC
reection and the SSFS signal can be considered to be equal and opposite. In
other words summing these two signal is equivalent to using the IMC reection
for the IMC locking, with the usual unity gain frequency of about 100 Hz.
In this way the laser frequency is kept locked to the mean length of thelong arms below some tens of kHz (by the SSFS loop) and the input modecleaner is simultaneously locked on the laser frequency. In this way however thereference cavity is no more locked and the common mode of the arm cavities isfree to swing at low frequency, mainly in correspondence of resonances of thesuspensions. This results in an equivalent change in the laser frequency, which
can be seen for example by the reference cavity error signal. Therefore this isused, with a bandwidth of about 1 Hz, to feed back a common mode correction
to the end mirrors.

In this way the mean length of the cavities and nallythe laser frequency are still locked to the reference cavity below 1 Hz.

Thislast control loop is usually called the CARM loop, since it acts directly on themechanical common mode of the end mirrors.

It must be distinguished fromthe SSFS loop which is the one controlling the laser frequency with high unity
gain frequency.
This control scheme is completed by the control of the dierential mode ofthe end mirrors (DARM). A suitable error signal is given by the transmissionof one of the two cavities, since any common mode motion is no more visible
there due to the action of the SSFS. Actually the demodulated signal comingfrom the west arm transmission is used to control dierentially the end mirrors.
The switch to common and dierential mode control and the engagement of
the SSFS is done quickly, without any mixing or blending of error signals. At
the same time the fringe oset is reduced from 0.5 to 0.4.

B.3. RE-ALIGNMENT OF POWER RECYCLING MIRROR

101
PR misaligned
PR aligned

Magnitude [a.u.]

100
10-1
10-2
10-3
10-4
10-5
50

Phase [deg]

0

-50

-100

-150

-200
102

103

104

105

Frequency [Hz]Figure B.4:

Measured open loop transfer function of the SSFS loop during

the second step of the lock acquisition sequence, when the PR mirror is still
misaligned (red) and during the third step, after the re-alignment of the PR
mirror (blue).

The SSFS control loop is implemented by means of analog electronics, sincedue to its high unity gain frequency it must be able to operate up to some
hundreds of kHz. This electronic system includes a limited remote control of
the gain which is necessary to cope with the great increase in the optical gain
during the dierent steps of the lock acquisition sequence.

Indeed when the

SSFS is rst engaged its unity gain frequency is around 160 Hz (see g. B.4)
while at later steps it increases up to tens of kHz.

B.3 Re-alignment of power recycling mirror
After the engagement of the SSFS control loop the power recycling mirror isrealigned from its position at 150 µrad in 15 seconds. During this transition thecirculating power inside the interferometer increases by a large amount due tothe recycling action of the aligned PR mirror (see g. B.5 right). Therefore all
signals undergo a large change in their optical gain, which is almost completely

219

APPENDIX B. LOCK ACQUISITION TECHNIQUE DURING VSR1

0.6

20
B5 power
PR misalignment

0
-20

0.4

-40
-60

0.3
-80
0.2

-100
-120

PR mirror misalignment [µrad]

0.5

B5 power [mW]

220

0.1
-140
0
190

195

200

205

210

-160
215

Time [s]Figure B.5:

Left:

scheme of the interferometer controls at step 3, after there-alignment of the power recycling mirror. Right: change in the power inside
the recycling cavity, as measured by a photo-diode on the B5 beam, during thepower recycling mirror re-alignment.

compensated by the use of suitable normalizations, as explained in the previous
sections. The only exception is the SSFS analog loop: the gain of the corrector
lter is changed to compensate the increase in the optical gain, and the new
UGF is set around 1 kHz (see g. B.4).

The corrector gain of the SSFS ischanged again during later steps to cope with the further increase of circulating
power.
At the same time the longitudinal control of the PR mirror is engaged using
the in-phase signal coming from the demodulation at 3Ω of the interferometerreection Pr_B2_3f_ACp (see g. B.5 left). The choice of this particular signal
instead of the normal demodulated one is motivated by the fact that the former isrobust against changes in the interferometer state, in particular against a change
in the fringe oset [46]. The normal demodulated signal strongly changes thegain and even the sign of its response to PRCL motion and it is therefore much
more dicult to use it during all the lock acquisition sequence.
This error signal is normalized using the value of the power inside the recycling cavity, as read by Pr_B5_DC. In this way the change in optical gain due tothe increase of the recycling gain during later steps of lock acquisition is almost
completely compensated.
This is the rst condition of the interferometer with all mirror aligned and all
longitudinal degrees of freedom controlled. Moreover during this step dierent
control lters are engaged for the MICH and PRCL loops to have more gain at
low frequency, allowing to obtain a better locking accuracy. Finally the tidal
control is engaged also for the PR mirror.

B.4. FINE PRE-ALIGNMENT OF PR MIRROR

221

50

8
B5 power
B5 2f power (x 400)
Lock step

7

40

Powers [mW]

35

6

30
25

5

20
4

15
10

Lock acquisition step

45

3

5
0
220

240

260

280

300

2
320

Time [s]Figure B.6:

Left:

scheme of the interferometer controls at step 6, after thetransition to dark fringe. Right: Evolution of total and sidebands power (see by
the Pr_B5_2f_ACq signal) inside the recycling cavity during the lock acquisition
sequence.

B.4 Fine pre-alignment of PR mirror
At this point the fringe oset is decreased to 0.2, and the control lter for the
DARM loop is changed in order to obtain more gain at low frequency.

Also

the input mirror coil drivers (see section 2.4) are switched to the low noiseconguration, even if they are not used for the control.

Afterward a pre-alignment of the power recycling mirror is performed, byusing global signals in a similar way to what is done at step 1 for the arm
cavities. The error signals used for these control loops (which have a bandwidthof about 3 Hz) come from wave-front sensors placed on the B5 beam (see chapter
4 for more details). These alignment loops are kept active until the power of the
sidebands inside the recycling cavity (measured by the Pr_B5_2f_ACq signal)
goes above a suitable threshold. At the end of this pre-alignment procedure the
PR angular control is switched back to the local sensors.

The fringe oset is then decreased to 0.1 and a ne tuning of several loop
gains is performed to maintain the stability of the interferometer. In this conguration the demodulated signals coming from the B5 beam are obtained byusing the photo-diode which receive a lower fraction of the total power (usuallycalled low power diode ), to avoid saturation due to the still large oset fromzero of the Pound-Drever-Hall signal for the MICH loop.

222

APPENDIX B. LOCK ACQUISITION TECHNIQUE DURING VSR1

B.5 Transition to dark fringe
The locking point of the MICH loop is nally moved to the dark fringe condition,
therefore setting the fringe oset to zero. This is done in two steps. The rstone changes the fringe oset from 0.1 to 0.05, still using as error signal theratio of the powers measured by photo-diodes placed in the B1p and B5 beams.
However as the locking point approaches the dark fringe condition, the slope of
this signal decreases more and more as B1p power becomes closer and closer to
its minimum. At the dark fringe this slope would be zero, and therefore thiserror signal could not be used any more. The last change in the oset from 0.05
to 0 is therefore obtained by using the quadrature demodulated signal of the
B5 beam (Pr_B5_ACq) instead of the ratio of powers.

This is done smoothly

by mixing the two signals with a time dependent ramp (this method is usuallycalled blending of the two signals).

After reaching the dark fringe, there are

no more risks of saturating the B5 demodulated signal and therefore they arereconstructed using the high power and low noise diode.
After all these actions the interferometer is locked close to its nal operating
point, with the arm cavities resonant as well as the power recycling one, and thedark point tuned at dark fringe (see g. B.6). However the error signals used
for the longitudinal control loops are not yet the nal one:

• the laser frequency is controlled by using the in-phase signal of the B5
beam, still using the low power photo-diode;• the common mode is controlled by using the reference cavity signal (thisremains the same even in later steps);• the dierential mode is controlled using the in-phase demodulated signal
obtained from the beam transmitted by the west arm;

• the power recycling cavity length is controlled by the in-phase signal coming from the reection of the interferometer, demodulated at 3Ω;• the Michelson d.o.f. is controlled by using the quadrature signal from thebeam splitter secondary face reection (this remains the same even in later
steps).

B.6 The thermal transient
B.6.1 Description
After reaching the dark fringe condition, the power build-up inside the recycling
cavity is close to its maximum: about 210 W are stored inside the recycling cavity and more than 4.5 kW inside the two Fabry-Perot cavities. In particular thePower [mW]

B.6. THE THERMAL TRANSIENT

50
45
40
35
30
25
20
15
10
5
0
0.4

223

B5 power
B5 2f power (x 400)

Carrier
Lower SB
Upper SB

Amplitude [a.u]

0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Unitary gain frequency [Hz]

160

DARM
PRCL
MICH

140
120
100
80
60
40
20
0
400

600

800

1000

1200

1400

Time [s]Figure B.7:
period.

Evolution of interferometric signals during the thermal transientTop:powers of carrier and sidebands inside the recycling cavity, as

measured by the Pr_B5_DC and Pr_B5_2f_ACq signals.

Middle:carrier andsidebands amplitudes at the dark port. These signals are obtained by elaboration of a scanning Fabry-Perot system output (see section 2.5). Bottom: unity
gain frequency of the three main longitudinal control loops, reconstructed using
a suitable calibration line.

224

APPENDIX B. LOCK ACQUISITION TECHNIQUE DURING VSR1
power that is transmitted through the input mirrors is high and even assuming
low losses in the substrate the absorption can be of the order of 20-30 mW (fromsimulations [94] and t to mirror temperature change [95]).
The heating of the input mirrors produce a thermal lensing eect whichchanges the geometry of the power recycling cavity and in conclusion the resonance conditions of both carrier and sidebands (see g. B.7). Initially, whenthe ITF is moved to the dark fringe, the power inside the recycling cavity suddenly increase to its maximum value (around 44 mW on Pr_B5_DC). After someseconds the system jumps down to a lower power state (around 30 mW). Thischange is induced moving the demodulation phase of the signal used to control
PRCL. If this jump is not performed, the ITF remains locked on the high powerstate, but systematically unlocks after few minutes.

