Noise from stray light in interferometric gravitational wavedetectors

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Graduate course in physics

University of Pisa

Phd Thesis:

NOISE FROM STRAY LIGHT IN INTERFEROMETRIC

GRAVITATIONAL WAVE DETECTORS

Candidate:

Jose M. Gonzalez Castro

Supervisor:

Prof. Francesco Fidecaro

June 19, 2018

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Acknowledgement

Along these years in Pisa I have lived the most exciting period in my life. For this reason, I want to express my gratitude to Prof. F. Fidecaro and the organizers of the GraWIToN project, M. Punturo, E. Cuoco and E. Morucci for the opportunity they gave me. Also, I thank A. Cisternino for the discussions about ray tracing and other computing science

problems I found. To the other GraWIToNs, Matthieu, Rudy, Zeno, Gr´egoire, Dani,

Im-ran, Omar, Marina, Shub, Gang, Serena and Akshat. It’s difficult to express how much happy I am to meet all you coming from so many different places and cultures. I hope that we’ll meet again soon. To the Virgo Pisa group, for hosting me and the other Phd students (or Msc) that “suffered” with me, Giovanni, Francesco, Lorenzo and Lucia. I also thank Sofia for the many times she helped me with the thesis and Italian.

To everybody I met at EGO, they were so nice and helpful to me when I had doubts about how to proceed with my research (or I was completely lost).

To my family, because they still do not have any idea what I do, but if I am happy, they are.

A todos los f´ısicos de la UAB, porque me hicieron pasar muchos buenos momentos

durante y después de nuestro periodo en la autònoma. A Emili, ¡espero verte más

fre-cuentemente! A Pablo, espero con ganas las birras que vendr´an despu´es de Pisa y Amberes.

A Parra, con quien no he parado de hablar este tiempo y espero que nos podamos re-ver

pronto. A Kike (@Lyzanor), son muchos a˜nos conoci´endote y aun me sale una sonrisa

cuando me acuerdo cuando te dije que iba a iniciar esta aventura en Pisa. A Carlos (ese maldito celuloide), me acuerdo como si fuese ayer de como perd´ıamos el tiempo en la

bib-lioteca. y a los dem´as f´ısicos Jorge, Turpin, Raul, Jordi, Sanahuja, Toni y Juanfran. La

mayor´ıa viviendo muy lejos de su casa. A los Erasmus que me encontr´e en mi tiempo en

la UAB, ellos plantaron la semilla que hizo que un d´ıa quisiera experimentar que es vivir

en otro pa´ıs. Entre estos Erasmus est´a Rux, a quien no tengo palabras de agradecimiento

por la amistad y la ayuda que me ha dado con la tesis. Igualmente, estoy muy agradecido a Mai Martin, una de las pocas personas que comprende el funcionamiento de mi mente.

Ai miei coinquilini che mi hanno aiutato tanto con l’italiano quando sono arrivato e

non riuscivo n´e a capire quando si dice ciao. E a tutti quelli che ho incontrato dopo per

aiutarmi a continuare questa avventura. A Rolando che sempre mi ha dato il suo sostegno. Agli amici che ho conosciuto in tutto questo tempo a Pisa, i ragazzi dell’AEGEE e quelli

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che ho conosciuto negli aperitivi linguistici, gli indiani che grazie a loro non ho dimenticato

del tutto l’inglese, specialmente a Mastan. A Sima, la cui amicizia `e tanto importante, mi

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Abstract

The era of the second generation of gravitational waves detectors has been very successful with the advanced versions of Virgo and LIGO. In September 2015, LIGO had detected the first gravitational wave the day before it started the first science run. After a success-ful first science run, LIGO upgraded both interferometers to improve the sensitivity and during the second science run Virgo joined LIGO in the search for gravitational waves. Many challenges have been faced in order to improve on the sensitivity of advanced detec-tors. One of those challenges has been stray light. Up to now, both Virgo and LIGO have been affected by stray light problems and this is expected to be an important problem to address in the future third generation of gravitational wave detectors.

At the same time, the search for stray light is probably the least studied aspect before the instrument is switched on. For this reason, it is important to develop new methods to predict and deal with stay light. This thesis focuses on the study and simulation of light in order to find the coupling between stray light and vibration of mechanical elements that spoil the sensitivity during operation.

The first section contains a short summary of the state of the art of gravitational waves detectors. The first chapter presents the General Relativity principle to generate grav-itational waves and summarises the astrophysical sources that emit gravgrav-itational waves, either they have already been detected or not.

The second section presents the computational methods used and developed to simulate stray light. The third chapter explains ray tracing as a tool to simulate stray light with-out frequency dependence. The fourth chapter contains two different extensions to ray tracing. The first is the development done to implement the coupling between mechanical elements and stray light. The second part describes different tests to accelerate the code using GPU and its results.

The third part is focused on the results. The fifth chapter contains simulations of stray light with the methods presented in the second part. Finally, the sixth chapter presents the studies done to analysing stray light in Virgo during the period of commissioning.

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Part I

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Chapter 1

Gravitational waves from General

relativity

In order to explain how a gravitational wave detector works, a simple introduction to general relativity (GR) and the mechanism that generate gravitational waves (GWs) will be presented in this introductory chapter. GR was presented in 1915, and one year later Einstein found that in his description of gravity, waves are generated from accelerated objects.

1.1 General relativity

In special relativity, Einstein showed that it is not possible to send any signal faster than the speed of light and established the concept of a four-dimensional space mixing time and space. With these concepts, he developed a new description of gravity based on the ideas obtained from special relativity, elaborating what is known today as general relativity. General relativity is a geometrical description of gravity. In this description, the geometry of space-time is curved by the presence of energy density. The description of space-time is written in the Einstein field equations, which are a set of 10 equations presented in a tensor form as:

Gµν+ Λgµν =

8πG

c4 Tµν (1.1)

On the left-hand side there is the Einstein Tensor , Gµν, which contains the curvature of

space-time and the cosmological constant Λ multiplied by the metric gµν. On the

right-hand side there is the stress-energy tensor, it quantifies the amount of energy density, flux of energy and flux of momentum in space-time.

Initially, Einstein thought that it was an error to add the cosmological constant and con-sidered that it must be 0, but further observations have shown that a positive cosmological constant explains the expansion of the universe. In the case that the cosmological constant was 0, we would see that the space is curved proportionally to the mass.

Finally, within Einstein equations it is possible to describe the dynamics of objects moving in free fall. Their movement must follow a geodesic line in the space-time, which

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Figure 1.1: Quadrupolar effect of deformation on a circular array of test masses produced by a GW. (image from [52]) is expressed as: d2xµ ds2 = −Γ µ βα dxα ds dxβ ds (1.2)

where s is the proper time and Γµ_βα are the Christoffel symbols.

1.2 Gravitational Waves

When a mass is accelerated, the deformation of space-time due to its presence is propagated at the speed of light. This maximum velocity at which the gravitational field can propagate gives an intuitive idea about the gravitational radiation that Einstein found in 1916 [28]. He first derived GWs from general relativity [32] in the approximation of linearized gravity,

in which the spacetime metric gµν is very close to the flat metric ηµν:

gµν = ηµν+ hµν khµνk 1 (1.3)

where ηµν is (diag(−1, 1, 1, 1)) and hµν to first order is the weak perturbation in the

metric.

