Studies of materials for future ground-based and space-based interferometric gravitational wave detectors

Transcription

1 Studies of materials for future ground-based and space-based interferometric gravitational wave detectors Stuart Reid, B.Sc. Department of Physics and Astronomy, University of Glasgow Presented as a thesis for the degree of Ph.D. in the University of Glasgow, University Avenue, Glasgow G12 8QQ. c [S Reid] [2006] July 21, 2006

2 Contents Acknowledgements xix Preface xxi Summary xxiii 1 Gravitational Wave Detection Introduction ProductionofGravitationalRadiation SourcesofGravitationalRadiation BurstSources PeriodicSources StochasticSources Experimental Techniques for the Detection of Gravitational Waves ResonantBarDetectors LaserInterferometricDetectors Limits to the Sensitivity of Interferometric Gravitational WaveDetectors SeismicNoise GravitationalGradientNoise PhotoelectronShotNoise RadiationPressure The Standard Quantum Limit ThermorefractiveNoise ThermalNoise Current Status of Interferometric Gravitational Wave Detectors PlansforFutureGravitationalWaveDetectors Conclusions i

3 Contents 2 Thermal Noise Introduction Brownianmotion Fluctuation-Dissipation Theorem Sourcesofdissipation Externalsourcesofdissipation Internalsourcesofdissipation-anelasticity Thermoelasticdissipation Internal dissipation applied to an harmonic oscillator Theinternallossfactor Thermal noise resulting from spacially inhomogeneous mechanicaldissipation Interferometersuspensionthermalnoisesources Brownian and thermoelastic noise associated with dielectric mirrorcoatings Combined thermal noise in a detector Influence of temperature and hydroxide concentration on the settling time of Hydroxy-Catalysis Bonds Introduction The chemistry of hydroxy-catalysis bonding Hydration Etching Polymerisation - a definition for the settling time Catalysis bonding procedure Silica surface preparation Bonding procedure Catalysis bonding dependence on hydroxide concentration Temperature dependence of settling time Experimental setup - temperature controlled environment Experimental setup - precise measurement of bond settling time Temperature dependent bonding results Temperature dependent bonding analysis Conclusions ii

4 Contents 4 Quality Factors of Selected Test Mass Mirror Substrates and Coatings Introduction Requirementsfortestmassmirrors Experimentalset-upandprocedure Bulk silica results Calculation of the substrate thermal noise in the GEO600 detector Bulk silica with HR coatings Bulk sapphire results Conclusions Silicon as a Mirror Substrate for Third Generation Detectors Introduction Experimentalsetup Resultsandconclusions Silicon as a Suspension Element for Third Generation Detectors Introduction Requirements for cooling pendulum systems Proposed sample design Silicon flexure fabrication ExperimentalProcedure Temperaturedependenceofmodefrequencies Lossasafunctionoftemperature Thermoelasticloss Mechanical loss in silicon flexures as a function of temperature Analysis of low temperature silicon results Surfaceloss Gas damping Bulkloss Modeled elastic loss in the clamping structure from a resonating flexure Clamping Loss Otherlosses Conclusions iii

5 Contents 7 Conclusions 142 A Aspects of Pendulum Dynamics 145 A.1 Introduction A.2 Silica Pendulum Setup and Results A.3 Analysis of Observed Pendulum Dynamics A.4 Further pendulum analysis A.5 Conclusions B Expected level of thermoelastic dissipation in the bulk silica samples tested 156 C Convergence issues in the F.E. modelling of the sapphire coins tested 158 References 160 iv

6 List of Figures 1.1 The effect of the two polarization states of a gravitational wave on a ring of free test masses. The propagation direction of the wave is perpendicular to the page Schematic of a Michelson interferometer consisting of one beam splitter and two end mirrors Schematic of a Michelson interferometer in a delay line configuration where the optical path length is extended by a folded optical path using (a) separate mirrors or (b) by an input mirror with a gap for the input and output beams Schematic of a Michelson interferometer utilising Fabry-Perot cavities in the interferometer s arms formed between the end test masses (ETMs) and inner test masses (ITMs) Schematic of a Michelson interferometer with the addition of a power recycling mirror at the input port, forming an optical cavity with the interferometer Schematic of a Michelson interferometer with the addition of a signal recycling mirror at the output (dark) port, allowing the length of the cavity to be tuned for better performance of the interferometer at different frequencies Transfer function of a 1 Hz simple pendulum with no damping term, i.e. only the real part. Inset plot illustrates the pendulum responding out-of-phase when above the pendulum s resonant frequency v

7 List of Figures 1.8 Left: LIGO gravitational wave detector at Hanford, Washington State in the USA with 4 km arms. Right: A LIGO optic consisting of a fused silica mirror and single wire loop suspension Sensitivity and noise budget for the 4km LIGO interferometer at the Hanford site, at the end of February Noise budget of VIRGO interferometer during the seventh commissioning run, September Left: GEO gravitational wave detector near Hanover, Germany with 600 m arms. Right: GEO optics consisting of a triple-stage pendulum with lower stage monolithic fused silica suspension and mirror GEO600 commissioning noise curve, February Predicted noise curves for the second generation of gravitational wave detectors, with GEO-HF (tuned for millisecond pulsars) and the AURIGA, SFERA and Dual Mo (molybdenum) bar detector Illustration of a damped simple harmonic oscillator Calculated levels of expected thermal and thermoelastic noise in a single suspended fused silica mirror illuminated by a laser beam of radius r 0 = 3.9 cm, assuming a test mass loss of [105], and an ion-beam sputtered coating formed from alternating multi-layers of SiO 2 and Ta 2 O 5 [106] Launch of the NASA Gravity Probe B Relativity Mission on April 20th The hydroxy-catalysis bonding technique was developed for the jointing of components for the telescope assembly to ensure the long-term stability of the components position The planned orbit of the LISA space-based gravitational wave antenna vi

8 List of Figures 3.3 The engineering model optical bench interferometer for the LISA Technology Package (LTP) which will fly on board LISA Pathfinder A simplified, 2D schematic of the surface of silica when hydrated Schematic of the stages in the etching process due to an hydroxide solution applied to the surface of silica Mechanism for polymerisation for the formation of siloxane chains from a solution containing silicate Graph of the rate of change of [OH] over time for two solution with initial OH concentrations labelled (a) and (b) Images of the major steps involved in preparing/cleaning of the silica sample surface prior to bonding Graph of ph (calculated ph =14 log 10 [OH]) against settling time Schematic diagram of temperature controlled environment constructed for carrying out hydroxy-catalysis bonding Picture of temperature controlled environment constructed for carrying out hydroxy-catalysis bonding Plot of settling time as a function of temperature for the hydroxycatalysis bonding technique when using 0.23 mol l -1 NaOH solution (9.2g NaOH per litre of water, a typical concentration used by Gwo et. al [107]) with exponential fit Picture of shaking and readout scheme used to measure the settling time for hydroxy-catalysis bonds Schematic of the sample and the mechanical oscillation used for measuring the settling time for hydroxy-catalysis bonds. Note that no vertical downward load was applied to the silica ear and the samples were held together during testing only by the surface tension of the bonding solution vii

9 List of Figures 3.15 Readout circuit diagram for measuring the amplitude of shaking of the upper silica sample as it is hydroxy-catalysis bonded to a lower clamped silica sample. The optical sensing was carried out using an infrared LED (810nm) with a beam angle of 20 used to illuminate two mm single-element silicon planar photodiodes separated by 500µm where a parallel shadow was cast by the 4 mm wide flag mounted on top of the silica ear sample holder Example of silica ear shaking amplitude as a function of time during bonding using readout scheme described Plot of settling time as a function of temperature for the hydroxycatalysis bonding technique when using 0.1 mol l -1 KOH solution with exponential fit The natural log of the settling time plotted as a function of 1/temperature for the hydroxy-catalysis bonding technique when using 0.23 mol l -1 NaOH solution (using the preliminary results - measured by hand) with linear fit The natural log of the settling time plotted as a function of 1/temperature for the hydroxy-catalysis bonding technique when using 0.1 mol l -1 KOH solution, using the readout scheme described, with linear fit Schematic diagram and picture of a typical test mass suspension Schematic diagram of the interferometer readout used for measuring the amplitude ring-down of resonant modes of a test mass Schematic diagram of pre-tensioning technique used for a suspension using silk thread Typical ringdown for Hemex sapphire sample of length 10 cm and diameter 3 cm viii

10 List of Figures 4.5 Plot of (a) measured and (b) predicted loss factors for the resonant modes of a Suprasil 311 fused silica cylinder of 65 mm diameter and 70 mm length, where the standard error of each point was calculated to be between 5-10% Illustration of 32,151 Hz and 70,357 Hz mode shapes for a Suprasil 311 fused silica cylinder of 65 mm diameter and 70 mm length where the relative displacements U = x 2 + y 2 + z 2 are plotted in dimensionless units Illustration of 39,645 Hz and 53,990 Hz mode shapes for a Suprasil 311 fused silica cylinder of 65 mm diameter and 70 mm length where the relative displacements U = x 2 + y 2 + z 2 are plotted in dimensionless units Plot of (a) measured loss factors of the resonant modes of a Suprasil 311SV fused silica sample 76.2 mm diameter and 25.4 mm length, (b) predicted loss factors for Suprasil 311 and 312 of the same dimensions and (c) predicted loss factors allowing an increase in the bulk loss term of a factor of two. The standard error of each point was calculated to be between 5-10% Plot of (a) measured loss factors of the resonant modes of a Corning 7980 fused silica sample 76.2 mm diameter and 25.4 mm length, (b) predicted loss factors for Suprasil 311 and 312 of the same dimensions and (c) predicted loss factors allowing an increase in the bulk loss term of a factor of 1.2. The standard error of each point was calculated to be between 5-10% Simplified schematic diagram of the optical layout of GEO600 comprising the near and far mirrors (NM and FM), beamsplitter (BS) and power recycling mirror (PR) ix