Otherwise the detector

switch to the dierent lock state, where both carrier and sidebands are not at
the best resonance condition.

The recycling gains for carrier and sidebands,

together with several other interferometric signals, evolve slowly with time, following the heating of input mirrors. After a period of time of the order of half
an hour all signals have reached a stable value, with again a high recycling gainof the carrier.

B.6.2 Simulations
The reason of this behavior has been understood by means of careful simulations, both computing the eect of thermal absorption in the mirror substrate[95] and the induced change in the resonance conditions [94]. The main eectof the power absorption is a thermal lensing in both the mirror substrate andcoating. These have two dierent eects on the carrier and sidebands resonanceconditions. The carrier power, as measured by Pr_B5_DC, returns to its maximum value with a time constant of about 200 s. The sidebands instead show aquite dierent behaviour, as it is well visible both looking at their power inside
the recycling cavity or at their amplitude at the dark port. After a longer period (of the order of 10 minutes) the sidebands power inside the recycling cavity
reaches a stable value, which is however lower by a factor 2 with respect tothe maximum. At the dark port the carrier amplitude stabilizes with a similar
time constant. Moreover the sidebands start being strongly un-balanced, withthe upper one almost vanishing. They tend to naturally balance with a time
constant of the order of several hours.
During the same period the response of all locking error signals also changes
strongly. In particular the Pr_B2_3f_ACp signal (used to control PRCL) deviates
a lot from a standard Pound-Drever-Hall signal (see g. B.8). Indeed, the zerocrossing point at the beginning of the thermal transient, when the input mirrors
are still cold, slowly disappears as the energy absorbed by the mirrors increases:
this explain why the initial high power state of the interferometer can not beB.6. THE THERMAL TRANSIENT

Figure B.8:

225

Evolution of the Pr_B2_3f_ACp error signal dependence on PRCL

motion, at dierent moments during the thermal transient, corresponding todierent power absorption in the input mirrors.

The circles show the evolu-tion of the zero crossing point corresponding to the high power state of theinterferometer. The squares show the point where the interferometer is locked
during the thermal transient. These results [94] has been obtained by mean of
frequency domain simulation using Finesse [61].

maintained for a long time.

If the demodulation phase of the error signal ischanged by a sensible amount (about 30-40 degrees) another zero crossing pointwith the correct sign appears during the transient. This is the point where theinterferometer jumps after reaching the dark fringe.

It slowly moves toward

the correct resonance condition for the carrier with the increase of the energy
absorbed by the input mirrors.

B.6.3 Locking strategy
The thermal transient period is maybe the most critical one for the stabilityof the interferometer, since it is mandatory to bring and keep it in the correctstate. Moreover the sensitivity of all error signals to the longitudinal degrees of
freedom changes strongly during this period. Therefore the rst ten minutes of
the thermal transient are spent mostly waiting for the mirrors to heat up andonly few actions are performed.
The rst one is to change the error signal used to control the dierential
degree of freedom. The demodulated transmission of the west arm is too noisy
and a better signal can be extracted from the dark fringe itself. Since the output
mode cleaner is usually not yet in resonance at this point of the lock acquisition
procedure, the in-phase demodulated signal from the dark fringe beam (B1p)

226

APPENDIX B. LOCK ACQUISITION TECHNIQUE DURING VSR1
that does not pass through it is used. The switch is performed blending together
the two signals.
One of the most important actions performed in the early part of the thermaltransient is the engagement of the automatic alignment control of the angulardierential motion of the cavity end mirrors (for more details about this alignment strategy refer to chapter 4).

The error signals are extracted from thedark fringe beam B1p itself by using wave-front sensors, similarly to the prealignment of the arm cavities.

The bandwidth of these two control loops areof about 3 Hz. Their engagement strongly stabilizes the dark fringe and power
recycling cavity powers and has proved to be crucial for the stability of thesystem.
During the thermal transient other alignment controls are engaged, usingdierent error signals and bandwidths: the input beam is controlled using signals from wave-front sensors placed in reection of the interferometer, in drift
control conguration, which means with a bandwidth of some tens of mHz;the end mirror common mode is controlled with full bandwidth (3 Hz) using a
combination of signals coming again from the wave-front sensors placed on thereected beam; the power recycling mirror is controlled using signals from theB5 beam, as during the pre-alignment procedure (with 3 Hz bandwidth); nallythe BS mirror is controlled using signals from the beam transmitted by the west
arm, with full bandwidth for pitch motions and drift control conguration foryaw. The remaining input mirror degrees of freedom are controlled only startingfrom later steps. See chapter 4 for more details.
Afterward the normalization of all error signals is changed from Pr_B5_DC to
Pr_B5_2f_ACq, which measures the sidebands power inside the recycling cavity.
This has experimentally proved to be the best choice to compensate for thechange in optical gains of the signals used for the control of the central part ofthe interferometer.
Even with this normalization the optical gains of all longitudinal control
loops change strongly during the transient period, as can be expected because of
the varying resonance conditions of both carrier and sidebands. For the DARM
loop, this variation proved to be too large: without any compensation the overall
gain soon changes more than the stability margin of the control system. For thisreason a slow servo has been implemented with the aim of reconstructing theunity gain frequency of the DARM control loop and stabilizing it to the desired
value. The same strategy is used for reconstructing the UGF of all longitudinalloops. A permanent calibration line is added to each one of the error signals
at a frequency slightly outside the bandwidth of the control loop (at 24 Hz for
MICH, at 62 Hz for PRCL and at 379 Hz for DARM). Using a demodulation
technique the ratio between the line amplitude in the error signal before andB.6. THE THERMAL TRANSIENT

227

120

-5
-5.5

110
-6
-6.5
-7

90

-7.5
80

-8
-8.5

70
60

DARM gain [a.u]DARM UGF [Hz]

100

-9

DARM UGFDARM gain

-9.5

50

-10
400

600

800

1000

1200

1400

Time [s]Figure B.9:

Evolution during the thermal transient of the reconstructed UGF

of the DARM loop and of the corrector lter gain. The former is the error signal
for the slow UGF servo, while the latter is the correction.

after the noise addition can be computed:g(f ) =

Amplitude before(f )Amplitude after(f )As explained in sec. E.1 this ratio gives the open loop gain at the line frequency.
Knowing the shape of the transfer function the UGF of the loop can be easily computed. The evolution during the thermal transient for the three main
longitudinal loops is shown in g. B.7.

During the rst part of the thermaltransient only the DARM loop UGF is stabilized.

This is obtained with theaction of a slow servo system. It computes the UGF of the loop averaging over
30 s and compares it with the desired value (100 Hz for DARM). If the dierence is greater than a xed threshold, the gain of the corrector lter is modied
accordingly (there is also a maximum allowed relative change in the gain, toavoid too big changes in a single step). The gain is then updated in the global
control parameters, where it is actually applied only after passing it througha low pass lter to have a smooth transition. The total eect is a slow servo
system that can maintain the unity gain frequency of the loop at any desired
value, with a bandwidth of the order of 10 mHz (see g. B.9).
At intermediate steps during the transient the way error the signals for
MICH and PRCL are reconstructed is changed. The matrix which reconstruct

228

APPENDIX B. LOCK ACQUISITION TECHNIQUE DURING VSR1
the error signals from the photo-diode demodulated outputs is initially diagonal:
the PRCL error signal is reconstructed only from Pr_B2_3f_ACp while the one
for MICH only from Pr_B5_ACq. An o-diagonal term is added with the aimof minimizing the contribution of PRCL motion to the MICH error signal. For
this reason the MICH error signal is given by
EM ICH = gM ICH

Pr_B5_ACq
Pr_B2_3f_ACp+ cM P
N1
N2

where Ni are normalization coecients build with powers.


(B.1)

The o-diagonalterm cM P is initially set to a xed value, close to the optimal one and afterward
a servo is engaged to keep it as close as possible to the good value, which
may change slowly with time.

The error signal for this servo is obtained bydemodulating EM ICH in correspondence of the PRCL calibration line (62 Hz).
If the o-diagonal term is optimal, the PRCL line should not be visible in the
MICH signal, and therefore the demodulation should return zero. If the term
is not perfectly tuned, some signal is visible at the line frequency, and thedemodulation contains both amplitude and sign information.
Close to the end of the waiting period, the global alignment control of the
input mirrors in engaged. This is a drift control system which uses the positionof the beams transmitted by the two arm cavities, as seen by cameras, as areference.
The nal action performed during the thermal transient waiting time is todigitally add an oset to the error signal used for PRCL control, changing thelocking point for this degree of freedom.

This is done in order to reduce thesidebands imbalance, since as will be shown in section 5.7 this depends strongly
on the operating point. At a later step another slow servo is engaged to control
the sidebands balancing at the dark port.

B.7 Locking of output mode-cleaner
During the waiting period of the thermal transient, the shutter in front of theoutput mode cleaner is opened, and the servo which nds and maintains its resonance is activated. In this way at the end of the transient the OMC is reasonably
close to its resonance and the DARM loop control can be switched from B1p(which is the dark fringe beam pick-up before the OMC) to the demodulated
signal obtained from the beam transmitted by the OMC, namely Pr_B1_ACp.
This signal is much better since the OMC lters out all non-Gaussian modes
which are usually modulated by mis-alignments. Moreover this beam contains
99% of the dark port power and therefore it has a much better signal-to-noise
ratio for dierential motion of the end mirrors. Initially a low power diode isused.

B.8. LOW NOISE CONTROL FILTER

Figure B.10:

229

Scheme of the longitudinal control system after the thermal tran-sient. The DARM loop is controlled using the beam transmitted by the OMC.

