Université Paris Diderot
Ecole Doctorale 517: Particules, Noyaux et Cosmos

Università degli Studi di Roma Tor Vergata
DOCTORAT
Physique: champs, particules, matières
DOTTORATO IN FISICA

Massimo Granata

Optical Development for
Second- and Third-Generation Gravitational-Wave Detectors:
Stable Recycling Cavities for Advanced Virgoand Higher-Order Laguerre-Gauss Modes

Thèse soutenue le 29·09·2011 au Laboratoire AstroParticule et Cosmologie (APC)

Jury
M.
M.
M.me
M.
M.
M.me

P. Binétruy,
M. Barsuglia,
V. Fafone,
G. Losurdo,
D. Shoemaker,
E. Tournefier,Président
Directeur de thèse
Directeur de thèse
Examinateur
Rapporteur
RapporteurTo Donatella“There are more things in heaven and earth, Horatio,Than are dreamt of in your philosophy.”“In our sky there is no limits
And masters we have none”

Acknowledgements
I first want to thank Fulvio Ricci and Yves Charon, who, by different ways, made this thesis
concretely possible. Then I sincerely thank Matteo for having supervised my thesis, and for histrust in me: I thank him for what he taught me, for his support, for being always present. Thanks
to him, I will always keep a good memory of this thesis. Thanks to Viviana, Alessio and all thegroup of Tor Vergata for the work together and the good time I spent in Rome, and thanks to
Alessandra and to the IDAPP (International Doctorate on AstroParticle Physics) programme forthis opportunity. Thanks to Christelle, who shared with me most of my time at the laboratory:
I really enjoyed working with her, as well as her company beyond the work. Thanks to Eric, for
his company, his advices and our discussions. Thanks to Robert, for his precious help, and thanks
equally to Hiro Yamamoto, Jérôme Degallaix, and Fabien Reversat and Stéphane Tisserand of Silios
Technologies. Thanks to Eden, Olivier, Pierre, Bérengère and Alain, for their help and friendship.
I acknowledge Pierre Binétruy and Yannick Giraud-Héraud, as well as all the people of thegroup Cosmologie et Gravitation and of the administration staff, for their warm welcome and theirsupport. It has been a great pleasure to work with all of them, I really loved my time at APC. Then,since they really made this time unforgettable, I want to thank all of my friends at the lab: Silvia, Michela, Lorenzo, Alejandro, George, Florian, Alex, Giulio, Margherita, Héloı̈se, Fabio, Dana,
Mariana, Marie-Anne, Gael, Sébastien, Romain, Alexandre, Josquin, Guillaume and Matthieu. A
special mention is for Mario and Emilie. Sincerely, thank you all.
I also would like to acknowledge the programme Particules et Univers of the Centre Nationalde la Recherche Scientifique and the programme Physique des 2 Infinis (P2I), as well as the Agence
Nationale de la Recherche (programme Blanc 2010 n. 041601), for the support to the experimental
work of this thesis.
Un ringraziamento molto speciale è per Filippo, mio padre, e per Marina, mia madre, per essermi stati sempre vicini: nei momenti più belli, e in quelli più difficili. Voglio infine ricordare
Piero e Valentina, i miei nonni, che purtroppo non hanno potuto condividere con me la gioia diveder realizzato questo lavoro. Questa tesi è dedicata anche a loro.

Contents
Acknowledgments

5

Introduction

11

1 Gravitational Waves
1.1 The Metric Tensor and the Geodesic Equation . . . . . . . . . . . . . . . . . . . . .
1.2 The Einstein Equations and the Wave Solution . . . . . . . . . . . . . . . . . . . . .
1.3 Emission of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Sources of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Coalescing Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.4 Magnetars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.5 Stochastic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.6 Multimessenger Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Indirect Measurement of Gravitational Radiation . . . . . . . . . . . . . . . . . . . .

15
15
16
18
19
20
21
22
23
23
24
25

2 Interferometric Detectors of Gravitational Waves
2.1 Geodesic Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The Michelson Inteferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Free-falling Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Sensitivity and Noise Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Radiation Pressure Noise and Quantum Limit . . . . . . . . . . . . . . . . . .
2.4.3 Seismic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Gravity Gradient Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.5 Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.6 Other Sources of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Thermal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Sensitivity Improvements with Optical Resonators . . . . . . . . . . . . . . . . . . .
2.6.1 Fabry-Perot Arm Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 Power Recycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.3 Mode-Cleaner Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28
28
30
34
35
36
37
39
39
40
44
45
45
46
49
53

7

CONTENTS

3 Present and Future Ground-Based Gravitational-Wave Detectors
3.1 The Network of First Generation Detectors . . . . . . . . . . . . . . . . . . . . . . .
3.2 Observational Results of the LSC-Virgo Network . . . . . . . . . . . . . . . . . . . .
3.3 Advanced Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Sensitivity and Science Case . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Third-Generation Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55
56
61
66
66
69

4 The Project of Advanced Virgo
4.1 Laser and Injection System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Optical Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Fields Inside the Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Thermal Compensation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76
76
77
78
80
84

5 Stable Recycling Cavities for Advanced Virgo89
5.1 Laser Modes and Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2 Motivations for Stable Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 How to build a Stable Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 Design Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6 Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo
109
6.1 Design Procedure and Simulation Tools . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.1.1 Ray-Matrix Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.1.2 Ray-Matrix Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.1.3 Fast Fourier Transform Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2 Design History: from the Baseline to the Vertical Layout . . . . . . . . . . . . . . . . 119
6.2.1 Baseline Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2.2 Telescopes in the Differential Part of the Michelson Interferometer . . . . . . 128
6.2.3 Long Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.2.4 Vertical Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.3 Simulation of the Vertical Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.3.1 Cold Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.3.2 Hot Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Conclusions and Perspectives - Part I

155

7 Reduction of Mirror Thermal Noise with Laguerre-Gauss Modes
161
7.1 Techniques to Reduce Mirror Thermal Noise . . . . . . . . . . . . . . . . . . . . . . 162
7.2 Thermal Noise Reduction with Laguerre-Gauss Modes . . . . . . . . . . . . . . . . . 165
8 A Table-Top Experiment to Test Generation and Interferometry of HigherOrder Laguerre-Gauss Modes
170
8.1 Techniques for Higher-Order Mode Generation . . . . . . . . . . . . . . . . . . . . . 171
8.2 Design of the LG33 Diffractive Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.3 Design of the Mode-Cleaner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.4 The Table-Top Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8

CONTENTS

8.5 LG00 Input Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.6 Mode Conversion: LG00 to Pseudo-LG33 . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.7 Alignment of the Pseudo-LG33 to the Mode-Cleaner . . . . . . . . . . . . . . . . . . . 203
8.8 Mode-Cleaner Lock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.9 Mode-Cleaner spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
8.10 Transmitted LG33 Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.11 LG33 mode conversion efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8.12 LG33 Michelson Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Conclusions and Perspectives - Part II

222

Bibliography

227

9

10

Introduction
The existence of gravitational radiation, known as gravitational waves, is one of the most important
predictions of the General Theory of Relativity. Gravitational waves are ripples in the metric ofspace-time that propagate at the speed of light, emitted by accelerating massive bodies. This
radiation has never been directly detected so far, but the discovery of the binary pulsar PSR1913+16
in 1974 provided the indirect evidence that gravitational waves do exist. The waves emitted by
nearby astrophysical sources, such as neutron star binaries, are expected to be detectable on Earth.
The very weak space-time strain of a gravitational wave might be detected as a phase shift
between two laser beams propagating in the arms of a Michelson interferometer. Thereby a basic
interferometric detector is composed of a laser, mirrors and a photodiode: the detection is performedby counting the photons of the laser beam interference fringe at the interferometer output. In
order to increase the sensitivity of the detector, Fabry-Perot cavities are usually implemented in
the interferometer.
During the last two decades, several interferometric detectors have been built and are now operating: LIGO, Virgo and GEO600 form a worldwide network of gravitational wave interferometers.
These first-generation instruments approximately reached their design sensitivities, successfully
demonstrating the technical feasibility of the detection principle, and completed several observational runs. No detections have been reported so far, but the data collected over the last years havebeen used to put constraints (either in terms of amplitude of emitted gravitational radiation or as
event rates) on several astrophysical sources in the nearby universe, such as binary systems, known
pulsars and burst sources. Also the amplitude of the stochastic gravitational-wave background of
cosmological origin has been constrained, in a limited region of its spectrum.
Advanced Virgo and Advanced LIGO are planned to increase the sensitivity of the initial detectors by an order of magnitude. The Large-scale Cryogenic Gravitational-wave Telescope (LCGT)
will attain a similar sensitivity, and will test for the first time the use of cryogenics and a new substrate material on a km-scale detector, in an underground site. These second-generation detectors
will allow the first direct detection of gravitational waves. The construction of Advanced Virgo andadvanced LIGO has already started, and their first joint scientific run is planned around 2016.
In the meanwhile, the project of a European detector of third generation, named Einstein Telescope, is presently in a phase of design study. The aim of this detector, which should be one
hundred times more sensitive than initial interferometers, is to initiate in the 2020 decade the eraof precision gravitational wave astronomy.
The sensitivity of future gravitational-wave interferometers of second and third generation will
be limited by mirror thermal noise in the central part of the detection band. The exigence ofsolving this issue led to spend a great effort in the design of future detectors, and triggered theinvestigation of new technical solutions.

11In advanced detectors the increase of the beam radius on the arm cavity mirrors will be necessary
to reduce mirror thermal noise, and this will increase mode degeneracy in the recycling cavities. As
already pointed out by the past experience with initial detectors, mode degeneracy can considerably
affect the operation of gravitational-wave interferometers. To solve this issue, Advanced LIGO
adopted non-degenerate recycling cavities as first, then initially Advanced Virgo embraced thissolution too. Non-degenerate cavities should make the interferometer more robust against figure
errors or thermal deformations of its mirrors.
The first part of this thesis concerns the optical design of non-degenerate recycling cavities for
Advanced Virgo. We worked on this subject since fall 2008 to spring 2011. In the beginning we
studied the general properties of non-degenerate cavities and how to adapt this solution to therequirements and constraints of the project. Then we developed a general design procedure to
draw the optical layout of the cavities, and several layouts have been proposed and analyzed. We
finally chose a configuration with the required optical features, the vertical layout, which might be
implemented in Advanced Virgo.
Mirror thermal noise might be reduced using non-Gaussian beams. Initially mesa and conicalbeams have been proposed, but they have the relevant drawback of requiring resonators with nonspherical mirrors. Higher-order Laguerre-Gauss modes have been proposed in 2006 in alternative,
since they should provide a comparable noise reduction but are eigenmodes of spherical resonators.
Laguerre-Gauss modes have never been used for applications in gravitational-wave detection. No
technique has been developed so far to generate these beams with reasonable purity and efficiency,
and with the capability of handling high-power laser beams that will be used in future gravitational
wave detectors. Moreover, and most important, the behavior of higher-order Laguerre-Gauss modes
in Fabry-Perot interferometers is not known at present.
The second part of the thesis concerns the reduction of mirror thermal noise in future detectors
through the use of higher-order Laguerre-Gauss modes. We assembled a table-top experimental
setup where we tested the generation of Laguerre-Gauss beams using a diffractive plate of fusedsilica and a Fabry-Perot mode-cleaner cavity. Subsequently we used these beams inside a Michelsoninterferometer, to start the analysis of their optical performances in a basic optical system.

Outline of the Thesis• Chapters 1, 2 and 3 are of introductory nature. Chapter 1 will describe the basis of
the theory of General Relativity and of gravitational waves and their astrophysical sources.
Chapter 2 will treat the fundamental principles of interferometric detection of gravitational
waves, introducing the Michelson interferometer, the Fabry-Perot cavities and illustrating thefundamental sources of noise that affect the instrument. In Chapter 3 we describe the networkof gravitational wave detectors of first generation, with a review of the observational results
achieved so far. We then introduce second- and third-generation detectors, with a particularfocus on the sensitivity and the science case of Advanced Virgo.
PART II - Optical Design of the Non-Degenerate Recycling Cavities
of Advanced Virgo• Chapter 4 will describe the Advanced Virgo project, focusing mostly on the optical layout,
the thermal compensation system and the suspensions of mirrors.

• Chapter 5 is dedicated to the design of the recycling cavities of Advanced Virgo. After abrief introduction about laser modes, optical resonators and mode degeneracy, we will describe
the motivations supporting the choice of non-degenerate recycling cavities. Therefore we will
illustrate how to design a non-degenerate cavity, and the particular design constraints ofAdvanced Virgo.
• In Chapter 6 we will describe the design procedure and the optical simulations that wedeveloped to design the layout of non-degenerate recycling cavities for Advanced Virgo. We
will present and discuss the various cavity layouts studied, and the one which has beenretained.
PART III - Use of Higher-Order Laguerre-Gauss Modes in Future Detectors• In Chapter 7 we will describe the available techniques to decrease mirror thermal noise in
future detectors, and we will show the sensitivity gain achievable using higher-order LaguerreGauss modes.
• Chapter 8 will illustrate the design and the assembly of a table-top experiment to test
generation and interferometry of LG33 beams, together with the results achieved.

14Chapter 1

Gravitational Waves
The theory of General Relativity is the geometric theory of gravitation published by Albert Einsteinin 1916 [1]. In General Relativity, the gravitational interaction is described as a geometric effectof curvature of space-time, which is determined by the energy-matter distribution of a system ofbodies. In this theoretical frame, gravitational waves are local deformations of space-time which
propagate with a finite velocity, being emitted by massive accelerated bodies. Since the coupling
between gravitational radiation and matter is predicted to be extremely small, only astrophysicalsources which are particularly massive and compact and move at relativistic speed are expected toemit gravitational waves of detectable amplitude. Thus the aim of gravitational wave detectors is
to observe on Earth the gravitational waves emitted by astrophysical sources in space.
In this chapter we will introduce the fundamental concepts of the theory of General Relativity,
and we will show that the prediction of the existence of gravitational waves is a natural consequenceof the theory. Subsequently, we will describe the astrophysical sources of gravitational waves
detectable on Earth. The last part of the chapter will introduce the indirect measurement ofgravitational wave emission through the observation of the orbital period decay of pulsars in binary
systems.

1.1

The Metric Tensor and the Geodesic Equation

In General Relativity, the line element ds which defines the distance between two points in spacetime in an arbitrary reference frame of coordinates xα (α = 0, 1, 2, 3) is
ds2 = gµν dxµ dxν ,(1.1)

where gµν is the metric tensor which describe the space-time geometry (µ, ν = 0, 1, 2, 3). The
equation of motion of a particle which is only subject to the influence of gravity is the geodesic
equationµν
d2 xα
α dx dx
+
Γ
=0,
(1.2)
µν
dτ 2
dτ dτ
where τ is the proper time in the frame where the particle moves and Γαµν are the affine connections(also referred to as Christoffel symbols), defined through the derivatives of the metric tensor:
 ∂g
∂gµγ
∂gµν 
1νγ
Γαµν = g γα
+
−
.
2
∂xµ
∂xν
∂xγ

15

(1.3)

Chapter 1. Gravitational Waves

Eqs.(1.2) and (1.3) are of fundamental importance, since they establish a relation between thegeometrical properties of space-time and the physical properties of the gravitational field. The
effect of the gravitational field can be interpreted as a curvature of space-time, forcing the bodies
of a system to move along lines called geodesics. In analogy with the law of universal gravitation of
Newton, the metric tensor gµν can be considered as the generalization of the gravitational potential,
as well as the affine connections can be considered as the generalization of the gravitational field
associated to the potential.
According to the Equivalence Principle, it is always possible to define a local inertial frame
where the effect of gravity is canceled by the choice of the frame coordinates. In the local inertial
frame of coordinates ξ α , the line element ds simplifies to the usual Euclidean distance of the flat
Minkowsky space-time metric,
ds2 = ηµν dxµ dxν ,
(1.4)
where


1 0
0
0
0 −1 00 ηµν = 
0 0 −1 0  ,
0 0
0 −1


(1.5)

and the relation between the metric tensor and the flat space-time metric writesgµν =∂ξ α ∂ξ β
ηαβ ,∂xµ ∂xν(1.6)

where ∂ξ α /∂xµ and ∂ξ β /∂xν are the matrices which describe the change of coordinates between
the arbitrary frame and the local inertial frame. In the local inertial frame the affine connections
are equal to∂xα ∂ 2 ξ λΓαµν =,(1.7)∂ξ λ ∂xµ xν
and the geodesic equation simplifies to
∂2ξα
=0.
dτ 2

(1.8)

Eq.(1.8) is analogous to the second law of motion of Newton, F~ = md2 ~x/dt2 , for a particle in asystem where forces are in equilibrium (F~ = 0). In other words, with a particular choice of thereference frame, the effect of gravity can be canceled so that the physics of the system is governedby the laws of Special Relativity.

1.2

The Einstein Equations and the Wave Solution

The curvature of space-time (and therefore the gravitational field) is determined by the energymatter distribution through the Einstein equations
18πG
Gµν = Rµν − gµν R = 4 Tµν ,
2
c

(1.9)

where c is the speed of light and G is the constant of gravitation from the law of universal gravitationof Newton. Gµν is the Einstein tensor which describes the space-time geometry, defined as the sum

16

1.2. The Einstein Equations and the Wave Solutionof two symmetric tensors: the Ricci tensor Rµν and the metric tensor gµν . The Ricci tensor derives
αfrom the contraction of the curvature tensor (also called the Riemann tensor) Rβµν
with the metrictensor, according to the relationαRµν = g kα Rkµαν = Rµαν.

(1.10)

The curvature tensor is obtained from the first and the second derivatives of the metric tensor,
through the affine connections:αRβµν= Γαβν,µ − Γαβµ,ν + Γασµ Γσβν − Γασν Γσβµ .

(1.11)

αRβµν
contains the information of how much a vector changes when it is parallel transported arounda loop in a curved space-time, thereby it is said to contain the information about the curvature ofspace-time.
Tµν is the stress-energy tensor which describes the sources of the gravitational field: matterand energy. In the simple case of a system of n non-interacting particles, each one with position
ξnα and four-momentum pαn , the T00 component of the tensor is the energy density of the system,
the T0i /c components are momentum densities, the Tik are the currents of momentum of the i-thcomponent through the unitary surface element orthogonal to the k-th axis. The stress-energy
tensor is a symmetric tensor.
The Einstein equations form a set of 10 non-linear partial differential equations, whose solution
is in general a complex problem. A simple solution may be obtained by applying a perturbative
approach in the limit of weak gravitational field in vacuum, where the metric can be written asgµν = ηµν + hµν(1.12)and hµν << 1 is the small gravitational perturbation. Within this approximation, the Einsteinequations are linear to first order with respect to hµν and it is possible to demonstrate that the
perturbation solution propagates in space-time like a gravitational wave. In fact, the freedom inthe choice of the coordinate system allows to impose a gauge where hµν is transverse and traceless(TT gauge), and eqs.(1.9) simplify to!
1 ∂2
2∇ − 2 2 hµν = 0 ,(1.13)c ∂tdescribing the propagation of waves at speed c. The solutions of eq.(1.13) can be written as planewaves,hµν = µν exp[i(ΩGW t − ~kGW · ~x)] ,(1.14)
where ΩGW is the angular frequency of the waves, ~kGW is the wave vector and µν is the polarization
tensor. In the TT gauge, the polarization tensor of a wave propagating along the z axis writes×µν = h+ +µν + h× µν ,×with +µν and µν being the basis tensors defined as
0 0 0 0
0 1 0 0 +µν = 
0 0 −1 0 
0 0 0 017

(1.15)

Chapter 1. Gravitational Waves


0

0
×
µν = 
0
0

0
0
1
0


0 0
1 0
 .
0 0
0 0

Thus gravitational waves have only two possible polarization states, which are + (plus) and ×(cross), both shown on fig.2.1. The important result that follows from eq.(1.15) is that the effectof a gravitational wave on the space-time metric has an intrinsic differential nature. The distance
between two nearby test masses in the metric perturbed by a gravitational wave is modulated at
the wave frequency with opposite sign in the plane transverse to the direction of propagation, as
shown on fig.2.1.

1.3

Emission of Gravitational Waves

Gravitational waves are emitted by accelerated masses, as well as electromagnetic waves are emitted
by accelerated charges. However, due to the conservation of total momentum of isolated systems,
the gravitational dipole is null and gravitational waves have a quadrupolar nature. Thereby theemission of gravitational radiation is associated to the variation of the quadrupolar moment tensor
of mass distribution, defined as
Z
1
Qij (t, ~x) = 2
T00 (t, ~x)xi xj dV ,(1.16)
c V
where T00 (t, ~x) is the time component of the time-dependent stress-energy tensor. From eq.(1.16) itis clear that systems with spherically symmetric mass distribution, which do not have a quadrupole
moment, cannot emit gravitational waves. In the far field and slow motion approximation (i. e.
far from the source and when the typical velocities of the physical processes of the source are muchsmaller than light speed), the amplitude of the wave due to the quadrupolar moment tensor ishij =2 G ∂2Qij (t − r/c, ~x) ,r c4 ∂t2(1.17)

where r is the distance between the source and the point of observation, t − r/c is the retarded time
representing the delay between the wave emission and the moment when it reaches the observer.
Thus the wave amplitude measured by an observer is inversely proportional to the distance traveledby the wave from the source.
The factor G/c4 ' 8 · 10−45 s2 /kg m in eq.(1.17) is extremely small, yielding a very weak
amplitude of the emitted wave. Starting from eq.(1.17), it can be demonstrated [2] that thegeneration of gravitational waves in laboratory with a controlled source is impossible. Consider for
example a system of two masses of 1 tonne each, at either ends of a rod 2 m long, spinning about anaxis orthogonal to the rod with an angular frequency of 1 kHz. The amplitude of the gravitational
wave generated by such a machine would beh ∼ 10−33 ·1,rwhich is of many orders of magnitude fainter than the amplitude measurable (h ∼ 10−21 − 10−23 )by any present ground-based detector, as we will see in the next chapter. It is for this reason thatthe experimental study of gravitational waves is directed towards astrophysical sources.

18

1.4. Sources of Gravitational Waves

The luminosity of the gravitational wave, that is the gravitational energy emitted by the source
per unit time, writes
3
G D X ... r ... r E
Qij t −
Qij t −
L= 5,
(1.18)
5c
c
c
i,j=1

where the average is performed over several wavelenghts since in General Relativity the energyof the gravitational field cannot be defined locally. Eq.(1.18) can be used to estimate with someapproximation the luminosity of an astrophysical source of mass M and characteristic radius R.
Let T be the characteristic time scale for the time evolution of the source, and let  be a factor
measuring the degree of asymmetry in the spatial mass distribution. Thus the quadrupole momentof the source is approximately equal to Q ∼ M R2 . The gravitational wave luminosity is thenG M 2 R4
L= 5
.
c T6

(1.19)

By introducing the Schwarzschild radius RS = 2GM/c2 , which is a length scale which measuresthe compactness of the source, and the characteristic speed v = R/T , eq.(1.19) can be rewritten as
c5
L=
G

RS
R

!2

 v 6c.

(1.20)

Eq.(1.20) indicates that the more the source is compact (R ∼ RS ), has an asymmetric mass
distribution ( ∼ 1) and moves or evolves with a relativistic speed (v ∼ c), the larger is thegravitational wave energy emitted per unit time.

1.4

Sources of Gravitational Waves

The sources of gravitational waves can be classified depending on the time evolution of the signal
they generate:• periodic and quasi periodic sources emitting a signal of superposed sinusoidal components,
whose frequencies are more or less constant over a time interval longer than the observational
time. Pulsars (pulsating neutron stars) and in general spinning neutron stars are expected
to be sources of periodic gravitational wave signals, as well as coalescing binary systems ofneutron stars or black holes during the early phase of their orbital motion• transient and impulsive sources (bursts), if the expected time duration of their signal is shorter
than the typical time of observation. This kind of signal is related to phenomena which
involve a violent gravitational collapse, like Supernovae, magnetars or coalescing systems at
the merging point• stochastic background, if the signal is expected to fluctuate statistically and last for a time
duration much longer than the typical observational time. This background can have an
astrophysical or a cosmological origin.

19

Chapter 1. Gravitational Waves

Figure 1.1: Typical chirp waveform emitted during the coalescence of a black-hole binary system, from [56].

1.4.1

Coalescing Binaries

A large fraction of the stellar objects presently known are binary systems. If in such a system thetwo stars are compact bodies, like two neutron stars, two black holes or a neutron star and a blackhole, the system is expected to lose a considerable amount of energy by emission of detectablegravitational radiation. The frequency of the emitted wave is the double of the orbital one.
In the early phase of the orbital motion the emission is small, and the emitted radiation is aperiodic signal with frequency of the order of mHz, too slow to be measured at Earth (in this band
ground-based detectors are blinded by seismic noise due to ground vibrations). However, when theemission of gravitational radiation increases, the orbital energy decreases together with the period
and the two bodies start spiralling around each other, in a phase named coalescence. The expectedsignal emitted in this phase of shrinking of the orbit has a characteristic shape called chirp, shownon fig.1.1, where the amplitude and frequency increase with time. By supposing that the system
has a circular orbit of radius R and a total mass M = m1 + m2 , the time scale of the inspirallingphase is [3]5 M  M −4tchirp =,(1.21)96 ν R
where ν = µ/M and µ = m1 m2 /(m1 + m2 ) is the reduced mass. It is clear from eq.(1.21) thatsystems with large mass ratios between the bodies can spend a long time in relativistic orbits,
whereas equal-mass binaries are expected to merge after only a few orbits in the relativistic regime.
The inspiral phase ends either when the bodies begin to interact and merge or when the distance
between the stars is roughly equal to the last stable orbit allowed. During this period the gravitational signal is expected to be burst-like, with frequencies around 1 kHz for systems composed of
two neutron stars. The characteristic amplitude of the emitted waves (for a system on a circular20

1.4. Sources of Gravitational Waves

sourcecoalescence rate R [Myr−1 MWEG−1 ]

NS-NS
NS-BH
BH-BH

100
3
0.4

Table 1.1: Predicted compact binary coalescence rates for systems of neutron stars (NS-NS), black holes(BH-BH) and black hole-neutron star (NS-BH), from [5].

orbit) at a distance r from the source is [3]1h ∼ M5/3 Ω5/3 ,r(1.22)M ≡ µ3/5 M 2/5(1.23)

withbeing the chirp mass of the system, and Ω being the angular velocity of the orbit. For example,
for a system of neutron stars at r = 10 Mpc with m1 = m2 = 1.4 solar masses and initial orbital
frequency 10 Hz, the amplitude of the emitted gravitational wave is h ∼ 2 · 10−22 .
A compact binary coalescence is the best known among the expected gravitational wave sources.
The detection of signals emitted in the inspiral phase of binary neutron-star systems would be adirect validation of the theory of General Relativity. Furthermore, a direct observation of blackhole coalescences would be a test of General Relativity in a strong field regime.
The black hole-neutron star or neutron star-neutron star collisions have the potential to shine
a light on the equation of state of neutron stars. In these events the neutron star is expected tobe disrupted by the tidal forces due the black hole field, and the emitted gravitational wave should
carry the information about the internal structure of the star. Moreover, neutron star coalescences
are one of the proposed mechanisms for producing the violent shocks that would generate γ-ray
burst events [4], which are flashes of gamma rays associated with extremely energetic explosions
that have been observed in distant galaxies. The detection of a gravitational wave in coincidencewith a γ-ray burst would be a fundamental test of the physical mechanism behind these events.
There are basically two methods to predict the number of binary neutron-star coalescence
events. The empirical method, based on a few observations of compact binary systems which will
coalesce on a timescale comparable with the age of the Universe, and the theoretical method, basedon models of binary star formation and evolution. For black hole-neutron star and black hole-blackhole events instead, it is only possible to use the same population synthesis models applied to theneutron star systems. In any case, the coalescence rates vary over about two orders of magnitude
because of the large uncertainties of the observations or the assumptions of the theoretical models.
The average values of coalescence rates that can be found in the literature [5] are reported in table1.1. Rates are reported in units of number per Myr per MWEG, where MWEG stands for Milky
Way equivalent galaxy.

1.4.2

Pulsars

Pulsars are neutron stars in rapid rotation which emit periodic pulses of electromagnetic radiation,and the period of the pulse sequence is equal to the rotation period of the star. Most of the known

21

Chapter 1. Gravitational Waves

pulsars emit in the radio frequency domain, but few cases of emission in the visible frequency range
or at higher frequencies are also reported (the Crab pulsar, for example). Presently more than
1500 radio pulsars are known, whose period varies from 10−3 s up to few seconds. The period
can be used to classify the pulsars into two classes of objects: the millisecond pulsars with period
P < 20 ms, which are presumably more aged objects, and pulsars with larger period of more recent
formation.
If a mass asymmetry is present at the surface of a pulsar, the star can emit gravitationalradiation at twice the rotation frequency. The star can be modeled as an ellipsoid of semiaxes(a, b, c), and its quadrupole moment can be expressed in terms of a dimensionless parameter , theellipticity (or oblateness), which measures the deviation from axisymmetry. If c is oriented along
the rotation axis, the ellipticity is2(a − b)≡.
(1.24)
a+b
The emitted wave is expected to have the following characteristic amplitude:−24

hc = 4.2 · 10

1 ms
P

!2

kpc
r

!

I3
38
10 kgm2

!


10−6

!
,

(1.25)

where P is the rotation period and I3 is the moment of inertia along the rotation axis. Theoreticalpredictions of  are affected by large uncertainties and can range from 10−7 up to 10−4 [3]. By
assuming that the typical radius of a pulsar is R = 10 km, an ellipticity of 10−7 corresponds to amass asymmetry of R = 1 mm. Despite the weakness of the signal expected from pulsars, theiralmost-periodic nature makes possible to√ integrate over a long period of time T to improve thesignal-to-noise ratio, which increases as T .
As for binary systems, the detection of continuous gravitational wave signals from galactic
pulsars would be a fundamental test of General Relativity. Moreover, since the strength of thesignal is proportional to the ellipticity parameter , the relationship between the gravitational
wave emission frequency and the rate of electromagnetic pulses would help discriminating among
competing mechanisms generating the axial asymmetry.

1.4.3

Supernovae

The general class of stellar collapse includes a rather wide variety of astrophysical objects, whichmight be roughly classified in two distinct categories on the basis of the dynamics of their formation.
According to the current accepted model, Supernovae of type I originate from the gravitational collapse of a white dwarf star in a binary system. The white dwarf accretes matter from the companion
until it exceeds the Chandrasekhar mass limit of 1.44 solar masses, undergoing a gravitational collapse. Following the collapse, the central core of the star reaches a temperature sufficient to trigger
the combustion of heavy elements like Carbon, and the the energy released in the reaction induces
a violent explosion. In most cases, the explosion causes the destruction of the star. On the other
hand, Supernovae of type II should originate from the gravitational collapse of a massive aged star,in the process which leads to the formation of a neutron star.
The emission of gravitational wave radiation from a Supernova explosion happens in the phaseof gravitational collapse of the star, and the amount of the emitted radiation strongly depends onthe degree of asymmetry of the collapse. A collapse which is perfectly spherically symmetric would

22

1.4. Sources of Gravitational Waves

not generate gravitational waves, whereas a highly asymmetric collapse might generate an intenseemission of waves.
It is hard to estimate the expected detection rate of gravitational waves emitted during thegravitational core collapse of massive stars. While the Supernovae rate in our Galaxy and the
local group of galaxies (i. e. up to distances of the order of 300 kpc) is rather low and probably
less than 1 event per two decades (the last observed event was the famous Supernova 1987a), there
might be one Supernova occurring about every 2 years between 3 − 5 Mpc from Earth [6]. A typicalSupernova explosion event should generate a gravitational wave signal in the 102 −103 Hz frequency
band, with a characteristic amplitude that depends on the total amount of energy released under
the form of gravitational waves (∆EGW , expressed in unit of solar masses M ) and is given by [7]hc ∼ 2.7 · 10−20



∆EGW
M c2

1/2 

1kHz 1/2
10M pc.
··fc
r0(1.26)

Recent estimates have shown that the gravitational wave energy emitted by such events might be
as low as ≤ 10−8 M c2 [6].
Although there is presently very little doubt that stellar collapse might generate gravitational
waves, since the modelling of these events is almost unknown it is difficult to accurately estimate
the features of the signal produced. The expected signal is taken to be burst-like, with a durationof few milliseconds. The waveforms of gravitational wave signals from Supernovae are currently
investigated through numerical simulations [6], with results of increasing confidence as well as themodels of gravitational core collapse are constrained by observations and theoretical developments.

1.4.4

Magnetars

Magnetars are thought to be isolated neutron stars featuring extreme magnetic fields, of the orderof ∼ 1015 G. The magnetar model can explain the observed properties of two classes of rare objects,
the soft gamma repeaters and the anomalous X-ray pulsars. These objects are compact sources of
X-rays which sporadically emit short bursts of soft gamma rays of the duration of ∼ 0.1 s. Less
than 20 objects with these properties are presently known.
Magnetars are likely to emit bursts of gravitational radiation through the damped pulsation ofstellar non-radial vibrational modes, excited by the repeated sudden energy flares released by thesestars. Depending on the different type of vibrational mode excited, magnetars may emit gravitational radiation with frequency within the 10−2000 Hz band, i. e. in the detection band of groundbased detectors. However, due to lack of theoretical understanding of the mechanisms triggering
the excitation of vibrational modes, quantitative predictions on the amplitude of gravitational-wave
emission associated with magnetar bursts are quite uncertain [8].

1.4.5Stochastic BackgroundA stochastic background of gravitational waves might have a twofold origin. The incoherent superposition of gravitational waves emitted by sources which are too faint or distant to be resolved
individually is expected to generate an astrophysical background. Sources of this background canbe divided into galactic sources, concentrated in the plane of our Galaxy, and extragalactic sources.
On the other hand, current cosmological models predict the existence of a relic stochastic
background, generated during the early phase of the Universe history [9]. Such a cosmological

23

Chapter 1. Gravitational Waves

background is expected to be isotropic, stationary and unpolarized. Its spectrum is expected tofeature the imprinting of early cosmology, carrying the information about the state of the Universe
at time scales t ∼ 10−20 − 10−26 s, or equivalently at temperature scales T ∼ 107 − 1010 GeV.
The intensity of the gravitational wave background is usually characterized through the dimensionless quantity
1 dρGWΩGW (f ) =,(1.27)
ρc d(log f )which represent the normalized energy density of the gravitational radiation in the Universe perlogarithmic interval of frequency. ρGW is the energy density of the background, f is the spectrum
frequency and ρc is the critical energy density value for a closed Universe. The critical density isdefined as
3H02
ρc =
,
(1.28)
8πG
H0 being the present value of the Hubble constant, parametrized as H0 = h0 · 100 km/s/Mpc.
The estimates for the background intensity are affected by large uncertainties. The strictest
limits on the ΩGW spectrum (for the cosmological background only) come from Big Bang nucleosynthesis model and observations [9],R∞ΩGW = 0 h20 ΩGW (f )d(log f ) < 5.6 · 10−6 (Nν − 3) ,(1.29)
where Nν is the effective number of neutrino species at the time of nucleosynthesis, and from theobservation of the cosmic microwave background anisotropies (CMB) measured by COBE [9]:h20 ΩGW (f ) < 7 · 10−11 H0 /f

2, 3 · 10−18 Hz < f < 10−16 Hz .

(1.30)

We will see in chapter 3 that at higher frequencies, in the detection band of the ground-based
gravitational-wave detectors of first generation (ranging from about 10 Hz to about 1 kHz), thislimits (taking a flat energy density spectrum) have now been beaten by LIGO measurements [10].

1.4.6

Multimessenger Astrophysics

A source of gravitational waves is reasonably likely to radiate in other ways too, and a detectablegravitational wave source should release a large amount of energy and/or be relatively close. Multimessenger astrophysics is the observation of the same astrophysical event or system made bydifferent (electromagnetic, particle, gravitational-wave) observatories. Electromagnetic or particle signals may provide more information about the gravitational wave source sky position, hostgalaxy type, distance, emission characteristics and related astrophysical processes. On the other
hand, additional coincident observations would increase the confidence in a putative gravitational
wave detection candidate. Furthermore, external triggers would allow the coincidence search forweaker gravitational wave signals of lower signal-to-noise ratio in the detector data.
A typical event that would be likely better understood through the complementary observationof a gravitational wave signal are γ-ray bursts. Long γ-ray bursts (or at least those with a softer γ
spectrum) might be associated with core-collapse Supernovae [11], whereas short bursts could be
related with coalescing binary neutron-star or neutron star-black hole systems [4].
If observed by a network of gravitational-wave detectors, a compact binary coalescence mightallow to reconstruct the distance of the source (from the observed waveform amplitude) and the24

1.5. Indirect Measurement of Gravitational Radiation

masses of the objects of the systems (from the time evolution of the waveform). If the event could
be associated with an astrophysical object, for example through a complementary electromagnetic
follow-up survey, it would serve as distance calibration, thus potentially competing with otherobservational methods for determining the cosmological Hubble constant H(t). As Supernovae oftype Ia are used as standard candles to determine the luminosity distance of the host galaxies,
the coalescence of binary systems might be used as standard sirens to estimate the distance ofastrophysical objects [12].

1.5Indirect Measurement of Gravitational Radiation

Although so far gravitational waves remains undetected, the discovery and the observation in 1974of the pulsar binary system PSR1913+16 by R. Hulse and J. Taylor [13] provided for the first timethe indirect evidence of the existence of gravitational radiation. The system is formed by a pulsar
and a dark companion, bound in an orbit of period P = 7h 45m .
General Relativity predicts such a system to lose energy and angular momentum via emission ofgravitational waves. This loss turns into a decrease of the orbital period of the pulsar, dP/dt < 0,which has been measured by Hulse and Taylor and compared to the value predicted by the theory.
The agreement between the measurement and the theory demonstrated indirectly the existence ofgravitational waves, and for this discovery Hulse and Taylor were granted the Nobel Prize in 1993.
The shift of the orbital period of PSR1913+16 has been measured since the system discovery
for more than 30 years, and is now known with great accuracy [14]:dP= −(2.4184 + / − 0.0009) · 10−12dt(1.31)The measured curve of the cumulated shift against time is shown on fig.1.2, together with thetheoretical curve predicted by General Relativity.
Since the discovery of PSR1913+16, other pulsars in binary systems of neutron stars have been
detected and observed [15]. Some of these binaries are PSR B1534+12 [16], PSR J1756-2251 [17]and PSR J0737-3039A/B [18], which is formed by two pulsars orbiting around each other. All ofthese binaries are relativistic systems which show a decrease of their orbital period, like the one
observed for 1913+16.

25

Chapter 1. Gravitational Waves

Figure 1.2: Orbital decay of PSR B1913+16, from [14]. The data points indicate the observed change inthe epoch of periastron with date while the parabola illustrates the theoretically expected change in epoch
for a system emitting gravitational radiation, according to General Relativity.

26

27

Chapter 2

Interferometric Detectors ofGravitational Waves
This chapter introduces the fundamental principles of interferometric detection of gravitational
waves, with the aim of describing all the main components of a typical gravitational-wave interferometer. To explain the choice of interferometry as the basic principle of detection, we will show insection 2.1 the effect of the passage of a gravitational wave on a set of free-falling test particles,
stressing the effect of the resulting differential strain of space-time. The Michelson interferometerwith suspended mirrors, described in section 2.2, is the most appropriate instrument to measure
this strain, since it has the optimal optical configuration to detect differential effects in its arms.
We will see in section 2.3 that mirrors must be suspended for a twofold reason: the suspended
mirrors of the interferometer can be considered (with a reasonable approximation) free-falling test
masses, furthermore they have to be isolated from seismic noise. Vibrations are a fundamental
noise which affects gravitational-wave interferometers, as well as thermal noise (due to the thermal
excitation of the particles in the mirror and in the suspensions) and photon counting noise. Together
with technical noises of the laser, fundamental noises limit the smallest signal detectable by theinterferometer. All these noise sources are described in section 2.4.
To increase the sensitivity of the interferometer, optical resonators can be used. Since thestrain sensitivity of the interferometer is directly proportional to the optical path length in its
arms, kilometer-scale Fabry-Perot arm cavities are used to increase the effective length traveledby the laser light. In order to decrease the photon noise, the power stored in the detector can beincreased with a power recycling cavity coupled to the rest of the interferometer. The laser noise at
the interferometer input can be filtered with an input mode-cleaner cavity, whereas the contrast at
the detector output can be increased by using an output mode-cleaner cavity. The basic working
principle of optical resonators will be illustrated in section 2.6.

2.1

Geodesic Deviation

From the theory of General Relativity we know that the motion of a single probe particle is notaffected by the field perturbation of a gravitational wave. To detect a gravitational wave, a system
of at least two test particles must be used. If δxλ is the distance between the two particles (λ =0, 1, 2, 3), the effect of the passage of a gravitational wave on such a system is described by the28

2.1. Geodesic Deviation

t=0

t=T/4

t=T/2

t=3T/4t=T

Figure 2.1: Effect of + (top row) and × (bottom row) polarized gravitational waves propagating along thez axis on a ring of free falling masses in the x − y plane. The strain is shown every quarter of wave period T .

equation of geodesic deviation,βνd2 δxλλ dx dx
=
R
δxµ ,
νβµ
dτ 2
dτ dτ

(2.1)

which shows that the relative acceleration d2 δxλ /dτ 2 of particles moving along nearby geodesics
λ . It can be demonstrated that if the gravitational wave hdepends on the curvature tensor Rνβµµν
perturbs a flat space-time metric, i. e. the metric tensor can be written as gµν = ηµν + hµν , eq.(2.1)
simplifies tod2 λ 1 λi ∂ 2 him m
δx = η
δx ,
(2.2)
dt2
2
∂t2
where hµν is the wave perturbation in the TT gauge (i, m = 1, 2, 3). The solution to this equationis1
δxλ = δxλ0 + η λi hik δxk0 ,(2.3)
2which is the sum of an unperturbed term δxλ0 and a term which depends on the perturbation of
the gravitational wave, η λi hik δxk0 /2. Thus the relative distance of the two particles varies in time
according to the oscillatory perturbation of the wave.
The result of eq.(2.3) can be generalized to more complex systems of bodies. For example,
fig.2.1 shows the effect of a polarized gravitational wave passing through a system of particles
arranged on a circumference. It is very clear from this figure the differential effect of the wave
perturbation in the (x, y) plane perpendicular to the propagation axis of the wave. This effect isthe key principle at the basis of gravitational wave detection with interferometers based on theMichelson configuration.

29

Chapter 2. Interferometric Detectors of Gravitational Waves

y

My
7

6
x

Mx

5

!!"

8
1

2

laser
4

3

BS

!#$%
PD
Figure 2.2: Optical scheme of the Michelson interferometer: the laser, the beamsplitter (BS), the arm
mirrors (Mx and My ), the photodiode (PD) and the field notation (ψi ) are shown.