11 List of Figures 4.11 Plot of the expected contributions to the effective (or measured) mechanical losses in the relevant GEO600 optics. The bulk mechanical losses have been estimated from the empirical formula for mechanical loss in fused silica [129], where the model has been verified experimentally in the case of the interferometer mirrors (see Figure 4.8) and modified in the case of the beamsplitter (see Figure 4.5) as detailed earlier. The surface loss was extrapolated from the empirical model for loss in fused silica Noise contributions in the GEO600 detector with the recalculated substrate thermal noise and updated total noise Measured mechanical loss values for a variety of coated silica samples where the standard error of each point was calculated to be between 5-10%. The frequency dependence of the measured losses are not seen to be dependent only on the mode shapes, as shown in Figure Illustration of khz, khz, khz and khz mode shapes for a fused silica test mass of 25.4 mm diameter and 76.2 mm length where the relative displacements U = x 2 + y 2 + z 2 are plotted in normalised, dimensionless units Loss measurements for one super polished and two commercially polished sapphire samples of length 76.2 mm and diameter 25.4 mm Surface profiles for one super polished and one commercially polished sapphire sample, where in 1-dimension Ra = 1 h(x) dx l where l is the length of sample studied and h(x) is the height above or below the mean and where R P V is the average height between the ten largest peaks and the ten lowest valleys on the sample surface Finite element models for measuring the stored strain energy in a damaged surface layer x l 0

12 List of Figures 4.18 Energy ratios in sapphire samples studied with varying thickness of surface layers for five of the resonant modes Plot of the thermal expansion coefficient of single crystal silicon over the temperature range 5 K 300 K where α 0at 125 K, with inset, magnified plot showing where α 0 at 18 K Plot of intrinsic amd thermoelastic noise of silicon as a function of temperature. (calculated intrinsic thermal and thermoelastic noise at 100 Hz in a single silicon test mass, sensed with a laser beam of radius 6 cm.) Setup of one of the suspensions used to measure the mechanical loss of one of the large silicon test masses Plot of the room temperature mechanical loss for a silicon coin (length 25.4 mm, diameter 76.2 mm) and two silicon rods of length 150 mm and diameter 53 mm, all (100) orientation, boron doped with resistivity 1 10 Ohm cm, where the standard error of each point was calculated to be between 5-10% Picture of the larger silicon cylinders tested Plot of the room temperature mechanical loss for two larger silicon cylinders (length 100 mm, diameter 98 mm), 1 (111) with resistivity 20 Ohm cm, 1 (100) undoped with resistivity 7000 Ohm cm, where the standard error of each point was calculated to be between 5-10% Schematic diagram of proposed silicon cantilever to be manufactured Plot of the calculated thermoelastic loss for the proposed silicon cantilever to be manufactured with the calculated resonant frequencies of the bending modes marked with arrows Schematic diagram of silicon cantilever tested xi

13 List of Figures 6.4 Plot of the calculated thermoelastic loss for the two tested silicon cantilevers, fabricated by wet chemical etching, with the calculated resonant frequencies of the bending modes marked with arrows Schematic diagram of cryostat Illustration of the prototype setup of the readout system for the cryostat using a split photodiode arrangement as shown in Figure 6.7 consisting of two 5 5 mm single-element silicon planar photodiodes separated by 500µm Schematic diagram showing the dynamic range of a split photodiode readout scheme where the differential signal from both photodiodes gives the readout signal Circuit diagram for the readout system used for reading out the cantilever amplitude in the cryostat. One mm singleelement silicon planar photodiode is used where the active length is 25.4 mm and the maximum active width after masking, labeled h, is 1 mm. The photodiode mask was made from standard black electrical insulating tape and the laser spot diameter was measured to be 5.0 ± 0.5 mm Responses of split photodiode readout (see Figure 6.6) and single photodiode readout (see Figure 6.8) systems for measuring silicon cantilever displacement angle Noise curves for single photodiode readout system with room lights on and off compared with noise curve for prototype split photodiode readout system with room lights off Picture of silicon cantilever mounted in the stainless steel clamp. 117 xii

14 List of Figures 6.12 Temperature dependence of the frequency of the third resonant mode of the cantilever of length 57 mm at 670 Hz. Measured data (o) and modeled (-). The curve is a best fit where E o is calculated from the resonant frequency when B (a constant related to the bulk modulus) and T o (related to the Debye Temperature) are known Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the second bending mode at 240 Hz, for cantilever of length 57mm Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the third bending mode at 670 Hz, for cantilever of length 57mm Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the fourth bending mode at 1320 Hz, for cantilever of length 57mm Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the fifth bending mode at 2185 Hz, for cantilever of length 57mm Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the sixth bending mode at 3265 Hz, for cantilever of length 57mm Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the third bending mode at 1935 Hz, for cantilever of length 34mm Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the fourth bending mode at 3788 Hz, for cantilever of length 34mm Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the fifth bending mode at 6265 Hz, for cantilever of length 34mm xiii

15 List of Figures 6.21 Plot of (a) the minimum measured loss of the third bending mode at f 670 Hz for the cantilever of length 57 mm compared with (b) the sum of the estimated surface loss and calculated thermoelastic loss Plot of the (a) 2-D surface profile and (b) 3-D surface profile for the upper (ground and etched) surface of the cantilever of length 57 mm. RMS surface roughness = nm Plot of the (a) 2-D surface profile and (b) 3-D surface profile for the lower (polished) surface of the cantilever of length 57 mm. RMS surface roughness = 3.99 nm Illustration of Ansys finite element simulation to calculate the elastic energy residing in the clamp as the cantilever flexes Finite element simulation of the elastic displacement in the stainless steel clamp and the silicon cantilever for the third bending mode of the cantilever of length 57mm at f 670 Hz Finite element simulation of the elastic displacement in the stainless steel clamp and the silicon cantilever for the sixth bending mode of the cantilever of length 57mm at f 6094 Hz Plot of the proportional elastic energy residing in the stainless steel clamp structure as the silicon cantilever bends Finite element simulation of the elastic displacement for the clamp-cantilever resonance at f 6984 Hz Plot of the average excess loss of the fourth bending mode at f 1.3 khz compared with the normalised magnitude of the signal from the piezo sensor. Sample length 57 mm Plot of the average excess loss of the fourth bending mode at f 2.2 khz compared with the normalised magnitude of the signal from the piezo sensor. Sample length 57 mm xiv

16 List of Figures 6.31 Plot of the average excess loss of the fifth bending mode at f 3.1 khz compared with the normalised magnitude of the signal from the piezo sensor. Sample length 57 mm Plot of (a) the response of the clamping structure to a constant driving signal and (b) the subsequent excitation of the modes of the cantilever of length 34 mm from this driving signal Plot of the response of the clamping structure to a constant driving signal at 240 K, 270 K and 280 K A.1 Motion applied to suspended mass in Heptonstall s pendulum setup A.2 XY Optical sensor comprising of two pairs of photodiodes each detecting orthogonal directions of motion A.3 Motion applied to suspended mass of the pendulum A.4 Output signals from orthogonal photodiodes showing observed beating in the pendulum motion A.5 XY plot of silica ribbon pendulum motion, over 20 seconds in air.150 A.6 XY plot of silica ribbon pendulum motion, over 40 seconds in air.150 A.7 Pendulum mode directions for the theoretical model A.8 Theoretical X-Y plot of pendulum motion, A=1, B=0.7, θ 1 = π 4, θ 2 = π 4, ω 1 =2π, ω 2 = π A.9 Plot showing a simulation of amplitude of the X and Y displacements due to the oscillation of a pendulum as a function of time. The pendulum was modeled as being suspended by a single ribbon fibre A.10 Phase-space plot for silica ribbon pendulum motion, over 2 minutes for the X degree of freedom A.11 Phase-space plot for silica ribbon pendulum motion, over 2 minutes for the Y degree of freedom B.1 Illustration of the fundamental drum mode of a cylinder. The arrows show the respective motion of this particular resonant mode xv

17 List of Figures B.2 Analytical calculation of thermoelastic loss as a function of frequency for Corning 7980, 311SV and 311 silica test mass (assuming drum mode only) C.1 Plot of the estimated levels of bulk strain-energy in a sapphire test mass as a function of the number of nodes used in the F.E. model xvi

18 List of Tables 4.1 Calculated residual losses for various coated test masses. Note that doping is only applied to the tantala component of the multi-layer Room temperature parameters for silicon, the variations in thermal conducitivity dependent on the doping [169] Room temperature parameters for silicon and stainless steel xvii

19 Quote When I was young, all I wanted and expected for life was to sit quietly in some corner doing my work without the public paying attention to me. And now see what has become of me. The New Quotable Einstein (Princeton University Press) xviii