This switch also improves the interferometer contrast. It has indeed been
observed that there is a relative oset between the locking point obtained with
B1p and with B1: this might be due to the contamination of the rst by higherorder modes generated for example by a mismatch between the laser beam
spatial mode and the cavities resonant mode.

B.8 Low noise control lter
From this point on the interferometer is locked in a robust conguration andthe operating point is very close to the nal one, the only dierence being theevolution due to the nal part of the thermal transient.

Therefore all action

performed in the later steps of the lock acquisition procedure are focused to
improve as much as possible the sensitivity of the detector by moving to theso-called low noise conguration.
The rst actions are devoted to reduce the noise re-introduced by the control
loops in the dark fringe signal. It has been experimentally found that the most
relevant limiting control noise source is the MICH loop. At this step of the lockacquisition a new corrector lter is engaged. It implements a steep roll-o atabout 50 Hz which strongly reduces the noise re-introduced outside the active
bandwidth of the feed-back system (see g. B.11).

APPENDIX B. LOCK ACQUISITION TECHNIQUE DURING VSR1

Magnitude [a.u.]

102
101
100
10-1
10-2

Acquisition
Low noise

10-3
150
Phase [deg]

100
50
0
-50
-100

Acquisition
Low noise

-150
10

-2

10

-1

0

1

10
10
Frequency [Hz]

10

2

10

3

106
105
4

10

103
Magnitude [a.u.]

230

2

10

Acquisition
Low noise

1

10

100
10-1
10-2
10-3

Figure B.11:

-280

-240

-200

-160
-120
Phase [deg]

-80

-40

0

Comparison of the MICH control lter used during the lockacquisition with the low noise lter. Top: Bode plot. Bottom: Nichols plot of
the open loop transfer function assuming a at optical response and the simple
pendulum mechanical model.

B.9. LOW NOISE ACTUATORSIn order to reduce even further the noise re-introduced by the MICH loop,
without making the control loop unstable, a noise-subtraction technique is used.
The correction signal of the MICH loop is sent, after proper ltering, to the end
mirrors in dierential conguration. This is the so-called alpha technique and it
is discussed in more details in section 5.4. A similar noise subtraction technique
is also implemented for the reduction of PRCL noise: the correction computedfor the PRCL d.o.f. is also added to the dierential correction of the end mirrors.
This is called beta technique. The overall gains of these two noise subtraction
paths are adjusted continuously by slow servos which uses as error signals thedemodulation of the dark fringe at the frequency of PRCL and MICH calibrationlines.
The sidebands balancing servo, described in section B.6.3 is engaged at thispoint, together with the unity gain frequency servos for MICH and PRCL. The
requested UGF are 100 Hz for DARM, 40 Hz for PRCL and 15 Hz for MICH.

B.9 Low noise actuators
A possible source of noise in feed-back control systems can come from the actuation noise, in particular the DAC noise introduced by the electronic boards
that converts the digital correction signals to the actual current applied to thecoils actuators.
At this step of the lock acquisition procedure, several actions are performed
to reduce this noise. First, the longitudinal correction applied to the end andinput mirrors is partially reallocated to the marionette actuators. To summarize, the longitudinal correction is split in several parts. The very low frequency
component of the correction is sent both to the reference mass actuators and to
the inverted pendulum controls. In this way the large low-frequency componentof the force is applied at the top level, greatly reducing the needed dynamical
range at the level of mirror. Moreover, the correction in the frequency region below roughly 7 Hz is reallocated to the marionette actuators. This is obtained byusing suitable low and high pass lters, which compensate also for the transfer
function from marionette to mirror due to the additional pendulum stage. This
re-allocation is performed for two main reasons: rst it reduces even further the
dynamical range needed for the reference-mass actuators; secondly any actuation noise at the level of the marionette is ltered by the additional pendulum
stage. Finally the components of the correction above 7 Hz is still applied using
the reference-mass actuators.
Since the dynamical range of the force that must be applied from the reference mass is greatly reduced in this way, the corresponding coil-drivers can beswitched to low noise mode (see section 2.4 for more details).
Finally the dark fringe signals are reconstructed using the sum of the outputs

231

APPENDIX B. LOCK ACQUISITION TECHNIQUE DURING VSR1
10-7
B2_3f_ACp
B2_ACpMix
10-8Spectral Density [W Hz-1/2]

232

10-9

10-10

10-11

10-12 -1
10

Figure B.12:

100

101
Frequency [Hz]102

Comparison of three dierent signals for the PRCL control.

Pr_B2_3f_ACp is the signal usually used during lock acquisition. The mix signal is obtained by combining this signal at low frequency (below 5 Hz) with

Pr_B2_ACp (above 5 Hz). The dierence at low frequency is due to the fact thatthe mixed signal is in-loop while the other two are not.

of two photo-diodes which receives each half of the beam transmitted by theoutput mode cleaner. In this way the shot noise limit is strongly decreased.

B.10 Last low noise steps
After the change in the MICH control lter the dominant source of noise is the
PRCL loop. The main reason is that the error signal used during lock acquisition (Pr_B2_3f_ACp) is very noisy at high frequencies, since it is limited bysensing (shot and ADC) noise above 30 Hz (see g. B.12). Another signal which
senses the same degree of freedom in a similar way is Pr_B2_ACp.

Therefore

the two signals are mixed together: Pr_B2_3f_ACp is low-passed through a rstorder lter with corner frequency at 5 Hz; the complementary high-pass lter
is applied to Pr_B2_ACp and the two signals are summed together. In this way
the behavior of the control loop is the same at low frequency and the noise
re-injected at high frequency is lowered by about one order of magnitude.

B.11. SCIENCE MODE CONFIGURATION

B.11 Science mode conguration
In conclusion, the state of longitudinal sensing and control in the nal conguration can be summarized as follows:• the DARM degree of freedom is controlled using the dark fringe itself as
error signal (Pr_B1_ACp) with unity gain frequency at 100 Hz;• the PRCL degree of freedom uses the Pr_B2_3f_ACp signal below 5 Hz
and Pr_B2_ACp above. The unity gain frequency of this loop is at 40 Hz.
• the MICH degree of freedom is controlled using Pr_B5_ACq. Its control
loop has a unity gain frequency at 15 Hz and a strong roll-o above 50 Hz;• the gains of these three longitudinal loops are stabilized by slow servo
using as error signal the demodulation of calibration lines;• the laser frequency is stabilized on the mean length of the arm cavities,
using Pr_B5_ACp as error signal and with unity gain frequency at about
25 kHz; at the same time the common mode of the arm cavities is lockedon the demodulated reection of the reference cavity, with a bandwidthof about 1 Hz;• all longitudinal actuators are in low-noise conguration: the very low frequency part of the correction is applied also at the top stage of the suspension, while for the two cavity end mirrors the correction is re-allocatedat intermediate frequencies to the marionette;• the re-injection of MICH and PRCL noise in the dark fringe signal isreduced by using noise subtraction techniques.

233

Appendix C
Linear systems
The simplest model of a physical system with one single input and one output
is a linear transformation:Z ty(t) =

h(t, τ ) x(τ ) dτ−∞

where the requirement of causality has been already taken into account. In general the integral kernel h(t, τ ) is a function of two variables, since the behaviorof the system can vary with time. If this is not the case, the system being timeinvariant, the above expression can be simplied considering that it is possible
to write h(t, τ ) = h(t − τ ) and therefore

Z 0h(τ ) x(t − τ ) dτy(t) =(C.1)−∞
This is the most general expression of a linear time-invariant system. The vast
majority of the systems involved in interferometric detectors are well describedby this kind of models.
A very convenient way to describe linear transformations, widely used in thetheory of control systems for example, is the Laplace transform [56]. Given asignal x(t), null when t < 0, its Laplace transform is dened asZ ∞x̃(s) =

x(t) e−st dt

0

Laplace transform

1
s
1
s−s0
ω02
ω
2
s + Q0 s+ω02

Time function (for t > 0)

1
s0 t

e
ω t

0
q ω0
e− 2Q sin
1
1− 4Q
2



ω0 t

q

1
1 − 4Q
2



Table C.1: List of anti-transform of common second order rational functions.

236

APPENDIX C. LINEAR SYSTEMS
It can be shown that this function of the variable s can be extended to a complex
function with s varying in a dened domain of the complex plane. Applying the
Laplace transform to eq. C.1 one obtainsỹ(s) = h̃(s) x̃(s)(C.2)

where the new function h̃(s) is usually called the (Laplace) transfer functionof the linear system.

The Laplace transform is an extension of the Fouriertransform, which can be recovered puttings = iω , if the imaginary axis iscontained in the domain of existence.
In most cases the transfer function of a linear system can be described using
rational functions

F (s) =P (s)bm sm + bm−1 sm−1 + · · · + b1 s + b0=
Q(s)an sn + an−1 sn−1 + · · · + a1 s + a0

If both numerator and denominator are factorized, the function can be described
as a list of poles and zeros.

Moreover, if

F (s) is the transfer function of aphysical system, all complex poles and zeros appear together with their complex
conjugate, since in the time domain the integral kernel h(τ ) must be real, being
both x(t) and y(t) physical real signals.
A rational function is characterized by its poles and can therefore be writtenas:F (s) =Xi

Ki(s − pi )ni

where ni are the pole multiplicities and Ki the corresponding residuals.

Theanti-transform of each of these terms in time-domain is proportional to an ex-

p tponential e i(see table C.1). This result is very useful when considering the

response of a system to a step function or to an impulse (usually called step re-sponse and impulse response ). In the latter case, being the Laplace transform of
an impulse at t = 0 a constant, from eq. C.2 follows that the impulse responseof the system is given simply by the anti-transform of the transfer function.
Therefore it will contain oscillating terms at frequencies given by the imaginary
part of each pole. If one pole has negative real part the oscillation is damped,
otherwise it has constant amplitude if the pole is purely imaginary, or it grows
up exponentially if the pole has positive real part. Therefore a system is stable,in the sense that its response to an impulse is asymptotically decreasing, if its
transfer function has only poles with negative real part.
In developing feed-back control system and in describing the properties ofmechanical and optical systems, it is customary to describe transfer functionsin terms of pole and zero frequencies and quality factors. A real (simple) zeroat a frequency f0 is given by

Z1 (s) = s − 2πf0

237

Magnitude [a.u.]

