Characteristics and Limitations of the Fiber Interferometer Readout for an Accelerometer

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Characteristics and Limitations of the Fiber

Interferometer Readout for an Accelerometer

Bachelor Thesis 02/09/2013 - 30/10/2013 by Dorine Schenk Student number: 6116760 under supervision of Ir. J.V. van Heijningen

Dr. A. Bertolini

Abstract

A sensor to measure the vibrations of quadrupole magnets in the future Compact Linear Collider (CLIC), with a sensitivity better than 1 pm/√Hz and a bandwidth from 5 Hz to 100 Hz, is built at Nikhef. There is no off-the-shelf sensor capable of detecting vibrations at this noise level in a high radiation, high magnetic field environment such as the designed CLIC. In the Nikhef design, the motion of the proof mass of the sensor is read by a fiber coupled Michelson interferometer. This way, the sensitive parts of the system, i.e. laser source and optical readout electronics, can be located outside the collider area. This report describes the measured characteristics and limitations of the new fiber interferometer used as the readout of the designed accelerometer. The measurements show that the bandwidth is 5 -475 Hz and the sensitivity is about 10−11m/√Hz. The expected sensitivity is about 50 fm/√Hz. Changes in the birefringence of the coupling fibers, caused by thermal fluctuations and mechanical vibrations result in fluctuations in the intensity of interferometer output signal and in a reduced sensitivity. The solution is proposed to reduce the interferometer arm length as much as possible. This should result in an improved sensitivity.

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Samenvatting

In de natuurkunde bestaan nog veel onbeantwoorde vragen. Zoals wat donkere materie is en of supersymmetrie (theorie waarin alle elementaire deeltjes een, veel zwaardere, superpartner hebben) bestaat. Theoretische modellen die hier voorspelling over doen, worden getest met behulp van hoog energetische deeltjesversnellers, zoals de LHC. Hier worden protonen met hoge energie¨en op elkaar gebotst. De modellen doen voorspellingen over deeltjes die dan ontstaan. Inmiddels is gebleken dat de LHC goede resultaten levert, zoals de bevestiging van het theoretische model dat er een Higgsveld met bijbe-horend Higgsdeeltje moet bestaan. Als ook elektronen en hun antideeltjes (positronen) op elkaar geschoten worden, zijn er meer en beter gedfinieerde begin energie¨en mogelijk. Hiermee kunnen modellen verder getest worden en wellicht worden er nieuwe deeltjes gevonden. Om elektronen en positronen te laten botsen is er een nieuwe deeltjesversneller nodig. Een voorstel van CERN is de Compact Linear Collider (CLIC).

Om het gewenste aantal botsingen in een bepaalde tijd teweeg te brengen moeten de bundels nauwkeurig gefocust worden. Om de bundels gefocust te houden tijdens het versnellen, wordt er gebruik gemaakt van magneten die een magnetisch veld veroorzaken dat de elektrisch geladen bundel op de juiste positie houdt. De schaal waarop de botsingen plaats vinden is zo klein, dat trillingen van de aarde de positie van de magneten kan verstoren. Het is dus belangrijk de magneten zo trillingsvrij mogelijk te houden. Om dit te verwezenlijken zijn trillingssensoren nodig die de vibraties van de magneten meten. Momenteel is er geen sensor op de markt die voldoet aan de gevoe-ligheidseisen van <pm en een bandbreedte van 5 Hz tot 100 Hz in combinatie met de mogelijkheid in hoge magnetische velden te opereren. Daarom wordt er op het Nikhef een trillingssensor met glasvezel interferometrische readout gebouwd. Dit betekent dat de trillingen worden waargenomen met een inter-ferometer in glasvezel; het licht in de glasvezel ondervindt geen effecten van hoge magnetische velden. Enkel de glasvezel bevindt zich dicht bij de mag-neten, de readout van het signaal wat daar gemeten wordt, gebeurt buiten de versneller, waar elektronica wel functioneert.

Het doel van dit project is om de kenmerken en begrenzingen van de hiervoor ontworpen interferometer in kaart te brengen. Op deze manier kan bepaald worden welke verbeteringen er doorgevoerd moeten worden om aan de eisen van de trillingssensor voor CLiC te voldoen. De interferometer blijkt nog veel last te hebben van temperatuur fluctuaties en mechanische trillingen. Deze effecten kunnen waarschijnlijk verminderd worden als de lengte van de gebruikte glasvezel ingekort wordt.

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1. Introduction

This report describes the bachelor project for the bachelor Physics at the University of Amsterdam. The project was carried out at Nikhef, with the Virgo research team. Among various other activities in the field of gravita-tional wave detection, this group is developing advanced seismic sensors. The proposed new high energy (7 TeV) electron-positron collider, called the CLIC (Compact Linear Collider) could only operate when the mechanical vibra-tions of the beam focusing quadrupole magnets will be kept at the nanometer level.

For this reason, active vibration isolation systems, making use of seis-mic sensors and piezoelectric actuators, are designed for each of the 3600 quadrupole magnets in the linear accelerator (linac) of CLIC and for the final focus magnets at the interaction point. Very low noise seismic sensors, with a sensitivity better than 1 pm/√Hz and over 5-100 Hz bandwidth are needed. The sensor also needs to be tolerant to large temperature drifts; the temperature in the tunnel can vary with as much as 8◦C. There is no seismic sensor on the market yet, which is capable of detecting vibrations at this sensitivity in the high magnetic fields and high radiation of the collider area of CLIC. Theretofore a new accelerometer, with fiber coupled interferomet-ric readout, is being built at Nikhef. The conceptual scheme of the sensor is shown in Figure 1.1.

Figure 1.1: Accelerometer with the suspended mass and fibers (in red) con-nected to the interferometric readout

To test the abilities of the interferometer, without being attached to the

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CHAPTER 1. INTRODUCTION 4

suspended mass, a setup is used in which the movements of the suspended mass of the accelerometer are mimicked under controlled circumstances. In chapter 4 this setup will be described. In chapter 5, the characteristics and limitations, like polarization effects in the fibers and the conversion factor of the interferometer are obtained. In chapter 5.3, the bandwidth, fringe visibil-ity, noise and expected sensitivity are determined. This information is needed to get a complete understanding of the interferometer. Then the noises and limitations can be minimized to optimize the interferometer. When this phase is completed the next step is to combine the optical readout with the accelerometer mechanics.

The goal of the project is to determine the characteristics and limitations of the interferometer setup. This is needed to improve the interferometer to perform according to the requirements needed for CLiC.

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2. Background

2.1 CLIC

On the fourth of July 2012, the Large Hadron Collider (LHC) discovered a previously unobserved particle with a mass of 125.6 GeV. In November 2012, this was beginning to look more and more like the theorized Higgs boson. At that time, the LHC, which collides two beams of hadrons, was operating at 4 TeV per hadron-beam [5]. This is the highest energy at which man-made particles colliders have been working. High energy physics is a field in which theory models about the most elementary objects, like the Standard Model and Supersymmetry, are tested.

At the moment, another high energy collider, the CLIC, is being designed. This is a study for a future electron-positron collider which works with a linear accelerator or linac (see Figure 2.1).

Figure 2.1: Schematic linear accelerator with the drift tubes increasing in length to accelerate the electrons. The black rectangles are the locations of the focusing magnets.