Expanding the Einstein tensor and changing the metric perturbation by setting ¯hµν =

hµν− 1/2ηµνh, it is possible to obtain the so called linearized Einstein equations1 as:

¯hµν = −16πTµν (1.4)

1

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Chapter 1. Gravitational waves from General relativity 1.2. Gravitational Waves

where = ∆ − 1

c2 ∂2

∂t2 is the d’Alembertian operation of the wave equation.

In the absence of matter, equations (1.4) become

¯hµν = 0 (1.5)

Finally, equation (1.5) can be solved with a superposition of plane waves similarly to electromagnetism which propagate at c.

The deformation produced by the GW in an array of test masses has a quadrupolar form as can be seen in Fig. 1.1.

1.2.1 GWs sources

Since accelerated masses are expected to produce GWs, there are many candidates to be sources of detected GWs. These sources can be classified depending on the frequency range (Very low frequency{f < 1nHz}, low frequency {µHz < f < 1Hz} and high frequency

{f > 1Hz })2 _{which are related to the type of experiment (pulsar timing array(PTA),}

space based interferometer or earth based interferometer)3 _[53].

Ground based detectors

Currently the most interesting sources for GWs signals are those accessible to ground based detectors since LIGO and Virgo collaborations have been taking data with their second generation detectors and did the first detection of GWs followed by more successful detections [4, 2, 3]. The detections are explained in Sec. 1.2.2, and the working principles of ground based interferometer detectors are explained in chapter 2. There are many GWs sources that can be detected.

- Compact binary objects: The inspiral of massive compact objects as binary black hole (BBH) and binary neutron star (BNS) systems have now been detected by Earth based

detectors. These are designed to detect binary systems ranging from one solar mass (M)

up to approximately 100 M. The waveform of the signals produced by GWs generated

during the coalescence of binary systems can be computed to a high level of precision. Up to now, the most common system detected have been BH-BH coalescence (see Sec. 1.2.2). - Continuous waves: If a rotating neutron star is not symmetric, it will be a source for continous GWs. The signal is expected to be almost monocromatic due to the high

stability of the pulsar frequency fpulsar, and fGW = 2 · fpulsar. The expected frequencies

are ranging between 20 − 103 _{Hz. Several candidates are being targeted as possible sources}

of GWs inside Ligo Virgo collaboration (LVC) data. Upper limits of GWs emission from pulsars were computed with the data taken during the first Science Run of LIGO [1]. The detection of continous GW might help to discard some equation of state (EOS) for

2_{At this point, we must be careful because on Earth based laser interferometer experiments, such as} those described in this thesis, the definition of low frequency and high frequency is done based on their sensitivity range. As is explained in the next chapter, the low frequency corresponds to the range from a few Hz to a few hundred of Hz.

3_{Resonant mass experiments are not considered here since the current experiments are close to being} decommissioned and no new experiments are expected to be developed.

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NS since each EOS produce different results for the deformations that a NS can have, hence producing different GWs signal.

Supernovae: It is expected that the core collapse of a star will produce a GW signal

between 102− 103_{Hz. This range of frequencies is within the detection band for either}

interferometer and resonant bar detectors, but the probability of detecting an event is low due to the fact that the expected signal would not be sufficiently intense and thus it is required that the supernovae explosion happens in our galaxy, which is a rare event (1/200 year).

Space based detectors

Again, the main source for GWs for space based detectors such as LISA [9] will come from binary systems . For these detectors, the range of masses is different from ground based interferometers in the case of binary black holes. Some possible candidates are:

Compact binary stars in the galaxy:These binaries are mainly form by white dwarfs, but also can comprise neutron stars and stellar-origin black holes. This is expected to be the most frequent source for LISA data and several candidates for detection have been identified by electromagnetic observations [60].

Massive black hole binaries: In the case of space detectors, the range of black hole

masses whose emission of GWs during merger can be detected is in the range of 104 −

107M, which corresponds to frequencies in the range of 10−4− 10−1Hz. It is expected

that the number of mergers detected per year will range from 10 to 100. The detected rate found by detectors such as LISA will help to estimate the population of massive black holes and understand their generation process.

Extreme mass ratio inspirals (EMRI): EMRI are binary objects formed when a com-pact object with a stellar mass inspires into a supermassive black hole. The estimation of events per year and the frequency with which they can happen is not clear since those events are related with a wide range of masses.

PTA detectors

In the case of PTA detectors, the sensitivity band is considered for frequencies lower than

1 nHz, and in this case the GWs detected will come from supermassive BBHs (≈ 109M),

which either can be resolved individually or form a stochastic background.

Also, it is expected that the stochastic background found with PTA will come from su-permassive BBHs at cosmological distances that cannot be resolved individually.

Unknown physics

There is a further kind of GWs that could be generated by early universe processes such as inflation. Also other types of GWs coming from exotic physics have been hypothesised, and some sources are expected to come from unknown physics, but the range of their frequency and strain for the signal is very unclear.

In any case, the unknown physics is the most exciting one, as happened with the first GW detection (Sec. 1.2.2), in which the mass of the black holes was completely unexpected.

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Figure 1.2: The signal from GW150914 was detected at both LIGO sites, being H1 on the left panel and L1 on the right. In the top row it is shown the detected signal, on the left L1 signals has been superposed with H1 data shifted in time and inverted, on the second row the detected signal is superimposed with the expected signal from numerical relativity. On the third row it is possible to see the residuals between the numerical relativity and the extracted signal. On the bottom row it is possible to see how the signal was seen on a time spectrogram.(image from [4])

1.2.2 GWs detections

The first experimental proof that GWs are real was the observation of the binary system PSR B1913+16 where the orbits of the binary system where spinning up because the

system was losing energy due to gravitational radiation [41, 78]. This discovery was

awarded with the Nobel prize in Physics in 1993, but there was no recording of the GW signal, only indirect evidence.

The first GW signal and posterior detections

The first detection reported by the LIGO-Virgo collaboration ocurred one hundred years after A. Einstein predicted GWs [4]. The first signal from a BBH coalescence was detected on the 15th of September 2015 in the two LIGO detectors. The detected BBH coalescence

had masses of 36+5₋₄ M and 29+4−4 M producing a black hole of 62+4−4 M. During the

process, 3+0.5_−0.5 Mc2 were radiated in gravitational waves. The signal arrived first to LIGO

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Figure 1.3: The signal from GW170814 was detected by the two LIGO detectors and Virgo. In the top row it is shown the SNR. Single detector SNR for H1, L1 and Virgo are 7.3, 13.7 and 4.4 respectively. On the second row it is shown how the signal was seen on a time spectrogram. On the third row there are the reconstructed signals for the coalescence in the three detectors. (image from [66])

time was used to estimate the location of the source.

Also, during the Advanced LIGO O1 run, there was another significant event, GW151226, and another event, however, not significant enough to be claimed as a detection called LVT151012 [2]. The first event was the signal from the coalescence of two blak-holes During O2, before Advanced Virgo joined LIGO in the search for GWs, GW170104 [3] was detected.