2.2

The Michelson Inteferometer

The basic optical configuration of all interferometric gravitational wave detectors is the Michelsoninterferometer [19, 20]. In its simplest form – depicted in fig.2.2 – the Michelson interferometerconsists of a source of light, a partially reflecting mirror called beamsplitter, and two end mirrors
located at a distance l from the beamsplitter along two orthogonal directions, forming two arms.
The light coming from the source (which is usually a laser) propagates to the beamsplitter, where
it is divided in two beams. One beam is transmitted through the beamsplitter, one is reflected.
Both beams propagate in the arms towards the end mirrors, where they are reflected backwards torecombine at the beamsplitter. A photodiode, placed at the output of the interferometer, converts
the light of the interference fringe in a current signal.
The function of the Michelson interferometer is to compare the amount of time the light beams
take to complete their paths in the two arms, or equivalently, to measure the phase difference
acquired by the light beams in their round trip in the arms. By computing the electromagnetic
fields inside the interferometer, it is possible to demonstrate that the output signal depends on
the phase difference of the two light beams interfering at the beamsplitter. Let ψin be the input
electromagnetic field, which for simplicity will be a plane wave. Each mirror is characterized byan amplitude reflectivity ri and an amplitude transmissivity ti (i = 1, 2), defined by the ratio ofreflected electric field to the incident field at the surface. For the propagation in the instrument,
the following conventions are used:• the field propagated over a distance L is ψ = ψin e−ikL• the field reflected by a mirror of reflectivity r is ψ = −rψin30

2.2. The Michelson Inteferometer• the field transmitted by a mirror of transmissivity t is ψ = tψin• the total energy is conserved: r2 + t2 + Lopt = R + T + Lopt = 1,
where λ is the light wavelength, k = 2π/λ is the wave number, Lopt is the amplitude loss coefficientof mirrors. The electromagnetic fields inside the interferometer are:
ψ1 = tBS ψinψ2 = exp(−ikL1 )ψ1
ψ3 = −r1 ψ2
ψ4 = exp(−ikL1 )ψ3

ψ5 = −rBS ψinψ6 = exp(−ikL2 )ψ5ψ7 = −r2 ψ6
ψ8 = exp(−ikL2 )ψ7 ,where ri and ti are the reflectivity and the transmissivity of the i-th mirror, respectively. At theoutput, the field is
ψOU T= −rBS ψ4 + tBS ψ8 == rBS tBS ψin (r1 e−i2ωL1 /c + r2 e−i2ωL2 /c ) ,(2.4)

where L1 and L2 are the lengths of the two arms, ω/2π is the frequency of the monochromatic lightsource (which is usually a laser), c is the speed of light. Thus the amplitude of the light leaving
the interferometer depends on the difference in phase φ = 2ω(L2 − L1 )/c cumulated by the lightbeams traveling in the two arms. The power on the output photodiode is
POU T = PIN RBS TBS (R1 + R2 + 2r1 r2 cos φ) ,(2.5)

with PIN = |ψIN |2 and POU T = |ψOU T |2 . The maximum output power of the interferometer is
PM AX = POU T (φ = 0), whereas the minimum output power is Pmin = POU T (φ = π). The output
power can thus be rewritten as
POU T = PIN RBS TBS (R1 + R2 )(1 + C cos φ) ,
where
C=

PM AX − Pmin
2r1 r2=PM AX + Pmin
R1 + R2

(2.6)

(2.7)

is the contrast of the interferometer. From eq.(2.6) it is clear that any change in the lengths l1
and l2 causes a change in the photodiode output current, so the Michelson interferometer acts as
a transducer from travel time difference to output optical power.
Assuming that the mirrors of the interferometer behave approximately like free-falling test
masses (we will see how in the next section), the detector can be used to measure the phase changeof light induced by the passage of a gravitational wave. The wave will change the optical path length
traveled by the light in the interferometer arms. Because of the differential nature of the strain
induced by a gravitational wave, shown on fig.2.1, the optical path variation will have opposite signin the two arms of the interferometer. Thereby the passage of the wave will modulate in time theinterference fringe at the output of the detector.
The phase shift due to an incident gravitational wave can be computed by making some simple
assumptions. Let the interferometer arms be equally long and oriented along the x and y axis,
with a symmetric beamsplitter (RBS = TBS ) at the origin. Then, since the paths taken by lightin each arm will have only either dx or dy different from zero, we will consider only the 11 and31

Chapter 2. Interferometric Detectors of Gravitational Waves

22 components of the metric. The light beam traveling in the x arm obeys the condition of nullgeodesic (i. e. the interval between two neighboring space-time events linked by a light beam isalways null),
ds2 ≡ gµν dxµ dxν
= (ηµν + hµν )dxµ dxν= −c2 dt2 + 1 + h11 exp[i(ΩGW t − ~kGW · ~x)] dx2 = 0 ,(2.8)

where ΩGW /2π is the wave frequency and ~kGW is the wave vector. The time the light takes to travelfrom the beamsplitter to the x end mirror at a distance L is
Z τ1
Z
Z

1 Lp
1 L
1
dt =
τ1 =
1 + h11 dx '1 + h11 exp[i(ΩGW t − ~kGW · ~x)] dx ,
(2.9)
c 0
c 0
2
0
and for the return trip a similar equation holds:
Z τ2
Z

1 0
1
τ2 =
dt = −
1 + h11 exp[i(ΩGW t − ~kGW · ~x) ]dx .
c L
2
τ1

(2.10)

Thereby the total round trip time in the x arm is
Z
Z
2L
1 L
1 0
∆τx =
+
h11 exp[i(ΩGW t − ~kGW · ~x)]dx −h11 exp[i(ΩGW t − ~kGW · ~x)]dx , (2.11)
c
2c 0
2c L
and an analogous expression can be written for the light round-trip time ∆τy in the y arm (recallingthat h22 = −h11 ):
Z
Z
2L
1 L
1 0
~
∆τy =
−
h11 exp[i(ΩGW t − kGW · ~x)]dy +h11 exp[i(ΩGW t − ~kGW · ~x)]dy . (2.12)
c
2c 0
2c L
Thus, due to the quadrupolar nature of gravitational of the impinging gravitational wave, there
will be equal and opposite perturbations to the light travel time in the two arms. For sake ofsimplicity, consider a plane wave propagating along the direction z perpendicular to the plane z0of the interferometer, and let the wave be + polarized with amplitude h+ ≡ h. Then eq.(2.11) and(2.12) yield the total time differenceLi
h h i2ΩGW L/c
2L sin ΩGWc∆τ (ΩGW ) = ∆τx − ∆τy =eiΩGW L/c ,(2.13)
e
−1 =h
ΩGW L
iΩGW
c
c
corresponding to the phase difference∆φ(ΩGW ) =2πc∆τ (ΩGW )λ(2.14)of the output field. Eq.(2.14) represents the frequency response of the interferometer, and its modulus is shown in fig.2.3. This figure puts in evidence the 1/f decay of the response for frequenciesΩGW /2π greater than c/2L, and the existence of frequencies for which no phase change is generated. This happens when ΩGW /2π = nc/2L with n ∈ N, i.e. when the phase variation period of
the gravitational wave is equal to the round trip time of light in the arms.

32

2.2. The Michelson Inteferometer

13

10

12

∆Φ / h [rad]

10

11

10

10

10

9

10

8

10

0

10

1

10

2

10
frequency [Hz]

3

10

4

10

Figure 2.3: Magnitude of the frequency response of a Michelson interferometer with 1-km long arms (green
curve) and 100-km long arms (blue curve), compared to that of a Michelson interferometer with 1-km long
Fabry-Perot cavities in the arms (mirror reflectivities: R1 = 88% and R2 = 1, red curve).

The response of the instrument is optimal when the frequency of the incident wave is smaller
than c/2L, that is
ΩGW
c
<<
.
(2.15)
2π
2L
When this condition is verified, the phase of the incident wave can be considered approximately
constant during the time any given wavefront is traveling in the interferometer, and the phasedifference of the light at the beamsplitter simplifies to
∆φ =

4π
hL .
λ

(2.16)

Eq.(2.16) clearly indicates that the response of the interferometer is directly proportional to thelength of its arms. For gravitational waves of given amplitude, the sensitivity of the interferometeris higher as the arms are longer.
In the case the wave has both polarizations and its arrival direction is not perpendicular to theplane of the interferometer, the response of the detector can be generalized to [2]!
4πL 1∆φ = h(1 + cos2 Θ) cos 2Φ cos 2Ψ − cos Θ sin 2Φ sin 2Ψ ,(2.17)
λ2where (Θ, Φ, Ψ) are the angles between the wave vector and the frame containing the detector.
Eq.(2.17) shows that the response, or antenna pattern, is function of the arrival angle. The pattern,shown on fig.2.4, is non-directional with nulls along the bisector of the arm axes. In these directions
the arm length changes are always equal in the two arms, yielding no response.
On one hand the broad angular response is an advantage, since the detector can survey almost
the whole sky. On the other hand, it is a disadvantage since such a response forbids the identification

33

Chapter 2. Interferometric Detectors of Gravitational Waves

z
z
Ψ
Θ

k

y
Φ

y
x

x

Figure 2.4: Antenna pattern of an interferometric gravitational wave detector. The arms of the interferometer are oriented along the x and y axes.

of the wave arrival direction, preventing the pointing of an electromagnetic receiver to collect any
potential complementary signal from the source. As we will see in chapter 3, the arrival direction
might be reconstructed by using the signal of the wave recorded by a network of detectors.

2.3

Free-falling Mirrors

The mirrors of a gravitational wave interferometer must be suspended like pendulums, for two
fundamental reasons: (i) suspended mirrors behave like free-falling masses in the frequency range
above the pendulum resonance, and (ii) as it will be shown in section 2.4, the suspension isolates
the optics from ground vibrations which are cause of noise.
Let us consider a mirror of mass m suspended to a massless wire of length l, moving withsmall oscillation amplitude and with no dissipation of energy. If we indicate by x(t) the mirror
position, by x0 (t) the position of the suspension point and by F (t) any force acting on the mirror,
the equation of motion is
F (t) − mω02 [x(t) − x0 (t)] = md2 x(t),dt2(2.18)

with ω02 being the resonant angular frequency. In the angular frequency domain, the solution toeq.(2.18) writes
ω 2 x0 (ω) + F (ω)/mx(ω) = 0.
(2.19)
ω02 − ω 2
From eq.(2.19) one can easily compute the transfer function
G(ω) =x(ω),
x0 (ω)

34(2.20)

2.4. Sensitivity and Noise Sources

which relates the input function x0 (ω) of the system to the output function x(ω). In absence of
external forces, F (ω) = 0, the transfer function isG(ω) =

ω02,
ω02 − ω 2

(2.21)

meaning that• below the resonance (ω << ω0 ) the motion of the suspension point (due for example toground vibrations) is entirely transmitted to the mirror: x(ω) ≈ x0 (ω)• above the resonance (ω >> ω0 ) the displacement of the mirror is attenuated likeG(ω) ≈

ω02.
ω2(2.22)

From eq.(2.19) we can see that the motion of the mirror for frequencies ω >> ω0 is described bymω 2 x(ω) + F (ω) ≈ 0 ,(2.23)

which is the equation of motion of a free mass. Under this condition, the effect of the gravitational
wave impinging on the detector is effectively described by eq.(2.8).

2.4

Sensitivity and Noise Sources

The output signal of a gravitational wave interferometer is affected by noise, which consists in random fluctuations of the measured output power. These fluctuations are associated to perturbationsof the mirror position and to noise sources that affect the optical read-out system (composed by
the laser and the photodiode). The fundamental noise sources that affect the detector are• the fluctuation of the number of photons impinging on the detection photodiode (shot noise)• the fluctuation of the laser radiation pressure on the mirrors• the seismic excitation of the ground• the stochastic fluctuations of the gravitational field around the detector, which couple directly
to the mirrors• the thermal excitation of the mirrors and the suspension wires.
The technical noise sources, instead, are• the amplitude, frequency and position (jitter) fluctuations of the laser beam
• fluctuations of the refraction index along the optical path of the laser• parasite diffused light that couples into the main laser beam generating phase noise• electronic noise in the main interferometer read-out and control systems

35

Chapter 2. Interferometric Detectors of Gravitational Waves

The common effect of these noise sources is the generation of a random time-dependent phaseshift δφ(t) which mimics the effect of an incident gravitational wave and limits the detection ofastrophysical signals.
The noise of the detector can be characterized by means of its power spectral density
2
S(f ) = lim
T →∞ T

2

Z T
A(t)e

−2iπf tdt,

(2.24)−T

where A(t) represents the time evolution of the measured physical quantity. Form eq.(2.24) one
can compute the linear spectral densityps̃(f ) = S(f ) ,(2.25)√
measured in [A]/ Hz, and the mean amplitude of noise fluctuationssZ∞

S(f )df .

Arms =(2.26)

0In order to quantify the noise in terms of the measurable gravitational wave amplitude, we can impose A(t) ≡ h(t) so that we introduce the equivalent strain noise spectral density h̃(f ) (defined only
for positive frequencies). For example, if x̃(f ) is the linear spectral density of mirror displacement
noise, the equivalent-strain noise isx̃h̃(f ) =,(2.27)
Lwith L being the interferometer arm length. In the following we will describe in detail the noise
sources mentioned above, giving for each of these the corresponding equivalent-strain noise h̃(f ).

2.4.1

Shot NoiseThe shot noise is a fundamental noise due to the quantum nature of light. Measuring the opticalpower at the output of the detector is equivalent to determining the number of photons impinging
on the photodiode during the measurement interval, which follows a Poisson distribution. Let us
recall the expression of the output power from eq.(2.6):
POU T = P0 (1 + C cos φ) .

(2.28)

Thus the average number of photons of energy ~ω detected in the time ∆t isN̄ =ηP0 ∆t(1 + C cos φ) ,~ω

(2.29)

where η is the photodiode quantum efficiency. The power fluctuation associated to the varianceσN̄ of the number of photons isr
ηP0 ~ω
δPshot =(1 + C cos φ) .
(2.30)∆t36

2.4. Sensitivity and Noise Sources

The passage of a gravitational wave yields an output power (to first order approximation) equal to
POU T (φ + δφGW ) ' POU T (φ) +dP
δφGW = POU T (φ) + δPGW ,dφ(2.31)

where δφGW is the phase perturbation andδPGW = −P0
CδφGW sin φ2(2.32)

is the power change due to the passage of the wave. Eq.(2.32) shows that the slope dP/dφ ismaximum when sin φ = 1, in the mid-fringe condition. If, however, we consider the signal to noise
ratio due to the presence of shot noise,rηP0 ∆tC sin φS|δPGW |
p==
δφGW ,(2.33)
N
δPshot~ω(1 + C cos φ)this ratio is maximum when√

1 − C2,(2.34)C
close to the dark fringe condition (destructive interference). We will see in section 2.6 that thethe dark fringe condition is also necessary to realize the recycling of the power circulating in thedetector. On the dark fringe, assuming C ' 1, the minimum detectable phase shift (S/N = 1) due
to shot noise iss~ωδφshot =,(2.35)
ηP0 ∆tcos φ =

−1 +which corresponds to an equivalent noise spectral density limit ofs√λ~ωh̃shot =[1/ Hz] .
2πL ηP0(2.36)The shot noise spectrum is flat, i. e. there is no preferred frequency scale for this noise: the arrivalof each photon is independent of the arrival of each of the others. For L = 1 km, λ = 1.064 µm
and P0 = 10 W the shot noise equivalent-strain is√h̃shot ∼ 10−20 [1/ Hz] .
(2.37)In a detector with Fabry-Perot arm cavities, as described in section 2.6, the spectrum of shot noiseis flat only below the frequency cut-off of the cavity. Eq.(2.36) indicates that shot noise could belowered by increasing the interferometer arm length, together with the laser power.

2.4.2

Radiation Pressure Noise and Quantum Limit

The power fluctuations associated to the quantum nature of light generate a radiation pressure
noise, which make the mirrors of mass m move with a frequency spectrum [2]r1~P0x̃(f ) =.
(2.38)
2mf
8π 3 cλ37eq. strain noise density [sqrt(Hz)]

Chapter 2. Interferometric Detectors of Gravitational Waves

−21

10

−22

10

−23

10

−24

10

−25

10

0

10

1

10

2

10
frequency [Hz]3

10

Figure 2.5: Equivalent strain spectral density of read-out noise (black curve), shot noise (blue curve) andradiation pressure noise (red curve) for L = 1 km, m = 20 kg, P0 = 1 MW.

The corresponding equivalent strain noise spectrum is
r
~P0
1
,
h̃rp =2mLf
2π 3 cλ(2.39)

which shows that radiation pressure noise increases as the power in the detector increases and is
proportional to 1/f 2 . Thus radiation pressure noise and shot noise can be considered as two aspectsof a unique optical readout noise, given (in the interferometer configuration under discussion) by
the quadrature sumqh̃or (f ) =h̃2shot + h̃2rp .

(2.40)

At low frequencies the radiation pressure term will dominate, whereas at high frequencies the shotnoise is more important, as shown by fig.2.5. We could improve the high frequency sensitivity byincreasing P0 , at the expense of increased noise at low frequency. At any given frequency f , there
is a minimum noise spectral density: this occurs when the power P0 is chosen to have the value theoptimal value that yields h̃shot = h̃rp . This optimal power is
Popt = πcλmf 2 ,(2.41)

which is typically quite large. For a set of parameters with m = 20 kg, λ = 1.064 µm and f = 100
Hz, the optimal power is 200 MW. This value is far higher than the power stored in present
detectors, which is of the order of 30 kW. Second and third generation detectors are expected tostore from about 1 to few megawatts, respectively.
In analogy to to the Heisenberg uncertainty principle, we can state that shot and radiationpressure noise set a quantum limit to the sensitivity of the interferometer. Shot noise represents
the minimum uncertainty related to the measurement of mirror position, while radiation pressure
noise causes the mirrors move in a random way. In this sense, these two sources of noise are38

2.4. Sensitivity and Noise Sources

conjugate phenomena. We can compute the quantum limit by replacing eq.(2.41) into eq.(2.40),
yieldingr~
1h̃QL =,(2.42)
πf L mwhich is the locus of the minima of h̃or (f ) at any given frequency. This limit does not depend on
P0 or any feature of the optical readout scheme.

2.4.3

Seismic Noise

Seismic noise is due to the vibrations of the ground, which couple to the mirrors via the suspension
system. There is no universal law describing this noise. Ground vibrations are caused by seismic andocean activity, wind, human activity. Thus seismic noise intensity can vary of orders of magnitude
from place to place, and in the same place from day to day. A general 1/f 2 behavior has been
observed worldwide in the 10 mHz−10 kHz region, following the empirical law [21]x̃s (f ) = 10−7 m Hz−1/2

 1Hz 2f,

(2.43)

which corresponds to an equivalent strain noise spectrum of x̃s (f )/L. To get rid of seismic noise,mirrors have to be isolated through a suspension system of suitable filters. For example, if N
pendulums were cascaded in a single chain, the transfer function of the system would be (recallingeq.(2.20))
N
Y
ωk2
x(ω)
=
,
(2.44)x0 (ω)
ωk2 − ω 2
k=1

where ωk2 /2π are the N resonant frequencies of the system. For ω >> ωk ∀k, the attenuation is
given by
N
Y
ωk2
x(ω)
'
,
(2.45)x0 (ω)
ω2
k=1

which shows that in principle an arbitrarily steep low pass filter for ground vibrations might be built,
by cascading a large number of oscillators. The filter cut-off frequency depends on the resonant
frequencies ωk of the system. The lower the frequencies, the wider the range where vibrations areattenuated. Nevertheless, due to technical limitations (for example, the number of pendulums andtheir length are limited by the space available for suspension), seismic noise below ∼ 1 Hz cannot
be attenuated and remains one of the main limitations to the sensitivity of ground based detectors.

2.4.4

Gravity Gradient Noise

Gravity gradient noise (also called Newtonian noise) is caused by fluctuations of the local gravitational field. These fluctuations are induced by the time variation of mass density distribution
around the mirrors, because of seismic waves in the ground [22] and sound waves and temperature
perturbations in air [23]. For instance, the equivalent noise strain due to seismic gravity gradientnoise is [24]x̃s (f )h̃GG (f ) ∼ 1.2 · 10−11 2 ,(2.46)
f39

Chapter 2. Interferometric Detectors of Gravitational Waves

where x̃s (f ) is the local seismic noise spectrum. The amplitude of noise due to atmospheric fluctuation is estimated to be of similar size to the seismic ground motion [23]. Gravity gradient noise
is a fundamental limitation for the sensitivity of ground-based detectors at frequencies below ∼ 10
Hz, since gravity field fluctuations bypass the suspension system and couple directly to the mirrors.
A suitable solution to decrease this noise might be the identification of a quiet site for the
installation of the detector, like an underground site where atmospheric perturbations are relatively
far from the mirrors, the environmental conditions are stable and seismic waves are attenuated [25].
Another option to mitigate gravity gradient noise might be the subtraction of the estimated noise
contribution itself, from the detector output [26, 25]. Since this noise cannot be measured directlywith the required precision, except than with a gravitational-wave detector, the noise contribution
should be predicted by monitoring the seismic field around the mirrors and feeding the data toa model of the sources. Both methods of mitigation might be implemented in third-generation
detectors, which will be presented in chapter 3.

2.4.5

Thermal Noise

Thermal noise consists in random fluctuations of a physical variable of a system which is in thermal
equilibrium with its surrounding. In gravitational-wave interferometers, thermal noise shows up
as fluctuations of the position of mirrors. According to the fluctuation-dissipation theorem, theamplitude of these fluctuations is related to the dissipative mechanisms of the system, which drive
the irreversible dissipation of energy into heat [27]. The power spectral density of mirror thermal
displacement can be calculated starting from the knowledge of the admittance Y (f ) of the system,
which characterizes the irreversible response of the system to applied forces:hikB T
ST N (f, T ) = 2 2 Re Y (f ) ,(2.47)
π fwhere κB is the Boltzmann constant and T is temperature of the system. This is equivalent to saythat the system is subject to a generalized thermal force with power spectrumhi|FT N (f, T )|2 = 4kB T Re Z(f ) ,(2.48)
with Z(f ) ≡ Y −1 (f ) being the generalized impedance of the system.
Since dissipations take place in mirrors and in their suspension wires, the global spectral densityof thermal noise is given by the sum of the spectral densities of mirror thermal noise and suspension
thermal noise. Suspension thermal noise is in turn given by the sum of pendulum thermal noise,
associated to the pendular suspension of mirrors, and of the thermal noise related to the transverse
vibrational modes of wires. Mirror thermal noise is given by the sum of substrate thermal noiseand coating thermal noise as well.
As we will see hereafter and later on, thermal noise is the dominant source of noise in thecentral region of the detection band of present interferometers, and is expected to severely affectthe sensitivity of future detectors. To give a global picture of all thermal noise contributions, wewill introduce hereafter the power spectral densities of suspension and mirror thermal noise. To
follow the notation used in the literature, we will introduce displacement power spectral densities
S(f ) instead of usual linear strain spectral densities h̃(f ). If L is the length of the interferometer
arms, the relation between the two is given byp(2.49)h̃(f ) = S(f )/L = x̃(f )/L .

40

2.4. Sensitivity and Noise Sources

Suspension thermal noise
Two main contributions of thermal noise originate in the suspension system. The first is pendulum thermal noise, where thermally induced motion associated with the pendulum mode of thesuspension directly affects the displacement of the test mass. In order to calculate the thermal
displacement due to pendular suspensions, the system can be treated as a damped harmonic oscillator with a frequency-dependent damping factor Φ(f ). If m is the mirror mass and k is the elastic
constant of the suspension, according to eq.(2.47) the pendulum thermal noise spectral density is
[28]
kB T
f02 Φ(f, T ) .
S(f, T ) = 3(2.50)
2π mf (f 2 − f02 )2 + f 4 Φ2 (f, T )The pendulum damping factor is [28]s
Φ(f, T ) =nY I
φ(f, T ) ,mgL2(2.51)

where φ(f, T ) is the loss of the suspension material, Y is its Young modulus, I is the cross-sectional
moment of inertia of wires, L is the wire length, n is the number of wires of the suspension. The
loss term itself consists of three main contributions:
φ(f, T ) = φbulk + φT E (f, T ) + φS .

(2.52)The bulk loss φbulk is the internal intrinsic loss of the material, φT E (f, T ) is the thermo-elastic
contribution, φS is a surface loss term. The thermo-elastic loss is due to the coupling between thetemperature field and the local strain of the suspension wires through the linear expansion coefficientα(T ). Because of the strain field, compressed region becomes hotter and extended region becomescooler. Energy is dissipated when irreversible heat flow driven by the induced temperature gradientoccurs, resulting in thermo-elastic damping. The frequency dependence of φT E (f, T ) is [29]φT E (f, T ) =Y α2 T
2πf τd,C 1 + (2πf τd )2(2.53)

where C is the heat capacity per unit volume and τd is the thermal diffusion characteristic timein the material. The surface loss φS depends on the surface-to-volume ratio of the suspension,according to the equation [30]S
φS = µ lS ,(2.54)
V
where µ is a geometrical factor of the wire and lS is the typical loss length of the material.
The second type of suspension thermal noise arises from the transverse vibrational modes ψnof the suspension wires under a tension mg, named violin modes. This type of noise can cause a
residual longitudinal displacement of the test mass at the end of the pendulum chain. The spectral
density of violin modes is [28]∞kB T Xψn2 φn (f, T )fn2
i ,S(f, T ) =(2π)3f (f 2 − f 2 )2 + φ2 (f, T )f 4
n=1

n

41

n

n

(2.55)

Chapter 2. Interferometric Detectors of Gravitational Waves

with mode amplitudes
ψn2 =

2ρL 1
π 2 m 2 n2(2.56)

and resonant frequencies

mg 2
n ,
(2.57)
4ρL2
where ρ is the linear mass density of the suspensions.
For an interferometer with mirrors of ∼ 20 kg each, suspended to 0.7-m long steel wires [31, 32](which is the typical configuration of present detectors), the pendulum mode is the dominant
contribution of suspension thermal noise from few Hz up to 50 Hz, where it is equal to ∼ 5 ·10−23 Hz−1/2 . The thermal noise of violin modes is larger for frequencies higher than 300 Hz,
which is the frequency of the lowest violin mode. However, beyond 500 Hz, thermal noise of violin
modes is below 10−24 Hz−1/2 (except for the narrow peak of resonances), so that other noise sources
like shot noise are much larger in this band. In summary, compared to all the other noise sources,
suspension thermal noise is typically the dominating source of noise around 10 − 20 Hz.
fn2 =Mirror thermal noise
The fluctuation-dissipation theorem provides a general approach for the computation of the amplitude spectral density of mirror thermal noise, in the case of a generic read-out beam transverse
intensity distribution I(x, y) [33, 34]. Let uz (t, x, y) be the time-dependent displacement of themirror surface along the optical axis of the beam. The equivalent displacement of the mirror isZZ
x(t) =uz (t, x, y)I(x, y) dxdy ,(2.58)
so the measurement of the mirror position is the average of the displacement field of the surface,weighted by the intensity distribution of the read-out beam. If F (t) = F0 ei2πf t is a generalized
oscillating force applied to the mirror, the response formula of the system is
Y (f ) = i2πfx(f ),F (f )(2.59)

where x(f ) and F (f ) are the displacement and the force in the Fourier domain. The oscillating
displacement caused by the generalized force isuz (f, x, y) = (1 − iφ(f ))uz (x, y) ,where φ(f ) is the dissipation in the system, so that the admittance Y (f ) is
RR
(1 − iφ(f ))uz (x, y)I(x, y) dxdy.
Y (f ) = i2πfF(2.60)(2.61)The spectral density of mirror thermal noise is then
4kB T Wφ(f ) ,
πf F 2(2.62)uz (x, y)P (x, y)dxdy(2.63)ST N (f, T ) =
where

1
W =
2

ZZ

42

2.4. Sensitivity and Noise Sources

is the elastic energy stored in the mirror, which is stressed by a generalized pressure distribution
P (x, y) = |F |I(x, y) that has the same profile of the read-out beam. By defining the strain energy
U ≡ W/F 2 , the spectral density of eq.(2.62) writes
ST N (f, T ) =4kB T
U φ(f ) .
πf(2.64)

Thus the mirror thermal noise can finally be estimated by computing the mirror strain energy U forthe desired read-out beam profile. Hereafter we will deal the case of Gaussian beams, whereas inchapter 7 we will see what is the mirror thermal noise due to some particular non-Gaussian beamprofiles, focusing our attention especially on Laguerre-Gauss modes.
Mirror thermal noise is the result of various contributions, due to different physical mechanisms that drive the dissipation processes in the optics. The dissipations are in general locatedin the coated surface as well as in the mirror substrate. When the dissipation is due to the internal damping of spatially distributed sources of mechanical friction, thermal fluctuations can
be classified as Brownian thermal noise. According to the fluctuation-dissipation theorem, thefrequency-dependent spectral density of substrate and coating brownian noise is [35]S(f, T ) =2kB T (1 − σ 2 ) Φ,
π 3/2 Y w f(2.65)

where w is the beam size on the mirror, σ and Y are the Poisson’s ratio and the Young modulusof the substrate, and Φ is the loss term, assuming different values in the substrate (Φ ≡ φs ) and in
the coating (Φ ≡ φc ).
As for the suspensions, thermo-elastic noise affects also the mirrors. Elastic deformations ina solid body lead to inhomogeneous distribution of temperature through the thermal expansion
coefficient α(T ), hence inducing dissipative fluxes of heat and losses of energy. For the mirror
substrate, the spectral density of thermo-elastic noise is [36]8kB T 2 w
S(f, T ) = √ α2 (1 + σ)2J[Ω] ,
κ
2π(2.66)

where κ is the thermal conductivity of the substrate material and Ω = f /fc is a dimensionlessparameter depending on the thermal properties of the sample, with
r Z∞
+∞
Z
2
2
u3 e−u /2 .
J[Ω] =
dudv 2
π(u + v 2 ) (u2 + v 2 )2 + Ω20(2.67)−∞The spectral density of thermo-elastic noise for a multilayer-coating of thickness l is [37]S(f, T ) =8kB T 2 l αs2 Cc(1 + σs )2 ∆2 g(f ) ,π 2 f w2 Cs2(2.68)Cc and Cs being the heat capacity per unit volume of coating and substrate, respectively, and αsis the thermal expansion coefficient for the substrate material. The explicit expressions of ∆ andg(f ), omitted for concision, can be found in [37]. It is worth noting here that both the Brownian
and thermo-elastic coating thermal noises also depend on the properties of the substrate material,
as well as both are inversely proportional to some power of the beam radius w.

43Chapter 2. Interferometric Detectors of Gravitational Waves

In transmissive optics, the rise of a temperature gradient additionally cause local randomchanges in the refractive index by means of its temperature dependence. This noise contribution iscalled thermo-refractive noise, and β(T ) = dn(T )/dT is the associated thermo-refractive coefficient.
For a coating of multiple alternate layers of two different materials, with β1 (T ) = dn1 (T )/dT and
β2 (T ) = dn2 (T )/dT , illuminated by a laser of wavelength λ, the thermo-refractive noise spectrumis [38]2 λ2
kB T 2 βeff
S(f, T ) = 3/2 2 √,(2.69)
π w ρκCf
where
βef f =

β1 n22 + β2 n21,
4(n21 − n22 )(2.70)

ρ is the mass density of the substrate. The validity of eq.(2.69) is limited to the case when, forevery layer of material, β(T ) >> α(T ). When this condition is not fulfilled, coating thermo-elastic
and thermo-refractive noises are correlated, giving rise to a global thermo-optic noise. The spectral
density of thermo-optic noise, not reported here for brevity, can be found in [39].
For an interferometer with mirrors of fused silica (SiO2 , currently used in present detectors)
with multiple-layer coatings of Ti-doped Ta2 O5 /SiO2 stacks (the best solution for low-loss coating
up to date), and with a beam size of 6 cm on the mirror surfaces, the dominant contribution to
mirror thermal noise is coating brownian noise for all frequencies within the 1 − 104 Hz band [40].
We will see in chapter 7 that coating brownian noise is expected to limit the sensitivity of future
gravitational-wave interferometers in the most sensitive region of their detection band, around 100
Hz, at about 3.5 · 10−24 Hz−1/2 .

2.4.6

Other Sources of Noise

In order to isolate the mirrors of the interferometer from acoustic vibrations, and to avoid phasenoise due to fluctuations of the refractive index along the laser beam path, the whole detector has
to be placed into a ultra high vacuum system of pipes and tanks. The residual gas pressure in thissystem is usually of the order of 10−7 − 10−8 mbar, which yields a negligible phase noise due torefractive index fluctuations.
Another important source of noise is diffused light, caused by photons of the laser beam thatare scattered around by mirror imperfections and are reflected back by the vacuum pipes. The
phase of scattered photons is modulated by the pipe vibrations, and when these photons recombinewith the main laser beam a phase noise arises. To get rid of this problem a set of mechanicallight absorbers can be placed along the vacuum pipe, avoiding the recombination of the spurious
photons with the main laser beam.
An additional noise source is the electronic noise in the main interferometer read-out system
and in the control system used to keep the interferometer in its working point. The electronicnoise can be introduced by both the sensors and the actuators of the feedbacks. The other noises
mentioned above (diffused light, etc.) can also pollute the auxiliary signals used in the control
systems and then degrade the interferometer sensitivity.
Also the laser source contributes to the total noise of the detector. Fluctuations of laser amplitude and frequency generate additional noise at the detector output, as well as the fluctuations of
the laser beam position (beam jitter) couple to mirror misalignments and induce phase noise.

44

2.5. Thermal Effects

2.5

Thermal Effects

Interferometric detectors of gravitational waves are also affected by thermal effects, which arise
when a fraction of the circulating power is absorbed by the optics. Absorption may take place atthe dielectric coatings and also inside the substrate of transmissive optics, like the beamsplitter
or, as we will see in the following, the coupling input mirror of Fabry-Perot cavites. The optics
are locally heated and consequently deformed, causing wavefront deformations of the reflected ortransmitted beam known as thermo-elastic effects.
In addition to the geometric deformation by local heating, a temperature gradient inside thesubstrate will introduce also a gradient of the index of refraction. Thus, if the laser beam istransmitted through a material with a temperature gradient, the heated region of the substrate
will act as a thermal lens, deforming the wavefront. The change in the refractive index induced by
the temperature gradient T (r, z) isδn(r, z) =dn
T (r, z) ,dT

(2.71)

where β = dn/dT is the temperature index coefficient. The thermal lens is equal to the integrated
excess optical path of the beam transmitted through the substrate, given bydnZ(r) =dT−h/2
Z

T (r, z) dz ,(2.72)

h/2

for a mirror of thickness h. The presence of the thermal lens causes a mismatching of the beam
which goes through it, resulting in coupling losses of the transmitted beam.
We will see in chapter 4 that, in order to mitigate the impact of thermal effects on the beam
wavefront, a thermal compensation system is necessary.

2.6

Sensitivity Improvements with Optical Resonators

We know from eqs.(2.16) and (2.36) that the sensitivity of the interferometer is limited by thelength of its arms and by the laser power. A ground-based detector cannot have arms longer thanfew kilometers. The present laser sources with the required specifications for gravitational wave
detection can generate laser beams of the order of ∼ 100 W.
The laser power and the optical path traveled by the beam can be increased by using opticalresonators. The optical path can be increased by replacing the end mirrors of the interferometerwith Fabry-Perot cavities, as shown on fig.2.6. The light inside the arm cavities is reflected backand forth by the input and the end mirrors, so that the optical path is folded in the cavity and isequal to the arm length times the number of the round trips made by the beam.
The laser power can be increased by coupling an additional mirror, also shown on fig.2.6, between
the laser and the beamsplitter to form an optical resonator for power recycling. Furthermore, theamplitude and frequency noise of the laser, as well as the beam jitter, can be filtered with an input
mode-cleaner, which is a particular Fabry-Perot resonator. At the output of the detector, an output
mode-cleaner might be used to filter the signal in order to increase the signal-to-noise ratio.

45

Chapter 2. Interferometric Detectors of Gravitational Waves

ETM
power recycling
cavityFabry-Perot
arm cavities

ITM
PR

!!"

BS

ITM

ETM

laser

!#$%
PD
Figure 2.6: Optical scheme of a power-recycled Michelson interferometer with Fabry-Perot arm cavities.

In the following we will describe the working principle and the main properties of the opticalresonators of the detector.

2.6.1

Fabry-Perot Arm Cavities

A Fabry-Perot cavity is a standing-wave resonator for the laser beam. In its simplest configuration,
the cavity is formed by two semi-reflecting mirrors, M1 and M2 , separated by a distance L. The
arrangement is schematically shown on fig.2.7, and the convention for propagated, transmitted andreflected fields is the same used in section 2.2. If ψin is a laser beam of wavelength λ, the equations
for the fields inside and outside the cavity areψres = t1 ψin − r1 ψbψb = −r2 ψres e2iφψt = t2 ψres eiφψr = r1 ψin + t1 ψb ,where φ = kL = 2πL/λ is the phase shift cumulated by the beam in a single trip from M1 to M2 .
The light that enters the cavity is reflected back and forth between the two mirrors, and is stored
in the cavity for a finite time equal to√2L r1 r2.
τs =c 1 − r1 r2

46(2.73)

2.6. Sensitivity Improvements with Optical Resonators

M1

M2

!!"

!#$%

!#

!&

!'

!
Figure 2.7: Optical scheme and field notation of a Fabry-Perot cavity.

Thus the explicit equations of the resonant, transmitted and reflected fields are, respectively,
1ψres = ψin 1−r1 r2texp(2iφ)

t1 t2 exp(iφ)ψt = ψin 1−r
1 r2 exp(2iφ)(2.74)

r −r (t2 +r2 ) exp(2iφ)

2 11
ψr = ψin 1 1−r
1 r2 exp(2iφ)

2 (1−L1 ) exp(2iφ)= ψin r1 −r.
1−r1 r2 exp(2iφ)

The cavity is resonant when φ = kπ, so that the incident light ψin can interfere constructively withthe light stored inside the cavity. The normalized power and the phase of the transmitted, resonant
and reflected fields are shown in figs.2.8, 2.9 and 2.10.
Transmitted power
The power transmitted by the cavity is
Pt (φ) = |ψt (φ)|2 = Pin



t1 t2 2
1,
4r
1
1 − r1 r2
1 + 1−r rr2 sin2 φ(2.75)

1 2

whereA(φ) =

1
4r1 r2
1 + 1−rsin2 φ1 r2(2.76)

is the Airy function. The separation between two consecutive resonances is the free spectral range(FSR), equal toc∆f = 2Lin terms of frequency(2.77)
λ∆L = 2 in terms of length .
The ratio of the free spectral range to the full width at half maximum (FWHM) of the resonancepeak is the cavity finesse,√
π r1 r2
FSR
F==.
(2.78)
FWHM
1 − r1 r2

47

Chapter 2. Interferometric Detectors of Gravitational Waves

Resonant power
The power stored in the cavity at the resonance is
Pres = |ψres (φ = π)|2 = Pin
where
Gopt =

T1,(1 − r1 r2 )2

T1(1 − r1 r2 )2(2.79)(2.80)

is the optical gain of the cavity. When r1 ' r2 ' 1, the optical gain can be approximated as
Gopt ≈ F/π.
Reflected field
The ratio between the reflected field and the input field defines the complex reflectivity of the
Fabry-Perot cavity:
ψr
r1 − r2 (1 − L1 ) exp(2iφ)=.
(2.81)
ψin1 − r1 r2 exp(2iφ)
Three possibilities are given:• optimal coupling: r1 − r2 (1 − L1 ) = 0
The reflected field at resonance is null. In this case the light that is promptly reflected by the
input mirror interferes destructively with the stored light leaking outside the cavity, throughM1 . If the optical loss of M1 is null (L1 = 0), this condition is verified when r1 = r2 .
• low coupling: r1 − r2 (1 − L1 ) < 0
The amplitude of the reflected field at resonance is negative, thus there is a phase shift of π
between the input field and the reflected field. This is verified when r1 < r2 , for L1 = 0.
• over coupling: r1 − r2 (1 − L1 ) > 0
The amplitude of the reflected field at resonance is positive, thus the input field and thereflected field are in phase. This condition is verified when r1 > r2 , for L1 = 0.
For small phase shifts δφ around the resonance, the phase of the reflected field varies according to
ψ 
dφr
d
(1 − L1 ) − r12
r
=
arg
= r2
.
dφ
dφ
ψin
(1 − r1 r2 )(r1 + r2 (1 − L1 )(2.82)In the particular case when r1 ' r2 ' 1 and L1 = 0, eq.(2.82) simplifies to
dφr
2F
'
.
dφ
π

(2.83)

Thus eq.(2.83) shows that for a small detuning in the proximity of the resonance (like the strain
induced by an impinging gravitational wave), the phase shift of the reflected field is amplified bya factor proportional to the cavity finesse. This is the reason why Fabry-Perot cavities are used in
the arms of the interferometer to increase the optical path length of the laser beam.

48

2.6. Sensitivity Improvements with Optical Resonators

Transfer function
In the low frequency condition ΩGW << c/2L, that is when the frequency of the impinging gravitational wave is much smaller than the typical frequency of a round trip cycle of the laser in thearms, it can be shown that the frequency response of the interferometer with Fabry-Perot cavites
is [2]8πc1p∆φ = hτs,(2.84)λ1 + (2ΩGW τs )2
with a pole at angular frequency fF P = 1/4πτs . Thus for frequencies ΩGW /2π >> fF P the phasesensitivity of the detector decreases, as shown on fig.2.14. Below the cavity pole (ΩGW /2π << fF P ),the ratio between the phase response of the Fabry-Perot detector of eq.(2.84) and the response of
the Michelson detector of eq.(2.16) is∆φFP
2L F 8πc  4π −1 2F
hL
=
=h
.
∆φMichc 2π λλπ(2.85)

This is another way to describe the improvement in sensitivity due to Fabry-Perot arm cavities,
alternative to that of eq.(2.83). The enhancement is once again evident: when Fabry-Perot cavities
of finesse F are inserted in the arms, the effective length of the interferometer arms is equal to
Leff = L2F.
π(2.86)For example, if L = 1 km and F = 100, the effective optical path length is Leff ' 64 km.

2.6.2

Power Recycling

If the interferometer is close to the dark fringe condition and the reflectivity of the arm end mirrors
is high, almost all of the fringe power is reflected back to the laser. This power can be stored in thedetector by adding a semi-reflective mirror between the laser and the beamsplitter, in order to form
an optical resonator for the power reflected towards the laser. The resonator will be composed by
the mirror itself and by the rest of the interferometer, considered as a composite mirror of equivalent
reflectivity rIT F . This particular resonator is a power recycling cavity, with optical gain GP R .
The recycling gain can be computed in the simple case where the power recycling cavity is
coupled to a single Fabry-Perot arm cavity of the interferometer. The equation of the resonant
field for the coupled cavities is thenψres = ψin

tP R
= ψin
1 − rP R rF P exp(2iφP R )
1−r



PR

tP R

,
r1 −r2 (R1 +T1 )exp(2iφ)exp(2iφ)F
P
PR
1−r1 r2(2.87)

where rP R is the reflectivity of the power recycling mirror, rF P is the equivalent reflectivity of the
Fabry-Perot cavity from the last of eqs.(2.74), φP R and φF P are the phase shifts acquired by thebeam in the power recycling cavity and in the arm cavity, respectively. The double resonance ofthe coupled cavities happens when φP R = mπ and φF P = nπ. The optical gain of the system canbe computed by taking the square modulus of eq.(2.87), and is equal to that of the power recycledinterferometer with both Fabry-Perot arm cavities.

49

Chapter 2. Interferometric Detectors of Gravitational Waves

FSR

Figure 2.8: Normalized transmitted power (top) and phase of the transmitted field (bottom) of a FabryPerot cavity with L = 1 km, no loss, r1 = r2 , for F = 10 (blue solid curve) and F = 100 (red dashedcurve).

50

2.6. Sensitivity Improvements with Optical Resonators

Figure 2.9: Normalized resonant power (top) and phase of the resonant field (bottom) of a Fabry-Perotcavity with L = 1 km, no loss, r1 = r2 , for F = 10 (blue solid curve) and F = 100 (red dashed curve).

51

Chapter 2. Interferometric Detectors of Gravitational Waves

Figure 2.10: Normalized reflected power (top) and phase of the reflected field (bottom) of a Fabry-Perotcavity with L = 1 km, no loss, r1 = r2 , for F = 10 (blue solid curve) and F = 100 (red dashed curve).

52

2.6. Sensitivity Improvements with Optical Resonators

2.6.3

Mode-Cleaner Cavities

The noise due to the laser source might be filtered out by an input mode-cleaner cavity, placed
between the laser and the power recycling mirror. A mode-cleaner is a Fabry-perot cavity with
optimal coupling, for the total transmission of the incoming beam. The mode-cleaner filters theamplitude and frequency noise of the laser like a low pass filter with cut-off frequency fF P = 1/4πτs ,where τs is the storage time defined in eq.(2.73). Moreover, the cavity filters the beam jitter andthe spatial defects of the beam [41], so that the Gaussian beam that enters the interferometer canhave an optimal coupling to the arm cavities of the detector. The spatial defects and the beamjitter are reflected by the mode cleaner as higher-order laser modes, which are described in detailin section 5.1.
Some spatial defects of the beam might also be present at the output of the detector, on thephotodiode, whereas the gravitational wave signal is only due to the phase modulation of theGaussian beam that resonates in the arm cavities. Thus an output mode-cleaner could be also used
to filter the spatial defects of the output beam, in order to increase the signal-to-noise ratio of apotential detection.

53

54

Chapter 3

Present and Future Ground-Based
Gravitational-Wave Detectors
Nowadays four gravitational-wave interferometers have been built and are operating worldwide:
TAMA300, GEO600, Virgo and LIGO. These instruments approximately reached their design sensitivities, demonstrating the technical feasibility of the detection principle. LIGO, Virgo and GEO600
form a network of detectors which have completed several joint observational runs. No detection ofgravitational waves has been reported so far. The observational results of the searches are upper
limits, either for the rate of binary coalescence or for the intensity of the gravitational radiation
emitted by other sources, like for example pulsars.
If we refer only to the coalescence of binaries (since the associated gravitational-wave signal
is the best known among those of expected sources), the predicted rate of detection for presentinterferometers is of the order of 10−3 /yr. Given this rate, the detection of a signal still remains
plausible, but is very unlikely. A tenfold improvement of the sensitivity (in the whole detectionband) of present interferometers might provide the needed increase of detection rate. Improving
the sensitivity of an order of magnitude would increase the detection horizon by approximately
the same factor and hence the accessible volume of the Universe by its cube. Assuming thatastrophysical sources of gravitational waves are isotropically distributed in space, this sensitivity
improvement would increase the present expected detection rate by a factor ∼ 103 , allowing thefirst direct detection of gravitational waves and leading to the detection of several events per year.
This is the motivation of the proposal of a second generation of interferometers, like Advanced
LIGO, Advanced Virgo and the Large-scale Cryogenic Gravitational-wave Telescope (LCGT). All
these projects have been presently funded and advanced LIGO and Advanced Virgo are currently
under construction.
In any case, the expected rate and signal-to-noise ratio of second generation instruments are
expected to be too low for precise astronomical studies of gravitational-wave sources. For thisreason, the project of a third-generation detector named Einstein Telescope has been proposed,
which should provide an increase of sensitivity of about one hundred times that of present detectors(corresponding to an increase of 106 times the present expected detection rates). The aim of thisproject, which is currently at phase of design study, is to initiate the era of precision gravitationalwave astronomy.
This chapter is intended to give a picture of the state of the art of present and future gravitationalwave interferometers, whose sensitivity evolution is significantly resumed by fig.3.1. To describe55

Chapter 3. Present and Future Ground-Based Gravitational-Wave Detectors

Figure 3.1: Sensitivity target (green dashed curve) of the Einstein Telescope interferometer, comparedto the sensitivity curves of first (LIGO, Virgo/Virgo+, GEO600) and second generation (advanced LIGO,Advanced Virgo, LCGT) gravitational-wave interferometers, from [75].

this first generation of detectors in section 3.1, we will take Virgo as example. This choice is motivated by the fact that part of this thesis work is dedicated to the optical design of Advanced Virgo.
The description of the present Virgo interferometer will be useful to put the emphasis on its future
upgrade, thus we will present its optical layout and its sensitivity curve. In the subsequent section
3.2, we will briefly summarize some of the most important observational results achieved by the
LIGO-Virgo-GEO600 network so far: we will present the results of the search for signals emitted bycoalescing binaries, burst sources, known pulsars (with electromagnetic counterpart observations)and coming from the stochastic background of cosmological origin. Section 3.3 will present a short
global overview of the projects of future detectors of second generation, and we will briefly discuss
there their science case. Again, we will assume Advanced Virgo as reference. At the end of thechapter, in section 3.4, we will describe the project of the Einstein Telescope, since the second partof this thesis work (about Laguerre-Gauss beams) is intended to be a contribution to the opticaldesign of this third-generation detector (besides the fact of possibly being part of an upgrade ofsecond-generation detectors).