20 Acknowledgements It is only right first of all to thank my supervisor Prof. James Hough for the opportunity to carry out the work detailed in this thesis and for his support and guidance through it all. I thank him also for his ability to push experiments forward and find the solutions to make them work. I would also like to thank both Prof. Sheila Rowan and Dr Geppo Cagnoli, who also supervised many stages of my work. I feel truly indebted not just to have one great supervisor, but to actually have three. I would also like to thank Prof. Jim Faller from the University of Colorado for his help and kind encouragement through the PhD. There are also many individuals within the IGR group who I would wish to thank. In the (current) office, I would like to thank Alastair Heptonstall for many useful discussions on our research work and for the encouragement he has given during these final stages of the PhD. I would like to thank him also for being willing to accept me taking over his lab in mid 2004 and moreover for forgiving me for accidently setting fire to it within six months! I would also like to strongly thank the other students, with whom I have spent significant time working in the labs: Peter Murray, Iain Martin, Alan Cumming, Alastair Heptonstall and Eoin Elliffe. Anyone who has carried out experimental physics will know just how valuable it is to have workmates who are all of the following; talented, reliable and (arguably most important of all) also enjoy being in the lab and facing the challenge of difficult, intricate experiments. And I am also grateful for the intermingled discussions, which I confirm did not inhibit any lab research(!), about God (Eoin), life, relationships (Alastair), food, model xix

21 Acknowledgements trains (Alan), photography (Peter) and even pet fish (Iain!). It has been my privilege to work with them all and the other members of the group. I would like to thank Russel and Mike for all their technical designs and drawings required for various aspects of the experiments. I would also like to thank technicians in the group, Colin and Stephen, for the nearly infinite number of speedy jobs done for me - they both never cease to amaze me at how beautiful a carefully machined metal component can look. I would like to thank the UK research council PPARC for providing the funding required to carry out all the research presented in this thesis. On a more personal front, I would like to thank my family for their unconditional support during my PhD and for being happy that I pursue what interests me. I particularly want to thank Mum and Dad, for being so understanding towards me as a child, when I dismantled almost every electrical item I owned and the preliminary experiments I carried out (most often without their knowledge, at least until after the event) such as testing the purity of homemade hydrogen balloons using a candle! I do not want to miss out my two brothers, Euan and Donald, who thankfully were never too discouraging towards my strange ways while we were (at least theoretically) growing up. I would also like to thank my friends Graham, Sheena, their son Jacob and Deirdre for also always being there and in particular the many beautiful weeks we spent on holiday together surrounded by the awesome scenery on the Isle of Skye. I would also like to thank my Church family back in Stirling and in particular for giving me the opportunity to develop my skills in speaking up front. This has certainly made the presentation aspects of my life in research much easier than it would otherwise have been. And finally, yet most of all, I thank Jesus for everything, always being there and making it all worthwhile. xx

22 Preface In Chapter One the nature of gravitational waves is discussed, methods for detection and the factors that limit the sensitivity when using interferometeric schemes. In Chapter Two aspects of the theory of thermal noise is described, how it can be quantified and therefore how it affects the detection of gravitational waves. Through understanding the processes that give rise to thermal noise, methods can be developed to help minimise it. In Chapter Three some investigations have been carried out into hydroxycatalysis bonding. The focus of this research was to develop techniques for extending the settling time for the bonding process, allowing increased time for aligning optical components. Contribution to the design of the controlled temperature environment came from J. Faller and J. Hough. The readout system used was modified to its final improved state through valuable suggestions from J. Faller, G. Cagnoli and S. Rowan. The experimental results were obtained by the author with the help of I. Martin and E. Elliffe. Chapter Four focuses on the measurement of the internal loss factor of bulk samples of pure silica, sapphire and silica coated with a mult-layer dielectric coatings. The experiments were performed by the author in conjunction with P. Murray. The finite element modelling was performed by the author, except in the case for the coating analysis where the modelling was carried out by D. Crooks. The calculation of thermal noise in the GEO600 detector was carried out by the author with the help of S. Rowan. The measurements of optical coating thickness using the EELS technique were performed by I. MacLaren xxi

23 Preface of the solid-state physics group. Chapter Five focuses on the measurement of the internal loss factor of bulk samples of single-crystal silicon. The measurements were performed by the author in conjunction with P. Murray and following discussions with J. Hough, S. Rowan and G. Cagnoli. In Chapter Six the measurements of the internal loss of thin samples of single-crystal silicon is detailed. The measurements were performed by the author. The analysis of the results was carried out by the author in conjunction with S Rowan and J. Hough. The finite element modelling was performed by the author. In Appendix A there is detailed some small investigations into the dynamics of pendulums. The experimental results were performed by the author with useful suggestions and discussions with G. Cagnoli, M. Hendry, J. Hough and A. Heptonstall. In Appendix B there is detailed the approximate level of thermoelastic loss expected in the bulk silica samples tested in Chapter 4. This work was carried out by the author in conjunction with G. Cagnoli and J. Hough. In Appendix C evidence is given that results obtained from finite element modelling were not dominated by convergence issues, that is, inaccuracies in the model as a result of modelling a system with an insufficient complexity (number of nodes/equations). This work was performed by the author. xxii

24 Summary Studies of materials for future ground-based and spacebased interferometric gravitational wave detectors In 1916 Einstein predicted the existence of gravitational waves as a consequence of his Theory of General Relativity. These can be considered as fluctuations in the curvature, or ripples, in space-time. Until now there has only been indirect evidence for their existence. However, many scientists around the world over many years have been developing ultra-sensitive measurement techniques that are expected to be capable of detecting these signals, with the hope of providing new information on the astrophysical processes and sources that produce them. Gravitational waves are quadrupolar in nature and therefore produce orthogonal stretching and squeezing (i.e. a strain) of space. These fluctuations in distance are very small with astrophysical events predicted to produce strains at the earth of the order of in the audio frequency band. One method for detecting such a strain is based on a Michelson Interferometer. The Institute for Gravitational Research in the University of Glasgow, under the leadership of Prof. J. Hough and has been actively involved in the research targeted towards the detection of gravitational waves for around 35 years. A strong collaboration exists with the Albert Einstein Institute in Hanover and Golm, the University of Hanover and the University of Cardiff. This collaboration has built an interferometer with 600 m arms in Germany called GEO600. GEO600 is designed to operate in a range of 50 Hz to a xxiii

25 Summary few khz, with strain sensitivities reaching the order of / Hz in the range 50 Hz a few khz. The work within this thesis describes various experiments carried out on materials and techniques used in current detectors and for the proposed future detectors. The principal aim is to cover various methods for reducing the levels of mechanical loss associated with the detector s optics and thereby minimising the impact of thermal noise on overall detector sensitivity. Chapter 1 starts by explaining the nature of gravitational waves, the sources that are expected to give the largest signals and details the development of resonant bar and interferometric detectors. This is followed by a discussion in chapter two of one of the principal limits to detector sensitivity, that of thermal noise. The reduction of thermal noise in detectors is a major challenge and care has to be taken in the construction of the suspension systems of the interferometric detectors to minimise mechanical loss effects, which adversely affect thermal noise. A particular technology for jointing components was used for the construction of the monolithic suspensions of the German/UK GEO600 interferometric detector, resulting in improved suspension thermal noise performance over other methods. This involved the use of hydroxycatalysis bonding. In Chapter 3 investigations into hydroxy-catalysis bonding are presented for determining the settling time dependence on hydroxide concentration and temperature. Quantifying these factors will help in the development of applications of this technology both in the area of future detectors and for space-based projects requiring precision optical sensing. Chapters 4 and 5 detail mechanical loss measurements on bulk mirror substrates for current and future interferometric detectors at room temperature. Fused silica, single-crystal sapphire and single-crystal silicon samples yielded values for the intrinic loss of φ mat-silica = , φ mat-sapphire = and φ mat-silicon = The lowest loss values obtained for Heraeus Type 311 fused silica appear to agree well with the semi-empirical xxiv

26 Summary formula for loss developed by Steve Penn at al. The calculated level of substrate thermal noise in the GEO600 detector is presented using this model for mechanical loss and indicates the levels to be around a factor of ten lower than previously calculated. This knowledge further strengthens the case of fused silica as a substrate material for the next line of future detectors, such as Advanced LIGO. The low levels of internal loss in bulk silicon samples is also of interest due to the silicon s high thermal conductivity. This is relevant when considering third generation detectors, that is, beyond the Advanced LIGO design, where significant thermal loading is anticipated due to the increased levels of laser power required for improved shot noise performance. Another significant source of thermal noise in the optics of current interferometers arises from the mechanical loss associated with the applied optical coatings required for high reflectivity. Results presented in Chapter 4 show that the mechanical loss is predominantly associated with the tantala component of the silica-tantala coating, and that doping the tantala with titania can reduce the mechanical loss by a factor of approximately two. Silicon may also be a useful material for the construction of low mechanical loss suspension elements. Chapter 6 details measurements of 92µm thick silicon flexures for use as suspension elements which show the mechanical loss to reach φ(ω) = at 85 K. The various sources of loss are considered, both internal and external to the samples, and presented. The level of internal loss is consistent at higher temperatures with thermoelastic effects and at lower temperatures to be dominated by surface loss in addition to perhaps some other loss mechanism. The measurements in Chapters 5 and 6 suggest that silicon may be a suitable material for both mirrors and suspension elements. xxv