Conceptually, a linac accelerates charged particles in a linear beamline by subjecting them to a series of alternating positive and negative charged conductors. As soon as the charged particle is halfway through the first conductor, the charge on the conductors will change, resulting in repulsion instead of attraction in the first conductor. Because the velocity of the particles increases and the conductors are attached to alternating current, with a fixed frequency, they start with short segments near the source of the particles and end with long segments near the target (see Figure 2.1), to maintain the change in charge halfway through the segment. The particles

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CHAPTER 2. BACKGROUND 6

get accelerated because the electric field of the conductor behind them push them away and the field of conductor ahead of them pull them forwards. In a collider, particles accelerated by two linacs are made to collide. In a linac like CLIC, the accelerator gaps are replaced by resonant cavities in which a 12 GHz alternating electric field can build up several MV/m, as shown in Figure 2.2. This field is a lot higher than what conductors in other accelerators provide. A cavity resonator is a hollow conductor, blocked at both ends. When a wave that is resonant with the cavity enters, it is will bounce back and forth in the cavity, without much loss. As more energy enters the cavity and will start to resonate as well, the intensity increases, resulting in high electric fields.

Figure 2.2: The distribution of electric fields in an 11.4 GHz prototype res-onant cavity.

The plan is to start CLIC at centre-of-mass energies of 500 GeV and ungraded it to maximum energy of 3 TeV. Its energy range is lower than that of the LHC, which had a record energy of 8 TeV and will go up to 14 TeV in the next few years. However, CLIC will collide electrons (e−) and positrons (e+_{) instead of protons. Since electrons are elementary particles and protons}

consist of two up quarks, one down quark and gluons, the e+e− collisions are cleaner. Furthermore, the initial states are more well defined, considering the only two particles colliding are electrons and positrons, instead of a mix of up quarks, down quarks and gluons; these cleaner collisions also result in better tuning of the particle energies. Since protons consist of three quarks and gluons, the energy of each individual quark/gluon is not the same as the energy of the beam. For elementary electrons, on the other hand, the particles energy is the beam energy.

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At lower energies, up to 100 GeV, the e+e−collisions in the former Large ElectronPositron Collider (LEP) proved to be an essential complement to hardon-hadron collisions. They helped establishing the validity of the Stan-dard Model, by precise measurements of cross-sections, thanks to their fine tuning ability. Therefore, it is expected that higher energy hadron-hadron collisions will not be sufficient to show a complete picture of the high energy physics and the e+e− colliders, like CLiC, will be needed to help unravel the TeV physics [2].

2.2 Quadrupole magnets

To get elementary particles to collide in the CLiC with the desired luminos-ity, the beam parameters are specified at the interaction point to be 1 nm vertical by 40 nm lateral. Thus, a small beam has to be created that is fo-cused to an area of about 42 nm2_{. To get this precision, along the linac, after}

each acceleration section, the beam needs to be refocused to prevent momen-tum dispersion. As can been seen in Figure 2.3) [12], one set of quadrupole magnets will focus the beam in one direction, but defocus it in the other direction. Therefore, the next quadrupole magnet will be rotated 90◦, with respect to the other magnet, to focus the beam in the other direction. This continues along the accelerator to maintain a focused particle beam.

The magnets are composed of permanent magnet elements combined with electromagnets, to meet the requirement of a very high field gradient together with a tuning capability.

Figure 2.3: Magnetic field lines of quadrupole magnets: the green arrow shows the forces on a particle

The beam will only get focused if is passes through the quadrupole mag-net, exactly along the focusing axis. Therefore, if there is an offset, the beam

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will get deflected. Since there are 3600 quadrupole magnets in the collider, a small vibration in the magnets will result in luminosity loss at the interac-tion point, caused by the build up jitter in the beam. Because these effects occur even when the vibrations are small, the requirements of the stability of the magnets are high. The toughest requirements are the ones for the last quadrupole magnets, the QD0s, which do the final focusing of the electron and positron beams, just before the collision. To meet the required focus, the pre-alignment of QD0s has to better than 10 µm and needs a stabilization of the position with an rms of 0.15 nm above 5 Hz is needed [2]. These require-ments for the alignment and focusing of beams of these sizes have never been met before. Therefore a new accelerometer, to measure the vibrations in the QD0 and other focusing quadrupole magnets, has to be developed. As part of the stabilization of the magnets piezos are used. These piezos can be con-trolled by a control system that responds to changes in the position of the magnets using the sensor output of the accelerometer with interferometric readout.

2.3 Radiation and Magnetic fields

When a collider like CLiC is in operation, the collisions produce radiation. This radiation is mainly bremsstrahlung and neutrons, created by interac-tions between photons with cavities, accelerator components and shielding elements. Both kinds of radiation can cause damage to electronic devices that are close to the linac. Bremsstrahlung results mainly in continuous degra-dation caused by the total ionization energy of the photons absorbed by the system during the exposure time (the total ionizing dose effects). Neutrons, on the other hand, are relatively heavy, compared to the photons, and can collide with other particles. Therefore they can cause lattice damage, which can result in lasting damage to devices exposed to this [9]. These effects can modify aspects of electronic devices, leading to failure of components or of the whole system.

There is also a pulsing high magnetic field in the detection area, which is caused by the Kickers. In the CLIC design the Kickers are placed next to each quadrupole and they are used to adjust the path of the beam at low frequencies (below 5 Hz). Whereas the quadrupole magnets steer the beam at frequencies above the 5 Hz. The kickers are dipole magnets which are activated in a pulse mode; their magnetic field is created by large, short current peaks. These magnetic field pulses are is strong enough to affect electric current; the devices, like an accelerometer, that are positioned close to the linac, should be able to handle this.

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It is clear that most electronic accelerometers cannot operate in the near surroundings of colliders. Hence, an accelerometer combined with an inter-ferometer, that will not be affected by the radiation and high magnetic field, is proposed. It uses fibers to attach the accelerometer to a interferometric read out outside the linac. Since fiber light traveling in fiber won’t be af-fected by the radiation or the magnetic field, it can transport signals out of the linac area without them being affected. The read out of the interfero-metric signals can be done outside the collider area, where electronics can be used. Thus, this interferometer in fiber is suitable for the surroundings of the collider where there are high magnetic fields and radiation.

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3. Theory

3.1 Michelson Interferometer

In 1887, most scientists believed light was a wave propagating through a medium called the aether. Two physicists, named Edward Morley and Al-bert Michelson, designed an experiment to prove the existence of this aether and the movement of the earth through it. Since it was possible to see the stars, light must be able to travel through space. Therefore, they thought the aether was not only present at the earth, but everywhere. If it would also be completely still, the velocity of the earth through space could be determined by measuring the velocity of light in several directions. The ’aether wind’, originated from the movement of the earth in the aether, would be different in every direction, thus the velocity of light would also differ per direction. Michelson built an interferometer to measure these different velocities of light in several directions to obtain the velocity of the earth through the aether.

In Figure 3.1, a simple Michelson Interferometer is shown. This interfer-ometer consists of a laser, a 50/50 beamsplitter, two mirrors and a photodiode detector. After the light leaves the laser it is passes through the 50/50 split-ter where half is reflected to mirror 1 and the other half is transmitted to mirror 2. Then both beams are reflected by the mirrors and sent back to the 50/50 splitter where they interfere. The outcome of the interference depends on the length of the two arms with respect to each other.