101

100

10-1

10-2

Real
Q = 0.5
Q=1
Q=3
Q = 1010-3150Phase [deg]

100
50
0
-50
-100
-150
100

101

102

103

Frequency [Hz]Figure C.1: Transfer function of systems described by a simple real pole or one
complex pole with dierent quality factors Q.

while a real pole is given by the inverse of this expression. A complex (double)
zero at frequency f0 and with quality factor Q is given by

Z2 (s) = s2 +

2πf0
s + (2πf0 )2
Q

The inverse of this formula gives the transfer function of a complex double pole(see g. C.1): this correspond to a system whose impulse response is a damped
oscillation at frequency f

= f0

p
Q
1 − 1/4Q2 , with time constant τ = πf
: the0higher the Q, the longer the oscillation will last. An example of such a system
is a simple mechanical pendulum with friction.

Appendix D
Signal analysis

D.1 Random processes
The main part of the work in commissioning and characterizing an interferometric detector deals with the analysis and interpretation of digital signals.
In general any physical quantity can be dened as a random variable depend-X(t, S) (a random process [96]) where S describes the state of
the physical system. A digital signal is a sampled version of this X(s) =(X(t1 , s), X(t2 , s), . . . , X(tn , s)). At a xed time t the cumulative probability

ing on time

distribution of the system can be dened as

FX (x, t) = P ({X(t) < x})and the probability density as its derivativefX (x, t) =∂FX (x, t)∂x

Similar denitions hold for probability distributions of the values of the randomvariable at multiple times:= P ({X(t1 ) < x1 } ∧ {X(t2 ) < x2 })∂ 2 FXX (x1 , t1 ; x2 , t2 )fXX (x1 , t1 ; x2 , t2 ) =∂x1 ∂x2

FXX (x1 , t1 ; x2 , t2 )In general the probability density at two times is not simply given by the productof the two at single times, unless the two random variables obtained at twodierent times are independent :FXX (x1 , t1 ; x2 , t2 )= FX (x1 , t1 ) FX (x2 , t2 )fXX (x1 , t1 ; x2 , t2 )= fX (x1 , t1 ) fX (x2 , t2 )It is possible to dene statistical moments for the signal, at xed time. The
mean value:ZµX (t) = E [X(t)] ≡x fX (x, t) dx

240

APPENDIX D. SIGNAL ANALYSISThe standard deviation:

i Z
h
2
2
σX
(t) = E (X(t) − µX (t)) = (x − µX (t))2 fX (x, t) dx
The auto-correlation:Z Z
RXX (t1 , t2 ) = E [X(t1 )X(t2 )] =x1 x2 fXX (x1 , t1 ; x2 , t2 ) dx1 dx2

The auto-covariance:

CXX (t1 , t2 )= E [(X(t1 ) − µX (t1 )) (X(t2 ) − µX (t2 ))]= RXX (t1 , t2 ) − µX (t1 )µX (t2 )In a similar way the correlation and covariance of two signals can be dened:

Z Z
RXY (t1 , t2 )= E [X(t1 )Y (t2 )] =CXY (t1 , t2 )= RXY (t1 , t2 ) − µX (t1 )µY (t2 )x1 y2 1, fXY (x1 , t1 ; y2 , t2 ) dx1 dy2A zero-mean, normal distributed or Gaussian process {Xi : i = 1, . . . , n} isspecied by the following probability density:f (X1 , . . . , Xn ) = p
X T Σ−1 X
exp −2(2π)n detΣ1

where Σ is the covariance matrix that completely characterize the process. If
this matrix is diagonal, random variables obtained at dierent times are independent.

D.2 Stationarity and ergodicity
A random process is called stationary if all its moments are independent oftime: this means that the statistical properties of the random variables at twodierent times are equal. For rst order statistics this implies that the mean isconstant in time. For second order ones, the auto-correlation is a function of
the time dierence only.
A moment of the process at a time t can be estimated by averaging over
several dierent realizations of the process, for exampleZµX (t) 'X(t, s) dP (s)This is usually not possible, since a detector output consists in one single signal, which is one particular realization of the random process. If the process isstationary, one can imagine to substitute the average over dierent realizations
with an average over time. In general the equivalence of the two means is notD.2. STATIONARITY AND ERGODICITY

241

true, unless the process is ergodic. In the following all random processes are assumed to be ergodic, therefore averages over realizations can be exchanged withaverages over time. For reference, the denitions of mean, standard deviation
and auto-correlation are given by:

Z +T
µX
2
σX

=
=

lim

T →∞

X(t) dt−T
Z +T

2(X(t) − µX ) dt

lim

T →∞

−T

Z +T
RXX (τ )=lim

X(t)X(t + τ ) dt

T →∞

−T

D.2.1 Power of a signal
The mean power density of a signal, also called root mean square value (RMS)
is dened as

Z +T
RM SX = lim

T →∞X 2 (t) dt−TIt can be easily shown that this is equal to the auto-correlation function at zerodelay and also thatRM SX = RX (0) ≥ |RX (τ )|It often turns out to be useful to separate the contribution to the signal
power coming from dierent frequencies.

For this reason the power spectral

density (PSD) or power spectrum of the signal can be dened as the Fouriertransform of the auto-correlation:Z +∞SXX (ω) =e−iωτ RXX (τ ) dτ

(D.1)

−∞
2

The physical units of the power spectrum are [X] /Hz.

If the signal X(t) isreal, the power spectrum is even SXX (ω) = SXX (−ω).
It is customary to dene the amplitude spectral density (ASD), or simply

spectrum, of the signal asX̃(ω) =p

SXX (ω)√whose units are [X]/Hz. This is not the Fourier transform of the signal, nora simple average of Fourier transforms.
The band-limited root mean square value (BRMS) of a signal in a frequency
band [ω1 , ω2 ] is simply the integral of the power spectral density in that band:

Z ω2
BRM SX [ω1 , ω2 ] =SXX (ω)dωω1

It is also possible to dene the cross power spectrum of two signals:

Z +∞SXY (ω) =−∞e−iωτ RXY (τ ) dτ

242

APPENDIX D. SIGNAL ANALYSIS
Another denition of power spectrum can be given as the following limit1
T →∞ 2T

Z +T

S(ω) = lim

2

X(t)e−iωt dt−TThis is equivalent to eq. D.1 if the signal satises the condition

Z +∞|τ R(τ )|dτ < ∞−∞which is usually true for most physical signals.

D.3 Linear transformation of signals
If a given signal X(t) is transformed in Y (t) = H[X(t)] with a time-invariant
linear transformationZ +∞h(τ )X(t − τ ) dτY (t) =−∞

the (Fourier) transfer function is dened as follows

Z +∞H(ω) =

h(t)e−iωt dt−∞
since it is the function that connects X and Y Fourier transforms:Ỹ (ω) = H(ω) X̃(ω)
The eect on correlations is easily expressed in terms of convolution integrals

RXY (τ )= RXX (τ ) ⊗ h∗ (−τ )RY X (τ )= RXX (τ ) ⊗ h(τ )RY Y (τ )= RXX (τ ) ⊗ h∗ (−τ ) ⊗ h(τ )and from these formulas the power spectra can be computed:

SY Y (ω)= |H(ω)|2 SXX (ω)

SXY (ω)= H ∗ (ω)SXX (ω)

SY X (ω)= H(ω)SXX (ω)

In particular the amplitude spectral density of the transformed signal is simply
the one of the input signal multiplied by the modulus of the transfer function.
From these denition it follows that the transfer function between two signals, supposed to be linearly related, can be estimated by the following ratio

T FX→Y (ω) =SY X (ω)= H(ω)SXX (ω)D.4. ESTIMATORS

243

Another useful quantity is the coherence :|SXY (ω)|2
SXX (ω)SY Y (ω)

CXY (ω) =If X and Y are linearly related, this is equal to 1. Therefore coherence is a good
indicator of the amount of linear relation between two signals.

D.4 Estimators
Since almost all signals are acquired digitally, the previous equations must betranslated in the discrete time domain. The discrete Fourier transform (DFT)
can be dened coherently with the continuous time version as:

N −1x̃k = DF T (x, ωk ) =

2πi
∆T X
xn e− N kn
N n=0

where ∆T is the length of the interval used for the computation and ωk =

(D.2)

k
N ∆T .

The discrete Fourier transform is periodic with period equal to N and for real
signals it has the reection property :∗DF T (x, ωk ) = DF T (x, ω−k )(D.3)

Therefore only the rst half of the discrete Fourier transform gives independent
results.
It is common to use window functions for the estimation of DFTs, to improve
the frequency resolution by reducing leakage between adjacent bins and to havea better convergence of the estimator [97]: the signal is multiplied by the windowfunction before computing the transform:

N −1

DF Tw (x, ωk ) =

2πi
∆T X
wn xn e− N kn
N n=0

This is equivalent to convolution in the frequency domain with the Fouriertransform of wn . A commonly used window is the Hann one:




1
2πn
hn = A
1 − cos2
N −1
where A is a normalization constant chosen in order to haveN−1X

wn2 = 1j=0
To estimate the power spectrum it is in principle possible to use its denition(eq. D.1), but this is quite a computationally heavy operation, and it is also nota good estimator, since it converges very slowly to the real value [98].