27 Chapter 1 Gravitational Wave Detection 1.1 Introduction The search for gravitational radiation remains one of the most difficult challenges faced by experimental physicists at this present time. Predicted by Einstein to be produced by the non-uniform acceleration of mass, as described in his general theory of relativity, at the time of writing they have remained elusive [2]. Early sceptical relativists considered gravitational radiation to be transformable away at the speed of thought since they have largely remained an enigma. However, the possibility of directly detecting gravitational radiation approaches as first generation interferometric detectors are online and nearing their design sensitivities. Gravitational wave signals from the heart of violent astronomical events would open up a new window in astronomy and significantly increase our understanding of the relativistic nature of the Universe. The first confirmation of their existence resulted in the award of the 1993 Nobel Prize in Physics to Hulse and Taylor for experimental observations and interpretation of the binary pulsar PSR [3], whose orbit decayed at a rate consistent with energy being lost due to gravitational radiation [4] [5]. Many groups around the world are now deeply focused on developing ultrasensitive detectors capable of directly observing gravitational waves. 1

28 1.2 Production of Gravitational Radiation 2 Gravity is by far the weakest of the four fundamental forces, with the relative strength of the gravitational, weak (radioactive decay), electromagnetic and strong (nuclear) forces being approximately 1 : :10 37 :10 39 respectively. However, gravitational effects are the most significant over very large distances due to the forces from all matter adding up. In fact, many new astrophysical sources have already been observed through gravitational effects, such as the existence of extra solar planets and black holes. Yet many questions relating to gravity remain, for example, why does the Universe appear to be accelerating rather than slowing down under gravity [6]? Clearly new discoveries that in turn lead to new theories in physics are required to explain such dynamics within the Cosmos. The detection of gravitational waves will give a significant opportunity to observe the distant Universe and probe deep into exotic and relativistic astrophysical systems, revealing the dynamics that govern them. 1.2 Production of Gravitational Radiation Electromagnetic waves, such as radio waves, are known to be produced by the acceleration of charge. In a similar way, gravitational waves are produced by the acceleration of mass. Conservation of mass, equivalent to conservation of charge in electromagnetism, denies the possibility of monopolar gravitational radiation. However, conservation of momentum also denies the existence of dipolar gravitational radiation. Dipolar radiation is possible in electromagnetism since two signs of charge exist, positive and negative, whereas in the case of gravity, gravitational radiation results from only one sign of mass, that is, positive mass. Therefore gravitational waves are only produced when matter is accelerated in an asymmetric manner. For quadrupolar radiation there are two orthogonal polarisations of the wave at 45 to each other, of amplitude h + and h, as shown in Figure 1.1. Gravitational waves simply can be thought of as ripples in the curvature of space-time, causing a change

29 1.2 Production of Gravitational Radiation 3 + L L- L L L+ L L x 0 π/2 π 3π/2 2π Figure 1.1: The effect of the two polarization states of a gravitational wave on a ring of free test masses. The propagation direction of the wave is perpendicular to the page. in the separation between two adjacent masses, where the gravitational wave amplitude h canbedefinedby, h = 2 L L (1.1) where L is the change in separation of two masses a distance L apart. There are currently a number of very sensitive detectors using laser interferometry with arm lengths in the km scale that are coming online. The largest project is LIGO (Laser Interferometer Gravitational-Wave Observatory) consisting of two 4km arm length interferometers at different locations in the US: Hanford in Washington State and Livingston in Louisiana. An additional halflength 2 km interferometer resides in the same vacuum system at the Hanford site. Other large-scale projects include VIRGO (Italy/France) with a 3 km detector located near Pisa in Italy, GEO600 (Germany/UK) with a 600 m detector located near Hannover in Germany, and TAMA-300 (Japan) with a 300 m detector located in Tokyo. The first generation of these detectors are expected to reach strain sensitivities down to / Hz. The specific optical schemes implemented by these detectors are detailed later in this Chapter.

30 1.3 Sources of Gravitational Radiation Sources of Gravitational Radiation Due to the nature of the gravitational interaction, detectable levels of radiation are only produced when very large but compact masses are accelerated in very strong gravitational fields. Therefore, large astrophysical systems become the primary candidates for generating detectable gravitational wave signals Burst Sources Supernova At the end of a star s lifetime the inward pressure due to gravity can significantly exceed the outward radiation pressure. This in turn leads to the possibility of the star imploding, and, due to the rapid collapse of the core, throwing off the outer stellar layers in a bright flash roughly one billion times the luminosity of our own Sun. If the collapse is perfectly symmetric, then no gravitational radiation will be produced. If the star s core has angular momentum (spins like our Sun) the collapse with be asymmetric and gravitational radiation should be produced. The dimensionless gravitational wave (strain) amplitude, h, expected from a supernovae can be shown to be [7], ( )( )( )( ) 1 h E 15Mpc 1kHz 1ms 2, (1.2) 10 3 M c 2 r f τ where E is the total energy radiated, M is the mass of the Sun, c is the speed of light, f is the frequency of the gravitational wave signal, τ is the time taken for stellar collapse and r is the distance from the source. The event rate for supernovae out to the Virgo cluster ( 15Mpc) is estimated as several per month [8].

31 1.3 Sources of Gravitational Radiation 5 Coalescing compact binaries The majority of stars exist in binary systems. In the case of compact binaries, this can consist of either neutron star/neutron star (NS/NS), black hole/black hole (BH/BH) or neutron star/black hole combinations (NS/BH). As already discussed, the binary neutron star system PSR exhibits a decaying orbital period as energy is lost in the form of gravitational waves. In the last few seconds before the coalescing stars merge, the frequency and magnitude of the gravitational radiation are expected to be observable by ground based gravitational wave detectors. The approximate strain amplitude expected from coalescing neutron stars can be shown to be [9], ( )( ) 5 ( ) 2 100Mpc h Mb 3 f 3, (1.3) r 1.2M 200Hz where M b =(M 1 M 2 ) 3/5 /(M 1 + M 2 ) 1/5 is the mass parameter of the binary, and M 1 and M 2 arethemassesofthetwostars. Although the population of NS/BH and binaries is expected to be lower than for NS/NS, the greater density and mass will produce larger amplitudes of gravitational radiation and may therefore be more visible to ground based gravitational wave detectors Periodic Sources Pulsars Neutron stars are made from matter typically 10 s of thousands of times the density of ordinary matter (ρ n-star : kg/m 3 [14][15]). Many have been observed by their characteristic periodic pulses of radio frequency electromagnetic waves, hence the name Pulsars. If the spin of such a star was to be asymmetric then gravitational radiation would be emitted. Asymmetry may result from the star having a mountain on the surface or elliptical geometry, similar to a rugby ball, possibly caused by the non-alignment of the spin and magnetic field axes.

32 1.3 Sources of Gravitational Radiation 6 The estimated magnitude of gravitational radiation due to the star s equatorial ellipticity is given by [16], ( ) 2 ( )( ) h frot 1kpc ε, (1.4) 500Hz r 10 6 where ε is the equatorial ellipticity and f rot is the rotational frequency of the star. The Crab pulsar, the remnant of a supernova documented by Chinese astronomers in 1054, is expected to be such a source of gravitational radiation at roughly 60 Hz. An upper limit to the gravitational wave amplitude has been set at h for an ellipticity ε =7 10 4, r 1.8kpcandf rot = 30 Hz [16]. Low-Mass X-Ray Binaries Low-Mass X-ray Binaries (LMXBs) are systems where matter from a normal star is pulled off by the strong gravitational field of a compact companion, in this case a neutron star. The process of accretion causes the neutron star to gain enough angular momentum to reach a limit known as the Chandrasekhar-Friedman-Schutz (CFS) instability point where its rotation becomes non-axisymmetric and will result in gravitational radiation emission. A neutron star at the CFS instability point is known as a Wagoner Star [17]. X-ray emission is observed as a result of matter being accelerated towards the neutron star s surface. A steady state is then reached where the viscous damping (due to the emission of electromagnetic (X-ray) and gravitational radiation) equals the increase in instability (due to continued accretion). The gravitational radiation amplitude may be defined in relation to the time averaged X-ray flux (l γ ) such that [17], ( )( 1kHz h mf l γ 10 8 ergs cm 2 s 1 ), (1.5) where m 4 is the mode number and f is frequency of the gravitational radiation.