The first triple detection

The first triple detection was GW170814 [66], and allowed for the first time to locate the event in the sky with a precision never found before, reducing the 90% credible region sky

area from 1160 deg2 using both LIGO detectors to 60 deg2 including Virgo. Also, it was

possible to observe for first time the two polarization of GWs.

The detected BBH coalescence had masses of 25.3+2.8_−4.2 M and 30.5+5.7_−3.0 M producing a

black hole of 53.2+3.2_−2.5 M. All the masses of first BHs discovered in GWs events until

GW170814 are higher to those of other previously BHs discovered from X-ray studies (see Fig. 1.5).

The first BNS detection

The latest result was the first detection of a GW generated from a BNS merger [67], which received the name GW170817. The masses of both neutron stars have been constrained

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Figure 1.4: Cumulative SNRs plot of the three detectors for the first triple detection. The cumulative SRN ratio shows the importance of the low frequency range (below 200 Hz) in which almost all the SNR was accumulated. (image from GraceDb)

between 0.91 and 2.15 M, at a distance of 40 ± 7 Mpc. The signal lasted 99 seconds

in both LIGO detectors with a total SNR of 32.4, being the loudest GW signal detected so far. Coherent with the mass measurement by LIGO and Virgo, which was indicating BNS, 2 seconds after the end of the GW signal, coincident gamma ray counterparts were detected by Fermi-GBM and INTEGRAL telescopes [65]. This detection had a complete multi-messenger follow-up which lead to the observation of a bright transient in the galaxy NGC4993 [68] confirmed by several teams was the first observation of a Kilonova. The time-delay between the GW and the GRB have been used to constraint the value of the

Hubble constant with an independent method giving H0 = 70+12.0−8.0 km/s/Mpc [64].

It is not clear whether the object formed from the merger is a neutron star or a black hole,

since it has a mass of 2.74+0.04_−0.01 M, which is more massive than the most massive neutron

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Figure 1.5: Population of BHs discovered with LIGO from GWs events (orange) compared with the ones discoverd by X-ray methods. [Image credit: LIGO/Caltech/MIT/Sonoma State (Aurore Simonnet)]

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Chapter 2

How to detect Gravitational

Waves

When GWs were predicted, it was thought that it would be impossible to detect them, since at that time neutron stars were not known. When it became clear theoretically that GWs were real in 1959, an intense on-going effort was invested in developing detection methods.

An introduction to the different methods that have been used in the search for GWs is presented in this chapter. The explanation is focused on Earth based interferometers, and specially on Virgo, this being the main aim of the thesis. The chapter starts with a general explanation of GWs detection systems, continuing on working principle of Earth based GWs interferometers. Finally a short review is given, presenting the challenges that are faced to control a complex instrument such as Virgo and the different sources of noise that can be found with a deeper explanation on stray light.

2.1 Detection methods

Resonant bars

In the decade of the 60s, the first experiments to reveal GWs were developed. Those exper-iments were carried out by J.Weber and were based on the assumption that a big cylinder bars should resonate under the effect of a GW [77]. During the first years there were even some claims that the experiments with resonant bars had found echoes of GW, but later they were considered not to be GWs due to the impossibility of finding new detection by other groups using more sensitive experiments [56]. Since then, many experiments using resonant bars have been developed and their sensitivity was increased through the use of criogenic temperatures to reduce the noise. However, more recently last years the interest on them has been declining due to the probability of GWs detection being very low. The main cause of this is the limited detection band on the kHz region and currently there are no resonant bars searching for GWs being Auriga [45] at LNL the last one to be shut down.

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Interferometer detectors

Few years after the first experiments with resonant bars began, two Russian scientists proposed a new method for detecting GWs using interferometric techniques [35]. The working principle of such devices is a Michelson interferometer, which under the effect of a GW, suffers a modification in the arm length proportional to the strain of the GW, producing a signal in the output.

In the following years, several interferometers were developed to test this technology, and finally the first proposals for interferometers with km arm length were made for Virgo [19] and LIGO [5]. The first generation was ready by the early of the 2000s and they were operating together with GEO600 (in Germany) and TAMA (in Japan). After an operating period of 9 years, between 2002 to 2011, Virgo and LIGO were stopped in order to implement upgrades in order to ensure increased sensitivity.

The era of the second generation of GWs detectors started in September 2015, when LIGO began taking data. During this first period, it detected for the first time a direct signal of GWs [4]. Advanced Virgo began taking data in May of 2017 and joined LIGO during its second science run since the 1st of August 2017. It is expected that, in a near future, Kagra detector in Japan and the future LIGO-India will be added to the network of GWs observatories.

All fore-mentioned detectors are Earth based, but it has been proposed that interferometric detectors could be used in space using a technique called time delay interferometry [70], since the arms of the interferometer would be around a million kilometres long. The current proposal for such a detector is LISA [9] which has been approved as a mission for the European Space Agency, to be launched around 2030 after the success of the LISA Pathfinder mission [11, 12] to test the technology that will be used in LISA.

Pulsar Timing Array

PTA detectors are expected to be sensitive to lower frequencies than other detectors. The working principle is based on the fact that the period of a pulsar is a regular clock and thus, the time of arrival on Earth of the pulse will be modified if a GW passes between the pulsar and the Earth. Using an array of pulsars, it will be possible to measure the passing of a GW in front of the array. Currently there are some collaborations trying to measure the effects of GWs in pulsars, being the International Pulsar Timing Array (IPTA) [72] the most important, but this field will go one step forward once the Square Kilometre Array telescope (SKA) [27] is completed giving a great improvement on sensitivity.

2.2 Interferometric detectors, VIRGO

The principle of interferometric detection of GWs is based on the effect that GWs have on the space they travel through, modifying the distance between two free-falling masses when a GW passes. Since the effect of a GW is differential, the effect the GW amplitude (h) on a distance (L) is

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Chapter 2. How to detect Gravitational Waves 2.2. Interferometric detectors

Figure 2.1: Comparative of the sensitivity-plots for different GW experiments and the sources that they are expected to detect. Sensitivities for current Earth based laser in-terferometric experiments are plotted, including LISA and SKA. Viewing sensitivities on a single plot helps to understand the complete spectrum that will be discovered with the future detectors. [53]

The ± sign refers to the contraction or expansion of space coordinates due to the effect of GWs. For measuring distances, the most precise transducers currently available are interferometers. In order to achieve this, each interferometer presents an optical gain G, dependent on the frequency response, translating length variations into phase variations

δφ = GδL (2.2)

thus, in order to measure distances more precisely, it is necessary to have a gain as high as possible.

Light description

In order to explain the working principle of an interferometer, the electromagnetic wave is considered to be (unless otherwise specified) in the plane-wave approximation, using complex notation as

Ψ(z, t) = A(z, t)eiφ(z,t) (2.3)

where the only important information are the amplitude of the field and the phase at each point, without taking into account the transversal profile or diffraction effects of Gaussian beams, since almost all the effects inside an interferometer are well described using plane waves. Also, the unit of the field amplitude A can be normalized in a way that the power

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Figure 2.2: Model of a Michelson interferometer. From the beam splitter (BS), the sym-metric port is defined as the beam that travels back to the Laser and the antisymsym-metric port is the beam going to the photodetector (PD).

of the field is just the square modulus of the field

P (z, t) = |Ψ(z, t)|2 (2.4)

and then the reflection and transmission effects on the field are

Ψr= irΨi (2.5a)

Ψt= tΨi, (2.5b)

Michelson Interferometer

Since GWs produce a differential change on perpendicular axes as explained in section 1.2, the interferometer that can measure changes in perpendicular axes is the Michelson interferometer. A Michelson transforms a phase change produced by the difference in length between its two arms, modifying the power leaving through the antisymmetric port as defined in Fig. 2.2.