When both arms are of exactly the same length, the distance the light travels is the same for both paths and the waves are in phase when they return at the beamsplitter. Then the interference results in constructive in-terference in the reflected arm and destructive inin-terference in the transmitted arm. Thus, all light goes into the reflected arm, while there is no signal in the transmitted arm, because of conservation of energy. This also happens when the arms differ a distance nλ₂ _{in path length with n ∈ N and λ as the} wavelength of the incoming light, because the wave in the longer path will have traveled twice an extra length of nλ/2 (nλ in total). Thus, the waves are again in phase. If the arm lengths differ a fraction (2n−1)λ₄ , then the total path length difference is nλ/2; the waves are in antiphase and interference causes all the light to go into the transmitted arm, while the reflected arm has no signal.

When the measurement mirror moves, the intensity on the detector will

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CHAPTER 3. THEORY 11

Figure 3.1: Michelson Interferometer: a light beam is split in two and those beams interfere after being reflected

change. If the mirror moves at constant velocity, the signal on the photode-tector will look like a sinusoid, as shown in Figure 3.2. The signal in the transmitted arm will always be in antiphase with the signal in the reflected arm, else there would be no conservation of energy. The difference between the maximum and minimum of the sinusoid refers to a path length differ-ence of λ/2 which corresponds to a mirror displacement of λ/4. This way, the interference signal coming from both arms tells us something about the movement of the mirror or the difference in length between both arms.

Michelson and Morley placed the interferometer in several directions to measure the different velocities of light. The differences in velocity were expected to result in different interference outcomes, even when both mirrors were fixed. However, Michelson and Morley did not measure these different velocities. The interference stayed constructive in the reference arm, no matter how they moved the interferometer around. The two scientist even repeated the experiment six months later to be sure of their results as they simply could not believe there was no proof for the aether. The results that light apperently did not use a medium to propagate was unexpected. It did however agree with Einsteins later theory of special relativity which

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Figure 3.2: The intensity on the photodiode changes as the length of the measurement arm is altered with a constant velocity. The points show the extreme values of the intensity corresponding to the mirror position.

postulates that there is no natural rest or relative frame in the universe and that any measurement of the speed of light in any inertial frame will always give the same result.

3.2 Open-air Sensor Setup

The setup with the interferometer in fiber is based on the open-air accelerom-eter with balanced output interferomaccelerom-eter. The schematic of this setup is shown in Figure 3.3.

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Figure 3.3: Setup of the accelerometer with interferometric readout in open air, based on the Michelson interferometer (on the right), but with two beam-splitters.

The interferometric readout of the open-air setup (and therefore of the fiber interferometer) is based on the Michelson interferometer, with the mov-ing mirror attached to the suspended mass. This way the relative motion of the mass compared to its frame can be measured. The difference with the simple Michelson interferometer described above is that there is a second beamsplitter placed between the laser and the first beamsplitter. This is an important addition, because this way, the transmitted arm and the reflected arm can both be read out by the two photodiodes shown in the figure. When the two photodiode signals are subtracted, this results in a differential signal. Because of the subtraction, common mode noise like laser power fluctuations, which is the same for both photodiode signals, is subtracted out. Resulting in a output signal of the sensor with less noise than it would have if only the signal of the transmitted arm was read out (as is the case for in a basic Michelson interferometer), which is shown in section 5.3.1. This improves the sensitivity of the sensor.

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4. Setup: Interferometer with

Feedback

4.1 Interferometer in fiber

The setup contains an interferometer in fiber of which one arm is part of a feedback loop, which makes it possible to determine the change in path length in the other arm. The goal of the final setup, is to get the feedback loop to respond to changes in path length in the measurement arm, caused by a moving mass. The setup is shown in Figure 4.1 and will be explained in the following paragraphs.

Figure 4.1: Setup of the fiber interferometer with feedback in loop

As can been seen in the figure, the interferometer part of the system starts with a 1550nm laser (RFLM-40-1-1550-1-U-0 Rock Fiber Laser Module). A 1x4 splitter (FCQ1315-APC) and a variable attenuator (VOA50-APC) are used to reduce the power of this laser, since the photodiodes have a saturation power of 70 µW and the lowest output power of the laser is 25 mW. After the

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CHAPTER 4. SETUP: INTERFEROMETER WITH FEEDBACK 15

power reduction, the light passes through a circulator (6015-3-APC) and is sent into a 50/50 splitter (10202A-50-APC), where the signal is split into two beams with the same intensity. After the splitting, both beams pass their own piezo stretchers (PZ1-SMF4-APC-E), are reflected by gold coated ends (FORF-31P-1300/1550-9/125-S-3A-1-0.5) and sent back through the fiber.

After the reflection, the beams are recombined and interfere at the 50/50 splitter. The interference happens as described in section 3.1. After the splitter, signal 1 is sent back into to previously mentioned circulator, while signal 2 goes through a second circulator (see Figure 4.1). Those circulators send the beams to two separate photodiodes (input+ and input- in the bal-anced photodetector (pdb440c-AC)) where the light beams’ intensities can be measured. The balanced photodetector converts the current of its pho-todiodes to a voltage, causing a gain of 104 V/A in the output signals of the photodiodes (monitor+ and monitor-). In the balanced photodetector, the two photodiode-signals are also subtracted to obtain a differential signal. This is done to get a signal without the common noise that occurs in both photodiode-signals. The gain of the differential output (RF output) is 5.1·104

V/A [19].

In the test setup both arms end in in-fiber mirrors, consisting of a cleaved fiber, on the end of which a high reflective (>95%) coating has been de-posited. This is to keep the path length fixed. When the interferometer will be attached to a suspended mass, the fiber of the measurement arm will end in a Gradient-index (GRIN) lens to couple the light out. The light of this arm will be reflected by a mirror attached to the suspended mass to detect the motion of the suspended mass with respect to its frame.

4.1.1 Splitter

The 50/50 splitter is fabricated by stripping the coating off two fibers and wrapping the cores with their claddings around each other. Then, this part is heated to melt the claddings together, while at the same time the ends are being pulled apart to get a thinner fiber (see Figure 4.2). During this heating, light is sent into input 1 and the intensity at output 1 and 2 are measured. First this will result in 100% of the beams intensity to go to output 1, but when the two fibers get melted together more and more light will be seen in output 2 as well. When the intensity measured in the outputs is 50/50 the heating stops and a 50/50 splitter is created. This method can make splitters up to one percent accuracy.

The reason the fibers are stretched is to get them thinner to give the light the possibility to cross over to the other fiber. When the fibers are not stretched the evanescent fields of the light (the part of the wave, outside the

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Figure 4.2: Making a splitter by melting and stretching two fibers.

core) will not overlap, as shown in Figure 4.3 a. After the stretching the peak of the light will still be in the core but the evanescent fields will overlap (see Figure 4.3 b). Because the two cores are wrapped around each other the light inserted into input one will cross over to the other core and this results in a splitter.

Figure 4.3: Evanescent fields in thick(a) and in thin(b) fiber.

4.1.2 Circulator

As shown in Figure 4.1, the setup contains two circulators. The first circu-lator transports the beam, coming from the attenuator to the 50/50 splitter and it transports the beam that comes back through the splitter, to PD1. The second circulator sends the other beam coming from the splitter to PD2 and it prevents the light that is reflected back from PD2 to get into the interferometer.