3.1

The Network of First Generation Detectors

Four large ground-based interferometers are currently operating around the globe. The Virgo interferometer [42, 43] is a detector located in the plain of Cascina, near Pisa in Italy. It has been built
by the Virgo Collaboration, a scientific collaboration between the laboratories of the Centre National de la Recherche Scientifique (CNRS) of France and the Istituto Nazionale di Fisica Nucleare563.1. The Network of First Generation Detectors

Figure 3.2: Aerial view of the Virgo interferometer in Cascina, Pisa (Italy).

(INFN) of Italy. Recently also groups from Holland, Poland and Hungary joined the Collaboration. The first proposal of Virgo was made in 1989, and the project has been inaugurated in 2003.
The experimental site is managed by a CNRS-INFN consortium, the European Gravitational Wave
Observatory (EGO) [44]. On fig.3.2 an aerial view of the detector is shown.
LIGO [45, 46] is a project of three kilometer-scale detectors, built at two sites in the UnitedStates. At Hanford, Washington, there are two power-recycled Fabry-Perot interferometers witharm cavities of 2 and 4 km, in the same vacuum tube. At Livingston, Louisiana, there is asingle detector with 4-km long arm cavities. LIGO reached its design sensitivity in 2005, with anequivalent-strain noise of ∼ 3 · 10−23 Hz−1/2 in the most sensitive frequency band, around 100 Hz.
A detector with 600-m long arms, GEO600 [47, 48], has been operational near Hannover, inGermany, since 2001, gathering research groups principally in Germany and Britain. Its maximum
sensitivity is ∼ 10−21 Hz−1/2 at 100 Hz. A Japanese detector with 300-m long arms, TAMA300[49, 50], is operating near Tokyo with a maximum sensitivity of ∼ 10−21 Hz−1/2 at 1 kHz. TAMA300
was the first large-scale interferometer to achieve a continuous operation, and is now used as adevelopment prototype. The detectors of LIGO, GEO600 and TAMA300 are shown on fig.3.3.
LIGO and GEO600 have worked together within the frame of the LIGO Scientific Collaboration(LSC) since the beginning of observational runs. The collaboration comprises research groups fromuniversities around the world, contributing to data analysis and technology developments. Virgo
has signed an agreement with the LIGO collaboration in order to pool the data in a joint analysis,
thereby creating a single worldwide network of long-baseline gravitational wave interferometers.
The simultaneous data taking among detectors located at separated sites is of fundamental
importance, for several reasons. The coincident measurement of a gravitational wave signal indifferent detectors would strongly improve the confidence of the detection. Since noise-induced
random triggers from a network of detectors are less likely to be consistent with each other, coinci-

57

Chapter 3. Present and Future Ground-Based Gravitational-Wave Detectors

Figure 3.3: Worldwide network of detectors. From top right, clockwise: aerial views of LIGO Hanford,
LIGO Livingston, GEO600, TAMA300.

dence analysis is a very powerful tool to veto out spurious events. Furthermore, the position of theemitting source in the sky might be reconstructed through triangulation, by looking at the delayin the arrival times of the signals recorded by several detectors [51]. The reconstruction of thesource position could allow for potential follow-ups of electromagnetic counterparts. The observation of an electromagnetic signal would improve the robustness of the gravitational wave detection
and would yield complementary informations about its source. All the observing instruments are
expected to receive the same gravitational wave polarizations, but the gravitational wave should
couple differently to each detector, according to the single antenna patterns. The result of thiscoupling might be modeled through a phase shift and a scale factor in the reconstruction of theobserved wave signal in each detector, from which it could be possible to infer the polarization of
the incoming wave [52].
To describe the technical features of present gravitational-wave interferometers we will use Virgo
as example, its sensitivity curve is similar to those of the other detectors of the network. This will
be useful for the following: the first part of this thesis work concerns the optical design of Advanced
Virgo, which is the project (illustrated in chapter 4) of upgrading Virgo to a more sensitive secondgeneration detector. Thus we will describe here especially the present optical layout of Virgo, toallow for the comparison with the changes discussed later on in this thesis.
Optical layout of Virgo
Virgo is a power-recycled Michelson interferometer with 3-km long Fabry-Perot cavities in thearms, its schematic optical layout is shown in fig.3.4. The two arm cavities (conventionally called
North and West), are composed of (North or West) input and end mirrors, and the first mirror

58

3.1. The Network of First Generation Detectors

WEwest cavity
L= 3 km F=50

Input mode-cleaner
144 m
F=1000

WI, R=88%

CW 20 W, 1.064 nm

PR, R=92%

BS (50%,50%)

G= 50

L= 3 km F=50

Laser
l=6 m
north cavity
NI, R=88%

Output mode-cleaner
2 cm
F=50

NE

Anti-symmetric
port

Figure 3.4: Optical layout of the Virgo interferometer.

encountered by the input laser beam is the power recycling mirror. The distance between the beam
splitter and the power recycling mirror is approximately 6 m, equal to the distance between the
beam splitter and the two input mirrors. Many upgrades have been made from the first installationof Virgo on 2003, here we report the original configuration of the detector.
The input beam (λ = 1.064 µm) is generated by a system composed of a high-stability solidstate Nd:YAG master laser and a Nd:YVO4 high-power slave laser, with a continuous power ofabout 20 W. Before entering the interferometer, the beam passes through the input mode-cleaner(IMC), a triangular Fabry-Perot cavity 144-m long with a finesse of about 1000 and suspended
mirrors. The main motivation for the input mode-cleaner is the filtering of the position noise of thebeam, but the cavity is also used for laser frequency stabilization. The beam then goes throughthe power recycling mirror (PR), which has a reflectivity R = 92%.
The laser light is split by the beamsplitter mirror (BS) in two beams, which enter the two long
Fabry-Perot arm cavities with a finesse of 50 (corresponding to an optical gain of about 31). This
finesse is obtained with a reflectivity R = 88% of the input mirrors and a reflectivity for the end
mirrors almost equal to 1. The optical gain of the power recycling cavity is about 50, the power
impinging on the beamsplitter is 500 W and the power stored in the arm cavities is about 7 kW.
In order to propagate along kilometer-scale distances with small divergence, the beam entering thearm cavities is large: the radius w is 2 cm on the input mirrors and 5 cm at the end mirrors.
The mirrors used in Virgo are made of high quality fused silica, with a diameter of 35 cm
and a mass of about 21 kg for power recycling and cavity mirrors. The beam splitter is slightly
smaller, with a diameter of 23 cm and a mass of 5 kg. All mirrors are flat, except the two end arm
mirrors which are concave with a radius of curvature of about 3.6 km. To increase the contrast of59

Chapter 3. Present and Future Ground-Based Gravitational-Wave Detectors

Figure 3.5: Virgo design sensitivity curve. The spectra of fundamental limiting noises are shown.

the interferometer, the output beam is filtered by an output mode-cleaner (OMC), a monolithic
Fabry-Perot cavity 2-cm long made of fused silica.
Each mirror of the layout is suspended from a chain of seismic filters, the Superattenuator [53],which provides the attenuation of ground vibrations. This is to date the most efficient seismic
isolation system of ground-based detectors, allowing the extension of Virgo detection band downto ∼ 3 Hz. The Superattenuator is even compliant with the low-frequency sensitivity requirementof Advanced Virgo and will be part of this project, thus it will be described in detail in chapter 4.
Sensitivity curve of Virgo
From 2003 to 2007 the Virgo interferometer underwent a commissioning period, which is the phaseof the project between the end of the detector construction and the beginning of data taking. This
phase involves mainly the development and the tuning of the control system of the detector, whichallows to keep the interferometer at its working point. The objective of the commissioning phaseis twofold: to reach the design sensitivity and to have sufficiently high duty cycle over long periodsof time, of the order of several months. The duty cycle of the detector depends mainly on therobustness of the control system.
Before reaching the sensitivity limit caused by fundamental noise sources (seismic noise, thermal
noise and shot noise), several technical noises (like laser noises and control noises) have to be
reduced. The progress marked by the commissioning (continued also beyond 2007) allowed todecrease the measured noise approximately to the design level, shown on fig.3.5, which is equal to∼ 3 · 10−21 Hz−1/2 at 10 Hz and ∼ 7 · 10−23 Hz−1/2 at 100 Hz.

60

3.2. Observational Results of the LSC-Virgo Network

joint data taking runs [LSC: Si − Virgo: VSRi]

S5 (Nov. 05 - Sept. 07) & VSR1 (May 07 - Oct. 07)

results to date [upper limits/triggers]
binary coalescence rates [55, 56]
burst event rates [57]burst emission from magnetars [8]emission of known pulsars [58]energy density of stoch. background [10]multimessenger events [59, 60, 61]

S6 (July 09 - Oct. 10)&
VSR2 (July 09 - Jan. 10)/VSR3 (July 10 - Oct. 10)

Vela pulsar emission [62]Table 3.1: Main observational results of the joint data taking runs of the LSC-Virgo network so far.

Since 2007, Virgo went through several technical upgrades with the aim of improving the sensitivity. The latest one, occurred on 2010, concerned the replacement of the steel wires of the mirror
suspension system with monolithic fibers of silica with low mechanical loss, in order to decrease thesuspension thermal noise [31, 54]. This upgrade led to an enhanced configuration of the detector,
called Virgo+, whose design sensitivity is shown on fig.3.1.

3.2

Observational Results of the LSC-Virgo Network

In 2005 LIGO began a two-year data taking run, S5, which ended on 2007. From 2009 to 2010
LIGO collected data during the S6 run. Since 2007, Virgo has accomplished three scientific datataking runs: VSR1, VSR2 and VSR3. VSR1 overlapped with the last months of S5, and VSR2 andVSR3 have been coordinated to be in coincidence with S6.
The analysis of the data acquired during the first S5-VSR1 joint run is now complete, and nodetection of gravitational waves has been reported. Data collected so far have been used to put
upper limits on the expected rates of coalescence of binary systems [55, 56] and on burst sources
rates and emission [57, 8], to constrain the amount of gravitational radiation emitted by known
pulsars [58], to constrain the spectrum of a stochastic background of gravitational waves [10] andto search for multimessenger events [59, 60, 61].
The analysis of the data of the joint run S6-VSR2/VSR3 is ongoing, but a first importantresult, constraining the gravitational-wave emission of the Vela pulsar, is already available [62].
Hereafter we review the main experimental results achieved by the LSC-Virgo network so far, also
summarized in table 3.1.
Search of signals from coalescing binaries
The method used for the search of signals from coalescing binary systems is coincidence analysis,
were data from each detector are analyzed separately by applying relevant filters and search algorithms. Then coincidences are searched in the multi-dimensional space of intrinsic (masses of
the component stars, their spins, etc.) and extrinsic (arrival times, reconstructed source location,etc.) parameters. Therefore data are analyzed only for times when at least two detectors of the
LSC-Virgo network are operating.

61

Chapter 3. Present and Future Ground-Based Gravitational-Wave Detectors

sourcerate R [yr−1 L−1
10 ]rate R [Myr−1 MWEG−1 ]

NS-NS
NS-BH
BH-BH

8.7 · 10−3
2.2 · 10−3
4.4 · 10−4

14.8 · 103
3.7 · 103
7.5 · 102

Table 3.2: Measured upper limits of coalescence rates for binary systems with total mass from 2 M
M , from [55].

to 35

The analysis is made on data that have passed a series of veto categories, based on data quality criteria. Environmental and instrumental monitoring channels of the detectors report about
identified noise transients, so that vetoes remove a limited time series of data in an interval centered around the transients. The list of gravitational wave detection candidates is made from theremaining volume of data.
Detection candidates are selected through search algorithms which are adapted to the expectedsignal waveforms, the templates. These latter are predicted by Post-Newtonian theory and numerical simulations. It is then possible to apply a matched-filtering technique for efficient search, wheretemplates are correlated to the data [55].
To estimate the significance of candidate events, the distribution of events due to backgroundnoise is measured. A sample of background noise events is created, by time-shifting data of one ormore detectors with respect to the others by time delays of no physical meaning (i.e. much larger
than the maximum time-of- flight of a gravitational wave between the detectors). Shifts typicallyrange from 1 s to a few minutes. Any triggers that are coincident in the time-shifted data cannot be
due to a true gravitational-wave signal. These coincidences therefore sample the noise background.
For all the searches, search algorithms are tested with hardware and software injections [55, 57].
At random times during the run, simulated waveforms are added to the detector output data, via
software, or are sent to the mirror position control system. Search algorithms work correctly if theydetect the injected signal and reconstruct the parameters of the simulated source. These injections
provide an end-to-end verification of the detector instrumentation, the data acquisition system andthe data analysis software.
The general search for signals from coalescing binaries in S5-VSR1 data has been conducted forsystems of total mass from 2 M to 35 M , for the following astrophysical objects: binary neutronstars (BNS), binary black holes (BBH), and black hole-neutron star binaries (BHNS). These objects
are expected to emit gravitational waves in the frequency band where LIGO and Virgo detectors
are most sensitive (40 − 1000 Hz). Since no detection candidate passed the severe criteria of data
analysis, upper limits on rates of binary coalescence have been established [55].
At present, there are significant uncertainties in the astrophysical rate predictions for compact
binary coalescences, which can vary over two orders of magnitude. Uncertainties arise from thesmall sample size of observed galactic binary pulsars, from poor constraints for predictions based on
population-synthesis models, and from the lack of confidence in a number of astrophysical parameters such as the pulsar luminosity distribution. Thus the direct measurement of coalescence rates
are important, since they might impose constraints on the theoretical models of stellar evolution
and compact binary formation.
The measured upper limits are reported in table 3.2. Rates are reported in units of number

623.2. Observational Results of the LSC-Virgo Network

per L10 per yr, where L10 is 1010 times the blue solar luminosity LB = 2.16 · 1026 J s−1 , derived
from the blue solar magnitude. Binary compact objects are usually produced via the evolution ofmassive stellar binaries, whose birthrate and evolution can be traced in a galaxy by measuring theamount of emitted blue light. Thus the blue luminosity of a galaxy should linearly scale with its
compact binary coalescence rate [63]. By using the conversion factor 1.7 L10 /MWEG [5], rates canbe converted in units of number per Myr per MWEG, also reported in table 3.2. Unfortunately,
these upper limits are still 2 to 3 orders of magnitude larger than astrophysical expectations [5],reported in table 1.1. These limits cannot yet provide any significant constraint to stellar evolution
models.
A separate search for signals from binary black-hole coalescences with total mass 25M ≤ M ≤100M , component masses 1M ≤ m1 and m2 ≤ 99M and negligible spin has also been performed[56]. No plausible gravitational-wave candidate event has been detected, and the coalescence rate ofnonspinning black holes with component masses m1 ≤ 19M and m2 ≤ 28M has been constrained
to be less than 2.0 Mpc−3 Myr−1 . This limit is still about one order of magnitude higher thanastrophysical estimates for such systems.
Search of signals from burst sources
The analysis performed for burst sources is similar to that of coalescing binary systems. The
method used is coincidence analysis, with the same class of veto categories for data quality selection. However, since burst sources are currently not understood well enough to generate accurate
waveforms, excess-power algorithms are employed instead of matched-filter searches with templates.
By measuring power in the data as a function of time and frequency, one may identify regions where
the output power is not consistent with the anticipated fluctuations of detector noise [57]. The estimation of background noise in the detectors is performed by time-shifting of the data, like in thecase of search for binary coalescence signals. Also software and hardware injections are used totest the efficiency of the search algorithms, though only by using approximated waveforms of the
expected signal.
No gravitational-wave event has been detected by the general search for burst source emissions.
The main result [57] is a rate limit R for strong gravitational wave signals, equal to 2.0 yr−1 in thefrequency band below 2 kHz and equal to 2.2 yr−1 for frequencies above 2 kHz. This rate is larger
than astrophysical estimations, which quote 0.5 yr−1 within ∼ 5 Mpc from Earth [6].
No evidence of a gravitational wave signal came also from a separated search for burst signals
from 6 known magnetars [8]. The search put upper limits on the gravitational wave energy emitted
for the lowest-order pulsational mode (the f-mode at 1090 Hz) and the white noise burst at 100−200
Hz. The two lowest limits of the search are estimated for the source SGR 0501+4516: these are
1.4 · 1047 erg (at a nominal distance of 1 kpc) for the f-mode and 3.5 · 1044 erg for the white noise
burst. This latter is, for the first time, comparable to the electromagnetic emission observed inmagnetar flares.
Search of signals from known pulsars
Within our Galaxy there are currently over 200 known pulsars with spin frequencies greater than
20 Hz, which therefore fall into the detection band of gravitational wave interferometers. Thus thesearch looks for signals from pulsars of known position and spin frequency in the S5-VSR1 data,assuming that: (i) pulsars are triaxial stars emitting gravitational waves at precisely twice their63

Chapter 3. Present and Future Ground-Based Gravitational-Wave Detectors

observed spin frequencies, and (ii) that gravitational waves are phase-locked with the electromagnetic signal. The observed pulsar phase evolution is modeled using a Taylor expansion about afixed epoch time t0 ,φ(T ) = φ0 + 2π[ν0 (T − t0 ) + ν̇0 (T − t0 )2 /2 + ...]
(3.1)
where φ0 is the initial spin phase, ν0 and its time derivatives are the pulsar spin frequency andspin-down coefficients at t0 , and T is the proper time of the pulsar. The expected signal in aninterferometer from a triaxial pulsar is then [64]h(t) =h0(1 + cos2 ι)F+ (t, ψ) cos 2φ(t) + F× (t, ψ)h0 cos ι sin 2φ(t) ,
2

(3.2)

where φ(t) is the phase evolution in the detector time t, F+ and F× are the detector antenna patterns
for the + and × wave polarizations, ψ is the wave polarization angle, and ι is the angle between therotation axis of the pulsar and the line of sight. The detector time differs from the pulsar proper
time by a relativistic Doppler effect, which is estimated and corrected before performing the data
analysis [64].
Data from gravitational wave detectors are heterodyned using twice the known electromagnetic
phase evolution of each pulsar. The heterodyne removes the rapidly varying component of thesignal and leaves only the daily varying amplitude modulation, due to the motion of the source
through the antenna pattern of the interferometer [65]. Heterodyned (complex) data are low-passfiltered and fed to the search algorithm, which estimates the Bayesian posterior probability of theunknown parameters (h0 , φ0 , cos ι, ψ) over the signal model of eq.(3.2). By using the posterior
probability on h0 , the search sets an upper limit on the gravitational wave amplitude emitted by
known galactic pulsars [58], with 95% degree of belief.
Amplitude upper limits can be used to constrain the spin-down limit amplitude hsd of eachpulsar, which can be calculated by assuming that the observed spin-down rate of the object isentirely due to energy loss through emission of gravitational radiation. For the sample of pulsars
considered in the S5 data search, the resulting median upper limit of the search on h0 is 7.2 · 10−26 .
For the Crab pulsar (PSRB0531+21, PSRJ0534+2200), the measured limit h0 ' 2 · 10−25 is afactor of 7 below the spin-down limit. This limits the power radiated via gravitational waves tobe less than ∼ 2% of the available spin-down power. For all of the other pulsars, the median ratio
to the spin-down limit is h0 /hsd = 108. For the millisecond pulsars of the sample, several of themeasured upper limits are only about an order of magnitude above their spin-down limits. Througheq.(1.25), these results can be interpreted as limits on the pulsar ellipticity, yielding a median valueof  = 1.1 · 10−6 .
One of the first results of the analysis of data from the S6-VSR2/VSR3 joint run is an upper
limit on the gravitational wave emission from the Vela pulsar (PSR B0833-45, PSR J0835-4510)[62], which is one of the nearest known pulsar at ∼ 290 pc from Earth. Vela is observed to pulsate
at ν0 = 11.19 Hz in radio, optical, X-ray and γ-ray radiation and is associated with the Vela
Supernova remnant. The frequency of the expected gravitational waves emitted by the Vela pulsar(ΩGW /2π = 2ν0 = 22.38 Hz) is within the detection band of Virgo, because of the very effective
seismic isolation provided by the Superattenuator. This frequency range is inaccessible to all othergravitational wave detectors to date, thus only Virgo data have been used in the search of signals
from the Vela pulsar.
Three independent methods of data analysis have been used in the search (all assuming thatgravitational wave emission follows the radio timing), the first being the Bayesian approach de-

64

3.2. Observational Results of the LSC-Virgo Network

scribed above, the others being two different frequentist approaches based on Monte Carlo simulations. The first method yielded an upper limit h0 = 2.1 · 10−24 , with 95% confidence. The othertwo produced the upper limits h0 = 1.9 · 10−24 and h0 = 2.2 · 10−24 , with 95% confidence. All
these limits are below the Vela spin-down limit of 3.3 · 10−24 , and correspond to a limit on the star
ellipticity of ∼ 10−3 . This is above the maximum ellipticity sustainable by a neutron star with astandard equation of state, but still comparable with the value allowed by some exotic equationsof state [66]. These results have also constrained the internal toroidal magnetic field of the Vela tobe less than 1016 G.

Search of signals from the stochastic backgroundThe search for the stochastic gravitational wave background is performed by cross-correlating datafrom pairs of interferometers of the network, in order to extract the background signal (supposed tobe the superposition of the astrophysical and the cosmological backgrounds) from the instrumental
noise of the detectors, which are supposed to be uncorrelated. In the frequency domain, the crosscorrelation between two interferometers is multiplied by a filter function [10]Q̃(f ) = N

γ(f )ΩGW (f )H02,
f 3 P1 (f )P2 (f )(3.3)

in order to optimize the signal-to-noise ratio. In eq.(3.3), ΩGW (f ) is the energy density spectrum ofgravitational waves defined in eq.(1.27), Pi (f ) is the noise of the i−th detector, γ(f ) is the overlap
reduction function, arising from the overlap of antenna patterns of interferometers at different
locations and with different orientations. This filter enhances the frequencies at which the signal of
the template gravitational-wave spectrum is strong, while suppresses the frequencies at which thedetector noise is large.
As in the network detection band most theoretical models are characterized by a power-law
spectrum, the search assumes a power-law template gravitational-wave spectrum with index α,ΩGW (f ) = Ωα (f /100 Hz)α . The normalization constant N in eq.(3.3) is chosen such that the
expected value of the optimally filtered cross-correlation is Ωα . The cross-correlation between
the data of the two most sensitive detectors of the network, the 4-km LIGO interferometers at
Hanford and Livingston, in the frequency band 41 − 169 Hz (which contains 99% of the sensitivity)
leads to the estimate for the frequency independent (α = 0) gravitational-wave spectrum Ωα=0 =(2.1 ± 2.7) · 10−6 . This corresponds to a 95% confidence upper limit Ω0 < 6.9 · 10−6 , and for othervalues of the power index α in the range (−3, 3), the 95% confidence upper limit varies between
1.9 · 10−6 and 7.1 · 10−6 [10].
Before these results, the most constraining bounds on the stochastic gravitational wave background around 100 Hz came from observations of Big Bang nucleosynthesis (BBN) and cosmic
microwave background (CMB) anisotropies, as reported in eqs.(1.29) and (1.30) [10, 9]. In theB < 9.5 · 10−6 ,LIGO frequency band and for α = 0, these bounds are ΩBBN
< 1.1 · 10−5 and ΩCM00which are about 5.2 and 4.5 times larger than the bound measured by the interferometer network,
respectively. This new limit is anyway many order of magnitude larger than the expected energy
density due to the standard inflationary cosmological model, which is Ωinflation∼ 10−14 .
065

Chapter 3. Present and Future Ground-Based Gravitational-Wave Detectors

Multimessenger astrophysics
A first multimessenger cooperation has been established between present interferometric detectors
and γ-ray satellites, for externally-triggered searches of gravitational wave bursts in coincidencewith γ-ray bursts [59, 60], though it yielded no detection. Multimessenger searches have been also
conducted the other way, by triggering the electromagnetic follow-up of sky regions identified by
the network of detectors [61], but so far no trigger has been confirmed by the observation of anelectromagnetic counterpart.

3.3

Advanced Detectors

In this section we introduce the projects of a second generation of instruments, which aim toachieve a tenfold sensitivity improvement with respect to the reference sensitivity of present kmscale interferometers. These projects are Advanced LIGO [67, 68] (hereafter aLIGO), Advanced
Virgo [69] (AdV) and – though with some difference with respect to the previous two – the Largescale Cryogenic Gravitational-wave Telescope [70] (LCGT). Thanks to the sensitivity improvementof one order of magnitude in all the detection band, each of these detectors will be able to scan a 103
times larger volume of the Universe than initial instruments. Expected detection rates (assumingthat sources are isotropically distributed in space) will increase correspondingly, leading to several
detections of gravitational-wave signals per year.
aLIGO, AdV and LCGT will be power-recycled Michelson interferometers with FabryPerot arm
cavities, with an additional mirror at the interferometer output. This signal-recycling mirror will
form an optical cavity with the interferometer, allowing the gravitational-wave signal sidebands
either to be re-injected and stored in the interferometer or extracted, depending on the cavity
resonance condition [71]. aLIGO and AdV will use their initial buildings and vacuum systems but
will otherwise consist of new instruments. The present three LIGO interferometers will be upgraded
and the arms of the shorter interferometer at Hanford will be extended to 4 km. According to thepresent planning of the two projects, the installation of the hardware at the sites will start at theend of 2011. Both aLIGO and AdV are expected to finish their commissioning phase in order tobe operative at design sensitivity for 2014.
LCGT will have 3-km long arm cavities, and will be the first km-scale interferometer to bebuilt underground and with a cryogenic system for the cooling of the arm cavity mirrors. Groundmotion of underground sites is expected to be sensibly smaller than that of surface facilities [72].
On the other hand, as it will be discussed in chapter 7, the cooling of the mirrors will reduce thethermal noise of the optics. The construction of LCGT is planned to end on 2015, and the datataking will start at the beginning of 2018, after a phase of commissioning.
We will describe in the following the science case of second generation instruments. Given that
advanced detectors will have comparable sensitivities in the same detection band, we will assume
AdV as a reference example, since the first part of this thesis work concerns its optical design.

3.3.1

Sensitivity and Science Case

The reference sensitivity curve of AdV with the main noise contributions is shown in fig.3.6. The
presence of the signal recycling cavity allows for tuning of the shape of the shot-noise-limited
sensitivity curve of the interferometer, a technique that can be used to optimize the detector
sensitivity for detecting expected astrophysical sources.

66

3.3. Advanced Detectors

Figure 3.6: Advanced Virgo design sensitivity (black solid curve) compared to the Virgo design sensitivity(black dashed curve), from [69]. The spectra of fundamental limiting noises are also shown.

For fig.3.6, the sensitivity has been tuned in order to maximize the sight distance for binary
neutron stars (NS-NS). The figure of merit that is used to evaluate the sensitivity of the detector
for coalescing binaries is the event horizon, which is the maximal distance for an optimally-oriented
observable source in the sky, in order to be detected with signal-to-noise ratio equal to 8. For
frequencies below ∼ 30 Hz, the sensitivity of AdV is limited by suspension thermal noise. Within
30 − 200 Hz, the dominating source of noise is the thermal noise of mirrors, due to brownian noiseof coatings. Between 200 Hz and 400 Hz, coating thermal noise is comparable to shot noise, though
this latter is expected to be larger. For f > 400 Hz, the noise curve is completely determined by
the shot noise of the laser.
The sensitivity curve might be tuned as well to [69]:• enhance the low-frequency response in order to increase the detectability of large mass blackhole binary systems (BH-BH)• enhance the sensitivity in a narrow frequency range, to target young pulsars• widen the detector response and increase the high-frequency sensitivity for burst search.
The sensitivity curves corresponding to different tunings are shown on fig.3.7 and summarized intable 3.3. Hereafter, for the sake of simplicity, we will only consider a sensitivity curve optimized
for a maximum binary-neutron-star detection horizon, as the basis for quantitative considerations.

67

Chapter 3. Present and Future Ground-Based Gravitational-Wave Detectors

Figure 3.7: Tuning of the sensitivity curve of Advanced Virgo (red curve), from [69]. Maximum detection
horizon for black hole-black hole binary systems (green curve), narrow-band tuning for pulsar search (cyancurve), wide-band tuning for burst search (blue curve).

sensitivity curve tuning

NS-NS range [Mpc]

BH-BH range [Mpc]√h at 1 kHz [1/ Hz]

Virgo (design)
AdV NS-NS (design)
AdV BH-BH optimized
AdV broadband

11
150
108
134

58
1112
1276
987

7.2 · 10−23
1.8 · 10−23
3.2 · 10−21
6.2 · 10−24

Table 3.3: Detection horizon and strain sensitivity for different tunings of the Advanced Virgo sensitivity
curve, from [69]. The design sensitivity of Virgo is reported for comparison.

683.4. Third-Generation Detectors

sourceAdV det. rate [yr−1 ]Virgo det. rate [yr−1 ]

NS-NS
NS-BH
BH-BH

17
4.7
7.6

4.5 ·10−3
1.2 ·10−3
1.9 ·10−3

Table 3.4: Expected detection rates of compact binary coalescences for Advanced Virgo, from [42]. Expected
detection rates for Virgo are reported for comparison.

The scientific motivations and the chances of detecting gravitational waves for different typesof sources of AdV (and more in general for advanced detectors) can be synthetically resumed asfollows:• coalescing binary systems: as the detection horizon increases accordingly with the increaseof sensitivity, by assuming the coalescence rates reported in table 1.1 the expected rate ofdetection is of the order of tens of events every year. Detection rates are reported in table
3.4, where also the estimated values for Virgo are reported for comparison. The improvementwith respect to the first generation of detectors is evident• pulsars: assuming 1 year of integrated observation, AdV is expected to beat by orders ofmagnitude the spin-down limit for about 40 of the pulsars presently known, mostly at low
frequencies [69]• core collapse Supernovae: as already mentioned in chapter 1, the gravitational wave
energy emitted by Supernova events might be lower than as previously expected, i. e. as low
as ≤ 10−8 M [6]. This fact strongly reduces the prospects of detection with second-generation
instruments. Advanced detectors could however put some constraints on the strength of thegravitational radiation emitted by this kind of events• multimessenger astrophysics: advanced interferometers will play a crucial role in thedevelopment of multimessenger astrophysics, since they will potentially observe the gravitational signal related to high-energy neutrinos from extragalactic sources, neutrinos fromSupernovae, optical (γ-ray bursts and nearby Supernovae), radio (bursts, afterglows) and
X-ray transients [69].
However, apart from extremely rare events, the expected rate and signal-to-noise ratio of secondgeneration instruments are expected to be too low for precise astronomical studies of gravitationalwave sources, and to yield complementary informations to electromagnetic observations in the studyof fundamental systems and processes in the Universe. This is why a third-generation detector withincreased sensitivity is presently under study.

3.4

Third-Generation Detectors

Einstein Telescope [73, 74] is a project of a third-generation detector, about one hundred times
more sensitive than present ones. This considerably large sensitivity improvement is motivatedby the aim of performing precision gravitational-wave astronomy. The project is presently at the69

Chapter 3. Present and Future Ground-Based Gravitational-Wave Detectors

phase of design study [75], supported by the European Commission through the 7th Framework
Programme of the AStroParticle ERAnet (ASPERA) network [76]. The science case of the Einstein
Telescope (hereafter, ET) project is vast and comprehensive:• to measure in great detail the physical parameters of compact objects in binary systems,
constrain the equation of state of neutron stars and solve the enigma of γ-ray bursts. Binary
systems of total mass M = 10 − 100 M are expected to be detectable with a signal-to-noise
ratio equal to 8 up to a luminosity distance DL ∼ 200 Gpc, corresponding to a cosmologicalredshift z ∼ 17. Depending on their spinning frequency and ellipticity , pulsars might bedetectedable up to several hundreds of kpc. For millisecond pulsars, the detection horizon is
expected to vary between 5 pc ( = 10−9 ) and 500 kpc ( = 10−5 ).
• to test General Relativity by comparing observations of massive binary systems with numerical relativity predictions, and to constrain alternative theories of gravity• to measure cosmological parameters (the matter content of the Universe Ωm , the energy
density ΩΛ and the equation of state w of the dark energy) from standard sirens, and probe theprimordial universe through the measurement of the gravitational-wave stochastic background• to probe the physics of core-collapse Supernovae, to which ET might be sensitive within few
Mpc with an expected event rate of ∼ 0.5 yr−1 .
The target sensitivity of ET is shown on fig.3.1, compared to the typical sensitivity of anadvanced detector. ET is planned to be a interferometer with 10-km long arms. The detector will
be dual-recycled, with power and signal recycling cavities, and will have a large detection band,spanning over four orders of magnitude in frequency, starting from 1 Hz. To achieve this target,
several limitations of the technologies adopted in the advanced interferometers must be overcome,and new solutions must be developed to reduce the fundamental and technical noises that will limit
the next-generation instruments.
From the technical point of view, the favorite arrangement for a detector with the target sensitivity of ET is a 2-band xylophone configuration, composed of two distinct interferometers withmaximum sensitivity at different frequencies, at an underground site. The low-frequency detector(ET-LF) could be a cryogenic interferometer with moderate optical power, whereas the high frequency detector (ET-HF) could use a high power laser [77]. The basic concept behind this designis the decoupling of conflicting technical solutions, like in the case of cryogenic test masses withhigh thermal load. The xylophone configuration, combining two detectors specialized on different
frequency bands, would allow to optimize the shot noise and the radiation pressure noise separately.
A possible realization of such a xylophone strategy is shown on fig. 3.8, while the main parametersof the xylophone detector are summarized in table 3.5.
The sensitivity requirement for the low-frequency part of the detection band is very challenging.
The region between 1 and 10 Hz is limited by seismic and gravity gradient noise [78], and will not
be accessible by second generation interferometers. This frequency band might become exploitableby realizing an underground observatory. Furthermore, with a subterranean installation, othernoises induced by external disturbances like wind, scattered light or temperature fluctuations will
be suppressed by the quietness of the site.
The shot noise at high frequency would be lowered by storing high laser power in detector. The
design study of ET plans to use an input power of 1 kW, which would be amplified by the power70

3.4. Third-Generation Detectors

Figure 3.8: Sensitivity of the Einstein Telescope, implemented by two frequency-specialized (low-frequency,
ET-LF, and high-frequency, ET-HF) detectors, with respect to a single wide-frequency range interferometer
implementation, from [77].

recycling cavity to 20 kW. Because of the high finesse cavities planned for ET, the power stored
in the arms of the detector will be about 3 MW. The increase of the laser power will cause theradiation pressure noise to increase below 20 Hz, therefore to lower this noise heavier mirrors ofM ∼ 200 kg could be used. Increasing the mirror mass will decrease the mechanical susceptibility
and hence cut down the effect of radiation pressure on the displacement of optics. To decrease
further shot and radiation pressure noise, squeezed light will be injected in the interferometer [78].

parameter

ET-HF

ET-LF

arm lengtharm power
temperature
quantum noise suppression
beam shape

10 km
3 MW
290 K
squeezing
Laguerre-Gauss

10 km
18 kW
10 K
squeezing
Gaussian

Table 3.5: Summary of the main parameters of the 2-band xylophone detector. ET-HF is the high frequency
interferometer, ET-LF is the low-frequency one.

71

Chapter 3. Present and Future Ground-Based Gravitational-Wave Detectors

In the most sensitive region of the detection band, the central region around 100 Hz, the highfrequency detector will be limited by mirror thermal noise. This limitation will be discussed inchapter 7, together with a list of possible solutions to decrease this noise. Among those, the use ofcryogenics to cool down the optics of the detector is a suitable option. Another solution might be
the use of beams with a wide radial distribution of the light intensity, Laguerre-Gauss beams.
In the second part of this thesis work we will present the expected benefits (expected decrease ofthermal noise and also of thermal effects) due to the use of higher-order Laguerre-Gauss beams infuture detectors, and we will propose a technique for the efficient generation of high purity beams
for future generations of gravitational-wave interferometers. We will present in chapter 8 the resultsof an experimental tabel-top setup which demonstrate the feasibility of this technique, which might
be in principle suitable even for the implementation in a later phase of advanced detectors.

72

73

PART II

Optical Designof the Non-Degenerate Recycling Cavities
of Advanced VirgoChapter 4

The Project of Advanced Virgo
This chapter is intended to give a technical overview of the AdV project, and we will focus ourdescription especially on those elements which will be relevant for the following of this thesis work.
We will present the optical layout of the detector, which features four major upgrades with respect
to Virgo: (i) the change of the arm cavity configuration, from plano/concave to bi-concave mirrors,(ii) the change of arm cavity finesse to 450, (iii) the implementation of signal recycling and (iv) the
introduction of compensation plates.
The arm cavity configuration is changed to increase the beam radius on the cavity mirrors, todecrease the mirror thermal noise which will limit the sensitivity of the detector in the central partof the spectrum. The bi-concave configuration is used to avoid the effect of tilt instability due tothe torque of radiation pressure [79], which is enhanced in a configuration with nearly flat mirrors.
We will see in the following chapters that the change of the arm cavity configuration has importantconsequences for the optical design of the recycling cavities.
Then we will describe the thermal compensation system, which will be used to compensate for
effects of wavefront distortion caused by thermal effects in the optics. The mitigation of thermal
effects is necessary because the aberrations in the beam wavefront can cause a loss of detector sensitivity, and also severely disturb the detector control. The compensation system will be composedof a series of sensors for the reconstruction of the phase front of the aberrated beam, which will
be used as error signal, and by two different types of actuators: ring heaters and CO2 lasers. The
ring heaters will act directly on arm cavity mirrors. The lasers will illuminate a transmissive optic,
the compensation plate, positioned between the arm input mirror and the beamsplitter, in order togenerate a phase pattern in the plane transverse to the beam. The pattern will be used to cancel
or attenuate the phase distortion caused by thermal effects.
Finally we will describe the mechanical suspension system, the Superattenuator, which is alreadyused in Virgo: it is composed of a chain of oscillators and of a last pendular suspension stage, thepayload. Its description will be useful for the understanding of some crucial topics presented in thenext two chapters.

4.1

Laser and Injection System

The laser of AdV will provide a power of 200 W, in order to have at least 125 W at the input of
the interferometer. The increase of input power (about 6 times larger than in Virgo) will lower the
shot noise, improving the detector sensitivity at high frequency. The design of the laser will be

76

4.2. Optical Layout

based on a completely new technology with respect to the present laser system of Virgo, a Nd:YAG
master oscillator with one- or two-stage fiber amplifier.
The optics between the laser and the power recycling mirror are called injection system. It will
include a Faraday isolator specific for the high-power laser beam, an electro-optical modulator, theinput mode-cleaner and a mode-matching telescope. The Faraday isolator will be used to protect
the laser system from backwards reflection, and the electro-optical modulator will provide the phasemodulation of the laser beam at radio-frequency, for the generation of radio-frequency sidebands
for sensing and control purposes. The input mode-cleaner will have the same 144-m long triangularconfiguration of the present one of Virgo and a slightly larger finesse, F ∼ 1200.

4.2

Optical Layout

The optical layout1 of the interferometer is shown in fig.4.1: the core of the interferometer will havethe same configuration of Virgo, with the short-baseline central Michelson interferometer formedby the beam splitter and the input arm cavity mirrors (ITM), and the 3-km arm cavities.
The major upgrades in the optical layout of AdV are the following:• the change in the arm cavity geometry, from a flat-concave to a concave-concave configuration.
The input and the end arm cavity mirror will have radii of curvature of 1420 m and 1683 m,
respectively. With this new configuration, the beam focus will have a radius w0 = 9.7 mm
and will be close to the center of the cavity. The beam will have a large radius on the cavity
mirrors: w = 4.9 cm on the input mirror, and w = 5.8 cm on the end mirror. The beam sizeon the optics of the arm cavity is maximized in order to reduce mirror thermal noise. The
larger size of the beam spot allows to average more effectively the surface fluctuations due tonoise, as discussed in chapter 7• the increase of the arm cavity finesse to ' 450 (yielding to an optical gain of about 286) [80],with a transmission T = 1.4% of input mirrors. This represents an increase of a factor ∼ 10with respect to Virgo, and ∼ 3 with respect to Virgo+• two transmissive optics, the compensation plates (CP), will be suspended facing the input
arm cavity mirrors, to be used in the thermal compensation system described in section 4.4• the implementation of signal recycling. A gravitational wave of frequency ΩGW /2π incident
on the interferometer generates two signal sidebands of frequencies (ω0 ± ΩGW )/2π. Since
the detection band of ground-based interferometers coincides roughly with the frequencies ofaudible acoustic waves, these sideband fields are called audio-frequency sidebands.
In a perfect interferometer operated on the dark fringe, the carrier and the signal audiofrequency sidebands come out of the arm cavities and are separated at the beamsplitter.
The carrier light is directed towards the power recycling mirror, whereas the sideband fields
induced by gravitational waves propagate towards the signal recycling mirror. This mirror
1
We will refer here to the optical layout as it is described in the baseline design document of AdV [69], with
non-degenerate recycling cavities. Recently marginally-stable cavities have been reconsidered, thus this design has
partially changed. The difference between these two configurations will be explained and discussed in the next two
chapters.

77

Chapter 4. The Project of Advanced Virgo

forms a coupled cavity with the arm cavities of the interferometer, for the recycling of theaudio-frequency sidebands [71].
The signal recycling cavity allows the signal sidebands either to be stored in the arm cavities
or to be extracted, depending upon the cavity resonance condition. The reflectivity of thesignal recycling mirror determines the finesse of the recycling cavity, and thus the storage timeof the signal. This latter sets the detection band of the interferometer, while the microscopic
tuning of the recycling cavity length changes the frequency of maximal sensitivity [81].
The baseline configuration for the signal recycling cavity of AdV is the resonant sideband
extraction, where the signal recycling cavity is used to widen the detection band of thedetector.
The semi-transparent power-recycling mirror (PRM1) will have a transmission T = 3.5%2 , yielding
a finesse F = 105 and a recycling gain of 40. The power stored in the arm cavities will be of
the order of 700 kW, while about 5 kW will be stored in the power recycling cavity. The signal
recycling cavity will have a finesse of F = 26, given by a transmission T = 20% of the signal
recycling mirror (SRM1). At the output of the detector there will be a short rigid mode-cleaner
cavity before the photodiode, to filter out the spatial defects of the beam. The parameters of thelayout are summarized in table 4.1.
The arm cavity mirrors will have a thickness of 20 cm and a weight of 40 kg, to decrease theradiation pressure noise. The quality of their reflective surfaces will be very high, with flatness < 5
nm (which is the rms of the difference between the perfect spherical surface and the actual surface
for low spatial frequencies, < 1 mm−1 ) and roughness < 0.1 nm (which is the rms of the difference
for higher spatial frequencies). We will see in chapter 6 that the quality of the mirror surface isvery important for having a high recycling gain of the sideband fields in the recycling cavities.

4.3

Fields Inside the Interferometer

For a clear understanding of the next two chapters it is important to distinguish all the fields thatwill resonate in the detector, thus we present here the following synopsis.
The beam entering the interferometer will be the sum of a carrier and radio-frequency sidebands,
which will be used to acquire and maintain the resonant condition of all the optical cavities [82].
Two modulation frequencies will be used. At each frequency a pair of sidebands will be generated,satisfying different resonance conditions: (i) the first pair will be resonant in the power recycling
cavity and almost anti-resonant (at half of the free spectral range) in the arm cavities and in thesignal recycling cavity, (ii) the second pair will be simultaneously resonant in both the recycling
cavities. These sidebands will be used in a standard Pound-Drever-Hall technique [83, 84] to extract
the longitudinal error signal for mirror positions. In summary:• the carrier field will resonate in the power recycling cavity and in the arm cavities, then it
will see the whole interferometer• one pair of radio-frequency sidebands, used for the control of the interferometer, will resonate
in both recycling cavities (but not in the arm cavities)
2

5% for the marginally-stable cavities, see the previous note.

784.3. Fields Inside the Interferometer

ETMy, T=5 ppm

west cavity
F=443
BS (50%,50%)

from input
mode-cleaner

ITMy, R=98.6 %P-RM1, R=96.5%
P-RM2

CP

ITMx, R=98.6 %

G=105
P-RM3
S-RM2

non-degenerate
power recycling cavity

F=26

north cavity
F=443

non-degeneratesignal recycling cavity

S-RM1,
R=80%
S-RM3

to detection

Figure 4.1: Optical layout of the Avanced Virgo interferometer.

79

ETMx, T=5 ppm

Chapter 4. The Project of Advanced Virgo

Advanced Virgo Optical Parameters
Light Powerpower on BS
arm cavity power
704 kW
Arm Cavity Geometry
cavity length
2999.8 m
input mirror RoC
1420 mend mirror RoC
beam size on input mirror
48.7 mm beam size on end mirror
waist size
9.69 mm waist position
Arm Cavity Finessefinesse450
round trip losses
input mirror transmission
1.4%
end mirror transmission
Power Recycling Cavity
recycling mirror transmission 3.5 %recycling gain
Signal Recycling Cavity
recycling mirror transmission 20 %finesseMirrors
input mirror diameter35 cm
end mirror diameter
input mirror thickness
20 cm
end mirror thickness

5 kW

1683 m
58.0 mm
1363 m75 ppm
5 ppm
40
26
35 cm
20 cm

Table 4.1: Parameters of the Advanced Virgo interferometer.