The field on both output ports will be

ΨASY = − 1 2 rxe−2ikLx + rye−2ikLy Ψi (2.6a) ΨSY M = i 2 rxe−2ikLx− rye−2ikLy Ψi, (2.6b)

which, in the case that r = rx= ry and

Lx = L + δL 2 Ly = L − δL 2

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Figure 2.3: Power output of the Michelson interferometer from symmetric and antisym-metric ports when the end mirrors are moved in opposite directions. The units are phase

shift in degrees. PLaser= 1W is assumed.

we have that the field can be rewritten as:

ΨASY = −re−2ikL(cos kδL) Ψi (2.7a)

ΨSY M = ire−2ikL(sin kδL) Ψi, (2.7b)

and the resulting intensities are plotted in Fig. 2.3.

Fabry Perot cavities

The problem of the Michelson interferometer is that it does not have enough sensitivity to detect any GW, so a different technique must be considered. An additional technique that can be used to increment the sensitivity of a Michelson is the use of resonant cavities such as Fabry Perot (FP) cavities in the arms.

The key point of the FP cavity is that when it is in resonance, the phase of the reflected field suffers a fast change, since the phase changes by π from the antiresonance condition to the resonance. For a FP cavity in resonance, the power inside the cavity is highly increased depending on the reflectivity of the mirrors which is equivalent to the Finesse (F ) of the the cavity. But the most important aspect of a resonant cavity is the change suffered by the fields when the cavity is close to constructive interference inside the cavity. At that moment, the phase suffers a fast change of π as can be seen on Fig. 2.4 in the case of the reflected field.

The phase changes when the cavity is close to the resonant point as [73]

dφ

dL =

8F

λ (2.8)

This means that the phase suffers a faster change around the resonant point when the finesse of the cavity is increased. Thus, it is possible to add two Fabry Perot cavities

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Figure 2.4: The upper image contains the circulant and reflected power for a Fabry Perot cavity 3 Km long, where HF stands for High Finesse and LF for Low Finesse. In the case of HF the cavity, mirrors have similar properties as the mirrors from AdV. The first mirror has a transmission of 0.98 for the first mirror with a radius of curvature of 1420 m. The second mirror has a reflectance of 0.996 with radius of curvature of 1683 m. It is considered that both mirrors does not absorb light. The lower image contains the phase

of the reflected field for the two different Finesse. We take PLaser = 1W

.

in the arms of a Michelson interferometer, and in consequence, the phase change due to different arm lengths will be enhanced. Even though this method in theory would permit to increase the response of the system, and consequently its sensitivity, as much as we desire just incrementing the Finesse of the resonant cavities, this cannot achieved in practice, because there are some limitations that must be considered when the Finesse is increased. The first limitation is the fact that the frequency response of the system is affected, reducing the sensitivity at high frequencies. This also affects the full width half maximum (FWHM) of the resonant peak, which is decreased, as can be seen in Fig. 2.4, incrementing the difficulty to control the cavities since the working point must be achieved with more precision. Another problem of increasing the Finesse is that the losses of the cavity are also increased.

Recycling cavities

Another important element that can produce a considerable increment in the sensitivity of an interferometer is the use of a power recycling mirror (PRM). This partially reflective mirror is located between the laser and the beam splitter and has the aim to recycle the light that comes back from the bright port of the Michelson interferometer. The addition of the new mirror creates a new resonant cavity formed by the PRM and the input mirrors of the FP cavities.

Another recycling mirror can be add in the dark port, this mirror is called signal recycling mirror (SRM). This mirror can create an optical gain which can be tuned to be more

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sensitive to a selected frequency in the RF spectrum.

Gaussian beams

Although plane waves are a sufficiently good approximation for describing the interfer-ometer basic working principles, it is important to mention that actually the transversal profile of the light has a Gaussian profile as the optics is diffraction limited. One can work in paraxial approximation, having the light decomposed in various Gaussian modes. Cavities are designed to be resonant for suitable Gaussian modes.

Gaussian beams can be modelled as an expansion of Gaussian modes, with the fundamen-tal mode being

Ψ(r, z) = Ψ0 ω ω(z)exp _−r2 ω(z)2 exp −i kz + k r 2 2R(z)− ψ(z) (2.9)

which contains the transversal dependence r and where ω is the beam waist, corresponding

to the diameter at which the amplitude is reduced by 1/e and being ω0 the position in

which the waist is minimal. ω(z) is the waist at position z, k is the wave number 2π/λ,

Ψ(z) is the Gouy phase, R(z) is the radius of curvature of the beam at position z and Ψ0

is the field at the origin. This fundamental transversal mode is also called T EM00.

In the case of high order modes, an Hermite-Gaussian or Laguerre-Gaussian model can be used assuming circular symmetry. Using high order modes allows to model any transversal form of the beam, but every higher order mode used requires more computational power to simulate the propagation of the beam inside the interferometer. A point is reached when the number of high order modes required to increment the precision of the results cannot be computationally afforded.

Virgo

The Virgo detector is located in Cascina, Italy and based on the elements defined until now, it consist on a doubled recycled Michelson with resonant arms of 3 Km. Currently, it is operating as Advanced Virgo (AdV), a version which has been designed to achieve 10 times better sensitivity than the original Virgo detector. During the first science run it reached an horizon for BNS of 27Mpc

The scheme of AdV can be seen on Fig. 2.5. The light, before entering on the main interferometer, is filtered by the input mode cleaner (IMC) which consist off a triangular cavity with a length of 144 m. Then the light enters the main interferometer through the power recycling mirror (PRM) and, after the beam splitter (BS), it goes to both arms formed by the FP cavities in the West and North directions. The light going to the dark port passes trough the signal recycling mirror (SRM) before arriving to the detection bench. Currently the SRM have an antireflective coating and only works as a convergent lens, but in the future it will be upgraded.

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Figure 2.5: Scheme of Advanced Virgo.

2.3 How to control the interferometer

Building an interferometer as explained in Sec. 2.2, with so many cavities that must be controlled with precisions below the wavelength, is a hard task. In order to operate the interferometer, it is required to use advanced techniques to control the length of the cavities and the alignment of the mirrors.

Control Signals

In order to be able to control the optical cavities we must be able to measure the phase of the field leaving through the cavities. There are some techniques to sense the state of the interferometer, all of them based on modulating the field with certain frequencies. Then the field will have the frequency of the carrier (laser frequency) and there will be sidebands specifically chosen to control the interferometer. There are two main techniques used to extract phase information from optical cavities, the so called Heterodyne detection and the Pound-Drever-Hall technique.