244

APPENDIX D. SIGNAL ANALYSIS
A more widely used method to estimate the power spectrum of a signal is
the Welch modied periodogram [99]. The time average of eq. D.1 in the limit
with T→ ∞ is replaced by an average over M dierent segments, possibly

overlapping. The discrete Fourier transform is then computed for each segment,
typically using dedicated algorithms like the Fast Fourier Transform (FFT)[100] which is particularly optimized for segments of data with a number ofsamples given by a power of two. The Welch estimator of the power spectrum
is given by:

Nj −1
M −1
2πi
1 X 1 ∆Tn X
S(ωk ) =wn X(tn,j )e− N kn
M n=0 ∆Tn Nn j=02

where ∆Tn and Nn are the length and the number of samples of the n−th data
segment.
The generalization of this method to the estimation of cross power spectrum,transfer function and coherence is straightforward.

D.5 Digital lters
Since almost all signals are acquired digitally, it is important to have a meanof applying linear transformation to discrete time signals. This kind of transformation are usually called digital lters and are at the basis of signal analysis
and of most of the control systems commonly used in interferometric detectors(see chapter E).
Given a digital signal

xn , the most general linear transformation can be

expressed by the following dierence equation [101]:NX

αp yn−p =

p=0

M
X

βq xn−q

(D.4)

q=0

or in a equivalent way:

#
" N
M
X
X
1
−
αp yn−p +βq xn−q
yn =
α0
q=0
p=1
The output signal at the n-th sample is computed from the previous M valuesof the input signal x and from the previous N − 1 values of the output signal
itself.

This kind of digital lters are usually called innite impulse-response(IIR) lters, since their response can be asymptotically not zero even when the
input is constantly null after a given time.
In dealing with continuous time system it is useful to dene the Laplacetransform. In discrete time system instead the z -transform is dened:X(z) =∞
X
n=0

xn z −n(D.5)

D.5. DIGITAL FILTERS

245

Time domain
Linearity
Time shift
ScalingTime reversal
ConjugationDierentiation
Convolution

axn + byn
xn−k
an xn
x−n
x∗n
nxn
xn ∗ ynz domainaX(z) + bY (z)z −k X(z)X(z/a)
X(1/z)X ∗ (z ∗ )−z dX(z)dz
X(z)Y (z)Table D.1: Some properties of the z -transform.

This has several properties similar to those of the Laplace transform (see tableD.1). In particular using the linearity and time shift properties, the dierence
equation in eq. D.4 can be written in the form of a transfer function connecting
the two z -transforms:

H(z) =Y (z)β0 + β1 z −1 + β2 z −2 + · · · + βM z −M=X(z)α0 + α1 z −1 + α2 z −2 + · · · + αN z −N

The typical situation in digital lter designs is to search for a discrete timesystem that best approximates a continuous time specication of the desired
frequency response of the lter, given in terms of a Laplace transfer function.
The most commonly used procedure is to employ the bi-linear transform, which
is a conformal mapping of the s-plane to the z -plane.
The starting point is the exact mapping of the s-plane to the z -plane, givenby the complex exponential:z = esT(D.6)

where T is the time interval between samples (the reciprocal of the sampling
frequency fs ). This expression can be approximated to rst order byz = esT =1 + sT
esT /22≈−sT/2e
1 − sT2

and in a similar way the inverse of eq. D.6 can be approximated by:

1
2
s = log z ≈
T
T

"

#

3
z−1 1 z−1
2 1 − z −1+
+ ... ≈
z+1 3 z+1
T 1 + z −1

The bi-linear transform uses this last rst order approximation to obtainthe z -domain transfer function Hd (z) from the Laplace transform Hc (s) of thesystem, by substitutings→

2 1 − z −1
T 1 + z −1or in other wordsHd (z) = Ha

2 1 − z −1
T 1 + z −1

(D.7)


(D.8)

246

APPENDIX D. SIGNAL ANALYSISTo determine the eect of this approximation on the frequency response of
the discrete time system, the resulting z -domain transfer function can be evaluated at points on the unitary circles that corresponds to physical frequencies

z = eiωT :
iωT

Hd (e


)

= Ha

2 eiωT + 1
T eiωT − 1

2 eiωT /2 − e−iωT /2= Ha
T eiωT /2 + e−iωT /2



ωT
2
= Ha i tan
T
2




This means that the imaginary axis of the Laplace plane, corresponding to physical frequencies, is mapped to the unitary circle in the z -plane with a warpingof the frequencies. The discrete time lter behaves at a frequency ω exactly likethe continuous time lter at a frequency given by

ωa =

2
tan
T



ωT
2


(D.9)

For frequencies much smaller than the sampling one this shift is negligible, but
it becomes more and more relevant as the frequency approaches the Nyquist
one.

This behavior can be corrected by pre-warping the frequencies of poles

and zeros of the continuous time lter according to eq. D.9.

Appendix E
Feed-back control systems
In an interferometric gravitational wave detector like Virgo, there are tens ofdegrees of freedom (mirror positions and pointing, input beam position, etc.)that must be kept close to well determined working points in order to maintainthe detector in the correct operating condition within the needed accuracy. Thisis a typical problem of developing suitable control systems, the most common
ones being those working in feed-back conguration. This appendix introduces
the basics concepts of control system theory.

The treatment is restricted tolinear control systems, since they cover virtually all of those used in gravitational
wave interferometers as long as the residual uctuations of the controlled degrees
of freedom are small enough.

E.1 Simple feed-back system
The simplest situation in a problem of control is a physical object (see g. E.1),
usually called plant, which can be described as a linear system with one single
input (for example a Fabry-Perot cavity and its length) and one single output,
the error signal (a photo-diode demodulated output for example). This system
is usually called in engineering books SISO (single-input single-output).
The goal of the control system is to keep the input degree of freedom as close
as possible to a given function of time, the reference. Being the system linear,
this can be taken as the value 0, and therefore controlling the input degree of
freedom is equivalent to keeping the error signal close to zero.

The responseof the plant can in general be described using a linear transfer function G(s)depending on the Laplace variable s. The error signal can be aected by somesensing noise which can be related to the sensor itself (for example shot noisefor photo-diodes, ADC noise, etc.) or can be injected on purpose to measure
the features of the system.
The feed-back is obtained passing the error signal through an ad-hoc correc-

248

APPENDIX E. FEED-BACK CONTROL SYSTEMS
tor F (s) and feeding the result (correction signal ) back to the plant by meansof actuators whose response is modeled by A(s) .

For example the length ofa Fabry-Perot cavity is controlled sending currents to coils attached to one of
the mirrors. In general the actuators themselves can introduce some additional
noise, called actuation noise, or as before an external excitation can be added
on purpose to the correction signal. There is a general convention that put aminus sign at the summation point of the correction and the system free motion.
In this way the correction signal is in-phase with the motion that the feed-back
suppresses.
The behavior of the full linear system can be easily computed:e(s)= G(s) [z 0 (s) − na (s) − A(s) (nc + F (s)ns + F (s)e(s))]z(s)= z 0 (s) − na (s) − A(s) (nc + F (s)ns + F (s)e(s))

where e(s) and z(s) are the error and input signal respectively. These implicit
equations can be solved to obtain:

e(s)

=

G(s)
G(s)
z 0 (s) −(na (s) + A(s)nc (s))
1 + G(s)F (s)A(s)1 + G(s)F (s)A(s)G(s)F (s)A(s)−ns (s)
1 + G(s)F (s)A(s)
(E.1)

z(s)

1
1
z 0 (s) −(na (s) + A(s)nc (s))=
1 + G(s)F (s)A(s)1 + G(s)F (s)A(s)F (s)A(s)ns (s)−
1 + G(s)F (s)A(s)(E.2)

The fact that the output of the system is used also as part of the input justies
the name feed-back and also explains why it is customary to refer to feed-backcontrol system as loops and to distinguish between the open loop conguration
when the feed-back signal is disconnected from the input and the closed loopconguration.
For simplicity the actuator transfer function can be considered as a part ofthe plant transfer function G(s) and the correction and actuation noise can betreated on the same basis.
All signals that are measured inside a feed-back loop are called in-loop (forexample the error and correction signals) while those measured outside or addedfrom outside are called out-of-loop (for example the free motion of the systemor any external added perturbation). As an eect of the feed-back all in-loop
signals are reduced, with respect to the free system, by an amount given by the

suppression factor

GCLT F =1
1 + G(s)F (s)E.2. RESPONSE TO VARIATIONS OF THE INPUT SIGNAL
also called closed-loop gain or closed-loop transfer function (CLTF) to dierentiate from the open-loop gain or open loop transfer function (OLTF) given
by

GOLT F = G(s)F (s)Usually the value of the open loop transfer function at a given frequency iscalled the gain at that frequency. A change in the overall gain of the transfer
function is simply a multiplication by a frequency independent coecient. The
frequency (usually unique) where the gain is equal to one is called unity gain

frequency UGF or band-width of the loop.
One widely used method to measure the open-loop transfer function of asystem consists in adding a known source of noise n(s) for example to the error
point.

If this noise dominates all the signals, in equations E.1 and E.2 only

terms proportional to n(s) are relevant:

G(S)F (S)n(s)
1 + G(S)F (s)1n(s)
1 + G(s)F (s)

e(s)

= −

e0 (s)=The OLTF is given by the opposite of the transfer function between the error
signal measured after and before the addition of the noise.
Both the error and correction signals contain information on the residual
motion of the system. At frequencies where the open-loop gain is higher thanone, the error signal gives a good estimate of the residual motion of the system,
apart from a calibration factor given by the plant transfer function.
At the same frequencies instead the correction signal is not a good estimate of
the motion. For example consider a resonant Fabry-Perot cavity with its length
controlled by applying a force to one of the two mirrors, using the demodulatedtransmission as error signal. If the input mirror is moving a lot, the correction
signal must be large enough to induce a motion of the end mirror such that thecavity length is constant. Therefore the correction can be quite large even if theerror signal and the residual uctuation of the cavity length are small.