• a second pair of radio-frequency sidebands, also used for control purposes, will resonate onlyin the power recycling cavity• the audio-frequency signal sidebands will resonate in the signal recycling cavities and partiallyin the arms.
The resonance conditions of carrier and radio-frequency sidebands in the detector are illustrated
on fig.4.2, where for the sake of simplicity recycling cavities are depicted with one recycling mirror.

4.4

Thermal Compensation System

The thermal compensation system will mitigate the thermal effects due to the absorption of laser
power in the optics3 . In Virgo this system [85] is composed of a phase camera [86] and a CO2 laserthat shines a heating pattern onto the reflective surface of both the arm cavity input mirrors. In
AdV, the compensating system will be upgraded to a blend of different sensors for the measurement and reconstruction of the aberrated beam wavefront, and a series of actuators to actively
compensate the wavefront distortions: the system is shown on fig.4.3.
Sensing
To lowest order, the degree of thermal aberration will be manifest in the main beam of the interferometer, which will be sampled at the output of the detector and through the extraction of3

As we will see in chapter 6, in marginally-stable cavities it will be necessary to use the thermal compensation
system to compensate also for mirror figure errors.

80

4.4. Thermal Compensation System

ETMy

ITMy
input field spectrum
carrier

PRM

BS

ITMx

ETMx

-f2 -f1 +f1 +f2

SRM

Figure 4.2: Resonance condition of carrier and audio-frequency sideband fields in Avanced Virgo.

sensing

RH

CP

BS

CP
sensingsensing

Figure 4.3: Scheme for sensing and actuation of the thermal compensation system: the sensing beams (indifferent colors), the compensation plates (CP) and the ring heaters (RH) are shown.

81

Chapter 4. The Project of Advanced Virgo

L

w w'

Δx

HP

CCD

Figure 4.4: Scheme of the Hartmann wavefront sensor: W is the reference wavefront, W’ the aberrated
wavefront, HP is the Hartmann plate and ∆x/L is the wavefront gradient.

secondary pick-off beams. These beams will be sensed by phase cameras, which can measure thespatial distribution of amplitude and phase for carrier and sidebands at the same time [86].
In addition, each optic with a significant thermal load will be independently monitored. Knowledge of the phase profile of optics is valuable, completing the information from the mode profileof the interferometer beam. The reflective surface of each arm cavity mirror (input and end) will
be monitored in reflection for deformation. The phase profile of input mirror and compensation
plates (described here below in the actuation system) will be monitored in reflection from the recycling cavity side. To date, the sensing of the beamsplitter is under investigation. Optics will bemonitored with beams of wavelength yet to be determined.
The sensitivity required for the wavefront sensor is at the level of 1.35 nm, corresponding toa maximum acceptable phase error of λ/593 at λ = 800 nm. A sensor that might achieve thissensitivity is the Hartmann sensor, shown on fig.4.4, composed of an opaque plate (the Hartmann
plate) with an array of apertures, and of a CCD detector. The Hartmann sensor measures thechange in the gradient of a wavefront, with respect to a reference wavefront. The gradient change
is then numerically integrated to give the wavefront change due to aberrations.
The basic operation of a Hartmann sensor is the following: the wavefront to be measured isincident on the Hartmann plate and divided into a set of rays, known as Hartmann rays, which
diffract from the holes of the plate. The rays propagate over a known distance L, where the patternof spots on the CCD is recorded as a digital image. The displacement of each spot from a previously
measured reference position is calculated, and used in the reconstruction of the wavefront. For AdV,a differential Hartmann sensor [87] with two CCD detectors at different positions has been proposed[69]. A prototype of this sensor has been already tested, and will be developed to be integrated in
the thermal compensation system.
Actuation
The system of actuators will compensate for thermal lensing and thermo-elastic deformation: theformer is expected to affect the recycling cavities, the latter will affect the recycling and the arm
cavities. Thereby the system will act on both input and end arm mirrors, as depicted in theactuation scheme shown in fig.4.3.

82

4.4. Thermal Compensation System

heating patternpattern from 1st axicon

pattern from 2nd axicon

Figure 4.5: Scheme of the double-axicon telescope for generating an annular heating pattern, from [88].

The wavefront distortions in the recycling cavities will be corrected by shining an annularheating pattern on a transmissive optic positioned between the beamsplitter and each arm cavity
input mirror, the compensation plate. Shining the pattern on the plates instead that directly on the
mirrors will avoid that power fluctuations of the CO2 laser induce a residual mirror displacement.
The pattern will be generated by sending the Gaussian beam of a CO2 laser through a telescope
based on a double-axicon system (an axicon is a special lens with one conical surface), whose opticalscheme is shown on fig.4.5.
The annular pattern is suitable to heat the edge of the mirror reflective surface, in order tocompensate for the radially-symmetric power absorption in the central area of mirrors. We will see
later in chapter 6 that relevant effects of wavefront distortion will come also from non-axisymmetriceffects, like mirror figure errors and absorption inhomogeneities. An additional adaptive system tocompensate for non-axisymmetric distortions is presently under study for the implementation in
the detector [89].
To compensate the radii of curvature of the test masses, a circular resistor named ring heater
will be placed around each arm cavity mirror. The ring will radiatively heat the barrel of the optic,
so that the additional corrective thermal load will flatten the radial profile of absorption (peakedaround the laser beam spot at the center of the mirror). The heater will be embedded in the recoil
mass which surrounds the mirror, and will be placed into the focus of a parabolic reflecting shield.
The shielding will be used to concentrate the heating radiation only on the edge of the mirror, thus
decreasing the total emitted power for equivalent required compensations. Ring heaters will also
compensate for aberrations due to mostly-symmetric absorption patterns in the optics.

83

Chapter 4. The Project of Advanced Virgo

4.5

Suspensions

AdV will adopt the passive seismic isolation system of Virgo, the Superattenuator [53]: this system allows Virgo to be presently the most sensitive gravitational wave interferometer in the lowfrequency region, and it will also be compliant with the√low-frequency sensitivity requirement of
AdV (a residual mirror displacement noise of 6 · 10−18 m/ Hz at 10 Hz). We will describe hereafter
the design of the Superattenuator, since it will be a relevant constraint for the optical design of therecycling cavity layout (as we will see in chapters 5 and 6).
The suspension system exploits the properties of attenuation of simple oscillators: the mirrors
are suspended to a chain of seven pendulums, about 8-m long. Because of the long pendulum chain,
all the normal modes of the system are confined below 2.5 Hz, in order to have an attenuation of
the seismic noise of the ground by more than 10 orders of magnitude in the horizontal (on axis)
direction, starting from a few Hz.
The suspension point of the chain is at the top stage of an inverted pendulum with a threeleg elastic structure. The legs of the inverted pendulum are about 7-m long, and seismic noise isfiltered above its resonance frequency of about 30 mHz. The inverted pendulum is used as passive
pre-isolation stage to reduce the seismic motion of the top suspension point. For the very low
frequency motion (∼ DC), the position of the suspension point is actively controlled through asystem of position sensors and actuators. The Superattenuator structure is shown on fig.4.6.
The vertical seismic vibrations are not attenuated by the pendulum chain, and are partiallytransferred to the longitudinal direction of the laser beam, by means of unavoidable mechanicalcouplings and because of the curvature of the Earth. Vertical attenuation is obtained by replacing
the upper five pendulum masses with seismic filters, shown on fig.4.6, each with a set of concentric
cantilever blade springs with low stiffness [90]. The blades support the next mechanical filter
through a 1 m-long steel wire, forming a chain of low frequency oscillators also in the verticaldirection. The blades work in parallel with a magnetic anti-spring system [90], assembled on eachfilter: it acts with two magnets on the steel wire as a spring with negative elastic constant. The
anti-spring system decreases the stiffness of the chain, thereby all vertical modes are below 2 Hz.
The two lower stages of the Superattenuator form the payload, shown on fig.4.7, composed ofthe mirror and two other mechanical elements [91]. The role of the payload is to compensate theresidual seismic noise and to steer the mirror, maintaining the relative position of the interferometer
optics. It is essentially a double-stage branched system with the marionette as first pendulum, anda recoil mass and the mirror as branches suspended from the marionette. The marionette is ananvil-shaped mechanical element used to steer the mirror and its recoil mass. This latter is acylindrical mass of metal that surrounds the mirror. The position of the mirror is controlled by
two sets of coil-magnet actuators, placed at the level of the marionette and of the recoil mass.
In initial Virgo the suspension wires of the mirror were made of steel and were wrapped around
its body in a cradle configuration. In Virgo+, to decrease suspension thermal noise, the wires havebeen replaced with fibers of fused silica with low mechanical bulk loss [54]. For steel wires, infact, the loss Φbulk (f ) is in the range of 10−4 –10−3 at 100 Hz, whereas for fused silica fibers theloss is Φbulk (f ) ∼ 10−7 at 100 Hz [31]. Fibers have a diameter of ∼ 280 µm and are fixed into
silica supports, the ears, which are chemically bonded to the lateral surface of mirrors to realize amonolithic suspension [31]. This technology is now tested in Virgo+, and will be used in AdV [69].
The design of the Superattenuator has a well-defined symmetry around the axis of suspension:
the inverted pendulum and the chain of filters are radially symmetric, and the elements of the84

4.5. Suspensions

top stage

inverted
pendulum~8m
filters

steering filter
marionette
mirrorFigure 4.6: Scheme (left) and picture (right) of a Superattenuator of Virgo.

85

Chapter 4. The Project of Advanced Virgomarionette

recoil mass

mirrorFigure 4.7: Scheme of the last suspension stage for the mirror: the Virgo payload, from [91].

864.5. Suspensions

marionette
recoil mass

marionette

mirrorrecoil masscompensation
plate

mirrormirrorFigure 4.8: Scheme of the last suspension stage for the mirror: the AdV payload (left) from [69], and thedesign of the mirror recoil mass with the compensation plate (right) from[92].

payload are suspended with the center of mass on-axis. This minimizes the coupling between the
degrees of freedom of the elements of the payload, and simplifies the control of mirror position.
We will see in chapter 6 that the symmetry of the Superattenuator (and in particular the on-axis
suspension of the payload) will be a critical element in the choice of the optical layout of therecycling cavities.
The design of the inverted pendulum and of the chain of filters will be substantially unchangedin AdV. The main modification will concern the control of the system of the suspension point at
the top stage: presently the inverted pendulum is controlled in four degrees of freedom only (thethree translations and the rotation around the vertical axis), whereas in AdV the Superattenuator
design will include the control of the tilt around the two horizontal axes [69].
More considerable changes will concern the payload, whose new design is shown on fig.4.8.
Possibly a new mechanical element will be added: a recoil mass for the marionetta, which could
greatly simplify the installation of the payload in the vacuum tank which host the mirror [69].
The design of the recoil mass of the arm cavity input mirrors will be adapted to accomodate thesuspension of the compensation plates [92].

87

88

Chapter 5

Stable Recycling Cavities for
Advanced Virgo
The experience with initial Virgo and LIGO demonstrated that the control of the detector canbe severely affected by higher-order mode contamination in the radio-frequency sidebands, due tothermal effects in the power recycling cavity. This happens because power recycling cavities of
initial detectors like Virgo or LIGO are degenerate, i.e. higher-order modes can resonate at thesame time with the fundamental mode, degrading the shape of the sideband fields.
This problem will even be enhanced for advanced detectors, where the increase of the beam sizein the arm cavities, necessary to decrease mirror thermal noise, will also increase the degeneracy.
Furthermore, advanced detectors will use higher input power, with consequent rise of thermal
effects. Also the signal audio-frequency sidebands are expected to be concerned by this issue, due
to thermal effects and mirror figure errors in the signal recycling cavity.
To solve the problem of mode contamination, aLIGO decided the implementation of nondegenerate or stable recycling cavities [93, 94]. A cavity is non-degenerate when its resonancelinewidth is much smaller than the frequency spacing of higher-order modes, so that when thefundamental mode resonates in the cavity, higher-order modes cannot.
The scope of our work within the Optical Design and Simulation (OSD) group was the study
and design of non-degenerate cavities for the requirements and constraints of AdV. We will showthat the recycling cavity layout should have a folded optical path with three mirrors, where two ofthem form a telescope for focusing the beam inside the cavity. Therefore two additional mirrors per
recycling cavity must be suspended. This layout must fit the present infrastructure and suspension
system of Virgo, which suffer of lack of flexibility: the Superattenuator was designed to suspend asingle mirror, in a vacuum chamber of narrow space. This technical constraint limits the choice ofsuitable layout configurations. Furthermore, the construction budget of the project does not allow
for large modifications of the infrastructure.
To design a layout which satisfies these constraints we developed a set of optical simulations.
We drew and studied several configurations (including some that had never been considered for
aLIGO), and finally we chose the vertical layout. In collaboration with the Thermal Compensation
System (TCS) group of Virgo, we also simulated the optical performances of this layout with respect
to realistic thermal effects and mirror figure errors in the power recycling cavity of AdV. The results
showed the expected sideband higher-order mode suppression in the cavity.
Only recently, budget and time-schedule constraints and progress in the development of the89Chapter 5. Stable Recycling Cavities for Advanced Virgo

TCS led to reconsider degenerate cavities. Even if finally the non-degenerate layout will not be
implemented in AdV, we believe that these studies will be useful for the optical design of thirdgeneration detectors.
In this chapter we will treat the design of the non-degenerate recycling cavities of AdV. Section
5.1 will introduce the properties of laser modes and resonators, with particular attention to therelevance of the Gouy phase in the resonance of higher-order modes. We will describe in section
5.2 the strong motivations towards the implementation of non-degenerate cavities. In section 5.3
we will present the conceptual design of non-degenerate recycling cavities, and we will show insection 5.4 the technical constraints of AdV concerning this layout. The work of optical simulationof recycling cavities will be presented in the next chapter.

5.1

Laser Modes and Resonators

We will introduce in this section the formalism and the fundamental notions used to describe laser
beams, optical resonators and their coupling. We will focus our attention especially on higher-ordermodes, Gouy phase and mode degeneracy inside a cavity.
Let E(~r, t) be a field component, function of space ~r and time t, which satisfies the wave equation


∇2 −

1 ∂ 
E(~r, t) = 0 ,c2 ∂t2(5.1)

where c is the speed of light. Since we are interested in the spatial part of the field, we will assume
that the component can be factored as E(~r, t) = u(~r)f (t), where the dependence on time and spacevariables has been separated. Then, the function u(~r) must satisfy the Helmoltz equation∇2 + k 2 u(~r) = 0 ,(5.2)
k = 2π/λ being the propagation constant and λ being the laser wavelength. For the beam travelingin the z direction, we can writeu(~r) = ψ(x, y, z)e−ikz ,(5.3)
where ψ(x, y, z) is a complex function which represents the non-uniform amplitude distribution andthe phase front curvature of the beam. We will assume that the paraxial approximation holds:∂2ψ∂ψ<< 2ik
,
2
∂z
∂z

∂2ψ
,
∂x2

∂2ψ
,
∂y 2

(5.4)

i.e. ψ is assumed to vary slowly across the direction of propagation. Thus the Helmoltz equation
can be approximated as∂2ψ ∂2ψ
∂ψ
+
− 2ik
=0,
(5.5)
2
2
∂x
∂y
∂z
where
r
ψ=

h −r2 i
h  
i
2P 1
r2 
exp 2
exp −i k z +− ΦG (z)π w(z)
w (z)
2R(z)

90

(5.6)

5.1. Laser Modes and Resonators

z

W
W0

θ
R

Figure 5.1: Parameters of a Gaussian beam.

is one of the possible solutions [95]. In eq.(5.6), P is the beam power, w(z) is the beam radius,R(z) is the radius of curvature of the phase front and r2 = x2 + y 2 . The radius w(z) is a measureof the decrease of the field amplitude with the distance from the axis, and indicates the distanceat which the amplitude is 1/e times the maximum. The beam radius and the front curvature havethe following dependence on z:r
 z 2w(z) = w0 1 +
,
(5.7)
zR
h
 z 2 iR.
R(z) = z 1 +
z

(5.8)

w0 = w(z = 0) is the beam size at its focus, called the beam waist, and zR = πw02 /λ is theRayleigh range, a characteristic length equal to the distance√ from the beam waist (in the propagation
direction) where the beam radius is increased by a factor 2. The width of the beam is an increasing
function of z, whose evolution is shown on fig.5.1. The beam has a flat wavefront at the waist (z = 0,
yielding infinite radius of curvature), which becomes more and more spherically curved as long as
it propagates.
The Rayleigh range determines the depth of focus, which is the length on the optical axis where
the beam is approximately collimated. For z << zR (in the proximity of the waist), the beam issaid to be in near-field regime, whereas for z >> zR the beam is in far-field regime and starts todiverge. The beam contour w(z) across z is a hyperbola with asymptotes inclined to the axis at
an angle called beam divergence,
λ
θ=
.
(5.9)
πw0
The term ΦG ≡ arctan(z/zR ) in eq.(5.6) is the Gouy phase, which represents the phasedifference with respect to the spherical wave. As shown on fig.5.2, the phase shift increases very
rapidly around the beam waist, in near field, whereas in far field the increment is almost null: wewill see in section 5.3 that this property will be crucial for the design of stable recycling cavities.

91

Chapter 5. Stable Recycling Cavities for Advanced Virgo

100
80

Gouy phase [deg]

60
40
20
0
−20
−40
−60
−80
−100
−50

−40

−30

−20

−10

0
z/zR

10

20

30

40

50

Figure 5.2: Gouy phase shift versus the z/zR ratio.

Eq.(5.6) describes a beam with a Gaussian intensity profile, called fundamental mode or TEM00 ,presently used in gravitational wave interferometers. It is characterized by the complex parameterq, defined as
1
1
λ
=
−i 2
,
(5.10)qR(z)πw (z)
or, in terms of distance from waist and Rayleigh range,q = z + izR .

(5.11)In the reference with the origin in the beam waist, z is negative for a beam converging to the waist,and positive for a beam diverging from it. The knowledge of the q parameter allows to computeall the other beam parameters like radius, front curvature and Gouy phase.
Higher-order modes
Eq.(5.5) has also other solutions than eq.(5.6), with similar properties. These form a complete
and orthogonal set of functions, called transverse electromagnetic modes of propagation (TEM) or
higher-order modes. Every arbitrary distribution of monochromatic light can be expanded in termsof these modes. In Cartesian coordinates (x, y, z), higher-order modes write [95, 96]q
h 2 2i √   √ 
x +y
y 2
x 2
P
1
umn (x, y, z) =exp−Hm w(z) Hn w(z)w2 (z)
πm!n! 2N −1 w(z)(5.12)
h  

i
2

2

+y
× exp −i k z + x2R(z)− (m + n + 1)ΦG (z),where Hk (x) are the Hermite polynomials, m and n are the transverse mode numbers and N = m+n
is the order of the mode. Eq.(5.12) defines the Hermite-Gauss modes (HGmn ), whose transverse
intensity patterns Imn (x, y) = |umn (x, y, z)|2 are shown on fig.5.3.

92

5.1. Laser Modes and Resonators

Figure 5.3: Transverse intensity patterns of HGmn modes, from [97].

For a system with cylindrical symmetry and coordinates (r, θ, z), higher-order modes are LaguerreGauss modes (LG`p ) with radial and azimuthal indices p and `, respectively, and order N = 2p + |`|.
Their complex amplitude is given by [95, 96]u`p (r, θ, z) =

q

h

2P 1
−r2
π w(z) exp w2 (z)

i

×

q

p!
(p+|`|)!



2r2
w2 (z)

|`|/2

|`|

Lp



2r2
w2 (z)

(5.13)


i
h
i
h  
r2
× exp −i k z + 2R(z)− (2p + |`| + 1)ΦG (z) × exp −i`θ ,where L`p (x) is the Laguerre generalized polynomial. Fig.5.4 shows the transverse intensity patterns
Ip` (r, θ) = |u`p (r, θ, z)|2 of the first LG`p modes up to N = 9.
LG`p beams with ` 6= 0 have p + 1 radial nodes and spiral phase fronts with spherical curvature,carrying orbital angular momentum of `~ per photon [98]. These are helical LG`p modes. The
spiralling phase distribution implies the presence of a phase singularity on the optical axis, also
called an optical vortex, and is characterized by a locally vanishing intensity. By changing themode base, the field of LG`p modes can also be written as(u`p ± u−`
p )=

q

h

2P 2
−r2
π w(z) exp w2 (z)

i

×

q

p!
(p+|`|)!



2r2
w2 (z)

|`|/2

|`|

Lp



2r2
w2 (z)

(5.14)

i  cos(lθ)
h  r2− (2p + |`| + 1)ΦG (z) ×× exp −i k z + 2R(z)sin(lθ),which are cosine/sine LG`p modes. Since helical modes are the most studied in literature, we willrefer only to this kind of modes throughout the rest of this thesis work.

93Chapter 5. Stable Recycling Cavities for Advanced Virgo

Figure 5.4: Transverse intensity patterns of LG`p modes.

The parameters w(z) and R(z) which appear in eqs.(5.12) and (5.13) are the same for all modesof the same set, implying that beam radius and phase-front curvature are the same and change
in the same way for modes of all orders. Then all modes can be characterized through the same
complex parameter q, and the fundamental Gaussian mode is a particular HGmn or LG`p mode withm = n = 0 or p = ` = 0, respectively. It is worth nothing that for higher-order modes, the value of
the Gouy phase shift is a function of the mode order N .
Fabry-Perot optical resonators
The beam resonance in a Fabry-Perot cavity is allowed if the cavity is stable. For a simple cavity
composed of two spherical mirrors M1 and M2 spaced by a distance L, like the one shown on fig.2.7,
the stability condition is satisfied if the laser that bounces back and forth between the mirrors of
the resonator experiences a periodic focusing action. Only certain ranges of values for the radiiof curvature (given L) would determine a stable resonator, in which periodic refocussing of theintra-cavity beam is produced. If R1 and R2 are the radii of curvature of the cavity mirrors, thestability condition translates in a geometric condition given by [95]

L 
L
0< 1−
1−
<1.
R1
R2

(5.15)On the contrary, if the cavity is unstable, the beam size will grow indefinitely.
If a cavity is a stable resonator, there exist a set of cavity eigenmodes (all with the same
parameters) which have a self-consistent field configuration and form the axial standing waves of94

5.1. Laser Modes and Resonators

the resonator: if an eigenmode can be represented by a beam propagating back and forth between
the mirrors, its beam parameters have to be the same after a complete round trip in the cavity.
The cavity eigenmodes are important since they form a complete base, which can be used
to perform the expansion of any arbitrary field in terms of the resonating modes. This base iscompletely determined by the size and position of the cavity waist, which in turn are functions of
the cavity geometry through R1 , R2 , L. The cavity waist size is given by [95]
 λ 2 L(R − L)(R − L)(R + R − L)
1
2
1
2
w04 =
,
(5.16)π(R1 + R2 − 2L)2and the waist distance from mirror M1 is
d1 =L(R2 − L).
(R1 + R2 − 2L)(5.17)

The waist position with respect to M2 can be computed from eq.(5.17) by replacing R2 → R1 andvice-versa (or alternatively by imposing d2 = L − d1).
The formalism developed for spherical resonators (i.e. for resonators with spherical mirrors)
can be also extended to the case of astigmatic resonators. This is of crucial importance since, as we
will see further in this chapter, the recycling cavities of AdV will be astigmatic. In the astigmatic
resonator, the incidence of the beam on the mirror is non-normal and the beam senses two different
radii of curvature for the two transverse directions (x, y). If R is the radius of curvature of themirror and θ is the incidence angle between the beam and the normal direction to the mirror, thebeam is focused by a radius R cos θ in the tangential plane that contains θ, and by a radius R/ cos θin the perpendicular sagittal plane. Thereby the fundamental mode that resonates in the cavity
can be expanded in the transverse (x, y) plane as
s
 1
 1
h

2P
k 
k 
ΦGx ΦGy iψ=exp −x2 2 + i− y2 2 + i+ i −kz ++,(5.18)
πwx wy
wx
2Rx
wy
2Ry
2
2
The two planes have independent sets of parameters: beam radius, Gouy phase, waist size andposition, Rayleigh range. Since wx 6= wy , the beam spot of the astigmatic fundamental mode
is elliptical. According to this formalism, the propagation of the astigmatic eigenmode in theresonator can be treated for the two planes independently (as long as both are stable), as it would
happen for two distinct beams.
Gouy phase and cavity degeneracy
The laser beam is resonant inside the cavity when the total phase shift experienced by the light in
a complete round trip is equal to 2π. For a generic mode of order N , the resonance condition is
2kL − 2(N + 1)ΦG = 2π ,(5.19)

which after some simple algebra becomeshN +1 iνN = ν0 1 +ΦG ,π(5.20)

where ν0 is the free spectral range defined in eq.(2.77) and νN is the resonance frequency of theeigenmode of order N (which is equal to m + n or 2p + |`| for HGmn or LG`p modes, respectively).

95Chapter 5. Stable Recycling Cavities for Advanced VirgoThus the resonance frequency is order dependent, and the frequency spacing between two modesof order N and N 0 is∆N∆ν = νN 0 − νN = ν0
ΦG ,(5.21)πwith ∆N = N 0 − N , and (for a given ν0 ) it depends on the value of the Gouy phase. If the phaseis too small (ΦG ' 0), or too large (ΦG ' q2π with q ∈ N), the frequency spacing ∆ν between themodes is approximately equal to a multiple of the free spectral range and the resonance frequencies
of all eigenmodes are very close to that of the fundamental mode. Thus if the cavity linewidth islarge, all eigenmodes can potentially resonate at the same time in the cavity, if excited. In thiscase the cavity is said to be degenerate or marginally stable, since the Gouy phase shift can also be
expressed in terms of the geometrical parameters of the cavity,
ΦG = cos−1p
(1 − L/R1 )(1 − L/R2 ) ,(5.22)

and when ΦG ' 0, according to eq.(5.15), the resonator is also close to instability.
The optical gain (or build-up) of higher-order modes could be suppressed by increasing the Gouy
phase shift in the cavity: according to eq.(2.74) for the resonant field, when the fundamental modeis resonant (N=0), the build-up for a mode of order N 0 is
Gmn (ΦG ) = 

tRM

2
h
i
 ,
1 − rRM rF P · exp −i2N 0 ΦG

(5.23)

which strongly decreases for increasing values of ΦG .

5.2

Motivations for Stable Cavities

In a degenerate recycling cavity, perturbations of the cavity geometry can transfer power fromthe fundamental TEM00 mode to higher-order modes. In general, to first order, the resonanceof higher-order modes has little effect on the carrier field, since it is resonant also inside the arm
cavities. Instead, as can be demonstrated [99], higher-order mode contamination degrade the modematching of sideband fields to the recycling cavity: their optical gain is reduced, and their spatial
distribution is aberrated [100].
The power recycling cavity of Virgo is degenerate, since the beam only gets ΦG = 0.5 deg1
inside it. The resonance linewidth of the cavity for sideband fields is 166 kHz, thus according toeq.(5.21) the first higher-order modes with ∆N = 5 can resonate with the fundamental mode.
Because of thermal effects, mode contamination perturbed the development of a stable control
system and slowed down the commissioning phase of Virgo [82]. During the VSR2 data taking run,the recycling gain of radio-frequency sideband fields was heavily reduced because of thermal effects,
and the shape of sidebands was highly aberrated [101], as shown on fig.5.5. A similar problem has
been encountered also for the degenerate power recycling cavity of LIGO [94].
1

The Gaussian eigenmode of the arm cavities of Virgo (plano-concave resonators with R1 = ∞ and R2 = 3.6 km)
has qarm = i1342 m and w0 = 2.1 cm on the input mirror, where (ΦG )arm = 0. The beam propagates across 12 m in
the recycling cavity, so that on the recycling mirror it has qRM = (12 + i1342) m, with z/zR ' 9 · 10−3 .

96

5.2. Motivations for Stable Cavities

Virgo
AdV

ΦG [deg]
0.5
0.1

∆N
5
17

Table 5.1: Comparison of marginally-stable cavities in Virgo and AdV.

Figure 5.5: Amplitude images of lower (left panel) and upper (right panel) radio-frequency sidebands inthe power recycling cavity of Virgo during VSR2, from [101]: because of mode contamination, the aberrated
shape is very far from the expected Gaussian amplitude.

97

Chapter 5. Stable Recycling Cavities for Advanced Virgo

Figure 5.6: Recycling gain of radio-frequency sidebands against the equivalent focal length due to thermal
lensing: AdV with marginally-stable power recycling cavity (blue curve), Virgo (red curve), AdV with
non-degenerate power recycling cavity (green curve), from [102].

The impact of degeneracy will be even larger in advanced detectors, if marginally stable recycling
cavities are used. In a cavity like the power recycling of Virgo, the eigenmode qarm 2 of the arm
cavities of AdV would get ΦG = 0.1 deg, which is even 5 times lower than the Gouy phase of Virgo.
In this case, the first higher-order modes with ∆N = 17 could resonate with the fundamental mode.
Table 5.1 summarizes the comparison between Virgo and AdV with marginally-stable cavities.
Therefore if AdV will have degenerate cavities, the relevance of thermal effects will be larger
than in Virgo. Fig.5.6 shows the predicted degradation of radio-frequency sidebands with thermal
effects in AdV, compared to that of Virgo: here we see a marked difference in the behavior, due tothe larger beam size in the arms of AdV. For a thermal lens with equivalent focal length of 200 km,
the recycling gain in AdV decreases down to about 35%, whereas for the same thermal lens, thegain in Virgo is above 95%. This is a strong motivating factor in the decision to use non-degeneratecavities.
2

In the baseline design of AdV, the arm cavity input mirror is a plano/concave thick diverging lens for thearm cavity beam: on its plane surface, the one facing the beamsplitter, the parameter of the beam is qarm =(960.830 + i134.636)m, with z/zR ' 7.

98

5.2. Motivations for Stable Cavities

Mode contamination will be a concern also for signal recycling. If we assume that at theinterferometer output the mode-cleaner filters out all higher-order modes in the carrier and signal
audio-sidebands fields, the power fluctuation due to shot noise is proportional to the square rootof the output carrier power in the fundamental mode:qδPshot ∝ P00|carr .
(5.24)The signal power comes from the beating of the signal sideband against the carrier, thus considering
only the fundamental mode for both we can writeqqPS ∝ P00|carr P00|sb .
(5.25)
By assuming also that shot noise dominates, we have that
SN R =qPS∝ P00|sb ,δPshot(5.26)

i.e. the signal-to-noise ratio is directly proportional to the signal sideband power in the fundamental
mode. If the signal recycling cavity is degenerate, thermal effects might reduce the signal-to-noise
ratio of potential detections. Moreover, aLIGO showed that also mirror figure errors can induce a
signal loss [94].
Non-degenerate recycling cavities show definitely a better behavior with respect to thermal
effects and mirror figure errors, since they suppress efficiently the resonance of higher-order modes.
Figs.5.7 and 5.8 show the build-up of the first higher-order modes of radio-frequency sidebands inthe recycling cavities of AdV, against the Gouy phase: it is possible to tune the value of the phase(20 ≤ ΦG ≤ 30 deg, for example) so that higher-order modes are suppressed of several orders ofmagnitude. This yields a robust optical configuration, as shown on fig.5.6: with non-degeneratecavities, the gain of radio-frequency sideband fields in AdV stays above 90%, up to a thermal lensof 20 km.
The implementation of non-degenerate recycling cavities in AdV could allow to achieve a stable
optical configuration and a robust control, and the detector could benefit of a higher signal-to-noise
ratio of detected signals. The stable configuration might ease and speed up the commissioning in
the early phase of operation, and will be necessary for the full exploitation of the potential of thedetector. We will see later in this chapter and in the following one that non-degenerate cavities
have also other important advantages:• they allow the simple extraction of secondary pick-off beams for the sensing and control
system of the detector. The beam size attains few mm inside the non-degenerate cavities
and about 1 mm on the recycling mirror. Thus secondary beams are small, and can be easily
separated from the main beam through the insertion of microscopic wedges (for example in the
beamsplitter) or from the reflection on tilted surfaces (for example by tilting the compensation
plates). This is not possible with marginally-stable cavities, where the beam always has aradius of ∼ 5 cm: thus large wedges (degrees) are needed to isolate secondary beams, which
can induce aberrations in the main beam
• they simplify the design of mode-matching telescope of the injection system: with nondegenerate cavities the telescope should match a beam of about 1 mm to the 2-mm beam99

Chapter 5. Stable Recycling Cavities for Advanced Virgo

0

10

−1

N’ = 1N’ = 2N’ = 3N’ = 4

Norm. optical gain

10

−2

10

−3

10

−4

10

−3

10

−2

10

−1

10

0

10
Gouy phase [deg]

1

10

2

10

Figure 5.7: Normalized build-up of sideband higher-order modes against the Gouy phase shift of the beam,in the power recycling cavity of AdV.

0

Norm. optical gain

10

−1

10

N’ = 1N’ = 2N’ = 3N’ = 4

−2

10

−3

10 −3
10

−2

10

−1

10

0

10
Gouy phase [deg]

1

10

2

10

Figure 5.8: Normalized build-up of sideband higher-order modes against the Gouy phase shift of the beam,in the signal recycling cavity of AdV.

100

5.3. How to build a Stable Cavity

inside the Faraday isolator, so a magnifying factor of 2 must be achieved. With marginallystable cavities, the telescope should match a beam of 5 cm and thus achieve a magnifying
factor of 25: this requires long telescopes and large mirrors to be placed on benches of limited
size. The use of a lens behind the power recycling mirror might be possibly necessary with
marginally-stable cavities, like it is presently in Virgo, with the risk of having problems of
spurious reflections on the bench. The same considerations hold also for the output telescopethat matches the beam to the detection photodiode• we can use small recycling mirrors of 10-cm diameter, which can be easily installed andpossibly replaced (to change the finesse of the signal recycling cavity, for example)• they allow to adjust the distances between the recycling cavity mirrors if errors in their radiiof curvature are present, to optimize the matching of the beam to the cavities. This is notpossible with marginally stable cavities.

5.3

How to build a Stable Cavity

In order to have non-degenerate recycling cavities in AdV, the resonant beam must undergo asufficiently large Gouy phase shift. We can estimate the minimum Gouy phase needed by making
the following assumptions:• cavities will have a length of the order of 10 m
• the first higher-order mode (∆N = 1) should be sufficiently spaced from the fundamental
mode. Three times the cavity linewidth, for instance, is a reasonably (arbitrary) safe spacing.
Let us consider the signal recycling cavity, which has a larger linewidth (FWHM = 536 kHz): then,by means of eq.(5.21), we can compute the minimum phaseΦG ≥ 3π FWHM' 19 deg.
ν0 ∆N(5.27)We also know from figs.5.7 and 5.8 that 20 ≤ ΦG ≤ 30 is a suitable choice, since it would allow
to suppress the first higher-order mode of several orders of magnitude. The phase should not belarger, in order to not excite accidentally the resonance of higher-order modes. In AdV, the reference value of the Gouy phase is ΦG = 20 deg.
To cumulate such a large phase shift, we need a new design of the recycling cavities:a) we can discard immediately the option of a layout with a single recycling mirror, shown on
fig.5.9a. In this case, in fact, km-scale recycling cavities would be needed: the arm cavity
eigenmode only gets ΦG = 3 deg over 500-m propagation. This is not feasible for AdV.
b) To achieve the needed phase shift we have to modify the parameter of the arm cavity eigenmode, by introducing one or more focusing elements in the layout of the recycling cavities,
like it is shown on fig.5.9b. We can use a lens (which might be also incorporated in the arm
cavity mirror) to increase the divergence of the beam and shorten its Rayleigh range, thus the101

Chapter 5. Stable Recycling Cavities for Advanced Virgo

accumulation of Gouy phase shift can occur on much shorter distances. The qarm parameterof the beam passing through a generic lens of focal length f changes according to [95]
1
1
1
=
− ,
q0
qarm f(5.28)

0 for the beam before
having defined the complex parameters qarm = z + izR and q 0 = z 0 + izR
and after the lens, respectively. Eq.(5.28) implies that for the converging lens of fig.5.9b, thefollowing equations hold:f (zf −z 2 −z 2 )z 0 = (f −z)2 +z 2RR(5.29)
f 2 zR0zR = (f −z)2 +z 2 .
RSince the eigenmode of the arm cavity is in far field, eq.(5.29) further simplifies toz
z 0 ' ff−z

(5.30)
0 ' f 2 zR .
zRf −z
0 ' z (f /z)2 . This is a reasonable approximation
If in addition z >> f , then z 0 ' −f and zRR
since we need to focus the beam over a distance of the order of f ∼ 10 m in the recycling
cavities, while z is of the order of 103 m. In synthesis, by inserting a lens we focus the beamover a distance approximately equal to f , and the Rayleigh range of the beam is shortened
0 ' 1.5 cm and the Gouy phase canby a factor (f /z)2 . By assuming f = 10 m, it becomes zR
be accumulated over a distance of the same order.

This layout has an inconvenient: it focuses the beam spot on a very small radius. If f ∼ 10,then the beam radius inside the Rayleigh range is w ∼ 102 µm. In the baseline design of
AdV, the power stored in the power recycling cavity will be 5 kW, so that the power density
on the recycling mirror will be about 16 MW/cm2 , higher than the usual damage thresholdof fused silica mirrors that will be used in advanced detectors (10 MW/cm2 ). Furthermore,this power concentrated on a small area can also cause strong thermal effects in the power
recycling mirror.
c) So we should introduce a second diverging lens in the layout, as shown on fig.5.9c, to enlarge
the beam size on the power recycling mirror by slightly defocusing the beam. With this setup
mm-size beam radius can be achieved, with still a reasonably Rayleigh range for the beam, of
the order of 1 m. It is then only necessary to tune the position of the recycling mirror within
the Rayleigh range of the beam, so to have ΦG = 20.
d) Finally, it is preferable to replace the lenses of the layout with mirrors of equivalent focal
lengths, in order to avoid spurious reflections of the beam from the anti-reflective coatedsurfaces of refractive components. Spurious reflections might recombine with the main laser
and become a source of noise, so whenever possible it is better to remove their potential sources
rather than blocking them after. Moreover, the use of mirrors also allows the extraction of
secondary pick-off beams from the rear surface of the optics. By introducing the mirrors in
the place of the lenses, the optical path inside the cavity becomes folded, completing theconceptual design of non-degenerate recycling cavities shown on fig.5.9d.

102

5.3. How to build a Stable Cavity

a)

RM

RM

b)

RM

c)

d)

P/S-RM1

ITM

ETM

ITM

ETM

ITM

ETM

P/S-RM2
w'0

w0
ITM

ETM

P/S-RM3

Figure 5.9: Evolution of the design concept of non-degenerate recycling cavities for AdV.

103

Chapter 5. Stable Recycling Cavities for Advanced Virgo

P/S-RM1

P/S-RM2

P/S-RM3

Figure 5.10: Evolution of the Gouy phase inside the non-degenerate layout.

In both cavities, P/S-RM3 (power/signal recycling mirror 3) is the focusing element of the layout,
so it is concave, and makes the beam converge. P/S-RM2 is used to reduce the divergence angle ofthe beam, in order to collimate it over a longer distance into the cavity. It is a diverging mirror,
thus it is convex. Mirrors 2 and 3 form a telescope that changes the beam divergence. The cavity
coupler is P/S-RM1, which is also convex since it matches a converging beam.
The beam waist w0 is behind P/S-RM1, but there is also a smaller virtual waist w00 behind P/SRM2, with a very short Rayleigh range. Figs.5.10 and 5.11 show the evolution of the Gouy phaseand of the beam radius inside the non-degenerate layout3 , respectively. Between the arm cavity
input mirror, P/S-RM3 and P/S-RM2 the beam is in far field and does not get sufficiently close
to its waist, so that its Gouy phase shift is negligible. In this layout, the phase shift is cumulated
mostly between P/S-RM2 and P/S-RM1. The beam size is effectively reduced to few mm inside
the cavity.
Because of the folded optical path, the incidence angle on mirrors 2 and 3 is not normal, so
the eigenmode of the recycling cavities is astigmatic. We will see in the next chapter that thisaberration may yield non-negligible coupling loss for the input beam to the recycling cavities.

5.4

Design Constraints

The non-degenerate layout of fig.5.9d has a crucial feature: two extra-mirrors per recycling cavity
have to be suspended. Since AdV will use the present vacuum infrastructure of Virgo, this posesa stringent constraint to the choice of the parameters of the layout: mirrors must be placed in thevacuum tanks, shown on fig.5.12, which have been realized to host a single Superattenuator for the3

We show here the parameters of the beam inside the vertical layout, which will be presented in chapter 6.

1045.4. Design Constraints

P/S-RM3

P/S-RM2

P/S-RM1

Figure 5.11: Evolution of the beam radius inside the non-degenerate layout.

suspension of only one mirror.
The vacuum system of Virgo lacks of flexibility, since it was not conceived to allow for further
modifications after its construction: each vacuum tank can be displaced by few tens of cm, otherwise
heavy building works are required. Furthermore, the infrastructure budget of the project is limited,
so little modifications can be done to the system.
There are also optical constraints to the choice of parameters, which concern• coupling: a set of radii of curvature and incidence angles should be found to make astigmatism
negligible, so to have the maximum beam coupling to the cavity• beam size on power recycling mirrors: since a few kW will circulate in the power recycling
cavity, we aim to have a beam radius on P/S-RM1 as large as possible• modulation frequencies: the radio-frequency sidebands of the control system have to be transmitted through the input mode-cleaner and must resonate in the recycling cavities. If fmod
is the modulation frequency used for the generation of sidebands, and ν0IMC , ν0RC are the free
spectral ranges of the input mode-cleaner and of the recycling cavities, respectively, thenfmod = NIM C ν0IMC ,NIM C = 2k, ∀k ∈ N

fmod = NRC ν0RC ,NRC = 2k, ∀k ∈ N.

(5.31)

This implies that NRC = NIM C LRC /LIM C , i.e. the recycling cavity length LRC must be aneven sub-multiple of the input mode-cleaner lenght LIM C = 144 m

105Chapter 5. Stable Recycling Cavities for Advanced Virgo

Superattenuatorbeamsplitter
input mirror(west)

input mirror(north)signal recycling
detection
injection

power recycling
mirrorFigure 5.12: Vacuum system of the central part of the Virgo interferometer (left) and tower vacuum enclosure
to host a Superattenuator and a mirror (right).

1065.4. Design Constraints• angular-to-longitudinal couplings: the curved mirrors of the recycling cavities can couple
transversal translation to displacement noise on the optical axis. To minimize this effect the
RoCs of mirrors should be as large as possible.
Because of all these constraints, the choice of the layout parameters is a complex issue. To solve
this problem we defined a general design procedure, based mainly on a set of optical simulation
tools that we developed. We could use this procedure to design several alternative layouts, which
will be presented in the next chapter.

107

108

Chapter 6

Simulation of the Non-Degenerate
Recycling Cavities of Advanced Virgo
We have seen in the previous chapter that the layout of the non-degenerate recycling cavities of
AdV should have a folded optical path with three mirrors, whereof two should form a telescope forthe focusing of the beam. We will present in this chapter the simulation work we carried out todetermine the design of the the non-degenerate recycling cavities.
During our work we have drawn and characterized many configurations, here we will present themost important four, following an historical perspective. We will see that the design process has
been heavily conditioned not only by the optics requirements, but also by the need of integrating
the optical layout with other subsystems like vacuum and suspensions. Finally, the configuration
called vertical layout has been retained.
The design and the study of the non-degenerate cavities has been achieved using two simulation
tools. First, a ray-matrix simulation has been developed to determine the set of parameters (radiiof curvature, incidence angles) of the cavity design, and to characterize its optical performances(astigmatism and coupling of the beam to the cavity) and tolerances. Then a Fast Fourier Transform(FFT) simulation has been used to study the astigmatism of the cavity design and its response tothermal effects and mirror figure errors.
In the ray-matrix simulation, the propagation of the beam inside the cavity is described bya product of 2 × 2 matrices which represent the optical elements of the system. This formalism
allows to compute simply and rapidly the behavior of the non-degenerate cavity as a function of
the layout parameters, especially when the number of parameters to be scanned is large.
In the FFT simulation the cavity field is expanded in the Fourier space of spatial frequencies,
where the propagation is equal to the product of the input field times a propagator of simple
mathematical form. The FFT code allows the simulation of the coupled system of the arm andpower (signal) recycling cavities with carrier and sideband fields, and the inclusion of phase maps
to simulate the defects of optics. It is thus suitable for the simulation of complex physical effects.
The FFT code we used has been developed inside the LSC collaboration by H. Yamamoto [103].
We will begin the presentation of our work by illustrating the general design procedure that
we used, then we will describe the basic formalism and the steps of the simulations. The opticalconfigurations that we studied will be presented subsequently.

109Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

6.1

Design Procedure and Simulation Tools

The design procedure to choose the parameters of the recycling cavity layout can be summarized
as follows:
1. choice of lengths, Gouy phase and of beam size on the recycling mirror
2. computation of the radii of curvature (RoCs) through an approximated analytical formula
3. computation of the exact RoCs through a loop which explores the space of parameters
4. computation and (possibly) optimization of the incidence angles
5. characterization of the layout: this step includes the computation of recycling cavity eigenmode astigmatism, beam coupling to the recycling cavity, parameter tolerances for cavity
stability
6. simulation of the response of the layout to mirror figure errors and thermal effects.
To perform steps 1 to 5 we developed a Matlab-based ray-matrix code. We chose to implement the
ray-matrix formalism because it allows the simple and fast computation of the stability conditionand of the eigenmode of the cavity, as a function of the parameters of the layout.
The computation of astigmatism and beam coupling of the eigenmode in step 5 is also performedin parallel with the FFT code SIS (Stationary Interferometer Simulation) [103], developed by H.
Yamamoto for the LSC Collaboration. We decided to use SIS in step 5 in order to study the
optical performances of the layout with an independent simulation, based on a completely different
formalism. We will see later in this chapter that the ray matrix code and SIS can give different
results if carrier or sideband fields are considered. This occurs because SIS can simulate theresonance of the carrier field in the coupled cavities of arm and recycling, whereas the ray-matrix
code assumes that the two cavities are independent.
As a consequence, SIS is used to compute the astigmatism and the recycling gain of the carrier inthe coupled cavities. For the radio-frequency sidebands, which resonate only in the power recycling
cavity, SIS and the ray-matrix code agree (in the limit of small astigmatism). Thus we computethe coupling of radio-frequency sidebands with the ray matrix code. We use SIS also to perform
step 6, since we want simulate physical effects in the mirrors that cannot be implemented into the
ray-matrix code.
In the following we will introduce the ray matrix formalism, and we will detail steps 1 to 5 ofthe design procedure through the description of the steps of the ray-matrix code. Then we will
introduce the basic working principles of FFT codes and some important functions of SIS that weused. We will present the results of step 5 in the next section, where we will cover the design history
of non-degenerate cavity, through the discussion of several cavity layouts. In the last part of thischapter we will describe step 6 for the configuration that has been retained, the vertical layout.
We will refer to fig.6.1 for the notation of the cavity parameters. Since the optical layout is thesame for both recycling cavities, for the sake of simplicity we will refer hereafter only to the power
recycling cavity.

1106.1. Design Procedure and Simulation Tools

PRM1

PRM2

L1

qarm

θ2
q1

d

L2
L3
PRM3

ITM

BS
Lx

θ3

Figure 6.1: Notation for the parameters of the non-degenerate power recycling cavity layout.

6.1.1

Ray-Matrix Formalism

For a light beam in the paraxial approximation, the path through a given optical system can be
described by a product of matrices with 2×2 elements [95], where each matrix represents an elementof the system: for example, the matrices




1
0
1 d
(6.1)
, L=
D=
∓1/f 1
0 1
represent the propagation in vacuum along the distance d and through the lens with focal lengthf (the ∓ sign changes for a converging or a diverging lens, respectively). Also mirrors can be
represented by a matrix, by simply replacing f = R/2, with R being the mirror RoC. If


A B
(6.2)
M = M1 M2 ...Mn =C D
is the matrix of a system with n elements, the parameter of a Gaussian beam passing through thesystem changes according to the formula (sometimes called the ABCD law)q0 =Aq + B,
Cq + D(6.3)

where q and q 0 are the parameter of the input and output beam of the system, respectively. Thereby,
the propagation back and forth of a laser beam within a resonator can be described by a periodic
sequence of identical systems. For example, the round-trip matrix product for the simple resonatorof two mirrors shown on fig.2.7 is (starting arbitrarily from the reflectivity surface of M1)





1
0
1 L
1
0
1 L
M = L1 D L2 D =,(6.4)−2/R1 10 1
−2/R2 1
0 1
where R1 and R1 are the RoCs of the mirrors, and where we unfolded the path in the resonatorthrough an equivalent sequence of lenses and propagations in space. The stability condition of the111

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

resonator is expressed through the elements of the matrix product by−1 <(A + D)
<1,
2

(6.5)

and the parameter of the resonant beam can be computed by postulating self-consistency andimposing q 0 = q after a round trip in the cavity. The eigenmode of the resonator is thenp
D − A ∓ i 4 − (A + D)2
1
=
.
(6.6)
q
2B
For the ray-matrix representation of the non-degenerate layout, we have to deal with the reflectionof beams on tilted surfaces: the reflection of a Gaussian beam on a surface with incidence angle θis represented by




1
0
1
0
Mt =
, Ms = 2 cos θ
,
(6.7)
2
1
R cos θ 1
R
where t stands for tangential the and s for sagittal. We used the following convention: if therecycling cavity layout lies in the same plane (x, z) of the interferometer, then the tilt axis of
mirrors is oriented along the y direction and (x, z) is the tangential plane, whereas (y, z) is thesagittal plane.

6.1.2

Ray-Matrix CodeWe simulate the recycling cavity coupled to a single arm of the interferometer, which might beeither the x arm (with the transmission through the beamsplitter) or the y arm. Therefore we donot simulate the differential effects of the layout.
In the first part of the code we compute the lengths of the layout, and we set the values ofGouy phase ΦG and minimum beam radius w on PRM1 that we want to obtain for the eigenmodeof the recycling cavity. Then we determine the RoCs of mirrors, {R2 , R3 }. In this phase we makethe assumption of null astigmatism, so we do not consider the incidence angles.
We first compute the RoCs {R2 , R3 } with an approximated formula of the telescope, then wechange {R2 , R3 } in a loop around these values to scan the parameters {ΦG , w} of the recycling
cavity eigenmode. At the end of the loop the code gives as output a list of exact RoCs {R2 , R3 }and corresponding beam parameters {ΦG , w}, and we select the RoCs that satisfy the initial choiceof phase and radius.
In the second part of the code we compute the incidence angles {θ2 , θ3 } on the mirrors of thetelescope. If the geometry of infrastructures allows it, we also compensate (even only partially) for
the astigmatism of the recycling cavity layout by optimizing the angle ratio θ2 /θ3 . At this step all
the parameters of the layout have been determined.
Finally we compute the tolerances of the layout: we change the parameters in a loop, in pairs,
in an arbitrary range, to compute the stability condition of the recycling cavity versus the percentage variation of parameters. At the same time we compute the parameters of the recycling cavity
eigenmode and the coupling with the input beam and the arm cavity eigenmode.
In the following we will give a detailed description of the steps of the code, following the same
sequence of the design procedure described previously.

1126.1. Design Procedure and Simulation Tools

1 - Lengths, Gouy phase and beam size
We first compute the total length of the recycling cavity through eq.(5.31) in order to satisfy theresonance condition of the radio-frequency sideband fields. Then we decide the position of the optics
in the vacuum tanks, so we determine the set of lengths {Lx , L1 , L2 , L3 }. We try to minimize thepotential astigmatism of the layout by maximizing the distance L2 , that is the lever arm of thetelescope: we will see in the following that for a given beam spacing d (determined by size of thebeam and the mechanical elements of suspensions), the larger is the lever arm, the smaller are theincidence angles on the telescope mirrors.
Then we set ΦG and w on PRM1: for the following we will assume the value of phase indicatedin the baseline design of AdV, ΦG = 20 deg, and we will consider w = 1 mm, which is a reasonable
beam size. Together with the arm cavity eigenmodearm
qarm = zarm + zR
= (960.830 + i134.636) m,{ΦG , w} represent the initial conditions of the computation that will follow: we need to determine
the RoCs of the non-degenerate layout, {R2 , R3 }. R1 is already bound, because it has to be equal
to the curvature of the beam wavefront on PRM1. We compute it, together with the other beam
parameters on PRM1, through the following equations:z/zR ' tan(ΦG )pw0 = w/ 1 + (z/zR )2
zR = πw02 /λ → z = zR tan(ΦG )h 2 iR1 = z 1 + zzR.
where we approximated the total Gouy phase shift in the cavity as the phase on PRM1. This is areasonable approximation, since as we have seen in chapter 5 most of the phase is cumulated along
L1 .
At this stage, step 1 of the design procedure is complete. We now have to determine {R2 , R3 }.
2 - Approximated computation of RoCs
We compute an initial pair of values {R2 , R3 }, through an approximated formula for the telescopeof the layout, which can be seen as a unique lens with equivalent focal lengthsfeq =zR (zarm + L3 )2
.
arm
zR

(6.8)

Eq.(6.8) is obtained by applying repeatedly eq.(5.28) for PRM3 and PRM2, then imposing zarm >>R3 , R2 . This is a reasonable approximation, since zarm is of the order of 103 m whereas R3 andR2 are of the order of 10 m. The equivalent focal length is known, since it is function of known

113

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgoparameters, so we can use it to compute the approximated values
2
R3 = z+L2Lfeq
1 −feq(6.9)
3 /2+L2 )R2 = − 2(Rfeq −R3 /2 feq ,which will be used in the next computation of the code.
3 - Fine tuning
At this stage {R2 , R3 } must be fine-tuned to have ΦG = 20. The code then executes the following
instructions:• the RoCs of PRM3 and PRM2 are scanned in a loop around the initial values R3 and R2• at every step of the loop, for every pair {R2 , R3 }, the code computes the eigenmode of therecycling cavity and its parameters {w, R, z, zR and ΦG } on every mirror of the layout. The
computation is carried out through the round-trip matrix product associated to the layout of
the recycling cavity,
A(Li , Rj ) B(Li , Rj ),(6.10)MRC =C(Li , Rj ) D(Li , Rj )whose elements depend on the layout parameters. We replace these matrix elements in eq.(6.6)
to compute the cavity eigenmode, then we use eq.(6.3) to compute the propagation of themode inside the cavity. In the same iteration we compute the stability condition of eq.(6.5),
and we discard the pairs {R2 , R3 } which make the resonator unstable• the code selects the pairs {R2 , R3 } which yield a phase in the range 15 < ΦG < 25 and w ≥ 1
mm on PRM1. We are interested in this range of values since we might want to tune thephase to avoid accidental resonances of modes of large order, and we would like to have thelargest beam radius possible. Then the code sorts a list of these parameters in an output fileand plots the results.
Finally we select the best set of RoCs through the output plot shown on fig.6.2, where we arbitrarily
chose R3 as independent variable to project the space of layout and beam parameters on the axes1 .
We always choose the values of the RoCs to be at least larger than 1 m, to minimize the riskthat low curvature induce spherical aberration of the beam. This may conflict with the requirement of having w ≥ 1 mm (R2 decreases where the beam radius w increases), and in this case, a
trade-off between these two requirements should possibly be made.
The choice of the RoCs marks the accomplishment of step 3 of the design procedure.
1

The plot shows an example of selection of parameters for the vertical layout that will be presented later in thischapter.

1146.1. Design Procedure and Simulation Tools

Figure 6.2: Output plot of the space of beam and layout parameters: beam radius w on PRM1 [m] (redcurve), RoC of PRM2 [m](green curve), RoC of PRM1 [m] (blue curve) and cumulated Gouy phase shift in
the cavity [deg] (black curve) versus the RoC of PRM3.

4 - Incidence angles
In this step we include the incidence angles in the layout. Of course the astigmatism increases with
the incidence angles, so we would like to choose the smallest angles possible. However there is alower limit for the choice of the angles, determined by the spacing d between the beams.
This limit changes according to the set of lengths and RoCs of the layout, but we can give ageneral estimate to show its order of magnitude. The beam is about ∼ 5-cm large between the arm
cavity input mirror and PRM3, whereas on PRM2 the beam radius is of the order of 5 mm. The
spacing is
d = 2.63(5 + 0.5) ' 15 cm,
where 2.63 is the ratio of the Gaussian beam radius to the mirror radius to have 1 ppm clipping
losses. For small θ3 we can approximate sin(2θ3 ) ' 2θ3 , thus the smallest incidence angle of thelayout is θ3 ' d/2L2 ∼ 0.4 deg for L2 = 10 m, and θ3 ∼ 0.9 deg for L2 = 5 m.
The minimum θ3 could actually be larger if we consider also the mechanical elements (marionettas and recoil masses) of the mirror suspensions in the computation of the the beam spacing.
In the case the allowable space in the vacuum pipelines and tanks was sufficiently large, insteadof choosing θ2 = θ3 we might optimize the incidence angles to compensate exactly for the astigmatism of the layout. This can be done by compensating the astigmatism of the concave mirror
PRM3 through the convex mirror PRM2, according to [104]
r
θ2 =

R2
zarm

θ3 .
R3 zarm 1 − 2L2 /R3 + L2

115

(6.11)

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

Usually the optimal θ2 is larger than θ3 by a factor 2 or more, then to fit the layout in the vacuum
pipeline (which has a limited diameter) we would like to choose θ3 as small as possible.
It could possible, in any case, to achieve just a partial compensation. For example, we will
see later in this chapter that for the vertical cavity layout we could choose θ3 = 1.06 deg. The
optimal θ2 for a prefect compensation of astigmatism would be 3.2 deg, but the geometry of theinfrastructure does not allow to choose such a big angle. Nevertheless we imposed θ2 = 2.53 andstill could achieve a good level of compensation.
When the incidence angles are computed, step 4 of the design procedure is accomplished.
5 - Characterization
In this step we compute the tolerances of the recycling cavity layout, the parameters of the cavity
eigenmode, the overlap integrals of the recycling cavity eigenmode with the input beam and withthe arm cavity eigenmode. The code executes the following instructions:• the parameters of the layout, {Lx , L1 , L2 , L3 , R1 , R2 , R3 , θ3 , θ2 }, are set as initial condition of
the computation• a pair of parameters (usually we start from {R2 , R3 }) is changed in a loop within an arbitrary
range. For every pair of values the code computes through eq.(6.6) the eigenmode of therecycling cavity on the transverse directions, using two distinct matrix products:
t
MRC
=




A(Li , Rj , θk ) B(Li , Rj , θk )A(Li , Rj , θk ) B(Li , Rj , θk )s,
, MRC =
C(Li , Rj , θk ) D(Li , Rj , θk )C(Li , Rj , θk ) D(Li , Rj , θk )



(6.12)

where t stands for tangential and s stands for sagittal. Then the eigenmode parameters{wRC|t , RRC|t , ΦG|t } and {wRC|s , RRC|s , ΦG|s } are computed on every recycling mirror, tostudy the propagation of the beam in the cavity. Because of the astigmatism, we have thatwRC|t 6= wRC|s , RRC|t 6= RRC|s , ΦG|t 6= ΦG|s .
In the same iteration, the code computes the following overlap integral
D
E
input
RC =
γinput ≡ T EM00
T EM00= √w

2s

input|t wRC|t winput|s wRC|s1
1+ 21 − iπ
λw2w
input|t
RC|t

s
×

1
w2
input|s

+

1
w2
RC|s

1
+R 1
Rinput|t
RC|t

1
− iπ
λ



1
+R 1
Rinput|s
RC|s(6.13)



input
between the input beam (coming from the injection system) T EM00
and the eigenmodeRC2of the recycling cavity T EM00 , taken on PRM1. Its square value γinput ∈ [0, 1] yields the116

6.1. Design Procedure and Simulation Tools

coupling of the input beam to the recycling cavity. We define the parameters of the input
beam aswinput|t = winput|s ≡ (wRC|t + wRC|s )/2
Rinput|t = Rinput|s ≡ (RRC|t + RRC|s )/2 ,i.e. the input beam has the average parameters of the tangential and sagittal planes of therecycling cavity eigenmode. This is the input beam that best matches the recycling cavity
eigenmode.
Always in the same iteration, the code computes the analogous overlap integral
D
E
arm
RC
γarm = T EM00
T EM00(6.14)

input
between the arm cavity eigenmode T EM00
and the eigenmode of the recycling cavity
RC , taken on the arm cavity input mirror.
T EM00

As for step 3 of the design procedure, all the quantities computed in this step are discarded
if the stability condition of eq.(6.5) is not respected. We require this condition to be fulfilled
by both transverse directions at the same time
• the code gives as output a stability plot, showing the cavity stability and the beam coupling2
γinputversus the percentage variation of the layout parameters. This plot is used to asses thetolerances of the parameters, so we repeat its computation for every parameter of the layout.
The stability plot is the main result of the code and closes the ray-matrix code: we will show thisplot in the next section for each layout that will be presented.

6.1.3

Fast Fourier Transform Codes

FFT codes are based on the expansion of the complex amplitude of a coherent electric fieldE(x, y, z), like a laser beam, in the Fourier space of spatial frequencies. This expansion is equivalent
to a decomposition of the field in a sum of elementary plane waves of the form~u(x, y, z) = u0 e−ik·~r = u0 exp[−i(kx x + ky y + kz z)],(6.15)q
where k = kx2 + ky2 + kz2 = 2π/λ is the wave vector. If we assume the paraxial approximation
along z as usual, kz >> kx , ky , then we can writekz =q
kx2 + ky2
k 2 − kx2 − ky2 ' k −' k − λπ(νx2 + νy2 )k(6.16)

with νx = kx /2π, νy = ky /2π being the characteristic spatial frequencies (number of wavelengths
per unit of length) of the wave in the transverse plane. If we propagate the wave over a distanced, we can writeu(x, y, d) = u0 exp[−i(kx x + ky y + kz d)] ' u(x, y, 0) exp[−i(k − λπ(νx2 + νy2 ))d],(6.17)

where u(x, y, 0) is the wave at z = 0. Eq.(6.17) is the fundamental equation of FFT codes: it
indicates that the propagation of the plane wave along a distance d is simply equal to the product117

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgoof the wave at the initial position times a harmonic propagator, which is a simple function of the
distance. In other words, the propagation is equivalent to a phase shift of the plane wave.
Eq.(6.17) allows to compute the propagation of any field E(x, y, z) through the following operations:• the field is expanded in the Fourier space as a sum of plane waves, through a 2-dimensional
Fourier transform:Z ∞Ẽ(νx , νy , 0) =E(x, y, 0) exp[−i2π(νx x + νy y)] dxdy
(6.18)−∞• each plane wave is propagated in the frequency domain through a phase shift term:Ẽ(νx , νy , d) = Ẽ(νx , νy , 0) exp[−i(k − λπ(νx2 + νy2 ))d](6.19)• the field is recomposed from the propagated plane waves through an inverse Fourier transform:
Z ∞E(x, y, d) =Ẽ(νx , νy , d) exp[−i2π(νx x + νy y)] dνx dνy .
(6.20)−∞

These operations are executed through the Fast Fourier Transform, which is an efficient algorithm
for the computation of the Fourier transform and its inverse.
The propagation of the field through optical elements is computed as a phase shift in the physicalspace of distances, according toEout (x, y, z) = E(x, y, z) exp[−ik∆Lopt (x, y)],(6.21)

where Lopt (x, y) is the optical path difference introduced by the optical element. For example,
referring to fig.6.3, the optical path difference introduced by the reflection of the field on thesurface of a mirror isp
Lmirr
R2 − ∆r2 ) ,(6.22)
opt (x, y) = 2∆s = 2(R −∆s being the change in sagitta, R the RoC of the mirror, ∆r2 = x2 + y 2 and the factor 2 indicatesthe forward and backward propagation of the beam. The reflection on a mirror with non-normal
incidence angles can be approximated by replacing R → R cos θ and R → R/ cos θ in eq.(6.22) for
the tangential and the sagittal directions, respectively.
Stationary Interferometer Simulation (SIS)In SIS it is possible to simulate a single arm cavity coupled to a recycling cavity. Among the various
options of the code, it is possible to simulate:• either a non-degenerate layout or a marginally stable layout• the power recycling cavity or the signal recycling cavity• the resonance of different fields in the cavities: the carrier, the radio- or the audio-frequency
sidebands

118

6.2. Design History: from the Baseline to the Vertical Layout

Δs
R
Δr
z

Figure 6.3: Computation of the optical path length difference introduced by a mirror.

• either perfect or realistic optics, by including measured or simulated phase maps of mirrors.
We use SIS in step 5 of the design procedure, to simulate the impact of astigmatism on carrier andradio-frequency sidebands in the non-degenerate power recycling cavity. In step 6 we use SIS tosimulate the shape and the recycling gain of carrier and radio-frequency sidebands in presence ofrealistic mirror defects and thermal effects, by using phase maps for the mirrors.

6.2

Design History: from the Baseline to the Vertical Layout

We applied the design procedure described in the previous section to explore several possible layoutsof the non-degenerate recycling cavities. We will present in the following the four most significant
configurations that we studied: they illustrate at best the concepts behind the different designs.
We will introduce the layouts following the chronological sequence of their conception, to illustrate
the evolution of the non-degenerate cavity design. Each of these layouts has been characterized interms of cavity stability, parameters of the resonant beam, astigmatism, coupling loss and parameter
tolerances.
The last configuration we analyzed, the vertical layout, is the one that was retained: it features
very low astigmatism, and tolerances comparable to those of other designs. Most important,
we will see that this layout does not require the off-axis suspension of multiple mirrors from a
Superattenuator, which instead is an important issue of the first two designs.

6.2.1

Baseline Design

The baseline design, shown on fig.6.4, is the first layout we studied [105]. PRM3 is placed in the
power recycling tank, PRM2 is in the same vacuum tank of the beamsplitter, PRM1 is suspended in
the injection vacuum tank, from the same Superattenuator which carries the bench of the injection
system. The beamsplitter tank also hosts SRM2 of the signal recycling cavity, thus three optics
have to be suspended from the same Superattenuator. The total length of the cavity is about 27.5m, the other parameters of the layout are reported in table 6.1.

119from input
mode-cleaner

injection tower

beamsplitter
tower

power recycling
tower

to signal
recycling

to armto armChapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

Figure 6.4: Baseline layout of the non-degenerate power recycling cavity, from [105].

120

6.2. Design History: from the Baseline to the Vertical Layout

length [m]Lx = 5.5
L3 = 6
L2 = 5.5
L1 = 10.5

RoC [m]angle of incidence [deg]R3 = 12.795
R2 = -2.042
R1 = -2.033

θ3 = 1.7
θ2 = 1.7

Table 6.1: Parameters of the baseline layout.

beam parameter
qarm
q1
ΦG
w on PRM1

x (tangential)(951.632 + i155.174)m(−0.768 + i0.986)m
47.5 deg
0.7 mmy (sagittal)(961.782 + i120.728)m(−1.561 + i0.858)m
25.1 deg
1.1 mmTable 6.2: Eigenmode parameters in the baseline layout.

Astigmatism and stability
Since the lever arm of the telescope is short and the spacing between the beams is large to accommodate the mirrors and their recoil masses, the angle of incidence on PRM3 should be large,
θ3 = 1.7 deg. In this configuration it is not possible to increase θ2 to reduce the astigmatism,
since the beam is already close to the edge of the vacuum pipe, so we take θ2 = θ3 . Thereby the
astigmatism of the layout is considerable: the parameters of the beam are largely different on thetransverse directions, including the Guoy phase shift, as shown by table 6.2.
Fig.6.5 shows the stability plot, which depicts the stability of the cavity with respect to thepercentage variation of the parameters on the axes2 . The white regions of the plot represent thesubsets of values which yield instability of the tangential or of the sagittal plane. Colored regions are2stable for both the transverse directions. The color scale represents the coupling γinput, computed
2 ,with the ray-matrix code through eq.(6.13). A similar plot can be obtained for the coupling γarm
computed through eq.(6.14). The point of coordinates (0%, 0%) represents the optimal values listed
2
2
in table 6.1. In this point, the couplings are γinput= 95.6% and γarm
= 97.5%.
Initially, by assuming that the arm cavity and the recycling cavity are completely independent,
we used to compute the total coupling loss for the carrier field as
2
2
Lcarrier = 1 − γinput
γarm= 6.8% .

(6.23)We used to consider eq.(6.23) as an upper limit, since actually the cavities are not decoupled andthe light which is not coupled into the arm cavity might still be recycled by the recycling cavity,2
then not being lost. We also expected γinput
to be the actual estimation of the coupling of theradio-frequency sidebands with the recycling cavity.
Then we studied the astigmatism of the baseline layout with SIS, which yielded unexpected
results for the carrier field. Results are shown on fig.6.6, where the loss of carrier recycling gain
(computed as the ratio of the power resonant in the astigmatic cavity to the power resonant with
θ2 = θ3 = 0) is plotted versus the incidence angles θ2 = θ3 . For small angles, below 1 deg, theloss is negligible (of the order of percent), and Lcarrier is overestimated. For larger values of the2

Here we show only the analysis of stability versus {R2 , R3 }, but we repeat the computation for all the parameters.

121

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo2
Figure 6.5: Stability plot of the baseline layout, γinput
versus {R2 , R3 }.

Figure 6.6: Loss of the carrier versus the incidence angles (θ3 = θ2 ).

122

6.2. Design History: from the Baseline to the Vertical Layout

parameter
L3 /Lx
L2
L1
R3
R2
R1

tolerance [%]± 100± 0.04± 4.0± 0.03± 0.3± 20

Table 6.3: Tolerance of parameters for a stable baseline design.

incidence angles the loss increase rapidly, attaining ∼ 30% for θ2 = θ3 = 1.7 deg. In this case
Lcarrier is largely underestimated.
The discrepancy can be explained by the fact that the coupling between the arm and the power
recycling cavity cannot be neglected for the carrier field. For low astigmatism the coupling reduces
the loss to a negligible level, for high astigmatism the coupling is affected and the loss is high. SIS
allows to simulate the coupling between the recycling cavity and the arm cavity for the carrier field,
while the ray-matrix do not. Thereby we realized that the upper limit Lcarrier was not a reliable
estimator for the loss of the carrier.
2
For the radio-frequency sidebands, instead, the results of SIS confirmed the coupling γinput
computed with the ray-matrix code. Thus we continued to characterize the astigmatism loss for2
the radio-frequency sidebands through the computation of γinput
and of the stability plot, which
will be presented hereafter for the other layouts that we studied.
Tolerances
The tolerances for cavity stability are derived from the stability plot: we measure the tolerance on
the plot axes as the minimum variation which makes the recycling cavity unstable, starting fromthe working point (0%, 0%). They are summarized in table 6.3.
The critical parameters of the layout are {L2 , R2 , R3 }. By analyzing other configurations, lateron in this chapter, we will see that this is a constant feature of the non-degenerate layout. This
happens because the Gouy phase in the cavity is determined essentially by the RoC of PRM2. If
this RoC is larger than the phase front curvature of the beam focused by PRM3, the field will
diverge and the cavity will be unstable: the Gouy phase in the cavity would be close to 0. Thus thecavity can become unstable for very small detuning of the telescope, either because of RoC errors
or because of errors of the distance L2 .
The most critical parameter is R3 , since its tolerance is more severe than the possible achievablemanufacturing error: with present technology, mirrors of ∼ 10-m RoC like PRM3 can be realizedwith an accuracy of ±0.1%, while in this case an accuracy of ±0.03% is needed.
Tuning
The instability of the cavity as well as manufacturer RoC errors could be compensated by tuning
the lever arm of the telescope. One of the interest of the folded design, in fact, is the possibility
to adjust the distance between the telescope mirrors in order to tune the focusing of the telescope
to recover the mode-matching [93]. Fig.6.7 shows that a percentage variation of L2 of the sameorder of the RoC error is needed to recover stability. This implies that in presence of an error of123

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

L2 tuning
R3 errorFigure 6.7: Tuning of the lever arm L2 in the baseline design.

+1 cm on R3 (corresponding to an error of about +0.1 %, making the cavity unstable), L2 shouldbe elongated of about 5.5 mm (corresponding to a variation of about +0.1 %).
The tuning of the telescope must be achieved by respecting the resonance condition of sidebands,
thus the total recycling cavity length has to be left unchanged. As a consequence, the tuning of
the lever arm L2 must be compensated by changing the distance L1 .
With the ray-matrix code we simulated an error of 0.1% on the RoC of PRM3, then we adjusted
the length of the telescope and the distance L1 : we could verify that after the adjustments werecovered the same parameters of the recycling cavity eigenmode listed in table 6.2. This confirmedthat the working point of the detector can be fully recovered after tuning the cavity matching.
Issues of the baseline design
The baseline design fits the infrastructure of Virgo, but features two important issues. The first is
the high astigmatism, which induces relevant losses for the carrier field, and makes the transverse
planes of the recycling cavity eigenmode very different for the sidebands. This could be in principle
solved by using parabolic mirrors, even though it is not obvious to obtain such mirrors with a goodsurface quality.
The second and most important issue concerns the suspension of three optics in the same
vacuum tank of the beamsplitter. This requires a modification of the suspension system: several
optics might be suspended from the same Superattenuator by implementing a new design of thelast suspension stage, a multi-payload for off-axis suspension [106], shown on fig.6.8.
The mechanical behavior of the multi-payload was studied by means of simulations. The results
showed that the off-axis suspension enhances spurious couplings between degrees of freedom ofsuspended elements, which instead are negligible in the original design of the payload [106]. The

124

6.2. Design History: from the Baseline to the Vertical Layout

last seismic filter

SRM2

PRM2
BS

Figure 6.8: Scheme of the beamsplitter multi-payload, from [106].

couplings highly increase the complexity of the control system of the multi-payload, which would
need the implementation of a large number of sensors and actuators and the development of a newcontrol strategy. The conclusion of these studies pointed out that the multi-payload design presents
a high technical risk and should be discarded [106].
Therefore the issue of the multiple off-axis suspension of mirrors triggered the design of a new
optical configuration.
Modification of the baseline design
We proposed a new layout to remove the off-axis multi-payload from the beamsplitter tank and tosuspend each element on axis with the Superattenuator [107]. This configuration lies in the verticalplane, as shown on fig.6.9. PRM2 is moved to the power recycling tank and PRM3 is moved in theinjection tank. PRM1, PRM3 and the optical bench of the injection system are suspended verticallyfrom the same Superattenuator, as shown on fig.6.10. The parameters of this new configuration
are listed in table 6.4.
In order to reduce the astigmatism, we moved the power recycling tank towards the beamsplitterto increase the arm lever L2 as much as possible, from 5.5 m to 6.5 m. At the same time we also
optimized the encumbrance of the mechanical elements of the mirror suspensions, to decrease thespacing of the beams. Therefore θ3 is decreased to 1 deg, so that the astigmatism is partially
optimized. The loss of the carrier, computed through a simulation with SIS, is reduced to 0.3%.
The coupling of radio-frequency sidebands, computed with the ray-matrix code and confirmed by2
the simulation with SIS, is γinput= 98.9%. Because of the lower astigmatism, the parameters of
the recycling cavity eigenmode on the transverse directions have closer values, as shown in table
6.5.

125

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

injection tankpower recyclingtank

beamsplitter
tank

PRM1
PRM2

from input
mode-cleaner

to arm cavity

BS
to arm cavity

injectionbench

to signal recycling

PRM3
vertical plane
Figure 6.9: Modification of the baseline layout in the vertical plane.

recoilmassesPRM1
injection/detection
bench

PRM3

Figure 6.10: Scheme of the payload for the vertical suspension of the elements in the injection vacuum tank,for the modified baseline layout. From [107].

length [m]Lx = 5.6
L3 = 11
L2 = 6.5
L1 = 6.5

RoC [m]angle of incidence [deg]R3 = 13.868
R2 = -1.009
R1 = -6.567

θ3 = 1θ2 = 1.7

Table 6.4: Parameters of the modified baseline layout.

126

6.2. Design History: from the Baseline to the Vertical Layout

inverted
pendulums

frame
seismic
filters

PRM1

PRM3
injection/detection
bench

Figure 6.11: Design of the multi-Superattenuator for the injection/detection vacuum tank, from [108].

127

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgobeam parameter
qarm
q1
ΦG
w on PRM1

x (tangential)(957.224 + i146.661)m(−3.115 + i3.279)m
28.6 deg
1.5 mmy (sagittal)(965.154 + i118.470)m(−4.214 + i3.149)m
21.2 deg
1.7 mmTable 6.5: Eigenmode parameters for the modified baseline layout.

However, the suspension of three elements from the same Superattenuator was considered technically challenging and was discarded. Thus we proposed to move the vertical configuration in thehorizontal plane, and to suspend each element in the injection tank from its own single Superattenuator [107]. The development of a multi-Superattenuator was proposed [108], where each element
is suspended from a single chain of filters by squeezing three standard Superattenuators into theinjection tank. The system is shown on fig.6.11: the three chains are suspended from the top stagesof three short inverted pendulums.
The multi-Superattenuator configuration represents the best solution to suspend multiple mirrors, since it follows the design concept of the original Superattenuator: each element is suspended
on axis, from a single chain of filters. Nevertheless, it requires the re-design of several parts of theSuperattenuator: seismic filters, for instance, have to be re-designed with a smaller volume to fit
into the tank. The development and construction process of this system was considered too longto meet the schedule AdV, and it was also discarded.
As we will see later on, in order to simplify the suspensions we proposed to move the injection
bench out of the injection tank (moving it in a small vacuum tank to be built beside), and wereconsidered the suspension of PRM3 and PRM1 in the vertical plane. This configuration, calledvertical layout, finally allowed to solve the issue of multiple mirror suspension. It will be presented
at the end of this section.

6.2.2

Telescopes in the Differential Part of the Michelson Interferometer

The second layout we present is based on a completely different concept from that of the baseline.
It has been conceived almost at the same time, and has been analyzed in parallel as an alternative
configuration [105]. In this layout, shown in fig.6.12, the telescope is placed in the differential partof the Michelson interferometer (formed by the beamsplitter and the arm cavity input mirrors):
PRM3 is suspended in the power recycling tank, PRM2 is suspended together with the arm cavity
input mirror (ITM) in the arm input tank, PRM1 is on the injection (detection) bench. The total
length of the recycling cavity is 41 m, and its parameters are listed in table 6.6.
This configuration has the advantage of allowing the compensation of thermal effects by acting
directly on PRM3 and/or PRM2, with CO2 lasers and ring heaters. Thus the compensation plates
could in principle be removed, with two important consequences: the first is the simplification of
the thermal compensation system, the second is the simplification of the design of the payloadof arm cavity mirrors. In the current design of the payload, compensation plates are suspended
off-axis with respect to the vertical of the Superattenuator, from the recoil mass of the mirror. The
removal of compensation plates could allow to avoid the off-axis suspension of optics in the payloadof arm cavity mirrors.

128to detection

from injection

PRM1 ininjection tower

power recycling
tower

BS

beamsplittertower

arm input
tower

ITM

to arm cavity

6.2. Design History: from the Baseline to the Vertical Layout

Figure 6.12: Layout of non-degenerate recycling cavities with telescopes in the differential part of theMichelson, adapted from [105].

129

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo2
Figure 6.13: Stability plot of the layout with telescopes in the differential part of the Michelson, γinput
versus {R2 , R3 }.

length [m]Lx = 6.6
L3 = 12
L2 = 12
L1 = 10.5

RoC [m]angle of incidence [deg]R3 = 27.960
R2 = -4.890
R1 = -4.616

θ3 = 1.9
θ2 = 1.9

Table 6.6: Parameters of the layout with telescopes in the differential part of the Michelson.

Astigmatism and stability
This design has also a lower astigmatism, since it has a much longer arm lever and larger RoCs.
The stability plot of the layout is shown on fig.6.13: because of the lower astigmatism, the size ofthe stable region is increased. Also the parameters of the beam, listed in table 6.7, indicate thatthe astigmatism of the layout is reduced. The coupling coefficient for radio-frequency sidebands in2
this layout is γinput= 97.7%. It is not possible to simulate this particular configuration with SIS,
so we could not study the recycling gain of the carrier field.
Tolerances and tuning
The tolerances for this layout are reported in table 6.8, and are similar to those of the baseline
design. Manufacturer errors on R3 may be recovered by tuning the lever arm of the telescopes, as
shown by fig.6.14: to recover a RoC error of +3 cm (corresponding to an error of about +0.1 %,yielding an unstable recycling cavity), L2 should be elongated of about 1 cm (corresponding to avariation of about +0.1 %).

130

6.2. Design History: from the Baseline to the Vertical Layout

beam parameter
qarm
q1
ΦG
w on PRM1

x (tangential)(957.435 + i136.642)m(−2.314 + i2.308)m
38.9 deg
1.3 mmy (sagittal)(965.829 + i103.582)m(−3.570 + i1.932)m
23.6 deg
1.7 mmTable 6.7: Eigenmode parameters in the layout with telescopes in the differential part of the Michelson.

parameter
L3 /Lx
L2
L1
R3
R2
R1

tolerance [%]± 100± 0.04± 4.0± 0.03± 0.3± 20

Table 6.8: Tolerance of parameters for a stable layout with telescopes in the differential part of the Michelson.

L2 tuning
R3 errorFigure 6.14: Tuning of the lever arm L2 in the layout with telescopes in the differential part of the Michelson.

131Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

to beamsplitter

power recycling
tank
PRM3

PRM2
new vacuum tankfrom input
mode-cleaner

PRM1
injection tank
Figure 6.15: Layout of long non-degenerate recycling cavities.

length [m]Lx = 5.6
L3 = 6
L2 = 80
L1 = 80.5

RoC [m]angle of incidence [deg]R3 = Inf (plane)
R2 = 153
R1 = -2.250

θ3 = 45
θ2 = 1.8

Table 6.9: Parameters of the long layout.

Issues of the layout with telescopes in the differential part of the Michelson
This configuration has two main drawbacks: first, the potential impact of differential noises introduced by the suspension of two additional mirrors per arm. More important, the P/S-RM2
suspension are close to the monolithic fibers of arm cavity input mirrors. Fibers are fragile andcan be destroyed by accidental contact with any other object. This issue was considered too risky,
and this configuration of the recycling cavity has been discarded.

6.2.3

Long Cavities

We studied a third configuration, shown on fig.6.15, in order to solve the issue of mirror suspensions. PRM3 is placed in the power recycling tank, PRM2 is placed in a new building to beconstructed (a new building must be constructed also for SRM2), PRM1 is suspended in the injection tank. Therefore in this configuration each mirror of the layout is suspended on axis from asingle Superattenuator.
The new building is positioned 80 m far from PRM3, so that the arm lever of this layout isvery large and the astigmatism is reduced. Because of the much larger cavity length (172 m), we
use only a single converging mirror, whose RoC must be as large as ∼ 150 m, to focus the beam
inside the cavity: PRM3 becomes a flat steering mirror and PRM2 becomes the focusing mirror of
the layout. The parameters of this configuration are listed in table 6.9.

132

6.2. Design History: from the Baseline to the Vertical Layout

NEW BUILDINGS

beamsplittertank
arm inputtank

arm input
tanksignal recyclingtank
detection tankpower recyclingtank
injection
tank

Figure 6.16: Construction of two new buildings for the long layout.

133

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgobeam parameter
qarm
q1
ΦG
w on PRM1

x (tangential)(957.077 + i137.866)m(−1.878 + i0.836)m
24.1 deg
1.3 mmy (sagittal)(960.982 + i123.823)m(−1.964 + i0.749)m
21.0 deg
1.4 mmTable 6.10: Eigenmode parameters of the long recycling cavity.

parameter
L3 /Lx
L2
L1
R3
R2
R1

tolerance [%]
± 100
± 56
± 0.3
−
± 0.3
± 12

Table 6.11: Tolerance of parameters for a stable long layout.

Astigmatism
Since PRM3 is a flat mirror, the astigmatism of the layout is entirely due to the incidence angle on
PRM2. Though θ2 = 1.79 deg is similar to the angles of previous layouts, astigmatism is very low
since R2 (determined by the arm lever) is very large. The beam parameters have closer values on
the transverse directions (see table 6.10), and coupling coefficient for the radio-frequency sidebands
2is larger: γinput= 99.9% with the input beam.

Tolerances and tuning
The values reported in table 6.11 and the stability plot of fig.6.17 indicate that RoC tolerances are
more relaxed in this configuration, of about an order of magnitude. For comparison, the central
area in the rectangle on fig.6.17 indicates the stability range for the baseline design (fig.6.5) andthe layout with telescopes in the differential part of the Michelson (fig.6.13).
R2 is a critical parameter: its tolerance, ±0.3%, is smaller than the achievable manufacturing
accuracy for a RoC of ∼ 150 m, which is ±1%. The plot of fig.6.18 indicates that to compensate fora RoC error of +1.5 m (corresponding to an error of +1 %, yielding an unstable recycling cavity)
L1 should be elongated of 0.8 cm (corresponding to a variation of about +0.1 %). The tuning of
L1 should be made by moving the suspension point at the top stage of the inverted pendulum of
the Superattenuator, either of PRM1 or PRM2.
However the inverted pendulum only allows displacements of of ±4 mm, thus the dynamics
needed to tune the matching of the cavity is larger by orders of magnitude than the achievableadjustment of the length. In order to tune the matching of this layout it would be necessary to correct the RoC of PRM2 by acting on PRM2 (SRM2) with ring heaters of the thermal compensation
system [109].

134

6.2. Design History: from the Baseline to the Vertical Layout

2
Figure 6.17: Stability plot of the long layout, γinput
versus {R1 , R2 }.

L1 tuning
R2 errorFigure 6.18: Tuning of the lever arm L1 in the long layout.

135

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

length [m]Lx = 5.6
L3 = 10.7
L2 = 6.8
L1 = 6.8

RoC [m]angle of incidence [deg]R3 = 15.098
R2 = -1.837
R1 = -1.870

θ3 = 1.1
θ2 = 2.5

Table 6.12: Parameters of the vertical layout.

Issues of the long cavities
This layout solves most of the issues of previous designs: it does not require off-axis suspension ofmultiple mirrors, has negligible astigmatism and larger tolerances. Its main technical drawback is
the need of ring heaters to tune the mode-matching of the cavity. However, there are two crucial
concerns with this layout:• the additional cost for the construction of the new buildings for P/S-RM2 and of the vacuum
tube and pumping system• the risk of delays due to the construction works of the new buildings
These reasons pushed us to explore other configurations.

6.2.4

Vertical Layout

In order to solve the problems related to the suspensions of the modified baseline configuration, andin order to limit the construction works necessary for the long cavity configuration, a new solution
has been proposed. The key elements are the following:• the recycling cavities lie in the vertical plane• the injection bench is moved to a smaller vacuum tank, to be built beside the injection tank(a new small vacuum tank has to be built also for the detection bench)• PRM3 and PRM1 are suspended vertically along the axis of the Superattenuator of theinjection tank
• PRM2 is suspended in the power recycling tower.
This configuration is shown on fig.6.19, and the parameters of the layout are listed in table 6.12.
The payload for the double vertical suspension of PRM3 and PRM1 is re-designed as on fig.6.20,
where mirrors are suspended from a new marionetta.
Astigmatism
To decrease the astigmatism of the layout, the optimization of θ2 is simulated with SIS and withthe ray-matrix code for the radio-frequency sidebands. Results are shown on fig.6.21, where theellipticity of the beam wx /wy is plotted versus the angle θ2 . The astigmatism is completely compensated (wx /wy = 1) if θ2 = 3.2 deg. Since the diameter of the vacuum pipe do not allow such
beam spacing, we chose θ2 = 2.5 deg: for this value, wx /wy > 0.9. The other parameters of the136

6.2. Design History: from the Baseline to the Vertical Layout

injection tankpower recycling
tank

PRM1

beamsplitter
tank

PRM2

from input
mode-cleaner

to arm cavity

BS
to arm cavity
to signal recycling

PRM3
vertical plane
Figure 6.19: Layout of vertical non-degenerate recycling cavities.

new marionette

PRM1

PRM3

Figure 6.20: Design of the Superattenuator PRM1-PRM3 payload for the vertical layout, from [110].

beam parameter
qarm
q1
ΦG
w on PRM1

x (tangential)(958.589 + i142.239)m(−1.360 + i0.833)m
26.3 deg
1.0 mmy (sagittal)(963.285 + i125.731)m(−1.492 + i0.751)m
22.1 deg
1.1 mmTable 6.13: Eigenmode parameters in the vertical layout.

137

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

total compensation

wx/wy ratio

ray matrix

SIS
SIS - θ2=θ3=1.25

SIS - θ2=θ3=1.5

incidence angle θ2 [deg]Figure 6.21: Simulation of astigmatism of radio-frequency sidebands in the vertical layout: the plot shows
the ratio wx /wy versus θ2 , computed for θ3 = 1 deg with the ray-matrix code (stars) and SIS (blue curve),
and for θ3 = 1.25 deg and θ3 = 1.5 deg with SIS (empty circles), from [107].

layout are summarized in table 6.12, and the parameters of the recycling cavity eigenmode arelisted in table 6.13.
The stability plot of this configuration is shown on fig.6.22, the coupling coefficient for the2
sideband is γinput= 99.8%, also confirmed by SIS. The loss of the carrier, computed with SIS, are
0.1%. These results confirm that the impact of the astigmatism is negligible in this layout.
Tolerances and tuning
Tolerances are of the same order of magnitude of those of the baseline layout, as shown by table 6.14:
R3 is the most critical parameter, and its tolerance is smaller than the achievable manufacturing
accuracy (0.1%). In presence of RoC errors for R3 , the matching of the cavity could be recoveredby adjusting the lever arm of the telescope, as indicated by the plot of fig.6.23: an error of +1.5
cm (corresponding to an error of about +0.1 %, yielding an unstable recycling cavity) could becompensated by adjusting L2 about +7 mm (corresponding to a variation of about +0.1 %). Thistuning should be achieved by moving the top stage of the inverted pendulum of the Superattenuatorof PRM2, which has a dynamics of only ±4 mm.
Moreover, the need of keeping the recycling cavity length constant after the tuning of thetelescope requires a larger dynamics. Since PRM3 and PRM1 are suspended from the same Superattenuator, in fact, the mode-matching of the cavity requires two steps [111]:• PRM2 is moved of ∆L to tune the arm lever L2• all the recycling mirrors (PRM1, PRM2, PRM3) are moved together by a length 2∆L in thesame direction than in the previous step.

1386.2. Design History: from the Baseline to the Vertical Layout

2
Figure 6.22: Stability plot of the vertical layout, γinput
versus {R3 , R2 }.

L2 tuning
R3 errorFigure 6.23: Tuning of the lever arm L2 in the vertical layout.

139

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgoparameter
L3 /Lx
L2
L1
R3
R2
R1

tolerance [%]± 100± 0.04± 4.6± 0.04± 0.4± 17

Table 6.14: Tolerance of parameters for a stable vertical layout.