The spectrum of the sidebands is in the radio frequency, typically the frequencies are between a few MHz to about 112 MHz. The frequencies are chosen in order to be resonant in the cavities that they control.

Then, following a demodulation process, it is possible to determine which is the phase of the output field. The demodulation process, in the simplest case, which corresponds for heterodyne detection, consist in dividing the signal into two components and multiplying

the output of one of them by sin(2πfM dt) and the other by cos(2πfM dt), being fM d the

modulation frequency. This will produce the so called in phase and in quadrature signal which corresponds to

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Chapter 2. How to detect Gravitational Waves 2.3. ISC

Figure 2.6: Scheme of the detection benches and convention for the beams of AdVirgo.

p(t) = P (t) cos(2πfM dt) = (A2+ B2) cos(2πfM dt) + AB[cos(4πfM dt) + cos φ]

(2.10a)

q(t) = P (t) cos(2πfM dt + π/2) = (A2+ B2) sin(2πfM dt) + AB[sin(4πfM dt) − sin φ],

(2.10b)

where A refers to the amplitude of the field leaking from the optical cavity being analysed

and B refers to the amplitude of the modulating field. Since fM d is slowly changing, it is

possible to determine which one is the phase φ of the leaking field.

Controlling Virgo

In the case of Advanced Virgo, many optical benches have been build to read control signals at different points of the interferometer as can be seen on Fig. 2.6. Those optical benches are build to control the different cavities of the interferometer. In order to read the signal at each detection bench, the carrier is modulated with frequencies of 6, 8, and 56 MHz. Those frequencies are resonant in different cavities of the interferometer and give information about the status of the different degrees of freedom of the interferometer. The signals are related to the degrees of freedom present in Virgo that are called:

DARM: Differential arm length. It is the difference in length between the two FP cavities, and is the channel used to measure the GW signal.

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in-terferometer composed by the input mirrors of the FP cavities.

CARM: Average arm length of the two FP cavities, or common node.

PRCL: Power recycling cavity length. It is the length of the cavity formed by the PR mirror and the input mirrors,

SRCL: Signal recycling cavity length. It is the length of the cavity formed by the signal recycling mirror and the input mirrors.

Feedback control loops

Finally, once a signal warns that the interferometer is going out of the desired configura-tion, it is necessary to create a control system that sends a signal to modify the interfer-ometer state. This process is done via feedback control systems in which the input signal R(s) is subtracted from a reference signal giving the so called error signal that is used to actuate in the system and control it. Those systems are considered to be linear systems. In the simplest way the are single-input-single-output (SISO). The linear system results in

Y (s) = H(s)R(s) (2.11)

where H(s) is the transfer function of the system and Y (s) is the output signal.

2.4 Sources of noise

There are many sources of noise that will limit the sensitivity of an interferometric detector. Those sources are mainly divided in fundamental sources and technical sources. The first ones are called fundamentals because they come from physical processes that can only be reduced and never completely eliminated. Fundamental sources of noise are the ones that define the design sensitivity curve of the detector. The second type of noises, as the name says, arise from engineering problems that cannot be directly related with a single physical principle, and they are characterized and suppressed during commissioning periods. Noises must be described in frequency domain since they change in time. For this reason it is analysed how the spectral density of each source of noise contributes to the sensitivity h(ν). To compute the contribution to the sensitivity it is necessary to first compute the power spectral density as [59, 79]

P SDh(ν) = lim

T →+∞ 1

T | F T (h(t)) |

2 _(2.12)

where F T (h(t)) refers to the Fourier transform of the noise, averaged over long periods of time. In order to obtain the final contribution to the noise budget we compute the amplitude spectral density as

h(ν) =p2P SDh(ν) (2.13)

When the sources of noise are defined, it is possible to compute the sensitivity curve which shows which is the minimum GW strain that is expected to be detected at design

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Chapter 2. How to detect Gravitational Waves 2.4. Sources of noise

(a)

(b)

Figure 2.7: Reference sensitivity curve for Virgo(a) [55] and Advanced Virgo(b) [26]. Both plots include the individual contribution of the main sources of noise. In the case of AdVirgo, the final reference sensitivity for the first science run can be seen on Fig. 2.8a

sensitivity. The design of Advanced Virgo had the objective to improve the sensitivity curve by a factor 10 from initial Virgo as can be seen on Fig. 2.7.

2.4.1 Fundamental noise sources Seismic noise

Seismic noise is generated in the range from mHz to tens of Hz and is a dominant

contri-bution in the low frequency region. It follows a shape of f−2 and the amplitude at 1 Hz

is around 1 µm; at 10Hz1 the amplitude is still many orders of magnitude larger than

distances to be measured. For this reason, it is required to suspend the mirrors within a superattenuator which consists of an inverted pendulum that is able to attenuate the ground noise for frequencies higher than the resonance frequency of the supperattenuator. Processes that generate seismic noise come from many geophysical phenomena such as wind, earthquakes and sea storms. However, is can also be generated by human activity, like road traffic (in fact, a road passes close to Virgo’s West arm).

Another fundamental noise closely related with seismic noise is Newtonian noise [38]. It is expected that at some point it will be possible to detect changes in the local gravitational field, such as the movements in the ground due to an earthquake, but also the movement of an object close to one of the detector mirrors. Currently, Newtonian noise is not a limiting source of noise for the sensitivity, but it is expected that it might be detected once the current generation of detectors reaches the maximum sensitivity and, in these circumstances it will be an important source of noise for the third generation.

1_{Even if the detection band of an Earth GWs detector starts at a few tens of Hz, seismic noise can} spoil the sensitivity at higher frequencies due to the upconversion mechanism explained within the technical sources of noise (Sec. 2.4.2), and then it is also important to attenuate seismic noise at frequencies lower than the minimum frequency that can be detected.

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Quantum noise: Shot and Radiation pressure noise

Quantum noise is a fundamental source of noise that is generated by the quantum fluctu-ations of the laser. Those fluctufluctu-ations are of two types, each one is dominant at different frequencies. The two types are radiation pressure and shot noise.

Shot noise arises from the fact that the photons produced by the laser are generated

follow-ing a Poison distribution. From this, the mean number of photons is known, ¯N =p2hpνP

and also it is possible to estimate the fluctuations with the standard deviation, √

¯

N . Then, the spectral density associated with the shot noise reads

h(ν) = π 2F L r ~cλ 4πP s 1 + ν νc (2.14)

where νc is the pole frequency which approximately corresponds to 60 Hz for Advanced

Virgo. In this formula, it is possible to see that there are two different regimes. Those regimes are defined by the time in which the light is stored inside a FP cavity; in the case of a cavity like the ones in Virgo, the storage time is of the order of µs. Then, at high frequencies, the number of photons measured is reduced and the measurement becomes noisier.

Thus, it can be seen that by increasing the power of the laser, it is possible to reduce the shot noise. However, due to the radiation pressure, there is a force applied on the mirrors

related to the power circulating inside the cavities. This force produces a pendulum

movement in the mirrors. The radiation pressure will produce a noise in the system that can be expressed as h(ν) = 1 mν2_L r ~Pin 2π3_cλ (2.15)

Radiation pressure and shot noise are complementary variables in a quantum sense, so the reduction of the uncertainty in the measure of one parameter is inherently related with the increase of the uncertainty of the complementary parameter from the Heissenberg uncertainty principle. This is the fundamental idea behind using the injection of squeezed light into the system to reduce the sensitivity below the standard quantum limit (SQL) [51]. Squeezing was first implemented in GEO600 and will be implemented in the future upgrades of Virgo.