33 1.4 Experimental Techniques for the Detection of Gravitational Waves Stochastic Sources A final source of gravitational radiation is expected from the superposition of a random background of signals from many sources. This could be observed as correlated noise between two or more gravitational wave detectors. One possible prediction for background gravitational radiation arises from the cosmic string model for galaxy formation, whose amplitude could be given by [18], ( )( ) 1 h H ( ) o Ωgw 2 3 f ( ) 1 2 B 2, (1.6) 75 kms 1 Mpc Hz 2Hz in a bandwidth B, at frequency f, whereω gw is the energy density required for a closed Universe and H o is the current value of Hubble s Constant. Another possible stochastic source is that of relic gravitational waves resulting from the early inflationary period of the Universe shortly after the Big Bang. 1.4 Experimental Techniques for the Detection of Gravitational Waves Since the interaction of gravitational waves with matter results in only a very small strain (h) in space, ultra-sensitive detectors must be constructed to allow the possibility of detection. Two designs of ground-based detectors are currently being developed; resonant bars at low temperature and laser interferometers. A space-based detector using laser interferometry is also being planned for reasons discussed in Section 1.6. Other methods for measuring the effects of gravitational radiation directly include Doppler tracking of spacecraft [10] [11] and studying the effect of low frequency gravitational waves on pulsar timings [12][13] Resonant Bar Detectors The first experiments for detecting gravitational waves were ground-based and carried out by Joseph Weber of the University of Maryland about 40 years

34 1.4 Experimental Techniques for the Detection of Gravitational Waves 8 ago. Initially the effect of low frequency gravitational waves were investigated by studying the normal modes of the earth [19], after which Weber moved to studying tidal strains in large aluminium bars at room temperature [20] [21]. These bars were resonant at 1600 Hz at which frequency Weber expected a broad peak in the gravitational radiation spectrum due to stellar collapse. Seismic noise from ground vibration and acoustic noise in the laboratory were reduced by suspending the bars from vibration isolation stages and placing them under vacuum. The sensitivity is then limited by the thermal motion of molecules (thermal noise, discussed briefly in Section and later in detail in Chapter 2) and readout noise from the sensors. Weber began to report many coincidences in 1968 that continued over the following years [22], thought to be from the detection of gravitational waves, between two detectors placed about 2 km apart. Similar experiments were set up by groups in the USA, Germany, Britain and Russia. However, none were able to repeat or confirm the claim by Weber. In fact, these initial bars were only sensitive down to strains of over millisecond timescales, far above the level predicted theoretically from likely astrophysical sources. With the focus on reducing the thermal noise through cooling, groups in the Universities of Rome [23], Padua [24], Louisiana [25] and Perth (Western Australia) [26] continued to develop Weber-like bar detectors reaching sensitivities approaching for millisecond pulses. Developments on spherical bars detectors, with improved performance particularly in directionality, are being carried out at the University of Leiden, Holland (MiniGRAIL [27]) and in the Mario Schenberg Detector, Brazil [28]. However, a significant disadvantage to the operation of such bar detectors is due to their narrow frequency bandwidth. Cerdonio and others [29] [30] have developed a scheme where it is possible to extend the frequency band of operation using a dual resonant mass detector. In this case, a full cylinder is nested within a larger, hollow cylinder where the larger cylinder has a resonant frequency two to three times lower than the

35 1.4 Experimental Techniques for the Detection of Gravitational Waves 9 smaller one. A gravitational wave that passes between the resonant frequencies of the two cylinders will excite the modes in anti-phase, thus enhancing the sensitivity. In principle, such a detector should reach strain sensitivities down to / Hz in a frequency band about 1000 to 3000 Hz. An alternative design of gravitational wave detector, providing the possibility of high sensitivity over a broad frequency range, is based on a laser interferometer Laser Interferometric Detectors Michelson interferometer A gravitational wave detector using laser interferometry uses test masses which are widely separated and suspended as pendulums to isolate against seismic noise. Laser interferometry then provides a way to sense precisely the differential motion, due to an incident gravitational wave, of the test masses end mirror in orthogonal axes, see Figure 1.2. The first work carried out on laser interlaser beam splitter end mirror Figure 1.2: Schematic of a Michelson interferometer consisting of one beam splitter andtwoendmirrors. ferometers in relation to detecting gravitational waves was by Forward [31] and Weiss [32]. Since the observed effect of a passing gravitational wave is a strain, i.e. fractional change in the distance under observation as seen in Equation 1.1, it is desirable to construct long-baseline interferometers to maximise

36 1.4 Experimental Techniques for the Detection of Gravitational Waves 10 the absolute dispplacement. The longest practical length for ground-based interferometers is several kilometers. The detectors, using interferometry, all operate in a scheme where their arm lengths are suitably adjusted so that the output port normally remains on a dark fringe for the laser frequency. This allows noise associated with the laser light to be minimised in the interferometer s output. Delay line interferometer Various methods may be implemented for increasing the effective arm length of interferometers by increasing the optical path of the light in the arms. Such a scheme, proposed by Weiss [32] in the early 70 s, is the delay-line interferometer. This uses a folded optical path, as shown in Figure 1.3 (a), or uses an input hole in a mirror close to the beamsplitter which allows the light to bounce between input and end mirrors several times before exiting again through the same hole [33], shown in Figure 1.3 (b). The folded delay line configuration is used in the GEO600 detector (German-UK collaboration) [34], effectively increasing the arm lengths of the interferometer from 600 m to 1.2 km. Delay lines were used in the early prototype interferometers at Caltech (40 m) and Garching (30 m). However, scattering was found to be a problem in these instruments and this ultimately limited their sensitivity. Fabry-Perot interferometer Another method for increasing the optical path length in the arms of an interferometer is the use of Fabry-Perot cavities in the arms, first developed for gravitational waves detectors in Glasgow [35] and shown in Figure 1.4. Fabry-Perot cavities are incorporated into the interferometer arms, with each cavity consisting of one partially transmitting input mirror and one highly reflecting mirror. The cavities are operated in resonant mode and the amount of energy stored is a maximum when the lengths are an integral number of half

37 1.4 Experimental Techniques for the Detection of Gravitational Waves 11 wavelengths of the laser light. This particular configuration is currently used in the LIGO [36], VIRGO [66] and TAMA [46] detectors. (a) (b) mirrors gap in mirror end mirror end mirror beam splitter laser beam splitter laser N=3 pass delay line Figure 1.3: Schematic of a Michelson interferometer in a delay line configuration where the optical path length is extended by a folded optical path using (a) separate mirrors or (b) by an input mirror with a gap for the input and output beams. ETM ITM ITM ETM laser beam splitter Figure 1.4: Schematic of a Michelson interferometer utilising Fabry-Perot cavities in the interferometer s arms formed between the end test masses (ETMs) and inner test masses (ITMs).

38 1.4 Experimental Techniques for the Detection of Gravitational Waves 12 Power recycling Other techniques have also been developed for enhancing the output signal of an interferometric detector. While other noise sources are dealt with later, it is perhaps intuitive that increasing the laser power will also increase the effect of the signal. In the case of power recycling, an additional mirror may be placed at the input port of the interferometer, allowing the light coming back out the interferometer through the beamsplitter to be reused, or recycled, shown in Figure 1.5. Similar to using Fabry-Perot cavities, power recycling allows a build up of light power in the interferometer, increasing the sensitivity to differential changes in the arms. laser power recycling mirror Figure 1.5: Schematic of a Michelson interferometer with the addition of a power recycling mirror at the input port, forming an optical cavity with the interferometer. Signal recycling As well as focussing on reusing the laser light at the input port of the interferometer, as seen in power recycling, it is possible to reuse the signal light at the output port. As mentioned, the interferometers are typically operated in a condition of destructive interference where the result of a differential change in arm length, caused by the passage of a gravitational wave, produces laser light at the output port. This light, or signal, may be recycled back into the inter-

39 1.4 Experimental Techniques for the Detection of Gravitational Waves 13 ferometer by use of a partially transmitting mirror, shown in Figure 1.6. An optical cavity is formed between the signal recycling mirror and the composite mirror formed by the interferometer. By careful positioning of this mirror, the frequency at which the interferometer is most sensitive can be changed. Variations of this technique in relation to interferometric gravitational wave detectors include broadband signal recycling, resonant sideband extraction and detuned operation, see [38][39][40][41]. laser signal recycling mirror Figure 1.6: Schematic of a Michelson interferometer with the addition of a signal recycling mirror at the output (dark) port, allowing the length of the cavity to be tuned for better performance of the interferometer at different frequencies Limits to the Sensitivity of Interferometric Gravitational Wave Detectors There are many sources of noise that may limit the sensitivity of interferometric detectors. These include seismic noise, fluctuations in gravity gradients, noise due to the optical sensing techniques (photo-electron shot noise, radiation pressure and ultimately the standard quantum limit ) and thermal noise, and these are discussed in turn.

40 1.4 Experimental Techniques for the Detection of Gravitational Waves Seismic Noise Seismic noise arises from a variety of sources, from ocean waves and earth tremors to various forms of human activity such as traffic and machinery. In a quiet place on earth the level of seismic (ground) motion in all three dimensions and at frequency f closely follows 10 7 f 2 m/ Hz. In order for the resulting disturbance of the interferometers mirrors to be less than m/ Hz, for example at 30 Hz, would require horizontal seismic noise reduction greater than 10 9 [42]. Isolation from seismic noise is therefore essential in the operation of any detector of gravitational radiation. Vertical isolation can be achieved by means of a spring, whereas horizontal isolation may be achieved by adopting a pendulum scheme. In the case of a simple pendulum system, the horizontal transfer function from the suspension point to the pendulum mass can be shown to fall off as f 2 above the pendulum resonance. The transfer function of a simple pendulum can be expressed as [179], x mass x clamp = f 2 o f 2 o f 2 (1.7) where x mass is the displacement of the pendulum s mass, x clamp is the displacement of the clamping point of the pendulum (in the simplest case the ground motion), f o is the resonant frequency of the pendulum and f is the frequency of ground motion. The degree of attenuation (or isolation) when the frequency of ground motion is significantly above the pendulum resonant frequency (f f o ) is approximately fo 2 /f 2. To further enhance this attenuation, a multistage pendulum can be constructed where the transfer function when f f o can be expressed where there are N pendulum stages as, x mass x g ( ) f 2 N (1.8) o f 2