A circulator consists of three birefringent crystals, two half-waveplate sets and two Faraday rotators, as can been seen in Figure 4.6 (a).

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Half-waveplates

The half-waveplate is made of a birefringent material. Therefore, the re-fractive indices are different for different polarizations, which results in the component of the polarized wave that propagates along the fast axis to go faster than the component that propagates along the slow axis. Thus, when a beam with the plane of polarization of the wave at an angle θ with re-spect to the fast axis travels through a waveplate, its horizontal and vertical components will travel with different velocities.

In a half-waveplate the component of the wave traveling along slow axis will lag the component along the fast axis by half a wavelength.

Figure 4.4: A wave with polarization plane at an angle θ with respect to the fast axis. It leaves the half-waveplate with its plane of polarization at an angle of 2θ.

When looking at the two components of the wave, it is shown in Figure 4.4, that when the light leaves the plate and the fast component of the wave has a maximum, the slow component will also have a maximum, but it lags behind half a wavelength and will therefore be 180◦ out of phase with the original slow component. Recombining the components results in a wave with its plane of polarization at an angle of 2θ with respect to the fast axis. As the wave continues, the slow component remains exactly 180◦ out of phase with the original slow component, relative to the fast component. This results in a rotation of the polarization orientation of the beam [13].

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Light traveling through a circulator

When a beam arrives at port 1 of a circulator, it is split into 2 beams with orthogonal polarization by the birefringent crystal, as shown in Figure 4.6 b). In birefringent material, the refractive index is different for different

po-Figure 4.5: Schematic optic circulator

Figure 4.6: a) components of an optic circulator. b) polarization of light traveling through the circulator components.

larizations, causing polarization orientations to split. The half-waveplate set rotates the upper beam 45◦ clockwise and the lower beam 45◦ counterclock-wise. Then they pass through the Faraday rotator where the polarization of both beams rotate 45◦ counter-clockwise, and the two beams are vertically polarized. When they pass through the second birefringent crystal there is no spatial position change because the polarization directions of the two beams

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match the ordinary ray direction of the crystal (the direction in which the beams suffer no birefringence). After passing through another Faraday rota-tor and half-waveplate set, the beam is recombined in the last birefringent crystal, which is identical to the first one, and leaves through port 2.

In the same way, a light beam entering into port 2 splits into two beams and passes through the half-waveplate set and the Faraday rotator. The two beams become horizontally polarized and are therefore spatially shifted by the second birefringent crystal because they match the extraordinary ray direction of the crystal (othogonal to the ordinary ray, resulting in a refractive index which depends on the direction). This spatially shifting causes the beams to go to the right side of the path (as in Figure 4.6 b is shown). After also passing through another Faraday rotator and half-waveplate set, the beam is recombined in the birefringent crystal and leaves at port 3 [13].

4.2 Feedback Loop

Feedback is used to track path length changes in the measurement arm, by acting on the path length of the reference arm by means of a piezo stretcher. In the current setup, the changes in the measurement arm mimic the move-ments of the suspended mass, to test the system. First, how the feedback signal is generated is described, followed by a section about how the piezos work.

4.2.1 Lock Box, High Voltage Amplifier and Low Pass

Filter

In general, a feedback electronic circuit (lock box) is used to keep a variable parameter of a system at a fixed value: the lock point. A variable parameter of a system is fed into the lock box and compared to a set lock point. The deviation of this signal to the lock point is called the error signal and is given by

e(t) = m(t) − LP. (4.1) were m(t) is the measured value and LP is the lock point. The lock box can be used to produce a signal that attempts to counter this error is sent back into the system.

To obtain the counter signal, the lock box uses a proportional-integral controller (PI controller). The proportional term in this system produces an output signal that is proportional to the error value. This term is given by

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where Kp is a constant. The integral term of the PI controller is proportional

to the magnitude and the duration of the error signal; it keeps memory of the initial set point of the system (at t=0). This is given by

Pout = Ki t

Z

0

e(τ )dτ, (4.3)

Ki is a constant. How these terms work for different values of Kp and Ki

can been seen in Figure 4.7.

Figure 4.7: The value of the variable plotted on the y-axis and the time in the x-axis. The PI controller effects for different values of Kp and Ki. High

values for Kpand Ki will overshoot the reference signal more, when adjusting

[20].

Because a non-zero error is required to drive it, using only a proportional controller generally results in a steady-state error. Steady-state error is the final difference between the process variable and lock point. The integral term eliminates this steady-state error that will occur, since it will contin-ually increase over time, unless the error is zero. Nevertheless, since the integral term is proportional to the amount of time the error is present, it can overshoot the lock point value. Therefore, both the proportional and the integral term of the PI controller are used in the feedback loop. It is not possible to determine the correct values for Kp and Ki automatically,

because at this point, too many factors that are difficult to model have to be taken into account. Thus the influence of the proportional and integral term are chosen manually.

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The input for the lock box is the differential signal obtained by the sub-traction of the two photodiode signals. When the length of the measurement arm is altered by the measurement arm piezo, resulting is a path length difference between the two arms, the lock box will detect a change in the dif-ferential signal. The deviation of the signal to the lock point is determined and a signal to make the reference arm follow the path length change of the measurement arm, is sent out by the lock box. This output signal is sent through a 20x high voltage amplifier (HVA) and a low pass filter (LPF). The LPF is added to the setup because the piezo resonates at high frequencies. Then the signal is injected into the reference arm piezo. It will cause a change of arm length in the reference arm that is equal to the change of arm length in the measurement arm. This change resets the differential signal to the lock point.

4.2.2 Piezo Stretcher

The goal of the feedback loop is to inject a signal into the refence arm’s piezo stretcher to respond to changes in the measurement arm. The piezo in the reference arm is part of the feedback and is therefore needed to determine the vibrations that will be measured. The piezo in the measurement arm is there for the tests that need to be done before the interferometer can be used. The deformations of this second piezo will mimic the vibrations of the suspended mass, which the sensor will have to be able to detect.

The piezo stretcher consists of a fiber, of about ten meter, wrapped around a piezo, a material that deforms when a voltage is applied. The deforma-tion of the piezo causes the fiber that is wrapped around it to change in length.When a positive voltage is applied, the piezo expands in the z-direction (see Figure 4.8); this results in an contraction in the radial direction. The fiber that is wrapped around the piezo will also contract, generating a de-crease in path length of the light travelling through the fiber. This way, the path length of the reference arm can be altered by sending an electric signal from the lock box to the piezo stretcher.

Figure 4.8: Schematic of a piezo stretcher: a piezo with about 10 meter of fiber wrapped around it. The expansion happens in the z-direction.

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The conversion factor shows how much the path length changes when a certain voltage is applied, it can be determined in m/V. To obtain the con-version factor of the piezos, the measurement arm displacement per applied volt was determined and results can be found in section 4.4.

Using the conversion factor and the fact that the deformation of the piezo in one arm is related to the change in path length of the other arm, the signal sent to the piezo of the reference arm can be used to determine the change in path length of the other arm.

4.2.3 Differential signal

The relation between the applied voltage and the deformation of the piezos is linear. Therefore, if a linearly increasing ramp signal is sent into the piezo of the measurement arm, the piezo will expand linearly. Thus a sinusoidal signal as described in section 3.1 will be seen in the photodiode output. The output signals of the two arms and the ramp signal are shown in Figure 4.9.