Thermal noise

Since all components (either mechanical or optical) are working at room temperature, there is a a source of noise generated by thermal fluctuations. The thermal noise can be computed using the fluctuation-dissipation theorem [48], as explained below.

In a system with a temperature different from zero, the atoms are vibrating. The

dis-placement of the atoms, ˜x(ω), due to thermal effects can be modelled as if there is the

influence of an external random force ˜F (ω). The relation between displacement and force

comes as

˜

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Chapter 2. How to detect Gravitational Waves 2.4. Sources of noise

where Z(ω) is called the impedance of a system.

The fluctuation-dissipation theorem states that the spectrum density of the force

respon-sible for thermal fluctuations, ˜S_FT hermal, is related to the real part of the impedance as

˜

S_FT hermal(ω) = 4kBT Re(Z(ω)) (2.17)

where the kB is the Boltzmann constant and T the temperature of the system.

Considering that the thermal energy is dissipated as damped oscillators, the impedance of the system is the defined as

Z(ω) = −im0

ω (ω

2_{− ω}2

0+ iω2φ(ω)) (2.18)

where φ(ω) is called the loss angle and depends on material properties.

Then, the displacement noise generated by the thermal noise can be found using the spec-tral density of the equation (2.16), which can be solved using the fluctuation dissipation theorem, explained in equation (2.17). Finally, adding the impedance defined by equation

(2.18), the displacement noise (in [m/√Hz]) results

X(ω) = v u u t 4kBT φ(ω) mω3h_{(1 − (ω} 0/ω)2)2+ φ2(ω) i (2.19)

The previous equation shows that thermal noise depends on the temperature and the material, giving two strategies to control the thermal noise. The first one is cooling the system, and the second one is selecting materials which dissipate faster the fluctuations of temperature.

Thermal noise is classified in two categories, depending on the location of the source. The two categories are the Suspension thermal noise and the Test-mass thermal noise which refer to the origin of the noise. Suspension thermal noise contains the contribution from the horizontal and vertical fluctuations and the suspension normal modes of the wires, being the first one called the pendulum modes and the last one the violin modes. The test-mass thermal noise contains the Brownian motion of the mirrors, the thermo-elastic fluctuations of the temperature and the termo-refractive fluctuations of the index of re-fraction.

In order to reduce thermal noise, it is expected that future generation of GWs interferom-eters will be cryogenic, being Kagra in Japan the first cryogenic interferometer that it is expected to operate in 2018, and followed by the 3rd generation of GWs detectors which, will include The Einstein Telescope [63].

2.4.2 Technical noise sources

Technical noises are the sources that generate the difference between the theoretical sen-sitivity curve designed with the fundamental sources of noise and the real one when the interferometer is running. They do not appear in the design sensitivity curve, because the expected sources of technical noise are designed to generate noise 10 times lower than the

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expected sensitivity. For this reason, before the interferometer starts to operate in science mode, during commissioning there is a period of noise hunting in which those sources are searched and suppressed as can be seen on Fig. 2.8 for Advanced Virgo and advanced LIGO.

There are many sources of technical noise. Sensing and actuation refers to all the noise generated in the sensing components or introduced in the control loops. Dark noise is the noise generated due to the dark current of the photodiodes. Digital noise is the noise introduced by all the digital electronics including the noise generated due to the transfor-mation from the analogical to digital signal. Phase noise refers to the difference in the phase between the carrier and the modulated frequencies. Scattered light (or stray light) is the the source of technical noise that in which the current thesis focuses so it is explained in the next section.

2.5 Stray light

Stray light (SL) in a system is all the light that rather than following the expected path, follows a different one after being dispersed. In a different way, it is all the light that is not expected to be found on the design. A simple example of stray light can be seen on Fig. 2.9 where the light can be seen from a camera installed in the laser room. It has been found to be an important issue spoiling the sensitivity on gravitational wave interferometers since early times [17].

Since the interferometer is working close to the dark fringe, meaning that few milliwatts of light reach the detection bench, and none of the external detection benches will be receiving more than few Watts of light between all of them, this means that almost all the light injected by the laser system will be lost inside the interferometer as can be seen on Fig. 2.11.

Then we can conclude that stray light is a main issue that needs to be addressed since it might be able to interfere with the main beam at the photodiode level modifying the detected intensity.

2.5.1 How Stray Light is Generated

There are many ways in which stray light can be generated, and depending on the process, the necessary actions to mitigate its effects will be different. The different mechanisms gen-erating stray light, these are secondary beams, scattering due to rough surfaces, scattering from point defects and diffraction.

Secondary beams

Secondary beams are generated by anti-reflective coatings (AR). After a beam hits an AR surface, a small fraction of the light is reflected. This means that the light reflected will continue inside the main beam, but it will suffer a different phase change with respect to the light inside the main beam.

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Chapter 2. How to detect Gravitational Waves 2.5. Stray light

(a) Evolution of the Sensitivity of AdVirgo during early commissioning. It is possible to see a considerable reduction on the noise levels between the 3rd of March and the 5th of March in which the noise generated by MICH was reduced due to the application of a new filter[15].

(b) Ligo Livingstone sensitivity changes over time until before final sensitivity was reached for O1 data run. As can be seen obtaining the expected sensitivity on a gravitational wave detector is a slow work in order to supress all the sources of noise. [16]

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Figure 2.9: Image from an IR camera observing the laser bench of VIRGO. The camera can capture an image of the room because the stray light from the laser system illuminates the room, if it was completely suppressed, the image would be completely dark.

Figure 2.10: Where is the light loss? Scheme of AdVirgo showing how much light is expected to be lost due to stray light issues once the laser input laser is upgraded to 125W .

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the reflected beam from the main beam. In the case of the beam splitter and the Pick-off-plate (POP), the separation of the secondary beam is used to control the interferometer. This can be an important issue for the sensitivity of a GWs detector and, in case that one secondary beam is predicted to recombine with the main beam, it must be dumped with baffles. If not, the sensitivity can be compromised, as was found during AdVirgo commissioning where the sensitivity for a wide range of frequencies was limited by a ghost beam of the POP, the identification and noise contribution is described in chapter 6.

Diffraction

As explained in Sec. 2.2the actual light from the laser is described by a Gaussian beam whose profile has an infinite surface, with the intensity decreasing with the radius as

I(r) = 2P

πω2e −2r

2

ω2 _(2.20)

where P is the power in the beam, ω is the waist diameter of the beam and r is the distance from the optical axis.

The intensity profile of the beam is very important because the optical elements have a finite size and this means that always some light is lost due to the clipping of the beam by the aperture of the optics. In order to estimate the appropriate aperture, it is known that with an aperture of r = σ, there is a transmission of the 86% of the light, and with r = πσ/2 a transmission of 99% is achieved.