41 1.4 Experimental Techniques for the Detection of Gravitational Waves 15 Below the resonant frequency of the pendulum the transfer function tends towards unity and therefore acts as a rigid coupling between the ground and the suspended mass. To increase the sensitivity of the interferometers at low frequencies, multi-stage pendulums with very low resonant frequencies are desirable. modulus of pendulum transfer function transfer function 50 0 in-phase response out-of-phase response frequency frequency (Hz) Figure 1.7: Transfer function of a 1 Hz simple pendulum with no damping term, i.e. only the real part. Inset plot illustrates the pendulum responding out-of-phase when above the pendulum s resonant frequency. Vertical isolation is required since the coupling of vertical noise through to the horizontal axis is typically 10 3, due to mechanical misalignments and since the earth s curvature results in the pendulums hanging under gravity and separated by several kms to be non-parallel. Some further considerations into the dynamics of a simple pendulum constructed using a fused silica ribbon are detailed in Appendix A Gravitational Gradient Noise A fundamental limit comes from direct gravitational coupling of seismic motions and other fluctuations to the test mass mirrors in the detectors. This ef-

42 1.4 Experimental Techniques for the Detection of Gravitational Waves 16 fect is referred to as gravity gradient noise, ornewtonian noise, see [43] [44] [45]. In this case, vibration isolation systems have no effect on reducing this noise. The noise spectrum from gravity gradients is below the sensitivity of initial LIGO, GEO600 and TAMA-300 interferometric detectors [44]. This is not the case for the VIRGO detector, which will not reach its original design sensitivity below 15 Hz, despite the various implementations of active and passive vibration isolation, due to gravitational gradient noise. A possible route for reducing this source of noise is to build a detector underground where gravity gradients (or fluctuations) are decreased. This is the case for the planned Japanese LGCT [47] detector and is also considered in the plans for a third generation detector in Europe [48]. Another approach would be to build a detector far away from the gravity gradients near earth, as planned for the space-based interferometric detector LISA [49] Photoelectron Shot Noise In order to detect gravitational wave signals, the output of the interferometers must be held at a particular interference condition, or fringe position. An obvious point at which to do this is half-way up a fringe where a differential change in arm length produces the strongest change in signal. To hold, or lock, the interferometer in such a condition requires monitoring the output signal with a photo-detector and feeding the signal to a transducer capable of changing the position of one of the interferometer mirrors. Information about changes in arm length as a result of a gravitational wave passing can therefore be obtained by monitoring this feedback signal. A source of noise, related to the statistical fluctuations in the number of photons detected at the output, therefore arises. Assuming the number of photoelectrons arriving in a time τ follow Poissonian statistics, the detectable strain sensitivity depends on the input power of the laser P of wavelength λ

43 1.4 Experimental Techniques for the Detection of Gravitational Waves 17 used in an interferometer of arm length L, suchthat[50], detectable strain in time τ = 1 ( ) 1 λ c 2, (1.9) L πpτ detectable strain/ Hz h = 1 L ( ) 1 2λ c 2, (1.10) πp where c is the velocity of light and is Planck s constant and we assume that the photodetectors have a quantum efficiency 1. For a laser operating at λ 1µm, a laser power of more than 10 6 W is required in order to reach the desired detector sensitivities. This gives further motivation for adopting some of the techniques outlined in Section for the reduction of photoelectron shot noise by increasing the effective laser power. Note that in practice the interferometers are locked on a dark fringe position. However the result remains valid Radiation Pressure As the effective laser power in the arms is increased, another phenomenon becomes increasingly important arising from the effect on the test masses of fluctuations in the radiation pressure. One interpretation on the origin of this radiation pressure noise may be attributed to the statistical uncertainty in how the beamsplitter divides up the photons of laser light [51]. Each photon is scattered independently and therefore produces an anti-correlated binomial distribution in the number of photons, N, in each arm, resulting in a N fluctuating force from the radiation pressure. This is more formally described as arising from the vacuum (zero-point) fluctuations in the amplitude component of the electromagnetic field. This additional light entering through the dark-port side of the beamsplitter, when being of suitable phase, will increase the intensity of laser light in one arm while decreasing the intensity in the other arm, again resulting in anti-correlated variations in light intensity in each arm [52] [53]. The laser light is essentially in a noiseless coherent state [54] as

44 1.4 Experimental Techniques for the Detection of Gravitational Waves 18 it splits at the beamsplitter and fluctuations arise entirely from the addition of these vacuum fluctuations entering the unused port of the beamsplitter. Using this understanding of the coherent state of the laser, shot noise arises from the uncertainty in the phase component (quadrature) of the interferometer s laser field and is observed in the quantum fluctuations in the number of detected photons at the interferometer output. Radiation pressure noise arises from uncertainty in the amplitude component (quadrature) of the interferometer s laser field. Both result in an uncertainty in measured mirror positions. For the case of a simple Michelson, shown in Figure 1.2, the power spectral density of the fluctuating motion of each test mass m resulting from fluctuation in the radiation pressure at angular frequency ω is given by [51], ( ) 4Ph δx 2 (ω) =, (1.11) m 2 ω 4 cλ where h is Planck s constant, c is the speed of light and λ is the wavelength of the laser light. In the case of an interferometer with Fabry-Perot cavities, where the typical number of reflections is 50, displacement noise δx due to radiation pressure fluctuations scales linearly with the number of reflections, such that, δx 2 (ω) =50 2 ( ) 4Ph. (1.12) m 2 ω 4 cλ The Standard Quantum Limit Since the effect of photoelectron shot noise decreases when increasing the laser power as the radiation pressure noise increases, a fundamental limit to displacement sensitivity is set. For a particular frequency of operation, there will be an optimum laser power within the interferometer which minimises the effect of these two sources of optical noise which are assumed to be uncorrelated. This sensitivity limit is known as the Standard Quantum Limit (SQL) and corresponds to the Heisenberg Uncertainty Principle, in its position and momentum formulation, see [51] [52] [53] [55]. This limit exists in any configuration of interferometer.

45 1.4 Experimental Techniques for the Detection of Gravitational Waves 19 Firstly, it is possible to reach the SQL at a tuned range of frequencies, when dominated by either radiation or shot noise, by altering the noise distribution in the two quadratures of the vacuum field. This effect can be achieved by squeezing the vacuum field. There are a number of proposed designs for achieving this in future interferometric detectors, such as a squeezedinput interferometer [53] [56], a variational-output interferometer [57] or a squeezed-variational interferometer using a combination of both techniques. This technique may be of importance in allowing an interferometer to reach the SQL, for example, when using lower levels of laser power and otherwise being dominated by shot noise. Secondly, if correlations exist between the radiation pressure noise and the shot noise displacement limits, then it is possible to bypass the SQL, at least in principle [56]. There are at least two ways by which such correlations may be introduced into an interferometer. One scheme is where an optical cavity is constructed where there is a strong optical spring effect, coupling the optical field to the mechanical system. This is already the case for the GEO600 detector, where the addition of a power recycling cavity creates such correlation. Other schemes of optical springs have been studied, such as optical bars and optical levers [58] [59]. Another method is to use suitable filtering at optical frequencies of the output signal, by means of long Fabry-Perot cavities, which effectively introduces correlation [60] [61] Thermorefractive Noise Another source of noise arises from the temperature dependence of the refractive indices of the mirror coatings of test masses and of the bulk material of the beamsplitter. This is known as thermorefractive noise [62]. Localised fluctuations in temperature alter the refractive index of the material resulting in a phase change of the transmitted or reflected light, causing phase noise in the interference point of the interferometer. Thermorefractive noise cannot be

46 1.5 Current Status of Interferometric Gravitational Wave Detectors 20 ignored and has been estimated to be of the same order of magnitude as the thermal noise from the suspensions and optics in the mid-band frequencies of the GEO600 interferometer Thermal Noise Thermal noise arises from the thermal energy of the atoms and molecules in the test masses and suspensions used in the interferometers. There are two forms of thermal noise: Brownian motion of the atoms and molecules proportional to their thermal energy (temperature), specifically 1 kt energy per degree of freedom, 2 where k is Boltzmann s constant and T is the temperature. The nature of the noise is dependent on the internal friction in the system. A low mechanical loss system will allow better thermal noise performance in the region of frequencies operated by the interferometric detectors. Thermoelastic thermal noise which results from the statistical temperature fluctuations in a system coupling through into mechanical movement. Thermal noise is one of the most significant noise sources at the low frequency end of the current interferometric detectors operating range [63]. Thermal noise is discussed in detail in Chapter Current Status of Interferometric Gravitational Wave Detectors As already discussed, laser interferometry provides a means of sensing the motions of test masses produced as they interact with a gravitational wave. Initial prototypes in Germany and the UK in the 1970 s were 3 m and 1 m respectively. In the early 1980 s these were followed by 30 m and 10 m interferometers

47 1.5 Current Status of Interferometric Gravitational Wave Detectors 21 in addition to a 40 m instrument being developed in Caltech as a spin-off from the UK (Glasgow) instrument. Two prototypes were also built in Japan, one using Fabry-Perot cavities, the other using delay lines. These instruments used multi-watt argon-ion laser and the majority achieved sensitivities better than m/ Hz over a frequency range of a few hundred Hz to a khz. After these developments, the technology was deemed suitably mature to consider the construction of significantly longer baseline detectors, which should have a significant chance of detecting gravitational waves. LIGO The LIGO detector facilities are located at two different sites, Hanford in WA and Livingston in LA. Both sites operate a 4 km interferometer with Fabry- Perot cavities in the arms, and the Hanford facilities house an additional 2 km interferometer within the same vacuum tubes [64] [65]. Each interferometer is constructed with 10.7 kg silica mirrors, suspended in a single loop wire suspension, see Figure 1.8. The suspensions are controlled by acting directly onthemasseswhichhavemagnetsattachedtothesidesandback. Figure 1.8: Left: LIGO gravitational wave detector at Hanford, Washington State in the USA with 4 km arms. Right: A LIGO optic consisting of a fused silica mirror and single wire loop suspension.