Thus finally PRM2 has to be moved of 3∆L, and PRM1 and PRM3 have to be moved of 2∆L each.
If ∆L = 7 mm, the displacements required for the suspension of each recycling mirror becomeslarger than 1 cm. To solve this issue we can measure the RoC of PRM3 with great accuracy, inorder to realize the pre-positioning of the suspension system before the installation of the mirror.
The RoC of PRM3 can be measured with 0.01% precision [112], corresponding to a tuning of
the telescope length of ∼ 1 mm. Nevertheless, the pre-positioning of the vacuum tanks is precise to
1mm, and the pre-positioning of the Superattenuator inside the tank (measured at the top stage ofthe inverted pendulum) can be achieved within 2 mm accuracy [112]. Therefore the pre-positioning
could not be achieved with an accuracy smaller than ∆L ∼ 3 mm. Knowing that PRM2 has tobe displaced of 3∆L, we can see that the needed adjustment is once again beyond the inverted
pendulum dynamics.
The tuning of the cavity mode-matching should be achieved with another actuator than the inverted pendulum. Two solutions have been studied in parallel: we might either use mechanicalactuators, or act with the thermal compensation system to change the RoC of PRM3.
Tuning - mechanical actuation
The mechanical actuators could be hydraulic jacks placed below the vacuum tanks. Their dynamics
is of the order of ∼ 4 cm [112] with an accuracy of the order of ∼ 0.5 mm [113], with mirrors andsuspensions mounted in the vacuum tanks. However, it is not clear whether with this kind of
actuators the vacuum tanks should be vented for safety reasons before moving.
Tuning - thermal actuation
The control of the RoC of PRM3 with the thermal compensation system could be realized withseveral different actuators [114]. In order to increase the RoC, a central heating pattern could beshined on the high-reflectivity surface, either with a CO2 laser or with an infrared source. For theCO2 laser, for instance, reasonable dynamics would be needed: a beam with 2 cm radius could
increase the RoC of PRM3 of 0.5% (5 times the achievable RoC manufacturing error for this mirror)
by shining 5 W on the mirror, 14 W would be needed to increase the RoC of the same amount with abeam of 6-cm radius [114]. The RoC could be increased also by shining an annular heating pattern(achievable either with a ring heater, a CO2 laser or an infrared source) on the high-reflectivity
surface of the mirror, but in this case the dynamics of the actuation would be too low, requiring
too much power [114].

140

6.3. Simulation of the Vertical Layout

In order to decrease the RoC, an annular heating pattern could be shined on the back of themirror (provided by a ring heater, or a CO2 laser or an infrared source), or a ring heater could heat
the lateral surface of the mirror. Both options could decrease the RoC of PRM3 of 0.5% requiring∼ 5 W [114], which is a reasonable amount of power.
The use of thermal actuators will induce residual mirror figure errors, whose impact on thematching of the cavity should be studied with SIS. Also the allowed level of astigmatism inducedby the thermal actuators should be studied. This will be the subject of future study.
Pick-off extraction
The vertical layout allows the simple extraction of four pick-off beams3 : two beams can be extracted
in transmission through P/S-RM2, where the beam has a radius w = 5.5 mm. The hardware forthe detection of pick-offs will be installed in vacuum, and will require the construction of two small
vacuum enclosures beside the tank of power (signal) recycling.
Issues of the vertical layout
The vertical layout has several advantages, which led us to retain this configuration. It solves the
problem of multiple mirrors suspension, limiting the infrastructure works to the construction ofseveral new vacuum chambers. Because of the astigmatism optimization, the losses for the carrier
and the radio-frequency sidebands are negligible.
Nevertheless, this layout also features two critical points. The first is the increase of cost due tothe construction of vacuum chambers. The second critical factor is the time required for developing
and testing the vertical double-mirror payload, which may delay the schedule of the project.

6.3

Simulation of the Vertical Layout

The vertical layout was extensively simulated with SIS, in order to study its optical performanceswith respect to realistic mirror defects in the coupled cavities of arm and power recycling. We
simulated the carrier and the radio-frequency sidebands to measure the recycling gain and thehigher-order mode content of the resonating fields. We did not include in the simulation also theaudio-sideband fields, thus this will be the subject of future study.
However, given the low astigmatism of the vertical layout and since the audio-frequency sidebands resonate in the signal recycling cavity and partially in the arm cavity, we reasonably expect
the audio-frequency sidebands to show an intermediate loss between the one of the carrier and theone of radio-frequency sidebands.
By applying phase maps of residual optical path on the surface of the arm cavity input mirror,
we simulated two different kinds of mirror defects. In one case we simulated the response of thecoupled cavities in presence of substrate index inhomogeneities (cold defects). In the other case,we simulated the coupled cavities in presence of residual thermal effects after compensation (hotdefects), due to uniform and non-uniform absorption patterns.
3

The system for the longitudinal and angular control of mirrors requires the extraction of secondary beams, or
pick-offs, from the beam of the interferometer.

141

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

In both cases we simulated the coupled cavities without the beamsplitter, thus the layout isthe one shown on fig.5.9d. The cavity parameters used in the simulation are those listed in tables
4.1 and 6.12. We used 1 W for the input beam, to have a direct measurement of the recycling
gain through the power resonant in the cavity. The modulation frequency used for the sidebands
is f = 7.47025 MHz.
Initially we simulated the coupled cavities with perfect optics and null incidence angles, in order to measure the recycling gain in the case of perfect coupling: it is GCARR = 40 for the carrier
and GSB = 112 for the radio-frequency sidebands. Then we included the incidence angles in thesimulation, though we can reasonably consider the effect of astigmatism negligible.
Finally we included the defects of the mirror, and we performed the simulations with the following
steps:
1. we put the carrier field in resonance with the coupled cavities: this is done in SIS with thecommand ‘lock’. The lengths of the two cavities refer to the ‘lock’ point
2. we inject the upper and lower sidebands. Because of the presence of defects, the resonanceof the sidebands is shifted with respect to that of the carrier
3. we change the length of the recycling cavity to find the resonance of the sidebands, thus we
make a scan of the cavity length. The carrier gets out of resonance, but we can neglect thiseffect since it is always possible to adjust the length of the arm cavity to put the carrier inresonance again. At each step of the scan we compute the resonant power in the coupledcavities, and we measure the recycling gain of the fields
4. we plot the recycling gain versus the scan length, for carrier and sideband fields, and wecompare the results with the case of perfect coupling. We also observe the shape of sideband
fields at the resonance peak and we measure the fraction of higher-order mode content.
We will present in the following the features of the phase maps we used and the results of oursimulations. In our analysis we will also compare the optical performances of non-degeneratecavities with those of marginally-stable cavities, for cold and hot defects. We will see that nondegenerate cavities are much more robust and yield a much better performance.

6.3.1

Cold Defects

For the simulation of cold defects we used a map of substrate index inhomogeneities of an aLIGO
mirror, measured at the Laboratoire des Matériaux Avancés (LMA) in Lyon. In the central area(within a radius of 10 cm), the inhomogeneity is about 0.03 ppm peak-valley for a mirror of 20-cm
thickness, yielding about 6 nm. The total area of the map is slightly smaller than the actual mirror
size (70 cm of diameter), but this has a negligible effect in the simulation.
The scan of the recycling cavity with cold defects is reported on fig.6.25. We can observe that:carrier the recycling gain is more than 99% of the maximum and the field has a negligible content
of higher-order modes

142

6.3. Simulation of the Vertical Layout

Figure 6.24: Phase map [rad] of substrate inhomogeneities of one arm cavity input mirror of aLIGO.

radio-frequency sidebands the curves of lower and upper sidebands are completely superposed,
showing that upper and sideband fields behave exactly the same way (we will see that this is
not the case with marginally-stable cavities). The resonance peak indicates a recycling gainof 104, i.e. 93% of the maximum, and is shifted of 1.8 nm with respect to that of the carrier.
The normalized intensity pattern at resonance is shown on fig.6.27: it has a Gaussian shape.
Its higher-order mode content is ∼ 2.7%, whereof 2.5% is due to LG01 mode.
These results indicates that the resonating fields in the non-degenerate recycling cavity are notappreciably affected by the presence of cold defects in the arm cavity input mirror.
Comparison with marginally-stable recycling cavities
We can make a comparison between our results and those of a simulation of the marginally-stable
power recycling cavity of AdV, reported in [115]. The comparison can be only qualitative, since
the two simulations have some different features: T = 5% was used for the power recycling mirrorof the marginally-stable cavity (yielding a lower finesse), and mirror surface defects were added tothe substrate inhomogeneities of fig.6.24. However, the presence of mirror surface defects only had
a little influence on the results of the simulation [115].
The simulation of the marginally-stable power recycling cavity and our simulation have been
performed with the same steps, therefore we can compare the results of the two cases one by one:the scans of the recycling cavity length, the recycling gain of the resonant fields, the shape ofsidebands.

143

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

100
90

recycling gain

80
70
60
50
40
30
20
10
−10

−8

−6

−4

−2
0
2
microscopic tuning [nm]

4

6

8

Figure 6.25: Scan of the recycling gain of carrier (blue curve), upper (red curve) and lower sideband field(green dashed curve) in the non-degenerate cavity with cold defects.

Figure 6.26: Scan of the recycling gain inside the marginally-stable cavity with cold defects, from [115].

144

6.3. Simulation of the Vertical Layout

Figure 6.27: Normalized intensity of the upper side-band in the non-degenerate cavity with cold defects.

Figure 6.28: Intensity of the upper side-band in the marginally-stable recycling cavity with cold defects,
from [115]. Units on the axes are [m].

145Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

Figure 6.29: Intensity of the lower side-band in the marginally-stable recycling cavity with cold defects,
with the annular heating pattern applied. From [116], units on the axes are [m].

Fig.6.26 shows the scan of the marginally-stable cavity (the scale of the axes is different from thatof fig.6.25):
carrier the recycling gain is more than the 99% of the maximum, which is GCARR = 36, thus
losses are negligible
radio-frequency sidebands the degradation is much larger than in the case of non-degeneratecavities. The losses in the TEM00 mode are around 90% (the maximum gain being 77),
whereof only 15% is due to the presence of surface defects. The two sidebands behave very
differently, their resonances show some structures and their TEM00 content is reduced to aminor fraction. Fig.6.28 shows the shape of the upper sideband at resonance: the Gaussian
pattern is almost totally lost.
The results of the comparison between the non-degenerate and the marginally-stable cavities are
summarized in table 6.15.
The impact of substrate inhomogeneities is so relevant that marginally-stable cavities will need
the thermal compensation system to compensate also for the cold defects of the interferometer.
The use of the annular heating pattern will improve the shape of sidebands, as shown on fig.6.29,
and will decrease the loss of their recycling gain to ∼ 45% [116].
In order to achieve a further improvement of the recycling gain and shape of the sidebands, a
non-axisymmetric thermal compensation system for the correction of asymmetric defects will benecessary, and is currently under study [117].

146

6.3. Simulation of the Vertical Layout

Figure 6.30: Phase map [rad]of residual thermal effects from uniform absorption.

With cold defects it will not be possible to sense the wavefront distortions with a Hartmann
sensor, since substrate inhomogeneities cause the aberration of the reference wavefront used for themeasure. Thus another wavefront sensor, like the phase camera, will have to be used.

6.3.2

Hot Defects

We used two phase maps provided by the TCS group of Virgo of Università di Roma Tor Vergata, tosimulate hot defects. Maps have been computed through simulations of thermal effects performed
with ANSYS [118] (a code for finite element analysis), assuming the interferometer running with
125 W of input power.
The group simulated two patterns of absorption: the first is uniform, the second is produced
from a real measurement of absorption of an aLIGO mirror (performed at LMA). In both cases
the annular heating pattern (generated by the double-axicon system of fig.4.5) has been applied
to compensate for thermal effects. Therefore the maps we used in our simulations represent the
optical path due to residual thermal effects. We will present in the following the results concerning
the two cases of hot defects.
We will also compare our results with those reported in [119], where the same hot defects havebeen simulated in a marginally-stable power recycling cavity with recycling mirror transmission
T = 1%. The sensitivity of marginally-stable cavities to hot defects depends on this parameter: thehigher is the cavity finesse, the the larger is the impact of the defects4 . Therefore the comparison
will be qualitative. Even though the simulated configurations are different, the conclusions willremain the same: as we will see, the non-degenerate cavities achieve better performances than themarginally-stable cavities.
4

For this reason, the reference value of the transmission of the power recycling mirror has been presently increased
to T = 5%.

147

Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

100
90

recycling gain

80
70
60
50
40
30
20
10
0

1

2

3

4
5
6
7
microscopic tuning [nm]

8

9

10

Figure 6.31: Scan of the recycling gain of carrier of carrier (blue curve), upper (red curve) and lower sideband
field (green dashed curve) in the non-degenerate cavity with hot defects from uniform absorption.

Figure 6.32: Scan of the recycling gain inside the marginally-stable cavity with hot defects from uniform
absorption, from [119].

148

6.3. Simulation of the Vertical Layout

Figure 6.33: Normalized intensity of the upper side-band in the non-degenerate cavity with hot defects from
uniform absorption.

Figure 6.34: Intensity of the upper side-band in the marginally-stable recycling cavity with hot defects from
uniform absorption, from [119]. Units on the axes are [m].

149Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

Uniform absorption
Fig.6.30 shows the phase map computed in the case of uniform absorption of 0.6 ppm, which thenhas a radial symmetry.
The scan of the recycling cavity length is shown on fig6.31. We can observe thatcarrier the recycling gain is higher than 99%, and the content of higher-order modes is less than
0.5%
radio-frequency sidebands the behavior of the upper and lower sidebands is the same, therecycling gain is 93%. The resonances of sidebands and carrier are spaced of 0.2 nm. The
shape of sidebands is illustrated on fig.6.33: they have a Gaussian shape. The higher-ordermode content is less than 0.5%.
The results of the simulation indicate that the residual thermal effects due to a uniform absorption
pattern do not affect considerably the fields resonating in the non-degenerate cavity.
Uniform absorption - comparison with marginally-stable recycling cavities
Fig.6.32 shows the results of the simulation of the same hot defects due to uniform thermal absorption in the marginally-stable power recycling cavity [119]:carrier the recycling gain is higher than 99% of the maximum, which is GCARR = 100
radio-frequency sidebands the recycling gain is 58% of the maximum, which is GSB = 380.
The 19% and 16% of the resonant power is transferred to higher-order modes for upper andlower sideband, respectively. The shape of sidebands is degraded, as shown by fig.6.34.
As expected, the behavior of non-degenerate cavities is better than that of marginally stablecavities, which show a degradation of sidebands in presence of axisymmetric hot defects. The
results of the comparison are summarized in table 6.15.
Non-uniform absorption
The map used in the simulation of non-uniform hot defects is shown on fig.6.35. It has been obtained from a measured absorption map of 0.57 ppm average level, with two point absorbers of 1.4
ppm and 1.9 ppm [120].
The scan of the recycling cavity length is reported in fig.6.36:
carrier the recycling gain larger than 99% of the maximum, the higher-order mode content is
negligible
radio-frequency sidebands upper and lower sideband behave equally, and their resonance isspaced of 0.4 nm from that of the carrier. The recycling gain is 103, equal to the 92% of the
maximum, and their higher-order mode content is less than 0.8%. The shape at resonance isthat of a pure Gaussian mode.
Therefore we can conclude that the carrier and the sideband fields resonating in the non-degeneratecavity are not appreciably affected by the presence of non-uniform residual thermal effects.

1506.3. Simulation of the Vertical Layout

Figure 6.35: Phase map [rad] of residual thermal effects from non-uniform absorption.

Non-uniform absorption - comparison with marginally-stable recycling cavities
The same non-uniform hot defects have been simulated in the marginally-stable power recycling
cavity [119]. The scan of the cavity length is reported in fig.6.36, where we can see thatcarrier the recycling gain larger than 99% of the maximum, which is GCARR = 100
radio-frequency sidebands the Gaussian mode of upper and lower sideband is almost disappeared due to the 96% of loss of recycling gain (the maximum being GSB = 380).
Thus also in the case of hot defects due to non-uniform absorption, we can conclude that the optical
performances of non-degenerate cavities are much better than those of marginally-stable cavities.
The results are summarized in table 6.15.
The results of the simulations with hot defects indicate that non-degenerate cavities will work
properly with 125 W of input power and with the axisymmetric heating pattern of the thermal compensation system, for any kind of absorption pattern in the arm cavity input mirror.
Marginally-stable cavities instead, will require the implementation of a non-axisymmetric thermalcompensation system to work with the same input power, in presence of a non-uniform absorption
pattern in the arm cavity input mirrors.

151Chapter 6. Simulation of the Non-Degenerate Recycling Cavities of Advanced Virgo

100
90

recycling gain

80
70
60
50
40
30
20
10
−10

−8

−6

−4

−2
0
2
microscopic tuning [nm]

4

6

8

Figure 6.36: Recycling gain of carrier (blue curve), upper (red curve) and lower sideband field (green dashedcurve) in the non-degenerate cavity with hot defects from non-uniform absorption.

Figure 6.37: Recycling gain inside the marginally-stable cavity with hot defects from non-uniform absorption,from [119].

152

6.3. Simulation of the Vertical Layout

defect/configuration
cold defects/NDRC
cold defects/MSRC
uniform hot defects/NDRC
uniform hot defects/MSRC
non-unif. hot defects/NDRC
non-unif. hot defects/MSRC

recycling gain [%]higher-order modes [%]

93%
10%
93%
42%
92%
4%

2.7
−
< 0.5
upper s.b. 19% / lower s.b. 16%< 0.8−

Table 6.15: Comparison of simulation results for the radio-frequency sidebands: non-degenerate recycling
cavities (NDRC) versus marginally-stable recycling cavities (MSRC, from [115, 119]).

153

154

Conclusions and Perspectives - Part I
The experience with initial Virgo and Ligo showed that higher-order mode contamination in
marginally-stable cavities can considerably affect the operation of the detector, slowing down thecommissioning phase and making the control complex and difficult.
The issue of mode degeneracy will be even more relevant for advanced detectors, where theincrease of the beam size on the arm cavity mirrors will be necessary to decrease mirror thermal
noise, which will be one of the major noise sources. Moreover, due to the implementation of signal
recycling, mode degeneracy will also cause a loss of signal of potential detections.
In order to remove mode degeneracy, the implementation of non-degenerate recycling cavities(proposed by aLIGO as first [93, 94]) was adopted in AdV as reference solution [69].
The achievement of the optical design of the non-degenerate recycling cavities of AdV took about
two years and a half, starting in fall 2008, and required several steps:• initially we studied the general features of non-degenerate cavities (Gouy phase and beamfocusing), and we confirmed the design originally proposed by aLIGO, featuring a folded
optical path with three mirrors. Two of these mirrors form a telescope for the focusing andthe collimation of the beam inside the cavity. This design has several advantages:– it allows to adjust the distance between the telescope mirrors, in order to tune themode-matching of the cavity– it allows the simple extraction of secondary pick-off beams, since the beam radius attains
few mm inside the cavity– it simplifies the design of the telescopes of injection and detection systems, which shouldmatch a beam radius of the order of 1 mm on the recycling mirrors– it allows to use small recycling mirrors which could be easily installed and replaced, if
it is necessary or to change the optical parameters of the cavity• then we studied the implementation of non-degenerate recycling cavities in AdV: the opticaldesign is limited by infrastructure and budget constraints (which are not present in aLIGO,
for instance)• subsequently we developed the optical simulations that we used to draw the design of thecavities. These simulation tools are flexible, so we used them to follow the design process ofvarious layouts• then we chose a configuration: the choice was determined not only by the optics requirements,
but also by those of the mirror suspension system155• finally we characterized the performances of the chosen design by means of optical simulations.
The vertical layout
We retained the so-called vertical layout, since it requires small changes to the infrastructures of
AdV and necessitate a limited re-design of the payload of one Superattenuator. We simulated thisconfiguration with the ray-matrix code and with the FFT code SIS:
• we optimized the astigmatism, reducing the coupling loss to 0.1% for the carrier field and0.2% for the radio-frequency sideband fields• we tested its robustness with respect to different classes of mirrors defects, cold (substrateindex inhomogeneities) and hot (residual thermal effects after axisymmetric compensation):– the recycling gain of the carrier is beyond 99% of the maximum (GCARR = 40) for all
the defects that we simulated: the loss is always negligible, as well as the higher-ordermode content– the recycling gain of radio-frequency sidebands is larger than 90% of the maximum(GSB = 112) for all the defects, and the higher-order mode content is always negligible:∗ with cold defects the gain is 93% and the higher-order mode content is ∼ 2.7%
∗ with hot defects due to uniform absorption the gain is 93%, and the higher-ordermode content is less than 0.5%
∗ with hot defects due to non-uniform absorption the gain is 92%, and the higher-ordermode content is less than 0.8%.
we did not simulate the audio-frequency sidebands of the signal, but we reasonably
expect them to show an intermediate behavior between those of the carrier and theradio-frequency sidebands.
The results show that our design is robust and achieves good optical performances with respect tocold and hot mirror defects.
However, the vertical layout incurs some budget and schedule risks, related to the infrastructure works and to the development of the vertical payload for the double-mirror suspension. Furthermore, the achievement of an optimal axisymmetric compensation pattern (realized with the
double-axicon system) for the thermal compensation system led in the spring of 2011 to reconsider
the choice of marginally-stable cavities for AdV.
Perspectives
The implementation of marginally-stable cavities will require a solution for two crucial issues. The
first is the extraction of pick-off beams for purposes of sensing and control, since the beam has aradius of several cm everywhere in the marginally-stable cavities. The second concerns cold andhot defects in the mirror.
Later analysis [115, 119] demonstrated that marginally-stable cavities will be highly sensitive
to substrate index inhomogeneities and non-uniform coating absorption, which will yield significant sideband degradation. For this reason the development and the implementation of a nonaxisymmetric thermal compensation system will be necessary for the detector operation [117].

156

Though non-degenerate cavities will not be implemented in AdV, we believe that they stillmight be part of a possible upgrade of the project, and that the optical design of third-generation
detectors may benefit of the studies developed in this work.

157

158

PART IIIUse of Higher-Order Laguerre-Gauss Modesin Future Gravitational-Wave Interferometers

Chapter 7

Reduction of Mirror Thermal Noise
with Laguerre-Gauss Modes
Mirror thermal noise is expected to dominate the thermal noise spectrum in the mid-frequency
range of all currently planned terrestrial detectors. Its reduction is of fundamental importanceto improve the sensitivity of the second-generation detectors (and to operate what are sometimes
called 2.5-generation detectors), and to build the third-generation detectors as ET.
The spectral density of mirror thermal noise shows a clear dependence on temperature, material
properties and read-out beam spot size. Several technical solutions may thus be adopted for its
reduction:• the cooling of mirrors using cryogenics• the choice of materials with good mechanical properties, for coatings and substrates• the increase of the beam size on mirrors• the reduction of the coupling of the laser beam with thermal noise fluctuations, using larger
or non-Gaussian beams
These solutions are not mutually exclusive in principle, and might be used jointly to reduce mirror
thermal noise. In section 7.1 of this chapter we will describe the solutions to decrease thermal noisewith better materials and cryogenics, pointing out the possible difficulties. In the same section wewill describe the limitations related to the increase of the beam size.
The option of using flat and wide non-Gaussian readout beams had been initially suggested for
so-called mesa or flat-top beams, and later on conical beams had been proposed as the optimal
beam shape for maximum coating thermal noise reduction. The main inconvenient of mesa andconical beams is that they resonate in cavities with non-spherical mirrors.
Higher-order LG`p modes have been proposed in 2006 in alternative to mesa beams. Being solutions of the paraxial wave equation, the LG`p modes are eigenmodes of spherical-mirror resonators.
Thus from the technical point of view, LG`p beams are more appealing since their implementationin gravitational-wave interferometers is in principle simpler.
Analytical calculations indicate that LG33 modes might decrease mirror thermal noise of a factor
1.76 in AdV and 1.83 in ET-HF, using the same mirrors. The reason of this decrease lies in thewider geometry of these beams, which averages out the fluctuations of the mirror surface, due to161

Chapter 7. Reduction of Mirror Thermal Noise with Laguerre-Gauss Modes

thermal noise, better than Gaussian beams do. LG`p beams have also an additional advantage:
they are expected to induce smaller thermal effects in mirrors, compared to those induced by thecurrently used Gaussian intensity pattern.
Some fundamental aspects have to be studied before LG`p beams could be used in km-scaleinterferometers. Topics to be addressed are the generation of LG`p beams and their optical performances in power-recycled Fabry-Perot interferometers. We need to analyze the feasibility ofhigher-order LG`p beams interferometry by studying the achievable contrast of an LG`p detector,
the compatibility of LG`p beams with the control scheme of gravitational-wave interferometers (forexample with respect to the extraction of longitudinal and angular error signals of mirror position),and the tolerance of LG`p beams to mirror misalignments and figure errors.
Studies on these subjects must be carried out by means of an experimental table-top setup andof optical simulations of real detectors (with realistic control schemes, defects of optics, etc.). In
the next chapter we will address some of these fundamental questions going through the results ofa dedicated experimental setup.

7.1

Techniques to Reduce Mirror Thermal Noise

Cryogenics
The linear spectral density of mirror thermal noise√ has a twofold dependence on temperature,through the fluctuation-dissipation theorem: its T dependence is determined by the thermalenergy of the system, and it is function of the mechanical properties of materials which can largely
vary with the temperature. If mirrors can be made of a material with low mechanical dissipation
at low temperature, cryogenics is one of the most appealing technologies to reduce thermal noise.
However, the fundamental technical issue to be solved in a cryogenic gravitational-wave interferometer is how to cool down the mirrors without introducing additional vibrations that would
spoil the low-frequency part of the detection band. Mirrors and suspensions are isolated in vacuum
enclosures, where the heat of the laser power absorbed by the optics cannot be evacuated neither
by convection of air nor by irradiation. Heat can only be extracted through mechanical-thermal
links between the mirror last suspension stage and the cold reservoir, which is usually a vibrating
pulse-tube cryostat. Therefore cryogenics is technologically challenging.
So far, it has been tested only in a small-scale detector prototype with no power recycling[121], thus with reduced circulating power. Among the second-generation detectors, only LCGT[70] will make use of cryogenics, testing for the first time its application to a kilometer-scale power
recycled interferometer. The use of cryogenics seems unlikely for the upgrade of advanced detectors(sometimes called 2.5-generation), due to the heavy infrastructures it requires. It is planned for
ET-LF, where low power is circulating, whereas ET-HF will be at room temperature [75].
Choice of materials
Currently, all gravitational wave detectors operating at room temperature use fused silica (SiO2 )as optical substrate material, due to a combination of excellent optical, thermal and mechanicalproperties. Therefore fused silica will be the substrate material also in advanced detectors [68, 69].
However, fused silica is unsuited for cryogenic use, because of its high mechanical loss at low temperature [40]. This is why sapphire has been selected as mirror substrate materials for LCGT [70],1627.1. Techniques to Reduce Mirror Thermal Noiseand silicon and sapphire are under investigation as mirror substrate material for third-generation
detectors [75]. Both materials feature a high thermal conductivity κ(T ) and a low thermal expansion coefficient α(T ) at low temperature, where sapphire has also low mechanical loss [40].
For what concerns the coating, the reference solution is the use of Ti-doped Ta2 O5 layers [69, 68].
The mechanical losses of this material are the lowest possible, so far. Several research projects areunder development for third-generation detectors, in order to assess the possibility of decreasing
further thermal noise with new coating technologies [78].
Beam size
The power spectral density of the substrate and coating Brownian noise is inversely proportional
to the beam radius w on optics. Thereby the detector sensitivity increases using a larger beam sizeon the mirrors of the Fabry-Perot arm cavities. This is the reason why the beam size on the arm
cavity mirrors of AdV has been increased with respect to initial Virgo.
However, the beam size on mirrors is limited by the cavity length L, because of the stability
condition of the resonator (see eq.(5.15)). Consider for instance the simple case where the RoCs of
the cavity mirrors are equal, R1 = R2 ≡ R. Then the beam size on the mirror is [95]w4 =

 λ 2πLR,
2 − L/R(7.1)

and the cavity is stable as long as R > L/2. We might choose R → ∞, but the beam size increases
slowly with respect to the RoC, thus it is very difficult to have large beams with very large RoCs.
For example, if we take L = 3 km like in the case of AdV, we should choose R = 25 km only to get
w = 4.6 cm. Furthermore, for too large RoCs we would start also to have problems of degeneracy:
the frequency spacing of higher-order modes (see eq.(5.21)),∆ν = ν0∆N
cos−1 (1 − L/R) ,
π

(7.2)

decreases as R increases.
Otherwise the beam size can be increased by taking R → L/2, but then the cavity gets close to
instability: we have to choose R with some safety margin, and the beam size is limited. Thereby thereference design of Fabry-Perot arm cavities in advanced and third-generation detectors is a tradeoff between the need of increasing the beam size on mirrors and the proximity to cavity instability.
Of course, as the mirror size increases, the technical difficulties in their realization (quality of thesurface, coating, etc.) increase as well.
Mesa and conical beams
Mirror thermal noise might be reduced by using mesa beams [122] and conical beams [123]. Historically, mesa beams have been the first to be studied for the purpose of decreasing mirror thermal
noise. They are large-radius, flat-topped beams with steep edges to constrain diffraction losses.
Mesa beams can be generated by overlapping many narrow Gaussian beams at waist over a circular surface D of radius b:
Zh (x − x )2 + (y − y )2 iA00Ψ(x, y, 0) = 2
exp −dx0 dy0 ,
(7.3)
πb D
w02

163

Chapter 7. Reduction of Mirror Thermal Noise with Laguerre-Gauss Modes

Figure 7.1: Intensity profiles of LG00 (red curve), mesa (green curve) and conical (blue curve) beams versus
the radial coordinate r, from [123].

p
where A is a normalization factor, and w0 = Lλ/2π is the optimal waist size for propagation
in a cavity of length L [122]. Let zR = πw02 /λ be the Rayleighrange ofpthe beam, and z be the√propagation distance. By defining Z ≡ 1 − iz/zR , w = w0 ZZ and r = x2 + y 2 , the normalized
amplitude of a mesa mode propagated over the distance z from waist is [34]2Z
Ψ(r, z) = √
b πM
where

Zb/w
h
i
exp − Z(r/w − x)2 exp(−2Zrx/w)I0 (2Zrx/w) dx ,

(7.4)

0

h
i
M ≡ 1 − exp(−b2 /w02 ) I0 (b2 /w02 ) + I1 (b2 /w02 ) ,(7.5)

In (x) being the modified Bessel functions. The intensity profile of a mesa beam is shown on fig.7.1.
These beams resonate in Fabry-Perot cavities with non-spherical mirrors, which have a shallow
bump in the middle and a steep outer rim to match the mesa beam wavefront [124].
However, even if mesa beams are expected to provide an effective decrease of mirror thermal
noise [34], their implementation in gravitational-wave interferometers presents some issues. As
demonstrated experimentally, the alignment tolerances for a mesa-beam cavity are expected to befrom 3 to 5 times more stringent than those of spherical resonators in advanced detectors [122, 124].
In a spherical resonator, a tilted mirror induces a shift and a tilt of the optical axis, and thecavity resonates the same transverse-mode spectrum centered on the shifted axis. In contrast, any
misalignment of the non-spherical mesa-beam mirrors destroys the cylindrical symmetry of thecavity. In this case the resonant beam senses a mismatching mirror profile, and the cavity modethus have a radically different intensity distribution and phase front [124]. Moreover, mesa beams
are particularly sensitive to mirror imperfections, especially in the central part of the mirror that
determines the resonant beam shape [122, 124].
Conical beams are another type of non-Gaussian beam, obtained by computing the optimal
beam shape which minimizes mirror thermal noise in a cavity with fixed mirror radius and for164

7.2. Thermal Noise Reduction with Laguerre-Gauss Modes

fixed beam diffraction losses [123]. In fact, conical beams are even wider than mesa beams, though
maintaining the same diffraction losses and total integrated power intensity. Their intensity profileis shown on fig.7.1: it extends more towards the boundaries of the mirror, but cuts off more sharply
at the edge, keeping clipping losses low. The phase front of these beams has a conical shape, likethe Gauss-Bessel beams of the form [34]h Z(r2 + θ2 z 2 ) i  krθ Ψ(r, z) = A exp −J0.
w2 (z)
Z(7.6)

Conical beams are expected to decrease mirror thermal noise even more efficiently than mesa
beams [123]. Nevertheless, a Fabry-Perot resonator with conical mirrors is expected to be sensitive
to mirror translations, misalignments and figure errors even more than a mesa-beam cavity [123].

7.2

Thermal Noise Reduction with Laguerre-Gauss Modes

LG`p modes have been proposed in alternative to mesa and conical beams [125]. For the samemirror diameter and for equivalent diffraction losses and beam power, higher-order LG`p beams
have a multi-ringed power distribution which is wider than the distribution of the fundamental
Gaussian LG00 mode. This is shown on figs.7.2 and 7.3: the first plot compares the intensity crosssections of LG00 , LG33 and LG55 modes, the second plot shows the integrated power of these modes
as function of the radial coordinate r.
Because of their wider intensity distribution, higher-order LG`p modes can decrease the impactof mirror thermal noise on the detector sensitivity. The beam size on optics should however be
adapted to limit the diffraction losses: for every given LG`p mode of radius w and mirror diametera, there is a ratio a/w yielding 1 ppm of losses [125], as shown on fig.7.4. For example, for the LG33
mode this ratio is 4.31, for the LG55 mode is 5.05.
The reduction of noise can be estimated by means of the mirror strain energy Up` associated to
the LG`p mode through eq.(2.64). Here we will limit our analysis to the case of substrate and coating
brownian noise, in the case of infinite mirrors. This approximation is useful to illustrate in a simple
way the decrease of thermal noise, when LG`p modes are used. This is also a good approximation
when the beam size is much smaller than the radius of the mirror, so that the diffraction losses are
negligible. The accurate computation of brownian noise and thermoelastic noise with finite mirrors
can be found in [34, 126].
We will describe the approach developed in [34]: for infinite mirrors, the strain energy Up` forthe substrate is
1 − σ2 `Up` = √
Γ ,(7.7)
2 πY w p
with σ and Y being the Poisson ratio and the Young modulus of the substrate material, respectively.
Γ`p is a numerical factor that comes from the integration of eq.(2.63) and decreases for increasing
indices p and `. For example:
Γ00 = 1
Γ33 = 0.31
(7.8)
Γ55 = 0.25 .

165

Chapter 7. Reduction of Mirror Thermal Noise with Laguerre-Gauss Modes

400
350

intensity [W]

300
250
200
150
100
50
0

0.01

0.02

0.03

0.04

0.05
r [m]

0.06

0.07

0.08

0.09

Figure 7.2: Intensity profiles of LG00 (red curve), LG33 (blue curve) and LG55 (green curve) modes versus theradial coordinate r, for a beam power P = 1 W and for 1 ppm clipping loss on a mirror with 10-cm radius.

0.9

integrated power [W]

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.01

0.02

0.03

0.04

0.05
r [m]

0.06

0.07

0.08

0.09

Figure 7.3: Integrated power of the LG00 (red curve), LG33 (blue curve) and LG55 (green curve) modes shownon fig.7.2, versus the radial coordinate r.

166

7.2. Thermal Noise Reduction with Laguerre-Gauss ModesNl,pFigure 7.4: Ratio a/w yielding 1 ppm of diffraction losses versus the order of the LG`p mode, from [125].

p
For larger values of p and `, Γ`p approaches asymptotically the value 2p + |`| + 1. By replacing
eq.(7.7) in eq.(2.64), for the power spectral density of substrate brownian noise we findSp` (f, T ) =2kB T (1 − σ 2 )φ(f ) Γ`p .
π 3/2 Y wf

(7.9)

Thus if we consider an LG33 mode, the linear spectral density of substrate brownian noise is lower
by a factor
s
s
S00
Γ00
=
' 1.8 .
(7.10)
3
S3
Γ33
Similarly, for an infinite mirror, the strain energy Up` of a coating layer of thickness l is [34]l(1 + σ)(1 − 2σ)χ(σ, σc , Y, Yc )Λ`p ,πY w2(7.11)

1 − 2σc hY (1 + σc )Yc (1 + σ) i+,2(1 − σc ) Yc (1 + σ)(1 − 2σ) Y (1 + σc )(7.12)

Up` =
where
χ≡

Yc , σc indicate the material properties of the coating, Λ`p is a decreasing numerical factor versus pand `. The first values for Λ`p are:
Λ00 = 1
Λ33 = 0.14
(7.13)
Λ55 = 0.10 .

167

Chapter 7. Reduction of Mirror Thermal Noise with Laguerre-Gauss ModesNl,pFigure 7.5: Relative reduction of the linear spectral density of mirror thermal noise versus the order of theLG`p mode (with w tuned for 1 ppm diffraction losses), with respect to the configuration of the arm cavity
input mirror of initial Virgo (LG00 with w = 2 cm on a mirror of radius a = 17.5 cm and thickness h = 10cm). From [125].

By replacing eq.(7.11) in eq.(2.64), we findSp` (f, T ) =4kB T (1 + σ)(1 − 2σ)lχ(σ, σc , Y, Yc )φ(f ) Λ`p ,π 2 Y w2 f

(7.14)

so that if again we consider an LG33 mode, the coating brownian noise is lowered by a factor
s
s
S00
Λ00
=
' 2.7 .
(7.15)
S33
Λ33
From eqs.(7.7) and (7.11) it is clear that (within the approximation of infinite mirrors) the reductionof mirror brownian noise with LG`p modes depends on the order N = 2p+|`| of the mode considered.
Higher N values lead to larger beams and thus are expected to yield lower thermal noise (fig.7.5).
In case of finite mirrors, the estimation of brownian noise is a complex calculation whose resultscannot be expressed with a simple and compact analytical form. We then report here only the
results from [126] about the detailed computation of mirror thermal noise decrease with an LG33
beam for finite mirrors, for AdV and ET-HF. We will justify the choice of the LG33 mode in the next
chapter. Considering the overall thermal noise contribution (brownian and thermo-elastic noise of
coating and substrate) of the four mirrors of the Fabry-Perot arm cavities, mirror thermal noise in
AdV is reduced by a factor 1.76 with respect to the case with an LG00 beam. In ET-HF this factor
is equal to 1.83 [126].

168

169

Chapter 8

A Table-Top Experiment to Test
Generation and Interferometry ofHigher-Order Laguerre-Gauss Modes
Before the implementation of LG`p modes in future gravitational-wave interferometers, mode generation and interferometry must be studied carefully through experiments and optical simulations.
We will address in this chapter the topic of generation of LG33 modes, together with some aspectsof LG33 mode basic interferometry, from the experimental point of view.
There exist presently many techniques to generate LG`p beams, originally developed for applications other than gravitational-wave detection. We need to understand wether one of these techniques could satisfy the requirements of gravitational-wave experiments, in terms of beam quality,
efficiency and compatibility with other technical solutions that will be adopted in interferometersof future generations. In section 8.1 of this chapter we will define a series of criteria to evaluate thegeneration technique to be adopted, and we will go through a review of all the generation methods
presently available.
Fused-silica etched diffractive plates are particularly interesting, since they could generate modesof reasonable quality, starting from an input Gaussian beam, and in principle they can handle thehigh-power necessary in future gravitational-wave interferometers. The quality of the generated
modes might be further increased by using a mode-cleaner cavity, to filter the spatial defects of thebeam. Thus we propose in this chapter the development of a new generation technique, suitablefor future detectors, by using a diffractive plate (a converter) and a mode-cleaner cavity (a filter).
We used our technique on a table-top setup to generate high-purity LG33 modes with good
power efficiency. Though initially the scope of the setup was only limited to the development of
the generation technique, we could accomplish a first step towards the assembly of a table-top
power-recycled Fabry-Perot interferometer with LG33 beams. We illuminated a simple Michelsoninterferometer with the generated high-purity LG33 mode, and we locked it on the dark fringe.
The chapter is organized as follows: in sections 8.3 and 8.2 we will describe the design of thetwo main elements of the setup, the mode-cleaner and the LG33 diffractive plate, respectively. The
main objectives and the detailed scheme of the experimental setup are presented in section 8.4,
and the results will be presented in the next sections. The mode-matching of the input LG00 beam
to the diffractive plate is described in section 8.5. The LG33 mode generated through the plate isanalyzed in section 8.6. The alignment of the optics of the setup is achieved following a procedure1708.1. Techniques for Higher-Order Mode Generation

described in section 8.7. Section 8.8 describes the servo loop for the lock of the laser frequency
on the resonance of LG33 mode-cleaner eigenmode, section 8.9 reports about the characterizationof the cavity spectrum. The high-purity transmitted LG33 mode is analyzed in section 8.10. The
global efficiency of our generation technique is discussed in section 8.11.

8.1

Techniques for Higher-Order Mode GenerationLG`p modes should be generated through a technique with the following features:• acceptable purity• capability of handling high-power• reasonably-high power efficiency• long-term stability.
According to techniques presently available, LG`p modes might be generated either inside the opticalresonator of a laser system or through the conversion of an input Gaussian beam. These techniques
for intra- and extra-cavity generation have been experimentally demonstrated in the past, for applications of LG`p modes in quantum optics experiments about entanglement of orbital momentumof photons [127], or tweezers for optical manipulation of microscopic particles [128].
The intra-cavity generation can be achieved by inserting phase plates [129] or cylindrical lenses[130] in the layout of a laser resonator. So far this technique has been demonstrated exclusively forLG10 modes and only qualitative analysis of the generated modes can be found in literature. Intracavity generation seems a complex solution with respect to the implementation in gravitational-wave
interferometers, since it would require a new design of laser sources. Thus we rather prefer to choose
the extra-cavity option, since it allows to use the well-established laser technology.
Extra-cavity generation is based either on astigmatic mode converters or on phase modulationof the input beam, through diffractive optics. Among these latter, we can distinguish spiral phase
plates, etched-glass plates, spatial light modulators and computer-generated holograms. The working
principle of phase modulation is the following: spiral plates or diffractive optics are used as phaseretarders, placed at the waist of the input Gaussian beam where the wavefront is plane. Here they
imprint a spiralling phase pattern to the input beam, which is converted into an LG`p mode by
self-diffraction along propagation. Hereafter is a brief description of the mentioned extra-cavity
generation techniques:• astigmatic mode converters [131] are systems of two cylindrical lenses which can convert
HGmn of arbitrarily high order in LG`p modes of the same order, with p = min(m, n) and` = m − n, by imposing a controlled Gouy-phase shift to the input beam.
• spiral phase plates [132, 133] are optics whose thickness along the optical axis increases
proportionally to the transverse azimuthal coordinate θ, as shown on fig.8.1. The induced
spiral phase delay isθφ(θ) = k(n − n0 ) hSP P = `θ,(8.1)
2π
where n is the refractive index of the plate material, n0 is the refractive index of the surrounding medium, k = 2π/λ is the wave number, hSP P is the period of the helix on the opticalaxis and ` is the winding number.

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Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

Figure 8.1: Design of a fused silica (n = 1.44963) spiral phase plate for the generation of a LG30 mode.

• spatial light modulators [134, 135] are made of translucent or reflective liquid-crystal
pixels, supported by a monolithic silicon circuit. An electrical signal flowing through thecircuit drives the orientation of the liquid-crystal molecules inside each pixel, and determines
the optical properties of the modulator surface. The driving electrical signal is steered throughan external digital controller, usually a computer, so that modulators can act as dynamic
optical elements.
• etched-glass diffractive plates [136, 137] are slabs of glass with patterns etched on thesurface, causing the diffraction of the incident wavefront in the plane (x, y) transverse to
the beam optical axis. Diffracted wavefronts propagate along a given distance z, and theresulting profile of the propagated beam is the superposition of the diffracted waves. Oneof the simplest example of etched diffractive plate is a transmissive diffraction grating, but
the pattern might also be more complex as well. Etched plates can work in transmission orreflection.
• computer-generated holograms [138] are digitally-computed patterns that contain a coded
record of an optical wave, including its amplitude and phase properties. The record is definedas the interference in the transverse plane between any desired object wavefront to be reproduced, and a plane (or more rarely a spherical) reference wave. Fig. 8.2 shows an example ofcomputer-generated hologram for the generation of an LG11 mode. These patterns are usually
printed on a thin-photographic film support and reduced to transmissive masks. The object
wave is then generated by illuminating the hologram with the plane reference wave.
Astigmatic converters, spiral phase plates and etched-glass diffractive plates satisfy the crucial
requirement of high-power handling capability, whereas holograms and spatial light modulators

1728.1. Techniques for Higher-Order Mode Generation

Figure 8.2: Phase pattern of a computer-generated hologram for the generation of an LG11 mode.

(due to the low damage threshold of thin films and liquid crystals) do not.
However, spiral phase plates can only generate LG`0 modes of lower order N , whose winding
number ` is also limited by the achievable thickness of the plate and the refractive index of thematerial. The use of astigmatic converters is limited by the ability to generate higher-order HGnm
input beams, requiring the implementation of an input laser system of unfeasible complexity. For
example, a mode HG63 would be needed to generate an LG33 mode.
Etched-glass diffractive plates are the most suitable technique for LG`p mode generation, fulfilling
all the fundamental requirements of gravitational-wave experiments. They are stable passive optics
which could handle high-power beams, and have been already used to generate LG`p modes ofrelatively high order, with reasonable results of generation efficiency and purity [137].
We propose here the development of the following generation technique, depicted schematicallyin fig.8.3: an input Gaussian beam is converted through phase modulation in an LG`p mode by afused-silica etched diffractive plate, and the generated mode is spatially filtered by a mode-cleaner
Fabry-Perot cavity to increase its purity. To test the performances of this technique we assembled
a table-top experimental setup [139], described in detail in section 8.4 of this chapter.
We chose to generate an LG33 mode, to test the extension of the diffractive plate technique tomodes of larger order with several radial intensity maxima. This mode can provide en effective
reduction of mirror thermal noise in future detectors, and features a moderate shape complexity.
Thus it represents a tradeoff between the need of generating modes of higher order, which may
provide larger noise reduction, and the increase of the complexity of the beam shape. However, inprinciple, this generation technique could be pushed further to generate any mode order.
In the next two sections we will describe the design of the two main elements of the experimental
setup, the LG33 diffractive plate and the mode-cleaner cavity.