But the problem comes from the fact that diffraction effects cannot be ignored and, for an aperture r = πσ/2 (T = 99%), the ripples generated by diffraction represent about 17% of the light power. For this reason it is important to consider larger apertures for the optics in order to avoid interferences processes that change the phase of the main beam. A safe aperture for the optics is considered to be five times the diameter of the beam.

Scattering due to rough surfaces

In the case of the mirrors, it is known that their optical surfaces are not perfectly flat. If we consider that the surface is smooth, i.e. that the changes in the surface are smaller than a wavelength (A λ), clean and reflective, it is possible to model the surface as a diffraction sinusoidal grating with a single frequency. Thus the light will be scattered off following

sin θn= nλfg (2.21)

where θn is the angle for the nth order of diffraction with respect to the normal of the

surface, fg is the spatial frequency of the diffraction grating and λ is the wavelength.

Thus, by modelling a surface as a diffraction grating with a set of sinusoidal surfaces, it is possible to consider the power spectral density (PSD) of the spatial wavelengths found in the surface profile. One property of the PSD for surface profiles is the relation with its root mean square (RMS or σ). RMS is the power of the PSD, being the PSD of a mirror

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Figure 2.11: The light is scattered off a mirror in all directions. In the FP cavity, the differ-ence between narrow and wide angle scattering is important since each type of scattering might recouple in a different way [34].

surface S(f ), the relation between PSD and RMS results

σ = 2

Z fmax

fmin

S(f )df (2.22)

where the factor 2 is applied because the PSD obtained from a surface map is real and

symmetric. If the PSD of the mirror is obtained from a surface scan, the values of fminand

fmaxdepend on the mirror diameter and the maximum resolution of the scan, respectively.

The RMS is important because it is closely related with the total integrated scatter (TIS), which is the total amount of scattered light normalized by the light specularly reflected. In the smooth approximation and for normal incidence, we have that

T IS ≈ Ps

RPin

= (4πσ/λ)2 (2.23)

In the case of Advanced Virgo, the mirrors from the main optics are superpolished and their RMS is specified to be lower than 0.5 nm giving a TIS of a few ppm [20, 21].

Scattered light can be classified in two different types, depending on the angle at which it is generated. The first type is narrow angle scattering and the second one wide angle scattering. The angle that separates the two regimes is considered to be around 0.1 radians, but depending on the situation, this angle can be even lower [34]

The main difference comes from the fact that narrow angle scattering can be analysed with wave optics models. Usually, the scattered light is expressed in the generation of high order modes from the Gaussian mode TEM00. Using wave optics, it is possible to obtain all the phase information, essential for interferometry, but it is necessary to generate high order modes of orders that are not possible to compute just by diverting small angles from the main beam. For this reason, it is not possible to compute high order modes of an order high enough to model wide angle scattering with wave optics.

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simulating the light as a particle. This will be presented in the second part of the thesis using a light ray approximation.

Scattering from point defects

Scattering due to point defects creates a diffracted wave. The defects can be a bubble in the mirror substrate or scratches in the coating, digs or dust contamination in the surface. In the case of bubbles in the substrate, this can be seen before the mirror is polished, so with the selection of the right substrate before polishing this issue can be solved. Other sources of noise present in the substrate can be local changes in the index of refraction or the presence of strain generated inside the substrate. In order to avoid these problems, the vendors apply annealing processes to the glass to reduce substrate imperfections. For surface defects, inspection for scratches and digs is done visually, so no visible imper-fection is accepted on their surface. To avoid dust on the mirror surface, the process of cleaning and installing the mirrors is very important since the stray light generated from dust on a mirror can be very intense compared with the light dispersed by the roughness of the mirror. In the past, a mirror has already been dismounted due to dust on the surface [76].

2.5.2 Stray Light recoupling

There are two similar mechanisms describing how stray light can recouple with the main beam on a surface . The first one is back-scattering in which the incident light in lenses, mirrors, photodiodes or beam dumps directly recouples into the main beam [22, 71, 6]. In this case we can also include back-reflection from an AR coating. The second case is when the scattered light is scattered off the main beam and then, through an external element, it is redirected again to the main beam [49, 34, 31]. Then, once the light is recoupled, it will introduce a phase noise in the main beam.

In the scattering process we can consider that a fraction fsc of the incident beam for any

optical element will be scattered and will recombine with the main beam. The fraction fsc

is defined in a similar way to T IS, as the ratio between scattered power and input power

fsc = Psc Pin

(2.24)

In the case of the back-scattered field, the amplitude is proportional to√fsc and its phase

has been changed by

φsc= 4π

λ(x0+ δxopt(t)) = φ0+ δφsc (2.25)

where x0 is the static optical path giving the static phase φ0, and δxoptis the displacement

of the scattering element. Since the elements are vibrating due to ground movement or the presence of vibrating devices, we have that the scattering element moves as

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In case that light has been scattered by more than one element, each element is considered to contribute equally, being the total modulation the sum of all the contributions, each

one with its own frequency fx, amplitude Ax and phase φx as shown in equation (4.4).

The phase noise generated due to recoupled light for the GW signal in the dark port is

hsc = K

p

fscsin φsc (2.27)

Where K is the coupling factor between the scattering source and the dark fringe. Coupling factors measured for Virgo+ and specified parameters for Advanced Virgo are given in [22].

The coupling factor can be simplified to include the scattering fraction as G = K√fsc.

The up-conversion case

In the case that the displacement of the scattering elements is small, it is possible to approximate equation (2.27) to the first order

hsc = Gφsc (2.28)

but for a large displacement, which is considered δxopt > λ/4π, the coupling is highly

non-linear, and the approximation does not stand. In this case, a displacement at a

low frequency will generate harmonics at higher frequencies. This happens because,

for a large displacement, the harmonics will be of a magnitude similar to the main fre-quency. This phenomena is called up-conversion shoulder and can be seen by expanding

the sin(Axsin(2πfxt)) as sin(aψ) using the Jacobi-Anger identity as

sin(a sin(ψ)) = 2 ∞ X n=1

J2n−1(a) sin[(2n − 1)ψ] (2.29)

where the higher harmonics have an amplitude proportional to the Bessel function of order 2n − 1. For a 1, the Bessel functions of order higher than 1, have a value close to zero. When a ≥ 1, the orders higher than 1 have an amplitude closer to the lowest order. This amplitude of the harmonics is similar to the amplitude of the fundamental frequency component responsible for the up-conversion shoulder.

The up-conversion phenomena is almost flat up to fmaxwhich can be obtained considering

how many times the moving object with its movement changes the phase of light by 2π.

If the optics moves in a sinusoidal way with a frequency fx and amplitude Ax, then the

maximum frequency is

fmax=

2Ax

λ 2πfsc (2.30)

Up-conversions in Virgo are investigated during noise injections (see chapter 6) where, at sensitive frequencies, the harmonics are also excited. One example was during a noise injection to identify possible sources of noise for the several lines present during AdVirgo commissioning, as can be seen in Fig. 2.12.