−2 −1 0 1 2 3 Mirror Displacement [nm] Am p li tu d e [V] Signal arm 1 Signal arm 2 Differential signal Ramp signal

Figure 4.9: The signal in the two arms changes as the piezo stretcher expands, because of the ramp signal

As can been seen in Figure 4.9, the differential signal is useful to get a linear signal through zero. This linearity is needed to get a linear response of the output of the accelerometer. The two photodiode signals have only a minimum at zero, their linear part is halfway up the fringe. However, when to two signals, which are in antiphase, are subtracted, the result is the differential signal with a linear part through zero. This signal is used

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as an input for the lock box and the point where the differential singal goes through zero is the lock point.

4.2.4 Transfer Functions

A transfer function can be used to show a system’s response to a certain input. It gives the gain of a system under test at different frequencies. The transfer function is usually defined in the frequency domain; it is the mea-sured output signal of a system divided to the input signal

G = Output

Input . (4.4)

G is the ratio between the input and output signal and is called the gain of the system. Since the gain can have a large range, the graph is more clear when put on a logarithmic scale. Therefore, the gain of the transfer function is converted to decibel [dB], using

G[dB] = 20 ∗ log₁₀(Output Input ). (4.5) 101 102 103 104 105 −30 −20 −10 0 10 20 30 Frequency [Hz] Am p li tu d e [d B ] _{Unity Gain} Amplitude

Figure 4.10: Example of a transfer function with the unity gain frequency at 2.5 kHz. Input with frequencies f < 2.5 kHz have positive gain and are amplified and with f > 2.5 kHz have negative gain and are attenuated.

To obtain a transfer function of a system, a sinusoidal signal with increas-ing frequency (a swept sine) is sent into the system. Then, the output signal

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of the system is measured and a Fast Fourier Transform (FFT) is performed on both the input and the output signal at all frequencies. Performing an FFT on this output signal can result in a peak in amplitude at the frequency of the input signal, if the transfer function at that frequency is not too low. Another possibility is to send a signal that contains equal power within any frequency band with a fixed width (white noise) into the system. When the FFT is performed, this will result in peaks in the output signal at the frequencies that are amplified by parts of the system.

After the FFT, the magnitude of the FFT at the frequency of interest of output and input signal are divided, to determine the gain. This results in a transfer function of the entire range, which can be presented as the gain in dB related to the input frequency. An example of a transfer function is shown in Figure 4.10.

When the transfer functions of two separate systems are known, the trans-fer function or gain of the systems combined can easily be determined [6]. For systems connected in series the total gain is

G = G1 · G2. (4.6)

4.2.5 Bandwidth of the Feedback Loop

To determine the bandwidth of the feedback loop, the transfer functions of the parts of the feedback loop have to be evaluated. The parts that contribute to the total transfer function of the setup are the lock box, the high voltage amplifier (HVA) the low pass filter (LPF) and the piezo stretcher. The transfer function of the HVA and LPF is measured separately from the transfer function of the piezo. To do so, the setups as shown in Figure 4.11 and 4.12 are used.

To determine the transfer function, a white noise signal was sent into the HVA and LPF and the produced output and the inserted input were measured. Labview produces a transfer function in decibel. The input and output signal where measured using a connector block, which also produced the input signal.

Transfer Functions

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Figure 4.11: Setup to measure the transfer function of the HVA and LPF, with the differential signal as output.

101 102 103 104 105 −60 −50 −40 −30 −20 −10 0 10 20 30 Frequency [Hz] Magnitude [dB]

HVA+LPF Transfer Function

Figure 4.13: The transfer function of the HVA and LPF, with the voltage at the output of the LPF divided by the voltage input of the HVA

As can been seen from the graph, the cut off frequency of the LPF is about 25 Hz. Furthermore, the graph starts at 26 dB, which is to be ex-pected, since the HVA amplifies the signal 20 times, and 20 · log₁₀(20) ≈ 26 dB.

In Figure 4.14, the transfer function of the measurement arm piezo is shown. The peak at about 50 kHz is the resonance of the piezo stretcher. The dip in the low frequency part of the graph is due fact that the system

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Figure 4.12: Setup to measure the transfer function of the Piezo, with the differential signal as output. Labview produces a transfer function using equation 4.4, with the gain converted to decibel using equation 4.5, resulting in a transfer function in decibel. The input and output signal where measured using a connector block (NI BNC-2090A) that was connected to the PC, which also produced the input signal.

was operating in loop. The feedback was correcting for the changes in the measurement arm, resulting in the dip. Because the bandwidth of the feed-back is about 475 Hz, its effect is absent in the higher frequencies. It was obtained while the setup was in loop, because the feedback loop was needed to reduce the effects of the drift.

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CHAPTER 4. SETUP: INTERFEROMETER WITH FEEDBACK 27 104 105 −10 0 10 20 30 40 Frequency [Hz] Magnitude [dB]

Piezo Transfer Function

Figure 4.14: The transfer function of the piezo, with a lock box gain of 9.5 dB.

The transfer function of the setup with the lock box, HVA, LPF and piezo can be obtained using equation 4.6. Because of the proportional term, the transfer function of the lock box is flat after about 10 Hz. The graph is shown in Figure 4.15. As shown in the plot, the unity gain frequency is determined to be approximately 475 Hz. 101 102 103 104 105 −50 −40 −30 −20 −10 0 10 20 30 Frequency [Hz] Magnitude [dB] unity gain

Open Loop Transfer Function

Figure 4.15: The open loop transfer function, with the unity gain frequency shown and with a lock box gain of 11.5 dB.

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4.3 Fringe Visibility

With the use of the measurement of the photodiode outputs, as shown in Figure 4.16, and equation 5.1, the fringe visibility is determined. Since the gain of the differential signal and the gain of the photodiodes differ a factor 5.1, the differential signal is divided by 5.1, to scale it to the photodiode signals. As shown in the plot, the maxima and minima of photodiode 1 are respectively 0.7790 V and 0.036 V. The maxima and minima of photodiode 2 have values of respectively 0.7442 V and 0.0251 V. This results in the following fringe visibilities for the two signals

F VP D1= 90.68% F VP D2= 93.22% −500 0 500 −0.5 0 0.5 1 Discplacement [nm] Amplitude [V] Ramp/5 PD1 PD2 Differential*5.1

Figure 4.16: The three output signals and the ramp signal

4.4 Conversion factors

To find the sensitivity of the setup the fringe visibility, the conversion factor and the noise have to be combined to calculate the expected displacement resolution. The conversion factor tells us how the piezo expands after a voltage is applied, resulting in a contraction of the fiber wrapped around it (see section 4.2.2). This relation is linear y(V ) = y(0) + cV , where y(0) is the size of the piezo when no voltage is applied and c is the conversion factor. Since the relation between the expansion of the piezo and the applied voltage

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is linear, the conversion factor can be determined using the setup shown in Figure 4.17.