48 1.5 Current Status of Interferometric Gravitational Wave Detectors 22 Figure 1.9: Sensitivity and noise budget for the 4km LIGO interferometer at the Hanford site, at the end of February VIRGO The French/Italian VIRGO detector [66] with arm lengths of 3 km is situated at Cascina, near Pisa in Italy and is close to completion [67]. The optics are suspended from a multistage pendulum system to permit lower frequency performance down to 10 Hz. TAMA The Japanese TAMA 300 detector, which has arm-lengths of 300 m, is operational at the Tokyo Astronomical Observatory [68]. TAMA, VIRGO and LIGO all have Fabry-Perot cavities in the arms of the detectors and also place a power recycling mirror between the laser and the beamsplitter to enhance the laser power in the entire interferometer, see section They also use standard wire sling techniques for suspending the test masses acting as mirrors.

49 1.5 Current Status of Interferometric Gravitational Wave Detectors 23 Figure 1.10: Noise budget of VIRGO interferometer during the seventh commissioning run, September GEO600 The German/UK GEO600 detector has arm lengths of 600 m and employs somewhat different techniques in its operation. A four-pass delay-line interferometer is used alongside advanced optical signal enhancement techniques by the addition of a signal recycling mirror, see section Thermal noise performance is also significantly improved since GEO utilises very low-loss silica suspensions for the test masses. These improvements should allow the sensitivity of GEO at frequencies above a few 100 Hz to be close to those of the longer VIRGO and LIGO detectors in their initial operation. Since Autumn 2001, the LIGO and GEO detectors have completed four data taking (science) runs. Analysis in the search for astrophysical sources in the first and second science runs is complete and most of the papers are now published. Analysis is in progress for the third and fourth science runs of which some runs were done in coincidence with TAMA (Japanese interferometric detector, also discussed shortly) and the Allegro bar detector. The fifth science run started on 4th November 2005 at Hanford (Livingston site started a few weeks later) and GEO joined in January 2006, initially for overnight data taking before operating fully. Around 18 months of coincident data will be

50 1.5 Current Status of Interferometric Gravitational Wave Detectors 24 Figure 1.11: Left: GEO gravitational wave detector near Hanover, Germany with 600 m arms. Right: GEO optics consisting of a triple-stage pendulum with lower stage monolithic fused silica suspension and mirror Displacement (m/sqrt(hz)) Figure 1.12: GEO600 commissioning noise curve, February 2006.

51 1.6 Plans for Future Gravitational Wave Detectors 25 taken between these three sites. LIGO is now at design sensitivity, detailed in Figure 1.9 and the proposed upgrades to LIGO are discussed in section 1.6. Current detectors give great hope for the possibility for direct detection of gravitational waves. This in turn drives research to develop future detectors with greater sensitivities. For example, reducing the noise levels by a factor of 10 will increase the possible number of detectable sources by a factor of Plans for Future Gravitational Wave Detectors Second generation There exists a well defined path for the future upgrade for both the LIGO and VIRGO systems, known as Advanced LIGO and Advanced VIRGO. In the case of Advanced LIGO, a number of goals are set out for improved displacement sensitivities: the seismic noise is to be reduced by a factor of 40 at 10Hz, thermal noise to be reduced by a factor of 15 and optical noise to be reduced by a factor of 10. To achieve such goals will require the adoption of silica fibre suspensions, signal recycling and higher laser powers in addition to improved seismic isolation. Advanced LIGO will start installation around 2010 and Advanced VIRGO around GEO600 at this stage, with its shorter arm lengths, will not be able to match the sensitivity of the Advanced detectors over their frequency range. Planned upgrades to GEO are still considered with the options of optimised tuning at low frequencies for network analysis or optimised tuning for high frequency sources (GEO-HF). Another function of the GEO site would be to perform developments and tests towards third generation detector technologies, materials and optical schemes. Third generation Discussions on the writing of a Whitepaper are underway for a third generation detector(s) in the USA along with VIRGO/GEO proposals for a future

52 1.6 Plans for Future Gravitational Wave Detectors 26 h (m/sqrt(hz)) Frequency (Hz) Figure 1.13: Predicted noise curves for the second generation of gravitational wave detectors, with GEO-HF (tuned for millisecond pulsars) and the AU- RIGA, SFERA and Dual Mo (molybdenum) bar detector. European instrument. There are also factors associated with the mirrors and their coatings which make further improvements to detector sensitivity difficult. Firstly, the mechanical loss, and therefore associated thermal noise, starts to set limits on the achievable sensitivity. Secondly, the optical absorption in the coatings causes heat to be dumped in the test masses from the high laser powers held in the Fabry-Perot cavities, resulting in thermal effects whose compensation is difficult. These detectors are likely to require non-transmissive optics to alleviate the problem of thermal distortion and use cooling of the test masses to reduce the level of thermal noise. Squeezed light techniques will also become increasingly useful as the other noise sources are reduced further. Longer baseline detectors in space The signals from some of the largest, and arguably most interesting, astrophysical events in the Universe, such as the formation and coalescing of massive ( M Sun ) black holes, will lie in the range Hz. The only feasible way of detecting such low frequency signals would be to avoid the Newtonian noise on the earth and to fly an interferometer into space. This

53 1.7 Conclusions 27 interferometer would use multiple drag-free spacecraft in orbit [69] [70]. A joint NASA/ESA mission is under development for such a space-based detector called LISA (Laser Interferometer Space Antenna) [71] [72]. LISA consists of three drag-free space-craft at the vertices of an equilateral triangle of length of side km. The spacecraft array would fly in an earth-like orbit around the Sun and approximately 20 behind the earth. Each space-craft would accommodate two proof masses, which would form the end points of three separate but not independent interferometers. Investigations into the jointing technology planned for the construction of the ultra-rigid and stable optical systems in LISA are detailed in Chapter Conclusions The experimental search for gravitational waves has become a worldwide effort, with several long-baseline detectors (LIGO, VIRGO, GEO600, TAMA300) now reaching sensitivities allowing there to be a real possibility of detection of astrophysical sources. Once the first detection is made, a whole new field in astronomy will open up. Like radio astronomy and X-ray astronomy, gravitational wave astronomy may allow the discovery of active and even exotic sources in our Universe currently unknown to us. However, to achieve the goal of large-scale gravitational wave astronomy will still require further improvements to detector sensitivities. Significant challenges involving the mechanical losses in the substrate and coatings [73] in addition to the thermal loading effects due to high levels of laser powers will have to be overcome. The work detailed in this thesis largely focuses on the measurement of the mechanical losses of various substrate materials proposed for future detectors; fused silica, sapphire and silicon. The last of these is investigated in detail in Chapters 5 and 6, in relation to third generation detectors operating with non-transmissive optics. Switching to a non-transmissive scheme with a material of high thermal conductivity, such as silicon, will allow

54 1.7 Conclusions 28 the issues of thermal loading to be overcome [74].

55 Chapter 2 Thermal Noise 2.1 Introduction As mentioned earlier in section , thermal noise associated with the mirror masses and lower stage suspensions is one of the most significant noise sources at the lower frequency range of the detectors. A mirror suspended as a pendulum is a mechanical system with many modes of oscillation (resonances) where thermal energy may reside. The Equipartition Theorem states that, in a classical system, the mean value in the energy of each quadratic term (often more generally referred as each degree of freedom) is equal to 1k 2 BT, through the application of the Boltzmann distribution for energy [75], where k B is Boltzmann s constant and T is the temperature in kelvin. This thermal energy is stored in the translation, vibration and rotation of the atoms in a mechanical system, and hence affects the coherent motion of the system as a whole. The thermal energy stored in the mirrors and suspensions of the interferometric gravitational wave detectors combine together to produce resultant displacement noise of the faces of the mirrors that are sensed. There are two ways in which this thermally induced motion may be reduced. One option is to cool the optics in the interferometer, therefore simply reducing the total thermal energy within the mirrors and lower-stage suspensions. However, cooling 29

56 2.2 Brownian motion 30 the optics whilst retaining a low-noise environment is nontrivial and is only being considered for future third generation detectors. Another option is to reduce the effects of thermal noise by using materials of low mechanical loss, or high Q. In this case, the thermally induced motion will lie predominantly within the narrow mechanical resonances of the system and thus, the higher the Q, the narrower the resonances and the lower the off-resonance thermal displacement noise. Exactly how this thermal noise is exhibited is detailed in the sections that follow. 2.2 Brownian motion The botanist Robert Brown observed that pollen grains left to float freely on the surface of water moved in a vigorous and irregular manner [76]. In 1905, Einstein showed that this motion resulted from the stochastic collisions of the water molecules with the pollen grain [77]. Einstein also showed that these impacts result in the pollen grain losing any initial kinetic energy, giving rise to a dissipation process, described by the viscosity of the water [78]. It is interesting to note that Einstein s three great publications in 1905 (Brownian motion, the photoelectric effect and special relativity) are so intertwined in the current search for gravitational radiation, originally inferred from Special Relativity and then formulated in his Theory of General Relativity in Fluctuation-Dissipation Theorem The Fluctuation-Dissipation theorem states that any dissipative linear system in thermal equilibrium exhibits fluctuations in its measurable parameters [79] [80] (a linear system is a system whose response is proportional to the applied force, or excitation). It states that there exists a relationship between the fluctuations in a system and the friction (which dissipates the energy within a system). The power spectral density of the fluctuating me-