Upconversion mechanisms are very important because even though the seismic noise is generated below 10Hz (see Sec. 2.4.1), outside the detection band, its amplitude is of the

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Figure 2.12: Longitudinal injections in the SDB2 bench performed during Advanced Virgo commissioning to search for back-scattering. Three different injections were performed with different pairs of amplitudes and frequencies of (600, 0.06 ), (600, 0.12 ) and (900, 0.12 ) in µm and Hz for hj1, hj2 and hj3, respectively. In this case it was also observed that, based on equation (2.30), the effective amplitude was different from the injected one. One hypothesis was the fact that the actual movement of one of the scattering elements was different from the sensor measuring the movement or that it might be dependent on the difference between SDB2 and SDB1, as can be seen in [40].

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Figure 2.13: Expected noise on AdVirgo during periods of intense sea activity. As can be seen, the upconversion is expected to be more important on the west end bench (WEB) during the periods of intense sea activity. This intense sea activity occurs approximately the 3% of the time and during this period might be a dead time for adVirgo [22]. A similiar excess noise was seen the 11 of August 2017 before the wind and intense sea activity unlocked the interferometer and could not be recovered for day[14].

order of several wavelengths and thus spoils the sensitivity band (see Fig. 2.13). On the 11 of August 2017 during the last science run, the high seismic noise produced due to sea activity and wind caused the interferometer to be unlocked during almost all the day [14], although it was not only a stray light problem, before the interferometer was unlocked, it is possible to see high upconversion shoulders in the sensitivity. This case is mainly studied in Sec. 5.3.

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Part II

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Chapter 3

Ray Tracing

The idea of ray tracing comes from the particle nature of light in which photons are modelled as point particles that follow straight paths between each interaction with the media. But currently, ray tracing is known more as an application for computer science rather than physics, because the first developments were related to how a computer could render an image. More exactly, the first works to apply ray tracing to rendering were published by Arthur Appel in 1968 [10] and Robert Goldsteins and Roger Nagel in 1971 [37], although the true breakthrough came with the work of Turner Whitted [80], who developed the first algorithm able to render an image tracing rays based in the lights that were illuminating the scene. This development was used to create the first computed generated video called The compleat Angler [81].

Realistic computer rendering of images is and has been for a long time an important topic in fields such as the film industry, where substantial investments were made to further develop such software. However, current rendering codes are actually based on the mathematical expressions describing the physical interaction between light and matter at the macroscopic level. The state of the art on ray tracing for computer graphics techniques can be found in [54].

This chapter presents the basic concepts of ray tracing that must be present in any ray tracing code. The chapter has been organized following the order in which the different modules of a ray tracing code are executed. The developed software has been coded in F#, a functional language that can be either compiled or ran as a script. The resulting code can be found in https://github.com/josesoyo/Ray-Tracing.

3.1 Basic Ray Tracing

In the most fundamental classification, ray tracing codes are classified in two categories, those which trace the ray from the sensor (camera) to the source of light and the second category is the tracers that trace the ray from the source of light to the sensor. The first one is called Backward ray tracing and the second one Forward ray tracing. Backward ray tracing is more often used since less rays are needed to generate an image, but at the same time, it cannot be used for stray light analysis. For this reason, the required ray tracing technique to analyse stray light is Forward ray tracing. In any case,

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both techniques use the same core algorithms and thus almost all the techniques described here can be applied to both cases and the same code contains a module able to render a simple image, as can be seen in Fig. 3.2.

Another classification for ray tracing is the difference between sequential ray tracing and non-sequential ray tracing. Sequential ray tracing is used for simple cases in which the ray is traced from the source to the target as a single ray. On the other hand, non-sequential ray tracing allows to split one ray into multiple rays, producing a set of rays that are traced from the source to the target or until a different termination condition is fulfilled.

3.1.1 Rendering Equation or Light Transport Equation

The rendering equation is the fundamental equation for ray tracing [43] . It expresses that the incident radiance on a surface is the same as the emitted radiance (if there is no absorption or emission). For this reason, the rendering equation is also called the Light Transport Equation (LTE). Conservation of the radiance is equivalent to conservation of energy and is expressed as

Li(p, ωi) = Lr(p, ωi) + Z

S

f (p, ωi, ωs)L(p, ωi)| cos(θj)|dωj (3.1)

where on the left-hand side of the equation there is the incident radiance and on the right-hand side there is the sum of the reflected radiance plus the integration along all the

solid angle of the scattered radiance. The term ωi refers to the direction of the incident

light, ωj to the direction of the scattered light which is integrated along all the solid angle

and f (p, ωi, ωs) is the bidirectional scattering distribution function (BRDF), which are

explained in Sec. 3.4.1. Finally, θj is the angle between the light output light and the

normal of the surface.

It is important to mention that the equation is based on the average transport of energy, wave optics effects based on the phase not being considered. Even though the equation assumes that the light does not enter into the material, it can be extended in a trivial way to include transmission, absorption and emission.

3.1.2 Basic algorithm

Ray tracing, as a field, presents many issues related with different areas of science as are mathematics, computer science and physics, that will be treated in the next sections. However, the basic algorithm of ray tracing can be simplified in a few simple building blocks (see Fig. 3.1).

First, it is necessary to preprocess the data in order to create the system, and then a set of rays, that are traced, are generated. Then, the most intensive process comes, in which rays are tested to intersect with all the elements present in the system. Once the closest intersection has been determined, the interaction between the ray and the intersected element is performed and new rays are generated after it interacts with matter. The process continues until each ray is terminated, either because it is extinct or it reaches a

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Chapter 3. Ray Tracing 3.2. Preprocessing

Figure 3.1: Basic algorithm for ray tracing.

sensing surface.

The process of ray tracing is a Markov chain since it does not have memory.

3.2 Preprocessing

Before performing a ray tracing simulation, it is necessary to set up all the elements needed in order to perform the simulation. This is very important because the definition of the elements will be the interface with the functions used to intersect rays. For this reason, its definition can modify significantly the code’s performance and intermediate structures can be created in order to optimize the code (Sec. 3.2.2). Finally, it is necessary to define the structure representing the ray of light and how it is generated within the system (Sec. 3.2.3).

3.2.1 Define a world

Even though one might think that creating a system using basic geometrical bodies as spheres, boxes or cylinders can be too simplistic for simulating a real system, joining many simple geometrical shapes can be used to create a complex geometrical structure. For example, a lens with its mount can be formed using a set of partial spheres and then, cylinders to form the optical mount.

Thus, the set of objects defined in a ray tracing simulation represents the ’scene’ and each different object contains a set of properties that are the physical properties of the material, which are further explained in Sec. 3.4.

Noise from stray light in interferometric gravitational wavedetectors

Graduate course in physics

University of Pisa

Phd Thesis:

NOISE FROM STRAY LIGHT IN INTERFEROMETRIC

GRAVITATIONAL WAVE DETECTORS

Candidate:

Jose M. Gonzalez Castro

Supervisor:

Prof. Francesco Fidecaro

June 19, 2018

Contents

Acknowledgement

Abstract

Part I

Chapter 1

Gravitational waves from General

relativity

1.1

General relativity

1.2

Gravitational Waves

Chapter 2

How to detect Gravitational

Waves

2.1

Detection methods

2.2

Interferometric detectors, VIRGO

2.3

How to control the interferometer

2.4

Sources of noise

2.5

Stray light

Part II

Chapter 3

Ray Tracing

3.1

Basic Ray Tracing

3.2

Preprocessing