E.2 Response to variations of the input signal
One of the most interesting properties of a control system is the ability to tracka varying input signal.

This performance of a linear system, described by atransfer function F (s), can be studied by analyzing the response to an input
step:(x(t) =

0
1

if t < 0
if t ≥ 0

249

250

APPENDIX E. FEED-BACK CONTROL SYSTEMS

Figure E.1: Scheme of a simple feed-back loop. See the text for details.

The Laplace transform of this signal is x̃(s) =1s and therefore the system output

is given byỹ(s) =F (s)sThe behavior of the anti-transform of this expression for large times can beobtained, assuming the system to be stable, by using the nal value theorem[56]:

lim y(t) = lim s ỹ(s)

t→+∞

s→0

In particular the response of the linear system to the input step for large times
is given by:

lim y(t) = F (0)t→∞Therefore the output of the system goes to zero only if its transfer function
includes a zero at s = 0.
In terms of feed-back systems, the input signal is an external disturbanceadded to the controlled d.o.f. and the output is for example the residual motionof the system. Therefore the transfer function of the system is given by

F (s) =

1
1 + GOLT F (s)and it has a zero at the origin only if GOLT F (s) includes a pole for s = 0 (apure integrator ). This is the reason why real poles at zero frequency are usually
added to the controller lter.

E.3. STABILITY251

The simplest case is that of a system with open loop transfer function given
by a simple pure integrator with unity gain at ω0 :GOLT F (s)

=

F (s)

=

ω0
s
1

=

1 + GOLT F (s)ss + ω0Its response to a input unity step is given by the anti-transform of

1
F (s)
=
s
s + ω0ỹ(s) =
which is exponentially decaying:y(t) = e−ω0 t
From this equation it is also possible to extract the time constant of the system,
which is the typical time scale of its response to variations of the input signal

τ=

1
1
=
ω0
2πf0

where f0 is the unity gain frequency of the system.
In a similar way the behavior of a system with two real poles at the origin
can be studied.

GOLT F (s)

=

F (s)

=

ỹ(s)

=

ω02
s2
1

=

1 + GOLT F (s)
F (s)
s
= 2
s
s + ω02s2
s2 + ω02

and the anti-transform of the last expression gives the time response to an input
step:y(t) = cos (ω0 t)
which is not exponentially decreasing in time.

Therefore the response of thissystem does not converge to zero when the input is perturbed, but rather apersistent oscillation at the unity gain frequency is present. This introduces the
subject of stability of a feed-back system.

E.3 Stability
From the general point of view, a linear system described by a transfer function

G(s) is proper if G(i∞) is nite and strictly proper if G(i∞) = 0. This is usuallytrue for all physical system, since for very high frequency the response is always
small.

252

APPENDIX E. FEED-BACK CONTROL SYSTEMS
Moreover, a system is stable if its transfer function is analytic in the closed
right half plane s ≥ 0 or in other words if it has no poles in that region. It is

asymptotically stable if there are also no poles on the imaginary axis.
As explained in chapter D, the anti-transform of the transfer function gives
the response of the system to an impulse (usually called impulse response of thesystem). Any pole in the half right plane will anti-transform to an oscillation
with exponentially increasing amplitude and any pole on the imaginary axis toa constant amplitude oscillation.

In the rst case the response of the system

to an impulse is therefore growing exponentially, while in the second case it
remains constant with time. This explains the given denitions of stability: astable system responds to an input impulse with a limited output signal, while
an asymptotically stable system with an output which goes to zero when time
tends to innite.

Since the system is linear, the behavior with any input iscompletely characterized by the impulse response.
Therefore any control system, in order to be able to maintain the controlled
d.o.f. close to the desired value must be stable or better asymptotically stable.
Otherwise any disturbance to the input will we amplied and will soon diverge
exponentially. The control will fail to work as soon as the motion is larger thanthe linearity range of the system.
For a feedback system, the requirements of stability and asymptotical stability translate in requirements on the closed loop transfer function: a feedback
system is asymptotically stable if and only if the closed loop transfer function

GCLT F =1
1 + F (s)G(s)has no pole for Re s ≥ 0.

E.3.1 Nyquist stability criterion
Usually the study of a system transfer function is carried out in the frequency
domain rather than in the Laplace domain, mainly because the values of thetransfer function on the imaginary axis s = iω are the only one that correspondto physical frequencies and that can be directly measured.

The link between

the behavior of the system transfer function and the requirements of stability
can be derived using standard complex analysis methods [102, 103].
The starting point is the well known argument principle [56]: the variation
over a closed contour

Γ of the phase of a function, analytic in the domainbounded by Γ except for a nite number of poles, is given by the following
equation:1∆Arg F (s) =2πi

F 0 (s)ds = nz (Γ) − np (Γ)Γ F (s)I(E.3)

where the contour is covered in counter clock-wise sense, np and nz denote thenumber of poles and zeros encircled by Γ. The integrand is called the logarithmic

E.3. STABILITY

253

Im
+i∞

Re

-i∞

Figure E.2: Contour in the complex plane used for the derivation of the Nyquist
stability criterion.

derivative of the function F .
The Nyquist criterion for the stability of a closed-loop feedback system canbe derived starting from the open loop transfer function GOLT F (s) and considering the contour in the Laplace s-plane shown in gure E.2: it consists of theentire imaginary axis, with a small half-circle around the origin if GOLT F (s)has poles there, covered in clock-wise direction from −i∞ to +i∞ and closed
by a half-circle at innity. The case of poles on the imaginary axis is not taken
into account here. If the system is stable in the open-loop conguration, thenthe function 1 + GOLT F (s) has no pole in the right half-plane. The closed-loopsystem is stable if and only if this function has also no zeros in the right halfplane. Using the argument principle, this is equivalent to the following integral
being zero:

G0OLT F (s)ds = ∆Arg (1 + GOLT F (s))
Γ 1 + GOLT F (s)

I
0=

(E.4)

The Nyquist plot of the open loop transfer function is the image of the Γ contour,
which is the curve obtained plotting the complex value of the transfer function
while varying the frequency from −∞ to +∞. Since the time-domain responseof the system must be real, the transfer function obeys the relation G(−iω) =G∗ (iω) and therefore it is enough to plot the transfer function for the positive
physical frequencies and to complete it with its mirror image with respect tothe real axis. The Nyquist plot must be covered in the direction corresponding
to frequency going from 0 to +∞ and then from −∞ to 0.
Since the contour Γ is covered in clock-wise direction, the right hand term of
eq. E.4 is given by the number of times the point −1 is encircled in clock-wise

254

APPENDIX E. FEED-BACK CONTROL SYSTEMS

f = -∞

f = -∞

f=0

f=0
-3

-2

-1

f=∞

0

1

2

3

-2

-1

0

1

2

f=∞

Figure E.3:

Examples of Nyquist plot for a system of the rst order (right) andof the second order (left).

sense by the Nyquist plot of the open-loop transfer function.
In conclusion the following criterion for the stability of closed-loop feed-backsystem is established:Nyquist stability criterion. Given a system with stable open-loop transfer
function, it is also stable in the closed-loop conguration if and only if its
Nyquist plot does not encircle the critical point −1 in clock-wise sense.
It is worth to better explain this result by means of some examples. If thesystem has no pole at zero frequency, the Nyquist plot has no point at innity
and the application of the criterion is straightforward. In g. E.3 the Nyquist
plots for very simple rst and second order system are shown:

F1 (s) =F2 (s)=1s − ω01
s2 − ωQ0 s − ω02

Both system are stable since the −1 point is never encircled.

Both transfer

functions go to zero when the frequency goes to ±∞ and therefore there is noneed to complete the Nyquist plot with the image of a circle at innite frequency.
In both cases if the sign of the transfer function is changed, which corresponds
to changing the sign of the control lter, the system becomes unstable, sinceboth Nyquist plots encircle the critical point.
As explained in the previous section, the corrector lters often includes one
or more pure integrators. In this cases the application of the Nyquist criterion
is made a bit more complex by the presence of the singularity in the origin. The
Nyquist contour Γ must include a semicircle around the origin, in the right halfplane. To understand the Nyquist plot at innity, it is sucient to consider theasymptotical behavior of the open loop transfer function, which is characterizedE.4. BODE AND NICHOLS PLOTS

255

only by the number of real poles at zero frequency. For a rst order system

G(s) ∼

1s

The small circle around the origin in the s-plane can be parametrized ass = εeiφwith φ varying from −ππ
2 to + 2 . The corresponding section of the Nyquist plot

describes a half circle with very large radius in clock-wise sense:

G(s) ∼
starting from the point at i

1 −iφ
e
ε

1
1
ε and ending at −i ε . This circle moves to innity

when ε goes to zero. In any case it never encircles the critical point. If instead
the sign of the open loop transfer function is changed, the portion at innity of
the Nyquist plot encircles the critical point and the system becomes unstable.
This allows to state as a general rule that for a system including a pure integrator
the phase of the open loop transfer function for frequencies going to zero mustbe −90 degrees.
A similar result can be obtained for a double pole at the origin.

It mustbe noted that a system whose open loop transfer function is described by only
a double real pole at zero frequency can not be stable, since whatever is thesign chosen, the Nyquist plot always pass through the critical point. Therefore
some more poles and zeros must be added to the corrector lter to stabilize
the system. However, for what concerns the behavior at innity of the Nyquist
plot, the same trick as before can be used. If the open loop transfer function
for small frequencies can be approximated byG(s) ∼1s2

then the part of the Nyquist plot at innity is a circle starting from a phaseclose to −180 degrees from above and ending again close to −180 from below.
This circle is covered in clock-wise direction.

As before, the system is stablewith the chosen sign, while it is unstable with the opposite one. In other words,
in presence of a double pure integrator, the phase of the open loop transfer
function in the limit of zero frequency must be −180 degrees.
In general, in presence of poles in the origin, the Nyquist plot must becompleted with a contour at innity consisting in one half clock-wise circle foreach pole, starting from the point at positive innite frequency and ending in
the one at negative innite frequency.