Figure 4.17: Setup to measure the conversion factor, a function generator is used the produce the input ramp signal and the output is read by an oscilloscope

A function generator sends a ramp signal into the piezo and this causes a path length difference in the interferometer. This results in a sinusoidal wave signal at the differential output of the interferometer which is read by an oscilloscope. To determine the conversion factor, the voltage difference between the maximum and minimum of the differential signal has to be measured, this difference accounts for half a wavelength and therefore for a path length difference of λ/4=1550/4=387.5nm. As can been seen from Figure 4.18 the voltage difference is 1.79 V. The ramp voltage measurement is taken directly from the function generator. The value is multiplied by 20, because the ramp signal goes through the 20x HVA before is is sent to the piezo stretcher. Thus the piezo input is 20 times the ramp signal. This leads to the conversion factor of 216.5 nm/V.

−100−4 0 100 200 300 400 500 600

−2 0 2 4

Change in arm length [nm]

Amplitude [V]

Ramp/20 Differential

Figure 4.18: Half a fringe of the differential signal and the ramp signal·20. This graph is used to determine the conversion factor.

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To obtain the total interferometer conversion factor, the gain of the inter-ferometer with piezo has to be determined. This gain is found by comparing the input into the piezo and the output signal, which is around the differ-ential signal lock point (since the system will operate there when in lock). The two signals differ in their slope and this difference corresponds to the ratio of the input to the output, as shown in Figure 4.19. The graph shows that the change in voltage for the differential signal is 1.77 V. The change in voltage of the ramp signal, in the same interval, is 0.28 V. This results in a interferometer gain of 6.32 V/V. This gain gives the output voltage of the interferometer (the differential signal) per applied volt on the piezo. When the piezo conversion factor is divided by this interferometer gain, the interfer-ometer conversion factor is determined. Thus the interferinterfer-ometer conversion factor is 34.25 nm/V. This gives the path length displacement per volt of the differential signal output, when operated around the lock point.

200 220 240 260 280 300 −1.5 −1 −0.5 0 0.5 1 1.5

Change in arm length [nm]

Amplitude [V]

Ramp/20 Differential

Figure 4.19: The linear part of the differential signal and the ramp signal·20. This graph is used to determine the interferometric conversion factor.

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5. Noise and Sensitivity

5.1 Polarization and Interference

In the setup, the beam travels between the different parts in single mode fiber (SMF). The light travels through the fiber with minimal loss, because there is total internal reflection. This means that when the angle of incidence exceeds a critical value, light cannot get out of the fiber, but is reflected back in. SMF has a narrow core (< 8µm) and one mode of transmission, therefore it requires a light source with a narrow spectral width, like the 1550nm laser that is used in the setup [7].

5.1.1 Polarization in Fibers

Only in an ideal circular-core fiber, the different modes of polarization will propagate with the same phase velocity. However, practical fibers are not perfectly circularly symmetric. Because of this slightly asymmetric form, the refractive indices are different for different polarizations, resulting in a slow and a fast axis inside the fiber. In section 4.1.2 birefringent crystal and half-waveplates were discussed, here birefringence happens in a non-controlled way. It can also be caused by environmental factors such as bend, twist, mechanical vibrations and temperature changes in the fiber. It results in the component of the polarized wave that propagates along the fast axis to go faster than the component that propagates along the slow axis, as shown in Figure 5.1. When you look again at the wave after it travelled through the fiber to distance L, the component along the slow axis will be retarded. Thus, when the two components are recombined, this will result in a different orientation of the total plane of polarization, compared to the starting po-larization, as can be seen in Figure 5.2. Thus, random birefringence results in changes in polarization when the wave travels through the fiber.

5.1.2 Influence of Polarization on Interference

The photons in the beam change polarization during their travel in the two separate arms,as a result of birefringence. The change in polarization influ-ences the interference. For instance, when the electric field components of two

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CHAPTER 5. NOISE AND SENSITIVITY 32

Figure 5.1: The two modes of polarization: along the fast axis(left) and along the slow axis(right.

photons both have a maximum, but there is an angle θ between them (as in Figure 5.3), the parallel components (blue arrows) of their planes of polariza-tion will have constructive interference, while their anti-parallel components (the red arrows) will have destructive interference. This decomposition of the parallel and anti-parallel components will happen for all the interfering photons in the beams that meet at the 50/50 splitter. So, when adding all the photon-photon interferences, total constructive interference of the beams will not happen.

Figure 5.3: Two waves with their polarization direction and the horizontal-and vertical-components of their polarization

Thus, there will always be light in both arms after the 50/50 splitter. This results in an decrease of the maxima and an increase of the minima, causing

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Figure 5.2: Shift in polarization of a wave, caused by birefringence

the minima to no longer be equal to zero. Therefore the fringe visibility is reduced, since it is given by

F V = max − min

max + min. (5.1)

The effects of the polarization on the interference effects the output signals of the interferometer. A decrease in fringe visibility results in flatter sine functions of the the signals measured at the photodiode. The differential signal is not a straight line, but noise causes it to fluctuate around a line, as shown in Figure 5.4. Therefore, when the sine is flatter, the point where the differential goes through zero is spread out and is thus less well defined. Therefore, lock point will also be harder to define. Thus it is essential to get the fringe visibility as high as possible to improve the feedback loop and therefore the interferometric readout. The fringe visibility is calculated using measurements of the output voltages (see section 4.3).

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Figure 5.4: Measurement of the differential signal around 0 V; for a flatter sine (a), this is less well defined than for a more steep sine (b).

5.2 Noise

In this section, the types of noise of the setup will be discussed. The noises have several sources and therefore different methods will be used to mea-sure and quantize them. The categories of noises that will be discussed are: electronic noise, intensity noise, shot noise, polarization noise and thermal effects.

5.2.1 Shot noise

Shot noise in the interferometer is an effect of the particle nature of light. The number of photons that reach the photodiodes is not constant over time. Since the intensity measured by the photodiodes is the number of photons per unit time, this results in tiny fluctuations in the output of the photodiodes. For a large amount of photons, as is the case with the interferometer, the arrival time of the detected particles can be approximated by a Gaussian distribution. The current that results from the shot noise is given by

in=p2eρPin,n, (5.2)

where e is the electric charge of an electron, ρ is the amount of ampere generated per watt, called the responsivity of the photodiode and Pin is the

incident optical power onto the photodiode [4]. The subscript n stands for the photodiode 1 or 2 (PD1 or PD2 as in Figure 4.1).

In the balanced photodetector, the current is turned into a voltage using a 50 kΩ resistor in the transimpedance amplifier (TIA). Therefore the total shot noise voltage in the detector output signal of the interferometer is the

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combination of the shot noise on PD1 and in PD2, times 50 kΩ

SNtotal =

q

(i1)2+ (i2)2· RT IA,RF (5.3)

The shot noise can be determined by measuring the light power impinging each photodiode, without interference (see appendix A). This is done by obscuring one arm, so only the light from other arm is used.

5.2.2 Electronic noise

To measure the electronic noise, the interferometer input to the photodiodes is cut off and the diodes are obscured. This way, the signal sent out by the diodes is not influenced by light, only by the electronic noise in balanced pho-todetector.The electronic noise can have several origins; a few are described below.

Dark current has its origin in the photodiodes. Even when no photons are incident on the photodiode, a small current is generated. This current is caused by electrons in the semiconducting material of the photodiode, which gain enough energy to move from the valence band to the conduction band. This creates an electron-hole pair, which cannot recombine because of the electric field inside the semiconductor. The electric field sweeps away the electron, resulting in a current[15].