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Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

diffractive plate
laser

LG00

mode-cleaner
pseudo-LG33

LG33

Figure 8.3: Design concept for the generation of a LG33 mode with a diffractive plate (mode converter) anda mode-cleaner Fabry-Perot cavity (spatial filter).

8.2

Design of the LG33 Diffractive Plate

The diffractive plate has been designed and realized by SILIOS Technologies, a company specialized
in the design and fabrication of micro-optical components [140]. We characterized the design of theplate by means of FFT optical simulations that we developed starting from the core of the FFT
code OSCAR [141, 142].
Simulations indicated that the mode generated by the plate would be reasonably pure. Thus,
after approval of the design, SILIOS took care of the fabrication of the plate. In the following we
describe the design procedure of the plate and the optical simulations we performed, together withthe results about the expected quality and efficiency of mode conversion.
Optical layout
The conversion from the LG00 mode to the LG33 mode might be carried out either with a single
phase-modulating plate [136] or with a pair of consecutive plates [137]. SILIOS initially studied
this latter option, but finally a single phase-modulating plate for an input LG00 beam with mm-size
waist was chosen.
The sole constraint of this layout is the alignment tolerance of the plate to transverse translations, to be controlled within ∼ 100 µm on the x and y. This is achievable with optical mounts for
precise alignment, available on the commerce.
Design of the phase pattern
The design of the plate pattern has been realized by SILIOS starting from the theoretical phasepattern of an LG33 mode at waist. The concept of the design is to use the phase plate to imprint
the phase pattern of an LG33 mode to a large-radius input Gaussian beam at waist, which can beapproximated as a uniform plane wave. Thus the input beam is forced by self-diffraction to assume
the shape of an LG33 mode along propagation.
The waist size of the LG33 mode pattern on the plate must be w0 = 705 µm, in order to obtainan output LG33 mode waist of w0 = 288 µm at the focal plane of a 600 mm lens, as shown on fig.8.4.
The theoretical phase of an LG33 mode is shown in fig.8.6 (in the transverse plane): it features the
typical spiral pattern with a phase discontinuity in r = 0 (the optical vortex), and three radialphase discontinuities at the roots of the Laguerre polynomial
L33 (2r2 /w02 ) = −(2r2 /w02 )3+ 3(2r2 /w02 )2 − 15(2r2 /w02 ) + 20 ,
6

174

(8.2)

8.2. Design of the LG33 Diffractive Plate

LG00

DP
LG33

mode-cleaner

600 mmFigure 8.4: Optical layout for the phase-retarding LG33 diffractive plate.

where the polynomial changes sign. The relative positions of the radial discontinuities have an
influence on the shape of the resulting mode, then these positions have been slightly adjusted ‘byhand’ in order to increase the generated mode purity.
Then a blazed grating with period Λ = 235 µm has been superposed to the adjusted LG33 phasepattern. In the transverse plane (x, y), the grating has a complex spatial harmonic transmittancetBG (x) = exp(−i2πxν)(8.3)

with characteristic frequency ν = 1/Λ. This is a standard technique to reduce the impact ofmanufacturing defects of the plate on the quality of the phase-modulated mode [134].
According to elementary Fourier optics, in fact, the pattern tBG (x) separates under an angle
θ = sin−1 λν the phase-modulated output LG33 beam (sent on the first diffraction order) from theresidual unmodulated input LG00 light (the 0th order), which is due to the presence of defects.
Thus the modulated beam has larger purity. The grating period chosen by SILIOS yields an angle
θ = 4.5 mrad in the (x, z) plane, shown on fig.8.4.
The resulting phase pattern of the plate, shown on fig.8.5, is analogous to that traditionallyused for LG`p mode generation with computer-generated holograms [138] (see for example fig.8.2),
and afterwards used also with spatial light modulators [134]. SILIOS indicated an input LG00 waistof w0 = 2.15 mm, to have a smooth varying intensity profile on the central area of the pattern.
As mentioned above, this is usually done to approximate the input Gaussian beam with a uniform
plane wave, in order to increase the effect of phase modulation [134].
Simulation of the generated pseudo-LG33 mode
The field generated with the diffractive plate is not a solution of the paraxial wave equation, since
it has the amplitude of a Gaussian mode but the phase of an LG33 mode. Such a mode does notfeature the structure stability of LG`p modes and can change its shape by propagation.
Therefore the phase modulation of the input Gaussian beam is expected to generate a pseudoLG33 mode of limited purity. To estimate this purity, we developed an FFT simulation of the wholelayout for mode conversion shown on fig.8.4. We simulated the passage of the input LG00 beam(with waist w0 = 2.15 mm) through the plate, which is assumed without any fabrication error, andthe consequent phase modulation of the beam.
Then the beam is sent through the 600 mm lens, and propagates in space (along a direction θwith respect to the input beam axis) to the focal plane of the lens. Here we compute numericallythe 2-dimensional overlap integral γ, between the theoretical LG33 field we expect (defined byeq.(5.13) with w0 = 288 µm), indicated as LG33 |theory , and the complex amplitude of the simulated

175

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

Figure 8.5: Design of the pattern of the phase-retarding LG33 diffractive plate, by SILIOS.

pseudo-LG33 mode, LG33 |sim , with the following operational definition:
R(LG33 |theory )∗ LG33 |sim dS
33.
γ = LG3 |theory LG3 |sim = qRR
2
2
LG33 |theory dS × LG33 |sim dS(8.4)

Then γ 2 is the fractional power content of LG33 mode, and the coupling losses are
L = 1 − γ2 .

(8.5)

The simulated pseudo-LG33 mode yields γ = 85% and L = 28%.
Optimal input beam size
In order to confirm the choice of w0 = 2.15 mm for the input Gaussian beam, we developed an
FFT code to simulate the phase modulation of input beams of different waist sizes.
The scope of the simulation is to investigate what is the optimal size w0 which yields the best
shaping of the pseudo-LG33 mode (and thus the maximum overlap integral), with respect to a phasemodulating pattern of a given size. This test has been introduced for computer generated holograms[138], then it has been used also for spatial light modulators [134].
Again, the code simulates the optical layout of fig.8.4. The size of the input LG00 Gaussian beam
is changed in a loop, and for each iteration the code computes the overlap integral between thegenerated pseudo-LG33 mode and the theoretical LG33 mode with w0 = 288 µm, at the mode-cleaner
input. The output plot of the code on fig.8.8 shows the overlap integral and the coupling lossesof the pseudo-LG33 mode versus the waist w0 of the input beam. The maximum overlap integral(corresponding to minimal coupling losses) is really achieved for w0 = 2.15 mm.

176

8.2. Design of the LG33 Diffractive Plate

Figure 8.6: Theoretical LG33 mode with w0 = 288 µm: normalized amplitude [a.u.] (left) and phase [rad](right).

Figure 8.7: Simulated pseudo-LG33 mode generated by the diffractive plate: normalized amplitude [a.u.](left) and phase [rad] (right).

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Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

0.8

0.5

0.75

0.7

0.4

coupling loss [%]purity − overlap integral [%]

0.6

0.65
0.3
1.2

1.4

1.6

1.8
2
2.2
2.4
2.6
Gaussian input beam waist w0 [mm]2.8

3Figure 8.8: Optimization of the Gaussian input beam on the diffractive plate: the overlap integral (bluecurve) and the coupling loss (green curve) are shown as functions of the input beam waist w0 .

Fig.8.9 shows the change in shape of the pseudo-LG33 mode profile for an input beam waist
varying of ±0.5 mm with respect to w0 = 2.15 mm, corresponding to a variation of the overlap
integral of ±4%. This variation sets a reasonable range for the tolerance on the size of the Gaussian
input beam.
Estimated power conversion efficiency
The plate is an etched slab of fused silica without anti-reflective coating, and the expected loss
due to reflection is 4% of the power on each surface of the plate. Thus the achievable efficiency islimited to DP = (0.96)2 = 0.92. Furthermore, the blazing must be considered.
In principle, as long as the blazing angle is small and the depth d of the etched pattern of theplate can be approximated as infinitely thin, the blazing of the grating pattern may send 100% of
the transmitted light on the first diffraction order [143]. Actually, when d is comparable with thegrating period Λ and/or errors of fabrication are present, the efficiency on the first diffraction orderdecreases to a lower value and some power is spread on other diffraction orders.
Thereby, the expected power conversion efficiency DP of the plate is defined as the ratio of thepower of the pseudo-LG33 mode on the first diffraction order to the available power of the input
Gaussian beam. This efficiency should be equal to 83% [147], which, multiplied by the losses due
to reflection on the two uncoated surfaces of the plate, should yield DP = 76%.
Production process
The fabrication process used by SILIOS for the manufacturing of the phase-retarding diffractive
plate is a recursive etching technique. The phase pattern is etched on the glass substrate throughmultiple steps of carving, each one removing a thin layer of material whose thickness is a given
fraction of the wavelength λ of the incident light. The layers to be removed are located on the178

8.3. Design of the Mode-Cleaner

1norm. amplitude [a. u.]

0.8

0.6

0.4

0.2

0
−3

−2

−1

0
r [mm]

1

2

3

Figure 8.9: Normalized amplitude profile of the simulated pseudo-LG33 mode, as a function of the Gaussian
input beam waist: w0 = 1.65 mm (light blue curve), w0 = 2.15 mm (reference value, grey curve), w0 = 2.65mm (dark green curve). The profile of the theoretical LG33 mode is also shown for comparison (dashed red
curve).

surface of the plate by applying a sequence of binary masks of increasing spatial resolution.
The pattern designed by SILIOS has been etched on the surface of a slab of Corning fused silica with 16 levels of phase and 2400×2400 pixels, each one measuring 5.9 µm.

8.3

Design of the Mode-Cleaner

A mode-cleaner is a Fabry-Perot cavity used in transmission, to spatially filter an input laser beam.
The injection of the beam into the cavity is equivalent to making the beam decomposition in thebase of the mode-cleaner eigenmodes. When the cavity resonates for one of those eigenmodes, it
only transmits the fraction of the input beam power with the same spatial distribution. The restof the power is reflected backward.
The transmitted fraction is the coupling coefficient of the beam, whereas the fraction reflected isthe coupling loss. In the following we will describe the optical design of the cavity, which determines
the filtering performance of the mode-cleaner. Then we will describe its mechanical design.
Optical layout
The optical layout of the cavity must have an even number of mirrors, because we want to transmit
helical modes with radial symmetry. To demonstrate this assertion, let us expand the field in termsof HGmn modes [144],
LG`p (x, y, z) =NX

ik b(m, n, k)HGN −k,k (x, y, z),

k=0

179

(8.6)

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

where p = min(m, n) and ` = m − n, and the real coefficients areb(m, n, k) =(N − k)!k!
2N m!n!

!1/2

1 dk[(1 − t)m (1 + t)n ]t=0 .
k! dtk

(8.7)

Let us now consider the first two coefficients of the expansion:b(m, n, 0) =b(m, n, 1) =



N!

1/2

2N m!n!

(8.8)(N −1)!
2N m!n!

1/2(n − m) .

If the azimuthal index ` is not null, then m 6= n and at least the amplitudes of the first two HGmn
modes of the sum are different from zero. In a cavity with an odd number of mirrors, these modes
resonate at different frequencies, and the radial symmetry of the helical LG`p modes is broken.
We chose a simple linear layout with two mirrors for the mode-cleaner of the experiment, inorder to transmit helical LG33 modes. Furthermore we chose a plano/concave configuration, where
the beam can be aligned to the cavity with two external steering mirrors. This configuration allows
to recover misalignments without acting on the cavity mirrors, and then simplify the design of thecavity itself, which can have a rigid structure.
Degeneracy
To achieve a good spatial filtering of the input mode, the mode-cleaner must be non-degenerate.
This means that the mode frequency spacing∆ν = ν0

∆N
ΦG ,
π

(8.9)

defined in eq.(5.21), must be sufficiently large, and we have to choose the cavity parameters accordingly. Let L be the cavity length, R1 the radius of curvature of the input mirror and R2 theradius of curvature of the end mirror. For the plano/concave configuration, R1 = ∞ and the waistof the cavity eigenmodes is on the input mirror.
We chose a cavity length L = 0.3 m, yielding a free spectral range of 500 MHz. This length
is reasonably small to fit on the optical table, and it is large enough to determine a cavity waistof about 400 µm for a radius of curvature of R2 of the order of 1 m, which is a reasonable beam
size. For such an optical configuration, R2 is the only free parameter that sets the base of theeigenmodes and their frequency spacing.
The optimal value of R2 must minimize the accidental degeneracy of modes of different orderwith the LG33 mode. If the mode-cleaner is resonant for the LG33 mode, which has N = 9, we
will have ∆N = N 0 − 9. As a consequence, the transmission of modes of order N 0 is attenuatedaccording to1T∆N = T33
(8.10)


2 ,
2
L
1 + π F sin 2π c ∆νwhere F is the cavity finesse and T33 is the transmission of the LG33 mode. Eq.(8.10) shows thatthe transmission of modes of different order, due to the expected spatial defects of the input LG33

180

8.3. Design of the Mode-Cleaner

Figure 8.10: Design of the mode-cleaner, choice of the concave mirror radius of curvature: on the y axis thefigure of merit for mode degeneracy (blue line) is shown. R = 0.5 m is the optimal choice, vertical red lines
indicate ± 1% mirror manufacturing tolerances.

mode, is strongly attenuated for a high cavity finesse and/or a large frequency spacing. The rightchoice of R2 can be made using a figure of merit of mode degeneracy, defined through eq.(8.10) asΓ(R2 ) =∞X(N 0 + 1)pT∆N (R2 ) .

(8.11)

N 0 =1

A local minimum of this figure of merit corresponds to a value of R2 that gives an effective separationof modes, minimizing the transmission of modes that are different from the resonant one. It is thensufficient to impose the resonant eigenmode of the mode-cleaner to be an LG33 mode. Fig.8.10
illustrates the behavior of Γ versus R2 for this set of parameters: L = 0.3 m, a reasonable value offinesse, F = 100, and a safely large order difference ∆N = 30.
The plot shows that R2 = 0.5 m corresponds to a local minimum of the figure of merit, and at
the same time this is a value that can be easily achieved by manufacturers, according to standard
optic catalogues. The vertical red lines show typical ± 1% manufacturing tolerances for R2 , stillreasonably close to the minimum degeneracy. Thereby R2 = 0.5 m has been chosen as reference
value.
According to this choice, eq.(5.16) yields a beam waist w0 = 288 µm on the input mirror, andconsequently the beam size on the end mirror is 455 µm. The ratio between the beam size and themirror radius for 1 ppm of clipping losses is 4.31 for LG33 modes, thus standard 1-inch mirrors arelarge enough to ensure negligible loss at input and end cavity mirrors.
FinesseIn order to confirm the choice of the cavity finesse, we run two FFT simulations of the mode-cleaner
with OSCAR. According to eq.(8.10) the spatial filtering improves for a larger finesse, thereby we

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Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

simulated F = 100 and F = 10001 . We injected the pseudo-LG33 mode of fig.8.7 into the cavity,
and we computed its spectrum and the purity of the filtered beam in the two cases.
The spectrum has been obtained by simulating a scan of the cavity length L. It is shown on
fig.8.11, where we can see the projection of the pseudo-LG33 mode on the mode-cleaner eigenmodebase: the modes have been identified through the relation
∆L =

λ ΦG
∆N ,
2 π

(8.12)

which is the homologue of eq.(5.21) for the tuning of the cavity length, and are indicated on theplot. We can see that all the resonances of other modes are well spaced from that of the LG33 mode.
The purity of the filtered mode has been computed by means of the overlap integral of eq.(8.4)
with the theoretical LG33 eigenmode transmitted through the cavity. It is γ = 99.8% for F = 100,and γ = 99.9% for F = 1000. Figs.8.13 and 8.12 show the filtered beam and the theoreticaltransmitted eigenmode, respectively, in the case of F = 100. The purity obtained with the lowfinesse seems sufficient for our purpose.
In summary, according to simulations, a high finesse of the mode-cleaner cavity is not determinant to achieve a good spatial filtering of the pseudo-LG33 mode. Thus we finally confirmed
F = 100, and then the transmission coefficients of the mirrors are T1 = T2 = 0.03.
Mechanical design
The resonance condition of the beam in the mode-cleaner might be perturbed by fluctuations of
the laser frequency or by microscopic fluctuations of mirror positions, changing the cavity length.
Mirrors might move because of the vibrations of the ground transmitted through the optical table,or because of acoustic waves in the surrounding environment. Thus the resonance condition has tobe stabilized through a control scheme, acting either on the laser frequency or on the cavity length.
The simplest technical solution is to build a cavity of fixed length, where the mirrors have nodegree of freedom for displacement (neither translations nor angular tilts), and to control the laser
frequency through a feedback loop for stabilization. The alternative would be to control the cavity
length by mounting one mirror of the cavity on a piezoelectric stack. This latter solution is more
complex: mirror mounts should be adapted to include the actuator, and the piezo should have alarge aperture to avoid the clipping of an LG33 beam, which is much larger than a Gaussian beam.
Moreover, the mount of the mirror with the actuator should have a negligible coupling between thelongitudinal and the angular displacements2 .
We decided to build a cavity of fixed length L with a monolithic design, where the mirror are
mounted at the ends of a rigid support, in order to have a stable mechanical separation of thecavity optics. The mechanical design of the cavity has been realized by the mechanical engineersof the AstroParticule and Cosmologie (APC) laboratory, and the model has been manufactured in
the laboratory workshop by the mechanical service.
The body of the monolithic mode-cleaner is a cylindrical mirror spacer of L = 0.3 m, screwed
on a rigid mount with four supports. The whole cavity (spacer plus supports) can be fixed rigidly
1
To have optimal coupling of the input beam to the cavity, we set equal transmissions for the input and the end
mirrors (T1 = T2 ), and we simulated perfect optics with no figure errors. No losses have been included as well, except
those due to the clipping of the mode on the mirror edge.
2
The mount with piezoelectric stack has now been realized, for the second phase of the experiment.

182

8.3. Design of the Mode-Cleaner

Figure 8.11: Spectrum of the pseudo-LG33 mode versus the tuning of the cavity length, as a function of
the mode-cleaner finesse: F = 100 (red curve) and F = 1000 (blue curve). The various components of
the input beam in the cavity base (black vertical dashed lines) are identified by the order difference ∆N ,the anti-resonance condition (close to 2.5·10−7 m) is also shown. The spectrum of a pure LG33 mode with
F = 100 (green curve) is reported for comparison.

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Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

Figure 8.12: Filtered pseudo-LG33 beam transmitted through the mode-cleaner, for F = 100: normalized
amplitude [a.u.] (left) and phase [rad] (right).

Figure 8.13: Theoretical LG33 eigenmode transmitted through the mode-cleaner, for F = 100: normalized
amplitude [a.u.] (left) and phase [rad] (right).

184

8.4. The Table-Top Experiment

plane mirror

concave mirror

Figure 8.14: Monolithic mode-cleaner cavity.

to the optical table. The mirrors are placed into two inner housings at the extremities of the cavity,
fixed by two annular caps of 21-mm aperture, screwed to the cavity body. Rubber joints are placed
between the caps and the mirrors, to uniform the pressure of the caps on the mirror surface.
The closed structure of the cavity avoids the effect of acoustic waves on mirrors. The cavity
and the mount are designed to be as stiff as possible, in order to move the first vibrational mode of
the structure to high frequencies, where the vibrations of the ground are supposed to be negligible.
A prototype of the mode-cleaner has been realized with aluminum (2017A T4), to test thetolerances of fabrication of manufacturing machines. This prototype has been used on the optical
table with the Gaussian mode of the laser, to check the global performances of the mechanicaldesign, showing extremely good results of stability. After setting the mode resonance in the cavity,
we observed a very slow drift (∼ 30 s) of the mode build-up from the resonance condition, indicatingthat the design satisfies the requirement of protecting the mirrors from external disturbances.
The final model of the cavity, shown on fig.8.14, was realized in Invar, which is a nickel steel
alloy with a very low thermal expansion coefficient: α = 2 · 10−6 K−1 . This material allows to
decrease any potential long-term drift of the cavity length, driven by a temperature change of thesurrounding environment, to a negligible level.

8.4

The Table-Top Experiment

The aim of the table-top experiment is: (i) to test the conversion of an input LG00 beam into ahigh-purity LG33 helical beam, using the LG33 plate and the mode-cleaner, (ii) to test the behaviorof a Fabry-Perot cavity illuminated by a (pseudo) LG33 mode. We will assess:

185

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes• the purity of the LG33 mode generated through our conversion-and-filtering technique.
We will demonstrate that the phase pattern of the diffractive plate can generate a pseudohelical LG33 beam with the expected initial purity, and we will characterize the purity andthe parameters of the filtered LG33 beam transmitted through the mode-cleaner• the global conversion efficiency of the technique, by measuring the power conversion
efficiency of the LG33 diffractive plate, and the transmission of the pseudo LG33 beam throughour setup and the mode-cleaner• the long-term stability of mode generation• the alignment tolerances of the LG33 beam on the mode-cleaner.
The scheme of the experimental setup is shown in fig.8.15. The Gaussian LG00 beam of the laser isfirstly passed through an electro-optic modulator, for the generation of radio-frequency sidebands
used for the servo control of the laser frequency. The beam then goes through a magnifying modematching telescope, to enlarge the beam radius on the LG33 diffractive plate up to mm size. The
generated pseudo-LG33 propagates to the mode-cleaner. At the resonance of the mode-cleaner LG33
eigenmode, the frequency of the laser is locked to the mode-cleaner length using a standard Pound
Drever-Hall locking scheme in reflection.
The generated-pseudo LG33 beam is pre-aligned to the mode-cleaner cavity using a Gaussian
pick-off, which propagates in the setup without going through the diffractive plate. Either the LG00
or the pseudo-LG33 beam can be blocked independently with a beam dump. The relative alignmentof the Gaussian beam and the pseudo-LG33 beam is achieved by using a setup of two CCD cameras,
placed in near field and far field on a parallel path with a flip-mount mirror.
We assembled the experimental-setup in a dedicated clean room of the APC laboratory, as
shown on fig.8.16, on an optical table of 150×180 cm. The room is a controlled environment, where
the circulating air is filtered to limit the number of dust particles, which may deposit on the optics
and perturb the measurements. This space is classified class 105 , meaning that the number ofmicroscopic dust particles between 0.5 µm and 1 µm size is less than 105 /cm3 . The temperaturein the room is stabilized within about ± 1 K.

8.5

LG00 Input Beam

The setup for generating and mode-matching the LG00 input beam to the diffractive plate includes
the laser, a Faraday isolator, three polarizers, a converging lens (L0) with focal length f = 300
mm, an electro-optical modulator and a mode-matching telescope composed of three lenses: L1,
L2 and L3.
Laser
The input LG00 beam is generated with a commercial continuous-wave laser, a model (Mephisto
by Innolight) widely used in present gravitational-wave interferometers as master laser. This laser
uses a Nd:YAG crystal as a monolithic nonplanar ring oscillator, and emits at λ = 1.064 µm. The
emitted power can be tuned from few mW up to 0.5 W, by changing the current that flows throughthe diode used as laser pump.

186

8.5. LG00 Input Beammode matchingtelescope
λ/4 λ/2

Faraday
isolator

λ/2

L0electro-opticalmodulator

L1

L2

L3

PPO

DP L6

laser

RF localoscillator

LG00 pathfrequency
servo

beam
dump

LPF

RF PD
CCD
cameraPD

CCD
cameraλ/4

far field
cameraλ/4filtered
LG33 beamPTnear fieldcameraPBS

monolithic
mode-cleaner

Setup forLG00/LG33
relative alignment

Figure 8.15: Experimental setup for the generating and filtering a pseudo-LG33 mode.

Figure 8.16: Photograph of the experimental setup.

187

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

The frequency of the emitted light, ν ' 2.8 · 105 GHz, can be tuned within a dynamic rangeof ±150 MHz, with a piezoelectric actuator glued to the crystal (gain of 1.5 MHz/V, maximum
voltage ±100 V). For a larger dynamics, a Peltier cell changes the crystal temperature, providing
a tuning range of 30 GHz (gain of −3 GHz/K, maximum tuning 10 K). The piezoelectric actuatorand the thermal actuator are used in the laser frequency stabilization feedback loop, as describedin section 8.8.
Faraday isolator and beam polarization
The laser beam is passed through a Faraday isolator, protecting the laser crystal from backwardreflections which could perturb the laser operation. The isolator is made of three parts: an input
polarizer (polarized vertically), a Faraday rotator, and an output polarizer (polarized at 45 degrees).
Light travelling forward is vertically polarized by the input polarizer, then the Faraday rotator
rotates the polarization by 45 degrees. The output polarizer is set to allow the beam transmission.
Light travelling backward becomes polarized at 45 degrees by the output polarizer, and the
Faraday rotator again rotates the polarization by 45 degrees and set it horizontal. Thus the lightpropagating backward is blocked by the input polarizer, which is vertically aligned. The isolator
acts as an optical diode, allowing the propagation of light in only one direction. The attenuationof backward reflections is called extinction factor.
The isolator used in the setup is an IsoWave I-106T-5, optimized for the highest attenuation
at 1.064 µm, with beamsplitter cubes as polarizers. Its extinction factor has been characterized
by measuring the ratio Pout /Pin between the output and the input beam power, with the isolator
placed in the reversed position with respect to the laser, as shown on fig.8.17. The beam poweris measured with a thermopile Gentec XLP12-3S-H2 powermeter, which measures the integrated
power of the beam on a circular surface of 12 mm diameter. The power measurements yield:!
Pout = 0.190 mW
extinction = 10 Log= −34.20 dB .
Pin = 500 mW
The measurement of the extinction factor appears to be limited by the sensitivity of the powermeter,
which can measure light power down to few hundreds of µW. According to the datasheet, theexpected isolation should be equal to -41.5 dB, corresponding to a power of about 35 µW.
The insertion of the Faraday isolator along the optical path of the beam induce a small power
absorption, which has been characterized by measuring the output-to-input power ratio in theallowed sense of propagation (see fig.8.17):
Pout = 448 mW
insertion loss = 10 Log
Pin = 500 mW!
= −0.48 dB .

The measured insertion loss is compatible with the expected value reported on the datasheet, equal
to -0.5 dB.
Before the isolator, a λ/4 wave-plate removes the residual elliptic polarization of the laser beamin order to obtain a pure linear polarization. The whole isolator is rotated such that the backwardreflections are deviated by the input polarizer cube to a beam dump, thus a λ/2 wave-plate before
the isolator is used to align the polarization of the beam to the input cube.

1888.5. LG00 Input Beam

Faraday isolator

Pin

Faraday isolator

Pout

Pin

Pout

Figure 8.17: Characterization of the Faraday isolator: measurement of the extinction factor (left) and of
the insertion loss (right). The allowed direction of propagation in the isolator is shown by the gray arrow.

After the isolator, another λ/2 retarder rotates the input polarization to make it s-oriented, i.
e. perpendicular to the plane of the optical table, since several optical components of the setup
are optimized for the operation with s-polarization. The fraction of light with s-polarization at theoutput of the last λ/2 wave-plate has been measured with a polarizer cube: the ratio between the
total power at the cube input and the s-polarized power at the cube output iss-polarized light =Pout = 349 mW= 99.7% .
Pin = 350 mW

Electro-optical modulator
The laser beam is focused using the lens L0 into the 2-mm input aperture of an electro-opticalmodulator, which modulates the phase of the beam to generate the radio-frequency sidebands
used in the feedback loop for the laser frequency control. The modulator is a NewFocus 4004 forbroadband frequency modulation, formed by a Pockels cell with a birefringent crystal of lithium
niobate (LiNbO3 ) of length l and thickness d. The operation of the modulator is based on the
Pockels effect, which is the linear dependence of the crystal refractive index on the applied electricfield. By applying a driving sinusoidal voltage, the change in the refractive index causes a periodic
phase shift in the input laser beam, which is equivalent to the generation of sidebands.
The effect of the field on the crystal index is described by a three-dimensional electro-optic tensor
rij . If ne is the unperturbed refractive index and E is the applied field, the induced refractive index
change writes
E∆n = n3e r33 .
(8.13)
2
The electric field is applied along the crystal axis, transversely to the direction of the laser beam
propagation, and is equal to the ratio of the applied voltage V to the crystal thickness: E ≡ V /d.
The total phase shift corresponding to V is the modulation index m,m=

2πl∆n = βV ,λ(8.14)

πl 3
n r33
λd e

(8.15)

whereβ=

in our case is equal to 15 mrad/V. The amplitude of the local oscillator applied to the electro-optic
modulator is V = 14 V, resulting in a modulation index m = 0.21.

189Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

Mode-matching telescope
The design of the LG33 diffractive plate requires a collimated LG00 input beam with waist w0 = 2.15mm, thus we had to realize a mode-matching telescope to magnify the beam coming out of theelectro-optical modulator.
The telescope is particularly important, since (as we have seen in section 8.2) the purity of thepseudo-LG33 mode generated by the diffractive plate depends on the parameters of the input LG00
beam. Therefore we need a tunable system, in order to adjust the matching of the input beam
to the plate. The limited space available on the optical table puts an additional constraint to thedesign of the telescope, requiring a design as compact as possible.
We decided to realize a layout composed of three lenses, where the first is used to tune theparameters of the beam and the two others provide its magnification. First we chose the parametersof the telescope (the relative distances of the lenses and the focal lengths) through an approximatedanalytical computation, then we developed a ray-matrix simulation to optimize this parameters andto study the tolerances of the layout. In the following we will describe the design and the realizationof the telescope, and the characterization of its output beam.
Telescope - input beam
As first step of the design process, we characterized the LG00 beam after the modulator3 through alinear fit of the beam radius w versus the on-axis position z. In the far-field regime, w is expected
to follow the relationw = θz ,(8.16)
where θ = λ/πw02 is the beam divergence. The measurements of the beam radius have been acquiredwith a Beamage Focus II beamscan by Gentec, which is a CCD camera with an active area of 20×15mm2 and pixels with 14.5 µm side. The beamscan is equipped with a dedicated software for beamdiagnostics, which gives the radius of the detected beam as output.
For each position on the z axis, the beam radius has been measured with five consecutive
acquisitions, and the average value and the variance of each sample have been computed. The
averages and the square root of the sample variances have been used in the linear fit as data points
and measurement errors for w, respectively. The order of magnitude of the error is few µm. The
origin of the z axis is set at the position of the first measurement, so the points are fitted with aslope that isw = θz + w̄ ,(8.17)and the position of the beam waist is computed as z0 = −w̄/θ, considering that the waist is locatedbehind the z axis origin. The fit, shown in fig.8.18, has been performed for the beam radius on thex and y axis separately, yielding the values reported in table 8.1.
The resulting position of the waist is close to the input aperture of the electro-optic modulator,
as expected. The plot of the fit and the large reduced χ2 values indicate that the fit uncertainties
are underestimated. By arbitrarily increasing the error bars to 10−5 m, the reduced χ2 values getscloser to 1. The new fit yields the results reported in table 8.1 for comparison.
3

The characterization of the beam has been performed after having verified the linear far-field condition with few
preliminary measurements of w and z.

190

8.5. LG00 Input Beam

−3

−5

x 10

2.6
2.5

X [data]
Y [data]
X [fit]Y [fit]

2.4
w [m]

x 10

X
Y

2

fit residuals [m]

2.7

3

2.3
2.2
2.1

1
0
−1

2
−2

1.9
1.8
0

0.2
z [m]

−3

0.4

0

0.2
z [m]

0.4

Figure 8.18: Characterization of the LG00 input beam: linear fit of the beam size w versus the position on
the optical axis z (left) and dispersion of fit residuals (right).

transverse direction

waist

distance to waist

reduced χ2xw0 = 229 ± 1 µm

z0 = −1.305 ± 0.005 m

4.5yw0 = 227 ± 1 µm

z0 = −1.306 ± 0.005 m

24.1

x

w0 = 229 ± 4 µm

z0 = −1.30 ± 0.02 m

0.9y

w0 = 228 ± 4 µm

z0 = −1.31 ± 0.02 m

2.2

Table 8.1: Characterization of the telescope input beam: results of the linear fit of fig.8.18 with statisticaluncertainties (∼ 10−6 m, top rows), and with arbitrarily increased uncertainties (10−5 m, bottom rows).

191Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

d

q0

q1
L1

L2

(w0)L1w0

Z0 = d0

q3

q2

Z1

L3

(w0)L2

d1

d12

Z2

(w0)L3Z3

d23

Figure 8.19: Layout of the LG00 beam mode-matching telescope.

Thus the parameters of the telescope input beam are known within ∼ 2% error: the telescope
has to magnify the beam size of about one order of magnitude, from w0 = 229 µm to w0 = 2.15mm.

Telescope - optical layout
The layout of the telescope is shown in fig.8.19: L1, L2 and L3 are three converging lenses, L1
is placed in far field with respect to the input beam waist w0 , L2 and L3 form an afocal system
which provides the magnification of w0 . L1 is used for tuning the parameters of the beam that goes
through the afocal system.
The working principle of this layout can be understood by computing the complex beam parameter q at different positions inside the telescope. We will refer to fig.8.19 for the notation usedhereafter. Let q0 = z0 + izR0 be the parameter of the telescope input beam. For L1 we havez0 >> zR0 , so we can approximate eqs.(5.29) to computez0
z1 ' ff11−z0(8.18)
f 2 zR0zR1 ' f11 −z.
1

2
If moreover z0 >> f1 , then z1 ' −f1 and zR1 ' zR0 f12 /z0 . This means that by tuning theposition z0 of L1 the size of the waist (w0 )L1 can be changed, while its position z1 remains almost
constant and equal to −f1 (z1 is negative because the beam is converging after the lens). In thecase of the afocal system formed by L2 and L3, where z2 ≡ f3 , eqs.(5.29) become
z3 = −f3(8.19)
zR3 = f32 /zR2 .

192

8.5. LG00 Input BeamThus, by choosing f3 = CzR2 , eq.(8.19) for q3 yields
z3 = −CzR2
zR3 /zR2 = C 2 ,(8.20)

so the waist (w0 )L3 is magnified of a factor C. L2 is used to focus the beam before L3 and shorten
its Rayleigh range zR2 , in order to keep the distance z3 = −CzR2 small.
The available space on the optical table allows to choose z0 = 75 cm, which according to thecharacterization of the telescope input beam (see table 8.1) yields z0 /zR0 ' 5, and is compliant
with the beam far-field condition that we imposed initially for L1 . By choosing f1 = 10 cm, also
the condition z0 >> f1 is fulfilled and (w0 )L1 is expected to measure about 35 µm.
f2 is chosen to be as small as f1 to focus the beam within the afocal telescope, so f2 = 10 cm.
The distance d1 is chosen so that also L2 is placed2 in the beam far-field range: with this further
approximation we can replace zR2 ' zR1 f22 /d1 in eq.(8.19) to find
f3 /f2 =√zR3 zR1 /d1 ,(8.21)

which is the relation used to compute the focal lens of L3. For d1 = 150 mm, f3 /f2 ' 1.5 so
f3 = 150 mm. The lengths d12 = z1 + d1 = 265 mm and d23 = z2 + f3 = 500 mm were fixed by
the layout of the telescope. The total length of this system (computed from w0 to the expected
position of (w0 )L3 ) is d = z0 + d12 + d23 + z3 = 1715 mm, and fits well on the optical table.
Telescope - fine tuning and tolerances
The design of the telescope has been finalized by optimizing the layout parameters, calculated so
far with approximated equations. The optimization has been performed by computing the beam
propagation throughout the telescope, with the ray-matrix formalism [95]. The whole system canbe represented by a matrix Mtel , whose elements depends on the layout parameters:
Mtel (f1 , f2 , f3 , d12 , d23 ) = L3 D2 L2 D1 L1 D0 ,
where




1 di
1
Di =
, Lj =
0 1
−1/fj


0
.
1

(8.22)(8.23)

Thus the magnified output beam is
q3 (f1 , f2 , f3 , d12 , d23 ) =Mtel (1, 1)q0 + Mtel (1, 2).
Mtel (2, 1)q0 + Mtel (2, 2)

Eq.(8.24) is equivalent to applying sequentially eq.(5.28) for all the lenses of the telescope.
Eq.(8.22) and (8.24) have been implemented in a Matlab script, where:• q0 is computed from the fit parameters of eq.(8.17)• f2 , f3 , d12 , d23 are automatically changed in a loop, within an arbitrary given interval• the waist size (w0 )L3 is computed at each step of the loop.

193(8.24)

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

telescope design parameters
f1
100 mmf2
100 mmf3
200 mmz0785 mm
114 mmz1
z2
197 mmz3
200 mmd1
203 mmd12
317 mmd23
397 mm228 µmw0(w0 )L1
32.5 µm(w0 )L2
31.5 µm(w0 )L3
2.15 mmTable 8.2: Parameters of the telescope layout of fig.8.19.

This script has been used first to tune the distances d12 , d23 in the design, in order to get (w0 )L3 =2.15 mm at the telescope output. The parameters that yield this waist size are listed in table 8.2.
Then we used the script to compute the tolerances of the system with respect to manufacturing
errors of the focal lengths, and errors in the positioning of the optics. The output tolerancescomputed through the script are shown in the plots of figs.8.20 and 8.21. In order to have (w0 )L3 =2.15 ± 0.05 mm, the tolerance for the focal length of f2 is ± ' 0.3 %, and ±1 % for f3 . Concerning
the distances, L2 must be positioned with a maximum error of ± ' 0.5 mm, and L3 with amaximum error of ±2 mm.
As declared by the manufacturer, the typical error for the focal lengths is expected to be ±1 %.
Moreover, the lenses have been positioned on the table with an error of ±3 mm, due to the difficultyof measuring accurately d12 and d23 with a tape measure. To compensate for manufacturing andpositioning errors, the optics have been mounted on optical rails with micrometrical translation
stages: the setup used is shown in fig.8.22. It allows to tune the telescope lengths after the optics
have been placed on the table, in order to obtain (w0 )L3 = 2.15 mm at the output of the system,
even in presence of errors.
The optical rails provide the possibility to tune the position of the optics with a minimum
displacement of few mm, on a large dynamic range of 20 cm. The fine tuning of the position isachieved with the micrometrical translation stages, which provide a longitudinal displacement of
few tens of microns over ±5 mm.
The telescope has been tuned by moving the lenses around their initial positions, while measuring the output beam size with the beamscan. The best tuning has been achieved for d12 = 345
mm and d23 = 386 mm, yielding a beam with (w0 )L3 = 2.19 mm and (w0 )L3 = 2.16 on the x andy transverse directions, respectively.
An image of the beam is acquired at z3 = 200 mm, where the collimated beam has its waist.
It is shown on fig.8.23: the beam features a low astigmatism, which entails an ellipticity of 0.99.
To estimate the purity of this beam, the intensity distribution is acquired with the beamscan at194

8.5. LG00 Input Beam

Figure 8.20: Tolerances of the telescope focal lengths: f2 [m] and f3 [m] are shown on the x and the y axis,
respectively, and the vertical/horizontal lines represent the baseline values. The color scale indicates the sizeof (w0 )L3 [mm], the contours show the region for a change of the waist size of ±0.05 mm.

Figure 8.21: Tolerances of the telescope lengths: d1 [m] and z3 [m] are shown on the x and the y axis,
respectively, and the vertical/horizontal lines represent the baseline values. The color scale indicates the sizeof (w0 )L3 [mm], the contours show the region for a change of the waist size of ±0.05 mm.

195

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

micrometrical mounts
L3

L2

L1

optical rails
Figure 8.22: Setup for the tuning of the telescope lengths.

the focal plane of L3, and the overlap integral defined in eq.(8.4) is used. The integral is computed
between the square roots of the theoretical and the measured intensity distributions, where thetheoretical one is given by the square modulus of eq.(5.13) with w0 = 2.15 mm. Since the phaseof the modes is neglected, this computation puts an upper limit on the purity of the LG00 mode,measured at the output of the telescope:γ = LG00 |theory LG00 |measure = 99.4% .

(8.25)We then decided that the quality of this beam is acceptable for our scope.

8.6

Mode Conversion: LG00 to Pseudo-LG33

The setup for the mode conversion is shown in fig.8.24: the output beam of the mode-matchingtelescope goes through the LG33 diffractive plate (DP in the figure), mounted on a three-axis translation stage for high-precision alignment. The diffracted beam is focused by a 600 mm lens (L6)on the input mirror of the mode-cleaner cavity. Before the plate, a pick-off (PO1) with reflectivity
R = 9.6% samples a fraction of the LG00 beam, and sends it to the mode-cleaner.
This setup is used to switch between the LG00 and the pseudo-LG33 modes, since each beam
can be blocked independently. The swap between the modes is needed for the pre-alignment of thepseudo-LG33 beam to the mode-cleaner, following a procedure explained in section 8.7. The pseudoLG33 and the LG00 beams recombine at a second pick-off (PO2) which has reflectivity R = 8.3%.
The pseudo-LG33 beam reflected by PO2 is sent to a CCD camera.
Alignment of the diffractive plate
According to the study of alignment tolerance performed by SILIOS, in order to generate theoptimal pseudo-LG33 mode shape, the diffractive plate has to be centered in the input LG00 beamtransverse plane with an accuracy of the order of 100 µm. For this reason the plate is mounted ona three-axis translation stage Thorlabs PT3 which allows a displacement accuracy of 10 µm.
The plate is put in place with the following procedure: first the input LG00 beam is analyzed
with the beamscan, in order to fix the position of the beam centroid on the CCD active surface.

196

8.6. Mode Conversion: LG00 to Pseudo-LG33

Figure 8.23: Telescope output LG00 mode: normalized measured intensity [a.u.] (left) and normalized
theoretical intensity [a.u.] (right).

PO2
beam dump

CCD

R = 8.3%

LG33

LG00

L6
DP

translation
stage

LG00
R = 9.6%

PO1

Figure 8.24: Setup for the generation of the LG33 beam.

197

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

The centroid is defined as the weighted average of the displacement vector on the transverse plane,RI(r, θ)~r dS~r0 = R,
I(r, θ) dS(8.26)and in this case it coincides with the geometrical center of the input LG00 beam. The cross hairpointing to ~r0 is recorded on the beamscan, which is kept in the same position during the alignment.
The plate is then roughly positioned on the path of the beam, before the camera, where it ispossible to observe the plate pattern covering the cross section of the beam. The alignment is tuned
with the translation stage, to superpose the center of the plate (clearly indicated by the diffractive
pattern) to the centroid of the beam. This procedure is illustrated on fig.8.25.
Successively the 600 mm lens is installed, and the pseudo-LG33 beam at the focal plane of thelens is monitored with the beamscan: the ultimate tuning of the alignment of the plate is achievedby optimizing the cylindrical symmetry of its power distribution. The residual tilt of the plate withrespect to the beam transverse plane is considered negligible.
Characterization of the diffractive plate
In order to characterize the behavior of the plate, we first measured the power conversion efficiency
DP . As expected, when we illuminated the plate with the input LG00 beam, we observed several
output diffraction orders, with decreasing intensity with respect to the order m. The intensity
distribution of orders up to m = ±2 can be clearly distinguished by eye with a beam viewer,
whereas higher orders are too faint to be seen.
These diffraction orders propagate along different directions, forming an angle θm with respect
to the optical axis of the input LG00 beam. Supposing that the input beam is perpendicular to thediffractive plate, the angle θm is defined through the grating equation,
Λ sin θm = mλ .

(8.27)The period of the grating being Λ = 235 µm, the first order should be spaced from the 0-th
order by 2720 µm at the focal plane of the 600 mm lens. For larger distances from the lens, thedifferent orders are more and more spaced, and the power of the first order can be measured witha power-meter placed behind an iris.
Fig.8.26 shows an image of the first diffraction order with the shape of the generated pseudo-LG33
mode, together with a faint spot of the 0-th unmodulated order of Gaussian shape. The distance
between the geometric center of the two beams is 2988 µm, compatible with the expected spacing.
On the first order, the efficiency of the LG33 diffractive plate isDP =(PLG33 )m=1 = 312 mW
PLG00 = 383 mW= 81.5% .

This value of efficiency is slightly higher than expected (76%, see section 8.2), possibly due to thepresence of some spurious light from other diffraction orders. In any case, the application of ananti-reflective coating on the non-etched surface of the plate is expected to increase this efficiency.

198

cross hair

8.6. Mode Conversion: LG00 to Pseudo-LG33

Figure 8.25: Alignment of the LG33 diffractive plate with the beamscan: (i) record of the beam centroid
position, indicated by the cross hair on the active area of the detector (top left), (ii) rough positioning of theplate before the beamscan (top right), (iii) superposition of the plate center on the beam centroid (bottom).

199

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

0-th

1st
LG00

LG33

DP

LG00

Figure 8.26: Diffraction orders of the LG33 plate: scheme (left) and measured zero-th and first orders (right).