E.4 Bode and Nichols plots
Together with the Nyquist plot, there are two other common ways to represent
the open loop transfer function of a system.

256

APPENDIX E. FEED-BACK CONTROL SYSTEMS

Figure E.4:

Example of Nichols plot of the open loop transfer function of astable system. The curves around the critical point show the minimum margin
allowed for an overshoot going from 1 to larger values, in steps of one unity.

The rst one is the Bode plot : it simply represents in two dierent panels theabsolute value of the transfer function and its phase as a function of frequency.
This plot is useful to get an idea of where the unity gain frequency sits andof the gain of the transfer function at dierent frequencies. However it is notstraightforward to infer the stability of the closed-loop system from this kind ofplot.
The Nichols plot of the system is obtained plotting the logarithm of theabsolute value against the phase of the open loop transfer function (see g. E.4)
when the frequency varies from 0 to +∞. The Nyquist stability criterion canbe translated in terms of the Nichols plot. If the transfer function includes pure
integrators, the plot must start for large amplitudes from the correct phase: −90
degrees for one pure integrator, −180 for a double integrator and so on. The
request that the critical point −1 is not encircled corresponds to the request that
the Nichols plot crosses the unity gain axis between the two points correspondingto -180 and 0 phases. If the unity gain line is crossed on the left of this segment,
the plot must return again to gains greater than one remaining on the left ofthe critical point.
The Nichols plot is particularly useful to easily obtain information on the
behavior of the closed loop transfer function near the unity gain frequency.

E.5. FABRY-PEROT CAVITY

257

Indeed, if the open loop transfer function passes close to the critical point, themotion of the system can be enhanced instead of reduced at that frequency,
since the closed loop transfer function:

GCLF T (s) =1
1 + GOLT F (s)can become even much larger than one.

A limit on this overshoot(E.5)

ε can beimposed:1≤1+ε
1 + GOLT F (s)Usually the system transfer function is much greater than one at low frequency
and the closed loop transfer function is smaller than one. It also decreases athigh frequency and the closed loop transfer function is then very close to unity.
Therefore to characterize the overshoot only the behavior around the unity gain
frequency is needed. If the open loop transfer function is close to one, it can beapproximated as

GOLT F (s) ≈ eM eiπ+iφ ≈ −1 − M + iφ
The request of eq. E.5 translates toM 2 + φ2 ≥1
(1 + ε)2

which is the equation of a circle in the Nichols plot, with radius given by

1
1+ε .

The circles corresponding to overshoots of 1, 2 and 3 are shown in g. E.4.
Moreover, in the Nichols plot a change in the overall gain of the open looptransfer function corresponds to a vertical translation of the curve. If the translation is large enough to make the plot cross the critical point, the system becomes unstable. The distances along the vertical direction of the critical pointfrom the Nichols plot are called gain margins since they give the maximum possible changes in the overall gain that leave the system stable. The two points of
the Nichols plot that intersect the −180 degrees vertical line give also the typicalfrequencies of the instability oscillations that arise if the gain of the control loopchanges too much.
Finally, the distance along the horizontal direction from the critical point to
the Nichols plot gives the phase margin which is an indication of the robustnessof the control against changes in the system transfer function.

E.5 An example: locking of a Fabry-Perot cavity
The simplest example of the kind of control system used in Virgo is the locking
of a single Fabry-Perot resonant cavity.

258

APPENDIX E. FEED-BACK CONTROL SYSTEMS

E.5.1 Pound-Drever-Hall signal
The optical behavior of the cavity is described in detail in appendix A. If thelength is not controlled, the resonance condition changes and the reected andtransmitted powers are not stable.
To keep the cavity on resonance it is necessary to nd a good optical signalthat can measure the length variation with respect to the resonant one. The rstidea could be to use for example the power in the beam transmitted throughthe cavity.

Even if this signal would be maximum at resonance, it decreases

both if the cavity gets longer or shorter. Therefore it does not contain enoughinformation for control purposes. One possible solution is the detuned locking :the cavity is kept slightly away from resonance and therefore the transmitted
power signal has a non-zero slope against length changes. A method very similar
to this one is used during the lock acquisition procedure for the full Virgointerferometer (see chapter 3).
The most common method used to lock a Fabry-Perot cavity at resonanceis the Pound-Drever-Hall technique [104, 105].

As in the frontal modulation

scheme for a Michelson interferometer (already explained in Chapter 1), thebeam sent into the cavity is rst modulated in phase at a given frequency Ω/2π .
The beam entering the cavity can be described in term of one carrier eld andtwo (upper and lower) radio-frequency (RF) sidebands (see app. A):
Ein (t) = E0 eiωt J0 (m) + J1 (m) eiΩt + J−1 (m) e−iΩt(E.6)

where the Bessel functions Jk (m) are used. Being the system linear, the elds
transmitted or reected by the cavity are still given by the superposition of acarrier and two sidebands, with dierent amplitudes and phases:

ET (t) = eiωt A0 + A+ eiΩt + A− e−iΩt



(E.7)

The power measured by a photo-diode placed in transmission or reection is

PT (t)

2

2

2= |A0 | + |A+ | + |A0 −|

(E.8)


∗

+ cos Ωt Re A∗0 A− + A0 A++ sin Ωt Im A∗0 A− + A0 A∗++2Re A∗+ A− e−2iΩtThe in-phase (ACp) and in-quadrature (ACq) demodulated signals are obtained by multiplying the output of the photo-diode by two sine-waves at frequency Ω/2π , time-shifted by one quarter of period, and afterward by low-passltering with a cut-o below the modulation frequency. Therefore the two demodulated signals are given by

ACp
ACq


= Im A∗0 A− + A0 A∗+
= Re A∗0 A− + A0 A∗+(E.9)(E.10)

259

8

8

6

6

4

4
Demodulated signal [a.u.]

Demodulated signal [a.u.]

E.5. FABRY-PEROT CAVITY

2

0

-2

2

0

-2

-4

-4

-6

-6

-8

-8
-0.4

-0.2

0

0.2
0.4
Length variation [units of λ]

0.6

0.8

-0.04

1

-0.02

0
Length variation [units of λ]

0.02

0.04

Figure E.5: Left: shape of the Pound-Drever-Hall signal for a Fabry-Perot cavity
similar to Virgo arm cavities. Right: zoom around the resonance.

The amplitude of the elds transmitted by the cavity are computed in appendix
A and the demodulated signals are given bysin(2kx)2

ACp= −2J1 (m)J0 (m) (tI tE ) rI rE

ACq

=

4

2

1 + (rI rE ) − 2(rI rE ) cos(4kx)0This is the general expression for the Pound-Drever-Hall signal, plotted ing. E.5.

In a narrow region around the resonance the response of the signal

to length variation is linear:ACp ≈ −8mF
xπλ

(E.11)

therefore allowing to reconstruct the displacement from resonance of the twomirrors and to act in order to correct it. In terms of control theory this is agood error signal for stabilizing the length of the Fabry-Perot cavity.

E.5.2 Frequency response of the cavity
To completely determine the behavior of the optical system, which is the plant tobe controlled, it is necessary to compute the frequency dependence of the cavity
response, which is the transfer function between one of the mirrors motion (theend one for example) and the transmitted signal [12]. In appendix A this transfer
function is computed:

2

H(ωx ) = −2iwhere fp =F(tI tE )2 π

1 − (rI rE )1ωx
1 − i 2πfp(E.12)c
4LF is the pole of the Fabry-Perot cavity. This transfer function(see g. E.6) is a low pass lter with a simple pole which for the Virgo arm
cavities is fp ≈ 500 Hz.

260

APPENDIX E. FEED-BACK CONTROL SYSTEMS
100
10-2
-4

10
Modulus [a.u.]

Modulus [a.u.]

100

10-1

10-6
10-8
10-10
10-12
-14

10

-2

10

-16

10

20

0

Phase [deg.]

Phase [deg.]

0
-20
-40
-60

-150

-80
-100
0
10

Figure E.6:

-50

-100

10

1

Left:

2

10
Frequency [Hz]

10

3

10

4

-200
10-1

100

101
102
Frequency [Hz]103

absolute value (top) and phase (bottom) of the frequency

response of a Fabry-Perot cavity demodulated transmission to longitudinal motion, for a nesse of F

= 50 and a length of L = 3000 m, corresponding to acavity pole at 500 Hz. The absolute value is rescaled in order to be equal to 1
at low frequencies.

Right: transfer function of the actuators, namely a simple

pendulum with resonant frequency at 600 mHz and very high quality factor.

E.5.3 Actuators and corrector
The last missing point is the frequency response of the actuators: for the locking
of a single cavity, acting only on the end mirror, this can be modeled by a simple
pendulum (see chap. 2), with resonant frequency at about 600 mHz and very
high quality factor (see g. E.6).
The corrector lter designed to control the length of the cavity is a very
simple one, see g. E.7, consisting in a double integrator, compensated at 10 Hz
by a complex zero and again at 800 Hz with a complex pole, to have a unity
gain frequency around 50 Hz. In this way below the resonant frequency of theactuators the system is a double integrator, capable of maintaining xed thelength of the cavity even in presence of big motions of the input mirror. The
pole and zero pair is a common method used to stabilize the system.
The Nichols plot of this system is shown in g. E.8. The loop margins canbe easily read from this plot: the overall gain can be increased or decreased bya factor 5.

The phase margin is 50 degrees.

This is a really robust feedback

system, capable of maintain the control of the cavity even with large changes
in the optical transfer function. This robustness is indeed needed in the rstmoments of the lock acquisition, when the system response is highly non-linear.

104E.6. MULTI-DIMENSIONAL CONTROL SYSTEMS

261

10

10

8

1014

10

13

10

6

10

Modulus

Modulus [a.u.]