Since the details of the photodiodes used in the balanced photodetector are not specified by the manufacturer. However, the dark current is charac-terized by a shot noise of 0.018 pA/sqrt(Hz), for this kind of photodiode.

Johnson noise is a current which occurs in resistors that are built-in in the photodiode readout circuit, including operational amplifiers and it is due to the random movement of electrons. Thermal energy is the reason for the random electron movements, which generate small currents. John Johnson was the first to measure this temperature related noise. He described his findings to Harry Nyquist who gave a theoretical explanation for this, which resulted in an equation for the Johnson noise current

I = r

4kBT ∆f

R , (5.4)

where ∆f is the bandwidth in Hz, T the temperature and R the resistance of the resistor in which the current is generated [11].

Another possible cause of op amp noise is flicker noise which is also called 1/f noise, since it increases as the frequency decreases. Its origin is up to now

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an unsolved problem. It is present in all active and many passive devices. The source might lay in imperfections in crystalline structure of semiconduc-tors, since better processing can reduce the noise.

In summary, the details of the readout electronics are not provided by the manufacturer but, according to the specifications, all these noise sources add up, at the balanced output of the device, to a minimum detectable optical power of 7 pW/sqrt(Hz) over a maximum input power of 70 uW on each photodiode.

5.2.3 Intensity noise

For macro scale applications the laser can be regarded as a constant light source. At the scale the interferometer has to work, the fluctuation in the intensity of the beam become significant. These fluctuations originate from technical noise sources such as vibrations of resonator mirrors and thermal fluctuations in the gain medium [16].

Fluctuations in the laser’s intensity influence the output of the interfer-ometer, because the intensity of the output is proportional to the intensity of the laser. Since a change in path length in one of the arms will alter the intensity as well, the difference between the movement of a mirror and intensity noise cannot be distinguished.

However, since both arms of the interferometer contain light of the same laser, the intensity fluctuations are identical for both arms. When the signals of the two photodiodes are subtracted to obtain the differential signal, the intensity noise will be cancelled out to some extent.

5.2.4 Polarization and Thermal effects

Changes in the polarization of the interfering photons cause fluctuations in the interference output. It occurs in fibers when environmental effects cause birefringence [7]. The several causes of this noise will be discussed below.

Mechanical noise occurs when the setup table is not free from vibra-tions. Because of this mechanical noise, the fibers feel vibrations. This results in birefringence, causing changes in the polarization. This noise is measured and discussed in section 5.3.3. Shorter fibers can reduce the effects of mechanical vibrations.

Thermal effects cause the refractive index of the fiber to change slightly, resulting in birefringence. This will cause the polarization of the light in the fiber to change its orientation. When the temperature changes are different for the two arms of the interferometer, the refractive indices will also vary,

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generating a difference in the orientation of the polarization planes for the two arms.

Thermal expansion is also a result of the temperature changes in the fiber. However, this effect is neglectable since the thermal expansion coefficient is an order of magnitude smaller than the thermo-optic coefficient that results from the change in the refractive index [17].

These thermal effects are measured and discussed in section 5.3.2. There, a solution to reduce these effects is also proposed and tested.

5.3 Measurements and Results

To quantify the noise in the interferometer, several measurements were done, which are described below.

5.3.1 Noise and Expected Sensitivity

The measured values for the intensity noise and the electronic noise are shown in Figure 5.5. Since the gain for the photodiodes and the differential signal differ a factor 5.1 (see section 4.1), the differential graphs are again scaled to the photodiode signals, to make it possible to compare them. The calculated shot noise of 5.12·10−7 V/√Hz is also given in the plot (the cal-culations can be found in Appendix A).

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CHAPTER 5. NOISE AND SENSITIVITY 38 100 101 102 103 10−6 10−5 10−4 Frequency [Hz] AS D [V/ √ H z]

Electronic Noise Plus Electronic Noise Minus Electronic Noise Diff Intensity Noise Diff Intensity Noise Plus Intensity Noise Minus Calculated Shot Noise

Figure 5.5: Plot of the intensity noise and the electronic noise of the plus and minus photodiodes and the differential (RF) signal. The subtraction causes the intensity noise of the differential to be an order of magnitude lower than the intensity noise of the photodiodes. The calculated shot noise is also given, but the electronic noise shows to be limiting for this setup.

The peaks in the electronic noise signal result from the mains frequency of the electric power grid. The alternating current of the electrical power generated by a plug, results in a peak at 50 Hz and the peak at higher fre-quency is caused by the harmonics of this effect. As shown in the graph, using the differential signal instead of the signal of one of the photodiodes, results in a decrease of the noise of almost two orders of magnitude which causes an improved sensitivity.

To obtain the expected sensitivity of the system the noise has to be mul-tiplied by the interferometer conversion factor. The expected sensitivity is limited by the electronic noise, as is shown in Figure 5.6. This results in a expected sensitivity of about 50 fm/√Hz.

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CHAPTER 5. NOISE AND SENSITIVITY 39 100 101 102 103 10−14 10−13 10−12 10−11 Frequency [Hz] AS D [m / √ H z]

Electronic Noise Plus Electronic Noise Minus Electronic Noise Diff Intensity Noise Diff Intensity Noise Plus Intensity Noise Minus Calculated Shot Noise

Figure 5.6: The expected resolution of the interferometer readout.

5.3.2 Thermal effects

The setup of the interferometer was on a tabletop in a non-temperature regulated room, where the components of the setup where up to 50 cm apart. This setup was likely to result in different temperatures at different parts of the setup. Temperature variations in the fibers can cause changes in the refraction index according to the optic coefficient. The thermal-optic coefficient for a single mode fiber and a 1550 nm laser is 40 rad

m·K [17].

This means that light passing through a fiber, it gets phase shifted by 40 rad/K per meter of fiber. The consequence is that if the two arms differ 1 cm in length a temperature change of 1 K creates a phase shift between the two interfering beams of 0.4 radians which corresponds to 23 degrees. Since the arms differ a few centimeter in the setup, even small temperature changes will result in a phase difference between the two arms which influences the interference of the beams. This results in a drift in the output signal of the two photodiodes. To reduce this thermal drift, a thermal-isolating box was fabricated and used to cover the part of the setup that was affected, namely from the 50/50 splitter up to the reflected ends. Only this part of the setup was covered, because that is the part where the beam is split in two, travels to the mirrors and interferes. The drift only occurs from effects of the temperature changes on the beams before they interfere. A new measurement was performed to determine the drift after the thermal box was placed.

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taken, with a ramp signal injected into the system (see Figures 5.7 and 5.8). Between the screenshots there is a time difference. This time difference is measured for the system before and after placing the thermal box. The results of the measurements are listed in table 5.1. The calculations done to obtain these values are given in Appendix B.

−5000 0 500 1000 1500 2000 0.1 0.2 0.3 0.4 0.5 Displacement [nm] Am p li tu d e [V] Ramp*5

Output signal PD1 t= 0 sec

Figure 5.7: Signal of photodiode 1 at t=0 sec.

−5000 0 500 1000 1500 2000 0.1 0.2 0.3 0.4 0.5 Displacement [nm] Am p li tu d e [V] Ramp*5

Output signal PD1 t=4.5 and 12 sec

Figure 5.8: Signal of photodiode 1 at t=4.5 sec before placing the box and at t=12 sec after placing the box.