Analysis of the generated pseudo-LG33 mode
The correct evaluation of the generated mode purity would be the computation of the overlap
integral γ defined in eq.(8.4), but in our experiment only the information about the intensity
distribution of the generated pseudo-LG33 mode is available. The purity can be thus computed
using the square root of beam intensities, instead of complex amplitudes, yielding an upper limit
estimation.
An image of the pseudo-LG33 mode intensity distribution is acquired with the beamscan in the
focal plane of the 600 mm lens, where the mode is expected to have its waist. The position of
the waist is found by manually moving the camera along the optical axis, close to the lens focal
plane, and looking for the minimum beam size during a continuous run of image acquisition. The
waist position is found at 600 ± 50 mm from the lens, where the error is due to the difficultyof appreciating by eye the small variations of the beam size observed with the beamscan. The
measured pseudo-LG33 mode yields a purity upper limit γ = 88%, compatible with the purity
predicted through FFT simulations (85%, see section 8.2). The coupling losses of the generated
mode are L = 1 − γ 2 = 23%, i. e. 23% of the converted power is not an LG33 mode.
Fig.8.27 shows the measured intensity distribution of the pseudo-LG33 mode, compared to the expected LG33 mode for w0 = 288 µm. Fig.8.28 illustrates the comparison between the cross-sectional
profiles of the expected and of the pseudo-LG33 mode intensities. There are several differences between the two. Their overlap is good only for the highest internal peaks, whereas the outer rings ofthe pseudo mode are shifted outwards, and the mode is globally larger. Some power of the pseudomode is left at the radial nodes, and the relative amplitude of the peaks of the theoretical mode is
not respected.
A least-square fit of the measured intensity profile versus the radial coordinate r is shown in
fig.8.29. The radial measured profile I(r) is obtained by carrying out an average over the azimuthal
coordinate θ in the transverse plane, as follows:
1. the two-dimensional transverse intensity distribution I(r, θ) of the mode is discretized withradial bins of the same size of the beamscan pixels, forming concentric annuli

2008.6. Mode Conversion: LG00 to Pseudo-LG33

Figure 8.27: Comparison between normalized intensity distributions at waist: measured pseudo-LG33 mode
(left) and theoretical LG33 mode for w0 = 288 µm (right).

1

norm. intensity [a. u.]

0.8

0.6

0.4

0.2

0
−1.5

−1

−0.5

0
r [mm]

0.5

1

1.5

Figure 8.28: Profile comparison: the horizontal (blue solid curve) and the vertical (green solid curve)
measured cross sections of the pseudo-LG33 mode are shown, together with the cross section of the theoretical
LG33 mode for w0 = 288 µm (red dashed curve).

201

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

fit [a. u.]

15000
10000
5000
0

0.5

1

1.5

1

1.5

residuals − intens. [a. u.]r [mm]
4000
2000
0
−2000
0

0.5
r [mm]

Figure 8.29: Least square fit of the pseudo-LG33 mode profile. Top panel: measured (blue dots with errors)
and fitted (red dashed curve) profiles in the r > 0 plane. Bottom panel: fit residuals, defined as the difference
between the data and the model.

2. the intensity associated to each bin is the azimuthal average of I(r, θ) over 2π, within the
single annulus. Due to the spatial defects of the pseudo-LG33 mode, the variance of theintensity in each annulus is not null. Its square root is taken as the error σ associated to eachazimuthal average.
The error on the radial position (not used in the fit) due to the discretization process is arbitrarily
chosen to be half of the pixel size. The fit of I(r) is performed over the r > 0 plane, and the fit
parameters are calculated by minimizing the function!2NX
I33 (ri ) measure − I33 (ri , A33 , B, w0 ) theory
χ2 (A33 , B, w0 ) =,(8.28)
σii=1

where I33 (r, A33 , B, w0 ) theory is the square modulus of the LG33 mode complex amplitude of eq.(5.13):"

A3
I33 (r, A33 , B, w0 ) theory = 23 exp
w0

−2r2
w02

#

2r2
w02

!3 "
L33

2r2
w02

!#2
+B .

(8.29)

A33 is the normalization factor for the LG33 mode, B is the background noise due to the potential
presence of residual environmental light collected by the beamscan, w0 is the beam waist. The fit
yieldsA33 = 7.8 · 10−5 [a. u.]
B = 780 [a. u.](8.30)
w0 = 318 · 10−6 m ,with a reduced χ2 equal to 24.9 (for 255 points and 3 parameters). The position of the innermost
and the outermost peaks of the generated pseudo-LG33 mode is well fitted by a theoretical mode

202

8.7. Alignment of the Pseudo-LG33 to the Mode-Cleaner

with w0 = 318 µm, which is only 10% larger than expected. The relevant difference between themodel and the measured profile (stressed by the large fit residuals and reduced χ2 value) is in therelative amplitude and in the position of the intermediate peaks. This indicates that the difference
between the theoretical and the pseudo-LG33 modes is not only due to a size mismatch, and thatthe pseudo mode has spatial defects induced by the plate.
Though the measurement of the intensity distribution of the pseudo-LG33 mode has been performed in a dark environment, some background can be also present, since it is not possible todarken the room completely. Nevertheless the environmental background light cannot completely
explain all the power left at the radial nodes, which is likely a spatial defects of the mode itself.

8.7

Alignment of the Pseudo-LG33 to the Mode-Cleaner

When we tried for the first time to align directly the pseudo-LG33 mode to the mode-cleaner, we
could not achieve the alignment only by looking at the shape of the mode transmitted through thecavity, as it was possible for some alignment tests with the Gaussian beam. As the input pseudoLG33 mode is misaligned with respect to the cavity optical axis, the shape of the all eigenmodes
resonating in the cavity was so distorted that they could not be distinguished from each other. We
then realized that we had to pre-align the beam to the cavity with the Gaussian beam, by using aprocedure described here below.
The setup we used is sketched in fig.8.30: a mirror installed on a flip mount, two near-field andfar-field CCD cameras for the mutual alignment of the LG33 and LG00 beams, two steering mirrors(SM) mounted on high-precision ultra-stable mounts, a movable beam dump. The procedure forthe manual pre-alignment of the pseudo-LG33 mode is the following:
1. both the pseudo-LG33 beam and the LG00 pick-off beam can propagate in the setup. The flip
mount mirror is switched up in order to deviate the beams to the path of mutual alignment(sketched with a dashed line in fig.8.30), where the two CCD cameras are installed. The
near-field camera is placed at the focal plane of L6, where is the waist of the pseudo-LG33
beam. The far-field camera is placed at 1.65 m from the position of the waist, about 7 times
farther than the Rayleigh range of the pseudo-LG33 beam, equal to 24.5 cm. The LG00 beam
is superposed to the pseudo-LG33 beam by turning the screws of the standard optical mountsof PO1 and M0. The superposition is carried out by observing the interference pattern of
the two beams on the cameras, and optimizing the radial symmetry of the pattern. The
superposition of the beams on both cameras guarantees that the beams are aligned over along baseline, which is large enough with respect of the typical lengths of propagation to themode-cleaner
2. the flip mount is switched down, and the pseudo-LG33 beam is blocked with the beam dump.
The LG00 beam is accurately aligned to the mode-cleaner with the steering mirrors, which aremounted on ultra-stable Thorlabs KS1D mounts. These mounts, whose calibration has beenmeasured before their installation, deliver an angular displacement of the beam of 660 µm per
revolution of the tuning screws. The alignment is tuned until the power of HG01 and HG10
modes that initially resonate in the cavity, due to misalignment, is decreased to a negligible
level

203

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

far-field
CCD
LG00

PO1

BS
LG00

near-field
CCD

M0

beam
dump

LG33

SM
transmitted field
CCD

flip mount

PD

PD

PBS
λ/4

reflected field
CCD

monolithic
mode-cleaner

λ/4

SM

Figure 8.30: Setup for the alignment of the pseudo-LG33 mode.

3. the LG00 beam is blocked with the beam dump and the pseudo-LG33 beam propagates to themode-cleaner, following the aligned path of the LG00 beam

After this procedure, the pre-alignment of the pseudo-LG33 beam is largely improved, so that it ispossible to lock the cavity. The fine tuning of the alignment is performed by optimizing manually
the radial symmetry of the reflected mode, monitored by a dedicated CCD camera. Fig. 8.31 shows
three images of the field reflected by the mode-cleaner, when the LG33 eigenmode is resonant inside
the cavity.

y

y

x

x

Figure 8.31: Field reflected by the mode-cleaner at the resonance of the LG33 cavity eigenmode: aligned
field (left), x misalignment (center), y misalignment (right). The reference frame on the transverse plane isshown by the red axes.

2048.8. Mode-Cleaner Lock

PBS

EOM

λ/4

monolithic
mode-cleaner

PDtransmitted
signal

laser
PD
mixer
local oscillator
LPF

servo

low pass filter

Figure 8.32: Pound-Drever-Hall locking scheme in reflection, for the lock of the LG33 mode resonance.

8.8

Mode-Cleaner Lock

The laser frequency is locked on the frequency of the mode-cleaner LG33 eigenmode through astandard Pound-Drever-Hall scheme [83, 84] in reflection, as sketched in fig.8.32: the laser beam
passes through the electro-optic modulator (EOM), where the laser frequency is phase-modulatedwith a radio-frequency signal used also as local oscillator for demodulation. Thus an upper and alower sideband fields are generated at the modulation frequency, which propagate with the carrier
field to the mode-cleaner. Here the fields are partially reflected, with a phase determined by theirfrequency. The reflected beam is picked off by a polarizer cube (PBS), and sent to a photodiode(PD). The modulation frequency is chosen large enough (10 MHz) so that the sidebands are poorlytransmitted by the cavity (half linewidth ∼ 2.5 MHz) when the carrier frequency is resonant, asshown on fig.8.33.
The reflected beat pattern of the carrier and the sidebands depends on the phase of the reflected
fields, and contains the error signal which is used to control the laser frequency. The error signal
is extracted through demodulation, by mixing the photodiode signal with the local oscillator, andfiltering the output with a low-pass filter. The signal is amplified by an analog servo, before it issent to the laser to control its frequency. Within the cavity linewidth, the error signal is linearly
proportional to the deviation from the resonance.
Pound-Drever-Hall error signal
In the following we will compute the theoretical Pound-Drever-Hall error signal, in order to compare
it to the measured one. Let the input field of the modulator be
E = E0 eiω0 t ,(8.31)and let the local oscillator be a sinusoidal function with angular frequency Ω,
VLO = V0 sin Ωt .

205

(8.32)

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

After the beam has passed through the Pockels cell of the modulator, its electric field has its phasemodulated in time, and becomes
E = E0 ei(ω0 t+m sin Ωt) .

(8.33)

According to the Jacobi-Anger identity, eq.(8.33) can be expanded in series using the Bessel functions of the first kind Jk (m),E = E0 eiω0 t∞
X

Jk (m)eikΩt .

(8.34)k=−∞

The field for k = 0, oscillating with the frequency of the input field ω0 , represents the carrier. The
light fields shifted in frequency by kΩ are the sidebands, which can be divided into upper (k > 0)and lower (k < 0) sidebands. The upper and lower sidebands with the same absolute value of k arecalled a pair of sidebands of order k. Eq.(8.34) shows that the carrier is surrounded by an infinite
number of sidebands. However, the Bessel functions decrease for large values of k, so for small
modulation indices, m << 1, the field after the modulator can be approximated by
E ' E0 (J0 (m)eiω0 t + J−1 (m)ei(ω0 −Ω)t + J1 (m)ei(ω0 +Ω)t ) .

(8.35)

Since J−k (m) = (−1)k Jk (m), the output field of the modulator can be finally written as
E ' E0 (J0 (m)eiω0 t − J1 (m)ei(ω0 −Ω)t + J1 (m)ei(ω0 +Ω)t ) .

(8.36)

Assuming that the input field is the sum of the carrier and the two first-order sidebands, the total
field reflected by the mode-cleaner is
Er = E0 (RM C (ω0 )J0 (m)eiω0 t −RM C (ω0 −Ω)J1 (m)ei(ω0 −Ω)t +RM C (ω0 +Ω)J1 (m)ei(ω0 +Ω)t ) , (8.37)
where RM C (ω) is the frequency-dependent reflectivity of the mode-cleaner cavity from eq.(2.81).
The corresponding reflected power is thus Pr = |Er |2 , equal toPr = PCARR |RM C (ω0 )|2 + PSBs (|RM C (ω0 − Ω)|2 + |RM C (ω0 + Ω)|2 ) +hi√∗ (ω Ω) − R∗ (ω )R(ωΩ)
cos Ωt ++2 PCARR PSBs Re RM C (ω0 )RM
+
0
−
M
C
C
MC
h
i
√
∗ (ω Ω) − R∗ (ω )R+2 PCARR PSBs Im RM C (ω0 )RM(ωΩ)
sin Ωt +
0
−
+
M
C
C
MC

(8.38)hi∗ (ω Ω) − R−2PSBs Re RM(ωΩ)
cos 2Ωt +
+
−
MC
C
h
i
∗ (ω Ω) − R−2PSBs Im RM+
M C (ω− Ω) sin 2Ωt ,Cwhere PCARR = J02 (m)|E0 |2 and PSBs = J12 (m)|E0 |2 . Eq.(8.38) shows that the total reflected field
is a wave with a nominal frequency of ω0 , but with an envelope displaying a beat pattern with
two frequencies. The Ω terms arise from the beating between the carrier and the sidebands, andthe 2Ω terms come from the sidebands beating with each other. The two terms oscillating at the206

8.8. Mode-Cleaner Lock−1transmitted power [a.u.]

10

−2

10

−3

10

−4

10

−5

10

−250

−200

−150

−100

−50

0
f [MHz]

50

100

150

200

250

Figure 8.33: Total transmitted field through the LG33 beam mode-cleaner (black solid curve). The carrier
field (blue dashed curve), the lower (red dashed curve) and the upper (green dashed curve) radio-frequency
sideband fields are shown.

3
2[a.u.]

1
0
−1
−2
−3
−30

−20

−10

0
f [MHz]

10

20

30

Figure 8.34: Expected shape of the Pound-Drever-Hall signal (red solid curve), superposed to the carriertransmitted signal (blue dashed curve).

207

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

modulation frequency Ω sample the phase of the reflected carrier, and must be extracted throughdemodulation to generate the error signal.
The beam reflected by the mode-cleaner is picked off and measured by a photodiode with gain
GP D , so that the signal of the modulated reflected field is Vr = GP D Pr . The Ω terms of Vr canbe isolated by feeding the signal of the photodiode to a mixer and a low-pass filter with cut-off
frequency ΩC << Ω. The DC and the 2Ω terms of Vr will be instead filtered off, since they will
result, after mixing, in a signal at frequencies higher than the cut-off. The only non-null terms arethose of frequency Ω.
The phase delay between the modulated signal and the phase oscillator can be adjusted to
extract the in-phase (ACp) or the in-quadrature (ACq) demodulated signals:hi∗ (ω Ω) − R∗ (ω )RACp ∝ Im RM C (ω0 )RM
(ω
Ω)
+
0
−
MC
C
MC
(8.39)hi∗ (ω Ω) − R∗ (ω )RACq ∝ Re RM C (ω0 )RM+
0
M C (ω− Ω) .
C
MCAssuming that the the error signal is contained in the in-phase demodulation, the Pound-DreverHall signal finally writes√
εP DH = 2 PCARR PSBs GP D GM IX GLP F
hi∗ (ω Ω) − R∗ (ω )R× Im RM C (ω0 )RM
(ω
Ω)
,
+
0
−
M
C
C
MC

(8.40)

where GM IX and GLP F are the gains of the mixer and of the low-pass filter, respectively.
The theoretical error signal is shown on fig.8.35, compared to the measured one. We remark
the good agreement between the expected and the measured error signals, even if the mode used
is not a pure LG33 .
Lock of the LG33 eigenmode resonance
The open-loop transfer function of the feedback is:
GOL = GM C GP D Gmix GLP F GS GP ZT ,(8.41)

where GM C is the optical gain of the mode-cleaner, measured in W/Hz, and GS is the gain of theservo. The gains and the bandwidths of the other electronic components of the experimental setup
are listed in table 8.3. The resulting global transfer function GOL has a unity gain at 7.75 kHz,with a phase margin of 57 deg. Its frequency dependence is the following:
GOL ∝ 1/f 5 DC < f < 72 Hz
GOL ∝ 1/f 4 72 < f < 904 Hz
GOL ∝ 1/f

(8.42)

0.904 < f < 30 kHz .

Initially the laser frequency is tuned manually with the thermal control of the laser crystal, in order
to set the resonance condition for the LG33 eigenmode in the mode-cleaner. Then the frequency of208

8.8. Mode-Cleaner Lockerror signal [V]

2

1

0

−1

−2
−15

−10

−5

0
f [MHz]5

1015

Figure 8.35: Pound-Drever-Hall signal of the pseudo-LG33 mode: measured (blue solid curve) and expected(red dashed curve) signals.

component
photodiodemixer
low-pass filter
piezo transducergain
GP D = 154 V/W
Gmix ' 0 dB
GLP F = −5.5 dB
GP ZT = 1.5 MHz/V

bandwidth
22 MHz200 MHz
2.5 MHzseveral tens of kHz

Table 8.3: Gain and bandwidth of electronics used in the servo control of the laser frequency.

209

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

resonant eigenmodetransmitted signal [mV]laser crystal temperature [◦ C]

LG33
LG`1
LG`4
LG`5
LG33
others (sum)872 ± 4065 ± 442 ± 464 ± 4872 ± 4042 ± 429.83 ± 0.01
29.85 ± 0.01
29.90 ± 0.01
29.97 ± 0.01
29.99 ± 0.01–

Table 8.4: Measurement of the pseudo-LG33 mode spectrum.

the laser is swept around the mode resonance with a periodic triangular signal, the ramp. The lockacquisition of the LG33 eigenmode resonance is performed manually: the loop is closed by turning
off the gain of the dithering ramp, at the same time that the frequency-dependent amplification of
the error signal is turned on. Following this procedure, the mode-cleaner has been robustly lockedon the resonance of the LG33 eigenmode for several hours.

8.9

Mode-Cleaner spectrum

The pseudo-LG33 mode can be projected on the basis of the mode-cleaner eigenmodes, as the sum of
a dominant LG33 mode contribution plus some spurious terms of LG`p modes of different order. Eachof this terms can resonate in the mode-cleaner, if the frequency of the laser is changed properly.
Thus, for the pseudo-LG33 mode, the plot of the power resonant in the mode-cleaner versus thefrequency of the laser (or, as it is equivalent, the cavity length) is expected to be a composite
spectrum of multiple resonance peaks, as shown on fig.8.11.
The amplitude of the resonance peaks of the spurious terms, if compared to the amplitudeof the LG33 eigenmode resonance, is an indirect measure of the purity of the pseudo-LG33 mode:the smaller the amplitude of the spurious peaks, the higher the purity of the mode. A roughmeasurement of the pseudo-LG33 mode spectrum has been carried out by scanning the free spectral
range of the mode-cleaner (500 MHz) with the thermal tuning controller of the laser, the frequencychange provided by the piezoelectric actuator (±150 MHz) being too small for this measurement.
The smallest temperature change achievable with the controller is ±0.01 K, corresponding to ∓30
MHz (the gain of the Peltier cell of the laser is -3 GHz/K).
As expected, LG`p modes of different order are observed in transmission with a CCD camera,as the frequency of the laser is changed. All of them have a radial node of null intensity at r = 0.
According to eq.(5.13), these modes are supposed to have ` 6= 0 and p + 1 radial nodes. Thereby
the radial index p of the modes could be identified directly by counting the number of visible rings.
The power of this modes transmitted through the cavity is measured with a photodiode, placed
after the mode-cleaner. Table 8.4 shows the result of the characterization of the resonant modecleaner eigenmodes, obtained from the measurement of their transmitted signals. Only the modes
who could be clearly visible on the CCD camera are reported in the table, whereas some very weak
and unidentifiable modes are only considered for their contribution to the total transmitted signal.
The uncertainty on the measured signals is equal to the maximum accuracy of the acquisition setup.
The images of the observed modes, acquired with the CCD camera, are shown on fig.8.36.
We used the spectrum of the pseudo-LG33 mode of fig.8.11 simulated with OSCAR, together

2108.9. Mode-Cleaner spectrum

Figure 8.36: Observed mode-cleaner eigenmodes, from the left: LG33 , LG`1 , LG`4 , LG`5 . Images have beenacquired with different filters before the CCD camera, the grey scale is not indicative of the relative intensities
of modes.

resonant eigenmode∆Nmeasured freq. spacing [MHz]expected freq. spacing [MHz]LG`1 → LG31
LG`4 → LG34
LG`5 → LG35

-4
-2
+4

-60 ± 30-218 ± 30-420 ± 30

-64
-210
-436

Table 8.5: Identification of modes in the pseudo-LG33 mode spectrum.

with eq.(8.12), to identify the azimuthal index of the modes. These are: {LG31 , LG34 , LG35 }. We
do not observe instead the modes identified by ∆N = 1 and ∆N = −1 on the simulated spectrum,
though they have an amplitude of the same order of magnitude of the observed modes. Other
modes in the simulated spectrum have negligible amplitudes.
The frequency spacings of identified modes have been computed and compared to the measured
values, as reported in table 8.5. Reported values are negative, since the frequency increases in
the opposite sense of the axis of cavity length tuning in fig.8.11. The comparison shows that themeasured values of the frequency spacing are in good agreement with the simulated values. In table
8.6 we show the reconstructed decomposition of the pseudo-LG33 mode in LG`p eigenmodes of themode-cleaner. Three quantities are reported here:• the fraction of transmitted signal of each mode, computed as the ratio of the single mode
signal to the sum of all the transmitted signals. This sum is equal to Vtot = 1083 mV,
including 40 mV of unidentifiable modes.
• the measured relative amplitude of each mode with respect to the LG33 mode, computed as
the ratio of the signal of one mode to the signal of the LG33 mode
• the simulated relative amplitude of each mode with respect to the LG33 mode, computed fromthe spectrum of fig.8.11.
We can observe from this table that there is a partial agreement between the measurement andthe simulation: the relative amplitudes of LG31 and LG35 modes are compatible within ∼ 15% error,
while the measured amplitude of the LG34 mode is larger than the simulated one by more than afactor of 2.

211

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

mode% of total signal

meas. % of LG33 mode peaksimul. % of LG33 mode peak

LG33
LG31
LG34
LG35

81
6
4
6

100
8
5
7

−
7
2
6

Table 8.6: Characterization of the pseudo-LG33 mode spectrum, compared to simulation.

Since the observed modes are also predicted by simulations where the input LG00 beam of theplate, the matching of the pseudo-LG33 to the cavity and the cavity mirrors are perfect, they arelikely due to the phase pattern of the diffractive plate.

8.10

Transmitted LG33 Beam

Mode transmission
When the laser frequency is locked to the resonance of the LG33 eigenmode of the mode-cleaner,
it is possible to measure the transmitted power with the powermeter and compare it to the powerof the pseudo-LG33 mode measured at the mode-cleaner input, as sketched on fig.8.37a). The ratio
between the output and the input power measures the transmission of the LG33 mode content in
the pseudo-LG33 input beam, through the mode-cleaner. The measured ratio is:
LG33 transmission =Pout = 155 mW= 58% .
Pin = 265 mW

However, this measurement is affected by the optical loss of the mode-cleaner mirrors. To correctly
characterize the throughput of the cavity, the measurement of the transmitted power has been
performed separately by using the LG00 beam of the laser.
The setup used for the measurement is shown in fig.8.37b): the LG00 beam has been deflected
after L0 on a parallel path, and the mode-cleaner has been positioned at the beam waist (previously
characterized for the mode-matching telescope). The input and the output power of the LG00 beam
are measured with the powermeter, and the output beam is also monitored with a CCD camera,
placed after a beamsplitter with measured reflectivity R = 63%. The monitoring of the output beam
has been used to check the presence of higher-order modes due to a possible mismatch, between
the beam waist and the cavity waist. In fact, only an LG01 mode is observed, with an amplitude
which is 9% of that of the LG00 beam. With this setup, the transmitted power is computed as theratio of the power measured by the powermeter and the reflectivity of the beamsplitter. The cavity
throughput is thenτ=

(PLG00 + PLG01 )out
(PLG00 )in=

365 mW + 33 mW= 91% ,
438 mW(8.43)

where (PLG00 + PLG01 )out is the total transmitted power and (PLG00 )in is the measured input power.
The measured throughput indicates that the mode-cleaner has a global optical loss of 9%, which2128.10. Transmitted LG33 Beama)b)
λ/4 λ/2

mode-cleaner
pseudo-LG33

Faraday
isolator

λ/2

L0

laser
LG33
LG00
LG00

Pin

Pout
Pout

Pin

Figure 8.37: Setup for the measurement of the LG33 mode power transmitted by mode-cleaner cavity: a)transmission of the LG33 mode, b) caracterization of the cavity throughput with the LG00 Gaussian beam.

must be taken into account in the measured transmission of the LG33 mode content of the pseudoLG33 mode. The correct estimation of the LG33 mode transmission is thus
LG33 transmission =Pout= 64% .
τ Pin

This result is roughly compatible with the expected coupling upper limit (γ 2 = 77%), estimatedthrough the overlap integral of the pseudo-LG33 mode computed in section 8.6. The measured
transmission is lower than expected, probably due to the difference between the phase of thepseudo-LG33 mode and the phase of the theoretical LG33 mode, neglected in the computation of theoverlap integral. The coupling might be also slightly decreased by some residual mismatch of thebeam waist to the cavity.
Analysis of the transmitted LG33 mode
The characterization of the transmitted LG33 mode has been carried out by computing the purity
and the least square fit, for an image of the beam acquired after the mode-cleaner. The imageis shown on fig.8.38: it has been acquired in far-field regime, with the beamscan at a distancez = 1.321 ± 0.003 m, equal to about 5.4 times the Rayleigh range of the cavity eigenmode. The
distance for acquisition is limited in this case by the size of the CCD detector of the beamscan,which cannot detect beams bigger than 15 mm in diameter.
For comparison, fig.8.38 also shows the theoretical LG33 eigenmode transmitted by the modecleaner and propagated at the same distance of the measured beam, simulated with an FFT codedeveloped from the core of the code OSCAR. The code simulates the transmission (at resonance)of a pure LG33 mode through the mode-cleaner, and its propagation in free space.
The purity of the transmitted mode is computed as the overlap integral between the squareroot of the intensities shown on fig.8.38, yielding
γ = LG33 |simulation LG33 |measure = 98.3% ,(8.44)

with L = 1 − γ 2 = 3.4% coupling losses. Thus the beam transmitted by the mode-cleaner is a highpurity LG33 mode, though with a lower purity than the one predicted by simulations (γ = 99.8%,

213

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

Figure 8.38: Comparison between normalized intensity distributions in far field: measured LG33 mode (left)
and simulated theoretical LG33 mode (right).

see section 8.3). These simulations indicate that the cavity should already filter all the spatial
defects of the pseudo-LG33 mode with a lower finesse, thus the measured purity might be likely due
to the quality of the surface of the cavity mirrors.
The comparison between the cross sectional profiles of the theoretical and measured transmitted
beams is shown on fig.8.39. The measured profile presents an overall correspondence with thetheoretical one: the position of the peaks and their relative amplitudes are those expected. This
also indicates that the divergence of the transmitted LG33 mode is close to that of the theoreticalcavity mode. The main difference between the cross sections is the presence of some power left at
the radial nodes.
The least square fit of the measured intensity profile, performed with the same method describedin section 8.6 for the pseudo-LG33 mode, yields
A33 = 1.8 · 10−3 [a. u.]
B = 479 [a. u.]
w = 1.944 · 10−3 m ,(8.45)

where the beam radius w replaced the beam waist w0 . The low residuals and the fit reduced χ2equal to 1.1 (for 512 points and 3 parameters) indicate that the model reproduces well the data.
This can be seen on fig.8.40: the radial position and the relative amplitudes of the peaks are very
well fitted by a theoretical mode with w = 1.944 mm, in good agreement with the theoreticalpropagation of the cavity eigenmode along z and through the (diverging) lens formed by the cavity
end mirror. The measured thickness of the end mirror is d = 6.50 ± 0.05 mm, with a curvedreflecting surface with R1 = 0.5 m and a flat rear anti-reflective surface with R2 = ∞. Assumingthat the refractive index of the fused silica of the mirror is n = 1.44963, the lens-maker equation
"
#
1
1
1
d(n − 1)= (n − 1)−+(8.46)
feq
R1 R2
nR1 R2214

8.10. Transmitted LG33 Beam1

norm. intensity [a. u.]

0.8

0.6

0.4

0.2

0

−6

−4

−2

0
r [mm]

2

4

6

Figure 8.39: Profile comparison: the horizontal (blue solid curve) and the vertical (green solid curve)
measured cross sections of the transmitted LG33 mode are shown, together with the cross section of thesimulated theoretical LG33 mode (red dashed line). The theoretical mode is not a fit, but it is obtained bypropagating the LG33 eigenmode of the mode-cleaner in space.

fit [a. u.]

10000
5000

residuals − intens. [a. u.]

0
0

1

2

3

4

5

6

7

4

5

6

7

r [mm]
800
600
400
200
0
−200
−400
0

1

2

3
r [mm]

Figure 8.40: Least square fit of the profile of the transmitted LG33 mode. Top panel: measured (blue dotswith errors) and fitted (red dashed curve) profiles in the r > 0 plane. Bottom panel: fit residuals, definedas the difference between the data and the model.

215

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

yields an equivalent focal length for the end mirror equal to feq = 1.112 m. The propagation of the
LG33 mode from the cavity waist, computed through ray-matrix formalism, returns w = 1.957 mm.
Thus the value of the beam radius obtained from the fit is in good agreement with the theoretical
expectation.
The power left at the radial nodes is interpreted by the fit as background light, but this is notcompletely true. Though some low background is present in the acquired picture, by looking at
the queue of the outermost peak of the mode in fig.8.40, it is clearly visible that the computed
background is overestimated. As mentioned previously, these spatial defects might be due to thesurface quality of the mirrors.

8.11

LG33 mode conversion efficiency

The mode conversion efficiency is a fundamental parameter, since it measures the power loss of the
LG33 mode generation technique. We define this parameter as follows:=(PLG33 )T(PLG00 )T

γ2 ,(8.47)

where• (PLG33 )T = 155 mW is the power of the LG33 beam transmitted by the mode-cleaner at theresonance of the LG33 cavity eigenmode• (PLG00 )T = τ ηPP O1 is the power of the LG00 mode transmitted through the experimental
setup, including the mode-cleaner:– PP O1 = 435 mW is the is the power of the LG00 beam at the input of LG00 pick-off,indicated by an arrow on fig.8.15– τ is the cavity throughput measured by eq.(8.43)– η is the LG00 beam transmission of the experimental setup without the diffractive
plate, from the input of PO1 to the mode-cleaner input mirror, as shown on fig.8.41.
η is measured as the ratio of the power PM Cin before the mode-cleaner input mirror to
the power PP O1 :PM Cin = 339 mW
η== 78% .
PP O1 = 435 mW
The optical loss of the setup are caused by the power fraction that goes into the LG00
pickoff, not used in the LG33 mode generation, and by the optics (mirrors, lens, polarizercube) placed before the mode-cleaner• γ 2 = 96% is the LG33 mode power content of the beam transmitted through the mode-cleaner,from eq.(8.44).
Eq.(8.47) defines a global conversion efficiency which takes into account the two steps of the generation technique:• the LG00 -to-LG33 mode conversion through the diffractive plate2168.12. LG33 Michelson Interferometer

PPO1

reflected field

L6

PO1

λ/4

LG00

laser
[DP]

(PLG ) T

PMCin
mode-cleaner

PBS

LG00 pick-off

Figure 8.41: Measurement of the LG00 beam through the experimental setup. The usual position of thediffractive plate (DP), removed from the setup for the measurement of the transmission, is shown with adashed line.

• the spatial filtering of the generated pseudo-LG33 mode.
This definition of efficiency is independent from the optical loss of the experimental setup, and only
depends on the design of the diffractive plate and on the transmission through the mode-cleaner.
It is equal to = 49% .
Of the 51% power lost in the conversion, about 20% is lost at the diffractive plate, going in scattered
light (due to the absence of the anti-reflective coating) and in higher diffraction orders. The restof the power is reflected by the cavity when it is locked on the LG33 eigenmode, in the form ofmodes of different order. A further increase of the efficiency could be achieved by optimizing theplate pattern and by tuning the matching between the beam and the mode-cleaner cavity. The
transmission η could be largely improved by replacing PO1 with a movable flip-mount, and byusing high quality, low loss optics in the setup.

8.12

LG33 Michelson Interferometer

The scope of the our experiment was initially limited to the test of our generation technique.
Nevertheless, as soon as we could generate high purity LG33 beams with our setup, we decided toextend the layout of the experiment to test also high-precision interferometry with the higher-order
LG33 mode. As a first step we chose to test a configuration of limited complexity, thus we assembled
a Michelson interferometer. The layout of this part of the experimental setup is shown on fig.8.42.
The LG33 beam coming from the mode-cleaner goes through a collimating lens with f = 1 m,
placed at 42 ±0.5 cm from the cavity end mirror. According to the theoretical propagation of thecavity eigenmode computed in section 8.10, the parameter of the beam after the lens is expected
to be q = (2.16 + i4.98) m, corresponding to a beam radius w = 1.3 mm. Thus we expect thebeam to be collimated over few meters, and this was checked by measuring the beam radius withthe beamscan at different positions after the lens. At ∼ 65 cm, where the beam encounters thebeamsplitter, the beam size does not change appreciably. The beam is collimated also inside theinterferometer, which has 16-cm long arms.

217

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modesfrommode-cleaner

PZT controllertranslation
stage
PZT
My

iris

16 cm
f=1m

lens

PZT controller

BS
LG33
16 cm

BS

Mx
flip mount

CCD
lens

CCD

DAQ anddigital control
systemPO
iris
far-field

PD

Figure 8.42: Layout of the Michelson interferometer illuminated with LG33 beams.

The beamsplitter, given the large size of the input beam (the diameter, taken at the outer ringof the mode, is of the order of 1 cm) and the non-normal incidence angle of the beam, is a 2-inches
optic to avoid the clipping of the beam. The high-reflectivity 1-inch end mirrors are plane. Facing
each mirror we put an iris to block the secondary beams, generated by a wedge of 30 arcminutes(according to the datasheet) in the beamsplitter, and also some spurious reflections coming from theother optics of the setup. We put the mirrors into two mounts Thorlabs KC1-PZ with piezoelectric
actuators, to remotely control the alignment. Each mount has three independent actuators torealize angular or longitudinal displacements, driven by a controller Thorlabs MDT693A withthree channels. The gain specified by the datasheet for each actuator is G = 6.1 · 10−8 m/V.
Under the end mirror of the y arm (the arm of the beamsplitter reflection) we put a translation
stage Thorlabs PT1, to tune the arm length within ±250 µm accuracy. We initially set the length of
the interferometer arms to (162 ± 2) mm. We then used the translation stage during the operationof the interferometer, as described further below.
At the output, the beam goes through a focusing lens (with f = 50 mm) and a pick-off (withR = 8%), then is detected by a photodiode. The pick-off beam is used to monitor the interferometer
output interference fringe (dark fringe) with a CCD camera. Facing the photodiode, another iris
blocks other spurious reflections.
Between the pick-off and the beamsplitter, we put a mirror on a flip-mount, to deviate the beam
on a parallel path for the pre-alignment of the beams in the arms. On this path we arranged asetup similar to that used for the relative alignment of the pseudo-LG33 mode and the LG00 pick-off,
described in section 8.7. A CCD camera is placed in near field, while we observe the beams in far
field against the wall of the room nearby, ∼ 10 m far. We pre-align the beams in the arms with218

8.12. LG33 Michelson Interferometer

Figure 8.43: Image of the interference fringe at the output of the LG33 Michelson interferometer (the image
has been acquired with a long exposition time in order to make it clearly visible).

this setup, following this procedure:
1. we block the beam in the y arm, before the end mirror
2. we align the reflection of the x-arm mirror on the input beam of the interferometer: we put
two iris on the path of the input beam, at two distant places, and we center the spot reflectedby the end mirror on both the iris
3. we align the beam of the y arm to the other. We make sure that the beams are well superposed,
by looking at the shape of the fringe in near and far field, at the same time. The long baseline
between the two points of observation ensures that the beams are well aligned also at any
point inside the interferometer.
When the pre-alignment is done, we lock the interferometer on the dark fringe, as shown on fig.8.43.
The error signal is generated by dithering the x mirror, by sending to the three actuators of themount the same sinusoidal signal at 1 kHz.
The dithering signal is generated by a digital acquisition and control system (DAQ, the same
presently used in Virgo), installed on a computer. The dithering induces a periodic phase modulation of the beam and thus generates sidebands. The beating pattern between the carrier andthe sidebands is detected by the photodiode. The signal is sent to the DAQ for demodulation,
performed by executing a series of coded commands in a script. To extract the error signal we
implemented in the code a standard demodulation scheme with a mixer and a low-pass filter.
Once the lock is acquired, we perform the fine alignment of the beams, using the controllersof the actuators to tune the angular position of mirrors. We optimize the alignment by looking
initially at the shape of the dark fringe acquired by the pick-off camera. When the change of thefringe shape cannot be appreciated any longer by eye, we continue the optimization of the alignmentby minimizing the output DC signal.

219

Chapter 8. A Table-Top Experiment to Test Generation and Interferometry of Higher-Order
Laguerre-Gauss Modes

We then compute the contrast of the interferometer from the measurement of the bright fringe
signal VBF , initially acquired when the interferometer is unlocked, and the dark fringe signal VDF
of the locked interferometer. Since the signals are directly proportional to the detected output
power, we can use eq.(2.7) to writeC=

VBF − VDF.
VBF + VDF

(8.48)

By applying iteratively the procedure of lock and fine alignment described above, we performed aseries of measurements of contrast adjusting the macroscopic length of the y arm, tuned with thetranslation stage. The best contrast achieved is C = 99.5% (VBF = 3.1 V, VDF = 0.0085 V). We
expect it should be possible to decrease the contrast defect (1-C) of about one order of magnitudein our setup, and this will be the subject of future study.

220

221

Conclusions and Perspectives - Part II
Mirror thermal noise will limit the sensitivity of all currently planned terrestrial detectors in thecentral region of the detection band. In order to decrease this noise, it has been necessary toincrease the beam size on the arm cavity mirrors of advanced detectors, and LCGT and Einstein
Telescope will implement cryogenics. The investigation is ongoing to find new solutions to the issueof mechanical loss of the coatings.
A relevant reduction of mirror thermal noise could be achieved by using non-Gaussian readout beams. The use of LG`p modes has been proposed, since they are eigenmodes of resonatorswith spherical mirrors. These modes had never been used before in applications of high-precisioninterferometry, thus we realized an experiment to study their optical performances.

Conclusions of the Experiment
We designed, assembled and operated a table-top experiment to generate LG33 modes and test basic
interferometry [139]. The results of the experiment may be summarized as follows:• the upper limit of the pseudo-LG33 mode purity is γ = 88%, which is compatible
with the prediction of FFT simulations of the plate (γ = 85%). The measured spectrum of
the mode features the resonances of three modes {LG31 , LG34 , LG35 } which were also predictedby simulations. Other modes predicted by the simulated spectrum, instead, have not been
observed• the upper limit of the LG33 mode purity at the output of the mode-cleaner is γ = 98%,
which is smaller than γ = 99.9% predicted by the FFT simulations of the cavity with perfect
optics. Thus this discrepancy is likely due to the surface quality of the mode-cleaner mirrors• the global conversion efficiency, inferred from our direct measurements of LG33 /LG00
power ratio (36 %) and setup transmission (η = 78%), is  = 49%. The power budget of the
setup is the following:– 18.5% of the initial LG00 power is lost at the diffractive plate. The majority of the loss islikely due to the efficiency of the blazed grating, while 4% is lost at each surface becauseof the lack of an anti-reflective coating. Part of the loss is also due to imperfections of
the plate phase pattern, caused by errors in the etching process of the plate surface– the FFT simulations of the plate indicate that the coupling of the pseudo-LG33 mode
should be γ 2 = (0.85)2 = 72%, but its measured transmission through the mode-cleaner
is 64%. Therefore 8% of LG33 power is lost, probably due to the combined effect ofastigmatism of the input LG00 beam and of mismatch of the pseudo-LG33 to the cavity222

– at the output of the mode-cleaner, a fraction L ≥ 1 − (0.98)2 = 4% is lost because of
the purity of the transmitted LG33 beam. As mentioned above, this should be due to thesurface quality of the mode-cleaner mirrors• the long-term stability of the LG33 beam generator is demonstrated by the robust
lock of the laser frequency on the LG33 resonance and by the constant transmission of themode, during several hours. The lock is realized through a standard Pound-Drever-Halllocking scheme in reflection: the shape and the amplitude of the error signal agree with thetheoretical predictions• the alignment of the pseudo-LG33 beam on the mode-cleaner cavity is more difficult
than with the LG00 beam, so we developed a pre-alignment procedure using the LG00 beam.
The procedure works correctly, and no fundamental obstacles have been encountered in the
manual alignment of the pseudo-LG33 beam on the mode cleaner cavity• we illuminated a Michelson interferometer with the filtered LG33 beam. We locked
the interferometer on the dark fringe and we achieved a contrast C = 99.5 %.
The measured conversion efficiency is a first acceptable achievement, and some elements of the
setup can be optimized to further increase its value. The estimated purity of the filtered LG33 beamis high, and might be likely increased as well. We have also shown that LG33 beams are compatible
with the technique for longitudinal error signal extraction used in gravitational-wave interferometers. The alignment and lock of the mode-cleaner cavity and of the simple Michelson interferometer
demonstrate experimentally the feasibility of higher-order LG`p mode basic interferometry.
LG33 beams of high purity have also been obtained using spatial light modulators [145]. However, the use of spatial light modulators is limited to applications with low-power laser beams,
making them unsuitable for the potential implementation in future gravitational-wave detectors.
Our technique has been developed to be specifically used in future interferometers, since the diffractive plate can in principle handle high-power laser beams.
Recently, a diffractive plate has been used to generate LG33 beams which have been injected ina 10-m long cavity of finesse F = 600 [146].

Further Developments
Efficiency and mode purity
The global power conversion efficiency could be increased by minimizing the loss of the diffractive
plate. In principle it should be possible to increase the efficiency of the blazed grating of the plate[143], then 4% of efficiency might be recovered by coating the plane surface of the diffractive platewith an anti-reflective layer. An additional ∼ 2% could be recovered by coating the etched surfaceof the plate [147].
Another improvement of the efficiency could come from the increase of the pseudo-LG33 mode
purity, in order to maximize its transmission through the mode-cleaner. For this purpose we could
optimize the shape of the input LG00 beam, as well as the matching of the pseudo-LG33 mode to thecavity. It should be possible to compensate for the astigmatism of the LG00 beam by adjusting the223lenses of the telescope. The matching of the pseudo-LG33 mode might be improved by optimizing
the position of the mode waist on the cavity. Therefore the resulting efficiency could be improvedby ≤ 8%. To increase much further the purity of the pseudo-LG33 mode, we should rather try to
improve the design of the diffractive plate.
There exist in literature some examples of complex-amplitude modulation patterns [148], which
are numerically computed to encode amplitude and phase modulation on a single phase plate, andmight potentially generate modes of very high purity. These patterns have been already tested on
spatial light modulators, yielding promising results in terms of achieved beam purity [135]. Also
the option of using two diffractive plates could be re-evaluated.
Finally, also the purity of the LG33 mode transmitted through the mode-cleaner might be further
increased, as shown by the FFT simulation of the mode-cleaner. By using high-quality cavity
mirrors, it should be possible to achieve a purity of the transmitted mode of more than 99%.
LG33 interferometry
In our experiment we operate a Fabry-Perot cavity and a Michelson interferometer with LG33 beams,
demonstrating that basic interferometry with LG33 modes does not pose any fundamental obstacle.
The next step should be the upgrade the Michelson interferometer to the full optical configurationof a gravitational-wave detector, by implementing Fabry-Perot cavities in the arms and for power
recycling.
The setup will be used to study the optical performances of an LG33 detector, to validate acontrol scheme for longitudinal and angular control, and to characterize the sensitivity of LG33
beams beams to mirror misalignments and figure errors. The setup could be used also to address
the issue of degeneracy of higher-order modes, described in the following, from an experimental
point of view.
Simulation of a real detector: the issue of mode degeneracyLG`p modes of the same order N = 2p + |`| are degenerate, i. e. they can resonate at the same−3
±5
±7
time in a Fabry-Perot cavity. For example, there exist 9 other modes, {LG±90 , LG1 , LG2 , LG3 ,±1
3
LG4 }, which have N = 9 and are degenerate with the LG3 mode.
The potential issue related to mode degeneracy had been already raised at the time LG`p modes
had been originally proposed [125], though without going into the details of its possible impact.
The issue has been successively confirmed by simulating the mode content of a km-long Fabry-Perotcavity with realistic state-of-the-art mirrors, illuminated with an LG33 beam.
The spatial defects of the mirrors transfer energy from the LG33 mode to other modes of thesame order. The contamination of non-LG33 modes is large, yielding an important decrease ofcontrast of the interferometer. For example, with a mirror surface roughness of 0.5 nm rms, thecontrast is decreased down to the order of 60% [149]. Especially low-frequency spatial defects, andin particular astigmatism, highly contribute to mode contamination [150, 149, 151].
In order to use LG`p beams, the quality of the mirrors should be drastically improved. For
example, by using ultra-high quality polishing for cavity mirrors, in order to achieve a surface
defect rms amplitude 0.2 nm and a cut-off of low-frequency spatial defects below 100 m−1 [149]. In
this case the expected mode contamination could be mitigated, and the contrast would be increased
up to 99% [149].

224

The application of a corrective pattern of rings on the surface of cavity mirrors (placed at theposition of radial nodes of the LG33 mode) has also been proposed [150, 152]. This would allow todamp resonant degenerate modes.
Beside the encouraging results of the simulations, we believe that mode degeneracy probably
remains the main issue with respect to the use of LG`p beams, so that a solution to this problem isneeded [152].

225

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