4

12

10

1011

10

2

10

0

10

-2

1010

10

109

10

-4

10-6
8

10

200

4

150
3

100
Phase [deg.]

Phase [deg.]

2
1
0
-1

0
-50
-100

-2

-150

-3
-4
10-2

50

10-1

100

101
Frequency [Hz]

102

103

104

-200
-2
10

-1

10

10

0

1

10
Frequency [Hz]

2

10

3

10

Figure E.7: Left: Bode plot of the corrector lter implemented for the control
of a single arm cavity.

Right: open loop transfer function for the control of asingle cavity. The unity gain frequency is set at 50 Hz.

E.6 Multi-dimensional control systems
So far only systems with one single degree of freedom has been described. In
practice, the most common case is to have multiple degrees of freedom to control
at the same time: for example the Virgo longitudinal control system deals with 4
d.o.f. (see chapter 3 for an extended discussion) and the angular control systemwith 14 d.o.f. (see chapter 4). The main diculty that can arise in this system isthe presence of cross-couplings between the various d.o.f.: this can happen at thelevel of the error signals, meaning that one signal is sensitive to more than one
d.o.f. at the same time, or in the actuation part, meaning that the corrections
can not be applied directly to single d.o.f., but only to combinations of them.
Such kind of systems is usually called MIMO (multiple-input multiple-output)
in control system literature.
This situation can be described in a way similar to the single d.o.f. case (seeg. E.9).

Let z denote the vector of free motions of the n system degrees of

freedom:




z1


 z2 
z=
 .. 
 . zn(E.13)and let e and c denote respectively the vectors of error and correction signals.
The plant can be described in the most general case with a matrix G(s) oftransfer functions, which converts the n motions of the d.o.f. to a given number

10

4

262

APPENDIX E. FEED-BACK CONTROL SYSTEMS
7

10

6

10

5

10

4

Magnitude

10

3

10

2

10

1

10

0

10

-1

10

-2

10

-400

-360

-320

-280

-240
Phase [deg]

-200

-160

-120

Figure E.8: Nichols plot of the control system for a single arm cavity.

m of error signals. Of course for the system to be controllable it is necessary thatm ≥ n. In the same way the corrector can be described with another matrix oftransfer functions F(s). The most common way to construct this matrix is towrite it as a product of a sensing one, which from the various signals reconstruct
as well as possible n independent error signals for the n system d.o.f.; a diagonal
matrix of correction lters, one for each d.o.f.; nally a driving matrix which
combines the corrections to obtain suitable signals to be sent to the actuators.
The computation of the eect of closing the feedback can be carried out ina similar way to the SISO case, taking care that all objects are vectors andmatrices, therefore the order of operators matters:−1e = [1 + G(s)F(s)]G(s) zand in particular the open loop and closed loop transfer functions are given by:

GOLT F (s) = G(s)F(s)GCLT F (s) =−1[1 + G(s)F(s)]For the system to be controllable, the open loop transfer function (which isalways a n × n matrix) must be invertible, otherwise there would be at least
one linear combination of the system degrees of freedom which has a correction
identically equal to zero: in other words the corresponding motion of the system
is not controlled. One necessary requirements is that the rank of all matrices
must be maximum, i.e.

equal to the number n of degrees of freedom to becontrolled.
The behavior of multidimensional coupled systems can be quite complex tounderstand and model. As already stated, an example of such a system is theE.6. MULTI-DIMENSIONAL CONTROL SYSTEMS

Figure E.9: Scheme of multiple-input multiple-output (MIMO) system.

263

z con-

tains the free motions of the system, e the error signals and c the corrections.

length sensing and control of the Virgo interferometer, which is described inchapter 3.

Appendix F
Non-stationary noise
couplings
It is quite common to permanently add some external perturbation to various
control loops, mainly in the form of single lines, in order to measure loop parameters and also for calibration purposes. It turned out to be quite useful to
track their evolution in time and also the variations of the ratio of lines as seenin dierent signals, since this can provide information on the time variations ofcontrol noise couplings.
In general, one calibration line at a given frequency f is added into oneof the control loops, usually to the error point.

The typical situation is to

track the coupling of the noise from the error point to the dark fringe.

Onepossible way is to measure the power in the line in both signals, using bandlimited RMS computations or line identication algorithms, and to estimate thecoupling by the ratio of the amplitudes in the two signals. In this way however
no information on the phase of the coupling coecient can be recovered.

Toovercome this limitation, the transfer function between the two signals can becomputed, using standard methods described in appendix D. The coherence
gives a quality check for the measurement
This method is however not the more convenient way to track a single lines,
since it implies estimation of full Fourier transforms to look at a single bin. Even
if it is possible to compute the Fourier transform at the line frequency only, amore convenient method to compute the amplitude of a line is to implement
a demodulation algorithm similar to what it is done in the frontal modulation
scheme for extracting control signals from an interferometer.
A rst kind of demodulation can be used to extract the amplitude and phaseof the line at frequency f in a given signal with respect to a reference pure sine
wave. Referring to g. F.1, the input signal x is rst passed through a resonantlter (made of a zero at 0 Hz and a pole with high Q at the demodulation

266

APPENDIX F. NON-STATIONARY NOISE COUPLINGS

Figure F.1: Scheme of the algorithm used for the demodulation of a signal at agiven frequency f using a internally generated reference sine wave.

Figure F.2: Scheme of the algorithm used for the demodulation of a signal at agiven frequency f with respect to another reference signal.

frequency f ), in order to strongly suppress all components of the signals far
from the interesting frequency. If the original signal contains a modulated line
at the f frequency, the result of this resonant ltering can be modeled asr(t) = A(t) sin (2πf t + φ(t)) = As (t) sin(2πf t) + Ac (t) cos(2πf t)
where both amplitude and phase modulations are considered. Here A(t), φ(t),As (t) and Ac (t) are changing slowly in time, meaning that their spectra aremainly conned to frequencies much smaller than f .
The resulting r(t) is mixed with a reference sine wave and with the corresponding cosine wave:

s(t)

=

sin(2πf + φ0 )c(t) =

cos(2πf + φ0 )

267

Figure F.3: Scheme of the behavior of P and Q values if there is (b) or not (a)a signal at the demodulation frequency. Refer to the text for an explanation of
how this dierence can be used to compute a quality check.

so that the two resulting signals contain both a component at low frequency
and one at 2f :r(t) · s(t)r(t) · c(t)1 − cos(4πf t)sin(4πf t)+ Ac (t)22sin(4πf t)
1 + cos(4πf t)= As (t)+ Ac (t)22= As (t)The amplitude of the line, divided into the in-phase and in-quadrature components, can be recovered by considering only the component at low frequency. For
this reason the two signals resulting from the mixing are passed through a notchlter at two times the demodulation frequency (a lter equal to one everywhere
except for a narrow dip at 2f ) and then low-passed with a second-order lterwith corner frequency well below the demodulation one. Finally the output signals are computed by down-sampling the results with simple averages. In thisway the two amplitudes As (t) and Ac (t) are recovered and if needed these canbe used to reconstruct the line amplitude and phase.
This procedure gives clearly an output even if there is no line at the demodulation frequency and therefore it is important to build a quality check, similar tocoherence. Suppose rst that there is no signal at the demodulation frequency.
In this case both As (t) and Ac (t) are varying randomly with zero mean on afast time scale. The signals obtained immediately after the 2f notch and before
the low-pass lter (called XP and XQ in g. F.1) have zero mean and non zero
RMS (see g. F.3a). The sum of the two squared signals is then computed andpassed through a low-pass lter, equivalent to averaging. On the contrary, thetwo signals after passing through the low-pass lter, are small compared to theAPPENDIX F. NON-STATIONARY NOISE COUPLINGS
101

Modulus [a.u.]+45 degrees+45 degrees
0

10

10-1

10-2 110

10Frequency [Hz]

2

10

3

102
Frequency [Hz]

103

100
80
60
Phase [deg.]

268

40
20
0
-20
-40
-60
-80
-100
101

Figure F.4: Bode plots of the phase lag lters used in the demodulation algorithm to add a dephasing to the input signals. The plotted lters are tuned fora frequency f = 100 Hz.

previously computed total RMS. Therefore the following quantity

2

C=

XP + XQ

2(F.1)

2
XP2 + XQ

is small when there is no signal at the demodulation frequency. Suppose now
that some signal is present (see g. F.3b). In this case both XP and XQ have
non-zero mean, greater than their standard deviation.

Therefore the sum of

their RMS is close to (and always greater of ) the sum of their squared meanvalues. From this follows that the quantity computed in eq. F.1 is close to 1: inconclusion this is a good estimate of the coherence between the signal and thereference sine and cosine waves, at the demodulation frequency.
A similar procedure can be used to demodulate a signal using another one
as reference.

A typical example is the injection of a calibration line and thedemodulation of the dark fringe signal with respect to the injected line itself.
In this kind of demodulation (see g. F.2) the two input signals are rst passed
through the resonant lter. The in-phase component of the demodulation can berecovered simply mixing the two resulting signals. To obtain the in-quadraturecomponents it would be necessary to add a delay of one quarter of period tothe reference signal: this can be done easily by shifting the samples only if therequired delay is a precise multiple of the sampling time. Since this is not often

269
the case, an alternative technique must be used. Two lters are developed inorder to produce a phase shift of ±45 degrees at the frequency f . This is doneby using a lter with a broad resonance around a frequency tuned to have thecorrect phase at f and unitary gain at the same frequency (see g. F.4). The
two resonant frequencies are given by

f± = √

2f
5∓1

One signal is passed through the +45 degrees lter only, while the other one ispassed through both +45 degrees and −45 degrees lters. Then mixing the rstsignals with the other two gives both in-phase and in-quadrature components.
The rest of the demodulation process is equal to that already described.

270

APPENDIX F. NON-STATIONARY NOISE COUPLINGS

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