As can been calculated using table 5.1, the velocity and frequency of the drift were reduced with a factor 2.67. The bandwidth needed to fulfill the

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requirements of the interferometer is from 5 Hz onwards.

Table 5.1: Drift Measurement Results

Before placing the thermal box

After placing the thermal box

Time 4.5 sec 12 sec

Drift Frequency 0.11 Hz 0.04 Hz Drift Velocity 0.349 rad/sec 0.131 rad/sec Temperature change 5.817 · 10−4K/sec 2.182 · 10−4K/sec

5.3.3 Mechanical Noise

With the setup as in Figure 5.9 the output signal of the feedback of the interferometer measured with a FFT analyzer, to obtain the noise in this signal.

Figure 5.9: Setup to measure the coherence between the noise in the output signal of the interferometer feedback and the mechanical vibrations on the table

This noise can be caused by mechanical vibrations that are different in the two fibers. To determine whether the cause of the noise is mechanical, a piezoelectric accelerometer (Wilcoxon Research, 731A) was used to measure the vertical motion of the table. Both the accelerometer and the feedback output signal of the interferometer are connected to a FFT dynamic signal analyzer (Agilent, 35670A). There, the two signals are compared and the coherence between them is shown (see Figure 5.10). The accelerometer mea-sures only vertical movements, but as can been seen from the plot, there is

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coherence between 10-40 Hz and around 100 Hz. This shows that the inter-ferometer, in this setup, acts as a microphone for mechanical vibrations.

100 101 102 0 0.2 0.4 0.6 0.8 1 Frequency [Hz] C o h er en ce

coherence between in lock signal and seismic accelerometer signal

Figure 5.10: Plot of the coherence between the differential signal and the vertical mechanical vibrations on the table

5.3.4 Sensitivity: In Loop Performance

To obtain the sensitivity of the interferometer, with interference, a noise measurement was done, while the system was in lock. The measured signal was the non-amplified signal sent out by the lock box; because the FFT analyzer cannot process the amplified signal. This signal is amplified twenty times, before it is sent to the reference arm piezo. Therefore, to obtain a noise in m/√Hz the conversion factor 20·216.5 nm/V = 4330 nm/V was used, where 216.5 nm/V is the conversion factor as obtained in section 4.4. Moreover, to correct for not measuring the signal after it passed through the LPF, the data was multiplied by LPF, where τ is determined using the cut off frequency of 25 Hz, LP F = 1 p1 + (2πfτ )2, and (5.5) τ = 1 2πf = 1 50π = 6.4 · 10 −3 s (5.6)

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CHAPTER 5. NOISE AND SENSITIVITY 43 100 101 102 10−12 10−11 10−10 Frequency [Hz] AS D [m / √ H z] In lock performance

Figure 5.11: The noise on the differential signal when the system was oper-ating in loop.

As can be seen, the sensitivity is around 10−11 m/√Hz, which is a factor 104 _{higher than the expected sensitivity. Presumably, this is caused by}

ther-mal effects and mechanical vibrations. Reducing these effects is possible and should result in a improvement of the sensitivity.

5.4 Thermal Noise

In the test setup the interferometer is not connected to a suspended mass. However, when it will be attached, the Brownian motion of the suspended mass will also result in noise. The Brownian motion is a small effect, but for the scale at which the sensor will be operating, it becomes relevant.

The thermal energy in the suspended mass and its suspension causes movement of the mass with respect to its frame, since the accelerometer, as mechanical oscillator, has an average kinetic energy of 3₂kBT (energy

equipar-tition theorem) resulting into a random motion of the proof mass. This in-fluences the outcome of the interference, since it causes changes in the path length of the measurement arm. If we model the accelerometer as a simple harmonic oscillator with mass M, spring constant k and damping coefficient γ the thermal noise power spectral density is

x2_th= 4kBT γ

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where kB is the Boltzmann constant, T is the temperature and ω is the

angular velocity. In Figure 5.12, it is shown that the thermal noise dominates in the low frequency range, whereas the sensor noise is dominant at higher frequencies [3]. 10−1 100 101 102 103 10−15 10−14 10−13 10−12 10−11 10−10

Frequency [Hz]

P

ro

je

ct

ed

d

is

p

lac

em

en

t

[m

/

√

H

z

]

Sensor noise Mass thermal noise Total

Figure 5.12: The thermal noise of the suspended mass, the sensor noise and the total noise. The thermal noise dominates at frequencies below 3 Hz. The expected sensitivity of section 5.3.1 is taken as the sensor noise dominates at frequencies above 3 Hz.

Therefore, the thermal noise is expected to be the dominating noise source below 3 Hz.

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6. Future Plans: Reducing

Fiber Length

Because the fibers are several meters long, the temperature changes between the measurement and reference arm and the mechanical vibrations in the the arms are significant. Making the fibers shorter and putting them closer together, could reduce these environmental influences and therefore, improve the fringe visibility and the sensitivity.

6.1 Piezo stacks

The longest part of the fibers (10 of the 15 meters of each arm) is wrapped around the piezo stretchers. To reduce this amount of fiber, another type of piezo is proposed; instead of piezo stretchers, piezo stacks (Piezomechanik, PSt 150/10x10/20) can be used. Piezo stacks consist of multiple blocks of piezoelectric material which are connected and stacked on top of each other. Applying a voltage results in the vertical expansion of the piezos, as the arrows show in Figure 6.1. This causes the fiber to be stretched in the vertical direction. Since the increase in path length of the fiber is now in the same direction of the expansion of the piezo, less fiber is needed to obtain the same increase in length.

To attach the fiber to the piezo, the secondary coating is stripped off and the fiber is pulled through the two small aluminium tubes, which are attached to the aluminium plates on the top and bottom of the stack. The part of the fiber inside the tubes has the primary coating removed as well and is glued to the tube. The tubes are used to glue to fiber to a larger surface this is needed to be able to distribute the shear force better.

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CHAPTER 6. FUTURE PLANS: REDUCING FIBER LENGTH 46

Figure 6.1: A stack of piezos which expand in the vertical direction when a positive voltage is applied. On the top and bottom of the stack, an aluminium plate, with a small tube, is attached and the fiber is glued in these tubes. The length of the fiber between the two plates is 18 mm.

Pre-stretching the fiber

The part of the fiber that will be stretched by the piezo is pre-stretched before it is attached to the plates. This is needed to use the compression of the piezo (which happens when a negative voltage is applied) as well as its expansion. If it would not be pre-stretched, it would deform instead of contract. The maximum expansion of the piezo is 20 µm in the vertical direction, and the maximum contraction is 8µm [14]. To be able to respond to this compression, the fiber will be stretched 10 µm. To gain this extra length a force has to be put on the fiber. This is done by hanging a weight at the end of the fiber. The stretch-factor is 0.8 mN/µ = 0.8 mN/(µm/m). Therefore, if the 18 mm long fiber will have to be stretched 10 µm,

10µm

18∗10−3_m = 555.6 µ

555.6 · 0.8 = 0.44 N =45.39 g,

a weight of 45.39 gram is needed to obtain the desired stretching.

Another way to reduce the amount of fiber in the measurement setup, it to decrease the length of the fiber to the reflective ends. To do this, the

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CHAPTER 6. FUTURE PLANS: REDUCING FIBER LENGTH 47

Characteristics and Limitations of the Fiber Interferometer Readout for an Accelerometer