57 2.3 Fluctuation-Dissipation Theorem 31 chanical driving force S F (f) is dependant on the dissipative (real) part of the mechanical impedance R[Z(f)] such that, S F (f) =4k B T R[Z(f)]. (2.1) Alternatively, the power spectral density of the mechanical displacement, x, is given by, S x (f) = k BT R[Y (f)], (2.2) π 2 f 2 in m 2 /Hz, where R[Y (f)] is the real part of the mechanical admittance. Until recently the choice of optics and suspensions has been predominantly based on models of the internal resonant modes as damped harmonic oscillators. The power spectral density of the displacement thermal noise, S x (f), of a mode of resonant frequency f 0 can be calculated to show that for Q>>1, S x (f) φ(ω), where Q is the Quality Factor and φ(ω) is the mechanical loss, both detailed later in Section 2.7. A resonant mode in a system with a high degree of dissipation will exhibit lower and broader spectral densities in displacement noise than a resonant mode in a system with low dissipation. It is this off-resonance thermal displacement noise which can set a limit to how accurately the position of the front face of the mirrors can be sensed. The test masses and suspensions in interferometric detectors have thus been fabricated from materials known to have a low loss factor, φ(f). For this reason, current detectors use fused silica for their mirrors, since the mechanical loss of fused silica is known to be very low. Measurements of φ(f) of silica and other materials at room temperature can be found later in Chapter 4. The GEO600 detector also uses fused silica suspension elements, where the other detectors use carbon-steel wires, allowing improved suspension thermal noise performance. However note that the analysis within Equation 2.1 and the following sections deals with single mode systems. When dealing with real, multi-mode systems, further considerations must be accounted for as discussed later in Section 2.8.

58 2.4 Sources of dissipation Sources of dissipation External sources of dissipation There are a variety of sources of external dissipation which may contribute to the level of thermal noise within an interferometric detector: Gas damping - where viscous damping is experienced by the suspensions and mirrors due to the friction of residual gas molecules Recoil damping - where energy may be lost from the pendulum suspension into a recoiling support structure Hysteresis and eddy current damping - due to any magnetic hysteresis loops and the passive eddy current damping implemented to reduce the level of motion of the upper pendulum stages of the suspension. Frictional (slip-stick) damping - at the suspension point and where the suspension elements meet the test mass Internal sources of dissipation - anelasticity Once all the sources of external dissipation have been suitably minimised, the dominant source of thermal noise results from the internal dissipation of the suspended optics. There are two kinds of (idealised) internal dissipation, viscous dissipation and structural dissipation. The internal dissipation can arise when a material is acted upon by a force and does not respond in an ideal elastic way (that is, obeying Hooke s law) but rather anelastically. When an ideal elastic material is acted upon by a force, a stress is produced (due to the material being acted on by a force) and this applied stress σ produces a resultant strain ε (stretching/compressing) within the material, related by the Young s modulus E, such that σ = Eε. In an

59 2.4 Sources of dissipation 33 elastic response, the stress and strain are directly related such that ε = Jσ, where J is the compliance of the material and is related to the stored energy due to the induced deformation. In the case of an anelastic material, there exists a finite relaxation time for the strain to fully develop after the deformation is applied. Therefore, the stress and strain can be modeled in their complex form, where the strain follows with a phase-lag φ such that, σ = σ 0 e iωt, (2.3) ε = ε 0 e i(ωt φ), (2.4) where φ is commonly referred to as the loss angle, where φ = 0 for an ideal elastic material. A phase-lag exists due to the system being acted on going through some transitional states, due to jumping via some intermediate variable (such as another internal degree of freedom, or some intermediate energy state) over a finite time. An example of anelastic relaxation arising from this coupling between the stress and strain can occur by the production and flow of heat in a system (thermoelastic damping) [81]. Due to the associated relaxation time in such a thermodynamic process, the internal dissipation will be distributed as a broad peak around a characteristic frequency (Debye peak). However, there may exist a number of distinct dissipation peaks over a range of characteristic frequencies associated with various anelastic relaxation processes. These Debye peaks may be separated in frequency by many orders of magnitude. When operating in a frequency regime far away from any of these peaks, the combined effect of the tails of the peaks is effectively constant with frequency [82]. Internal dissipation in the form where φ is approximately constant over a very broad range of frequencies has been noted by various authors [83] [84] [85] [86] [87] and has often been attributed to friction from lattice dynamics, such as dislocations [88] or point defect migration. The frequency band of gravitational wave detectors from a few Hz to a few 10 s of khz, usually lies in such a region.

60 2.5 Thermoelastic dissipation 34 Another possible phenomenon that would describe a frequency independent dissipation is to use a viscous term where the viscous coefficient varies as 1/ω. As mentioned earlier, thermoelastic dissipation is one particular example of an anelastic process, where an additional internal degree of freedom plays a part in the relationship between the stress and strain, in this case the temperature. This is due to the temperature being coupled to the strain through the non-zero thermal expansion coefficient. There exist other sources of internal damping that are not anelastic, such as creep, where the system exhibits a plastic response. An example of this would be when a system changes shape or structure and subsequently does not return to its original state when the applied force is removed. 2.5 Thermoelastic dissipation Thermoelastic loss was studied by Zener [89] and later quantified by Nowick and Berry in the study of heat flow across thin flexing beams or fibres [81]. This dissipation mechanism can therefore be directly applied to the suspension elements in the interferometers and the resulting thermal displacement noise calculated. However, this dissipation mechanism is also relevant to the test mass mirrors, as pointed out by Braginsky et al [90]. The thermo-mechanical properties of many crystalline materials, such as sapphire or silicon, results in the statistical fluctuations in temperature causing displacements in the front surface of the mirror due to the coefficient of thermal expansion, α. In fact, the level of this thermoelastic displacement noise can dominate over the other sources of thermal noise within the gravitational wave detection band in ground-based detectors. The level of thermoelastic thermal noise, S TE (f), in a test mass modeled as half-infinite (assumed where the laser beam diameter is considerably smaller than the dimensions of the mirror) at room temperature is given by, S TE (f) = 2 π 5/2 k B T 2 α 2 (1 + σ) 2 κ ρ 2 C 2 r 3 0f 2, (2.5)

61 2.6 Internal dissipation applied to an harmonic oscillator 35 where α is the linear coefficient of thermal expansion, C is the specific heat capacity, κ is the thermal conductivity, ρ is the density and r 0 is the radius of the laser beam where the intensity has fallen to 1/e of the maximum intensity. Thermoelastic dissipation will be discussed in greater detail in Section 6.8 in the analysis of the mechanical losses in thin silicon flexures. 2.6 Internal dissipation applied to an harmonic oscillator We have already discussed that internal damping can be modeled by harmonic analysis where there is a phase-lag associated between the stress and strain. For a simple harmonic oscillator this results in a complex spring constant. In the simplest picture, the resonant modes of the suspended optics in the interferometers, whether the modes associated with the pendulum or in the test mass, can be modeled as a harmonic oscillator using Newton s equation (F restoring = kx) subject to internal friction such that, F (ω) =mẍ + k(1 + iφ(ω))x, (2.6) where φ is the loss angle describing the phase-lag at which the strain follows the restoring force of the oscillator. Applying this model into the fluctuation dissipation theorem shows that the power spectrum of the thermal motion is given by, S x (ω) = 4k BT ω φ(ω)ω 2 0 m[ω 4 0φ 2 (ω)+(ω 2 0 ω 2 ) 2 ], (2.7) where ω 0 = k/m is the resonant frequency of the oscillator. The calculated level of thermal motion is also dependent on the nature of the dissipation. For example, as discussed earlier, the dissipation may be expressed as a viscous dissipation, resulting from a resistive force proportional to the velocity. In this case the loss factor φ(ω) is proportional to the frequency,

62 2.7 The internal loss factor 36 ω, intheformφ(ω) = βω, whereβ is a constant. In this case the frictional force can be written as F friction = bv where b = kβ and equation 2.6 would then become, F (ω) =mẍ k(1 + iβω)x, (2.8) where v = iωx. The calculated spectrum of thermal noise is also dependent on the frequency dependence of the dissipation φ. 2.7 The internal loss factor The internal loss factor, φ(ω), is a very important parameter for the correct estimation of the level of internal thermal noise. The loss factors for the pendulums and test masses in interferometric detectors are very low and therefore extremely difficult to measure directly at all the frequencies under consideration. Therefore we must look more closely at the form of the internal loss factor to investigate methods by which it may be calculated. F m k+iφ Figure 2.1: Illustration of a damped simple harmonic oscillator. Using Equation 2.6, the internal driving force, F (t), may be written as, F (t) =mẍ + k ( 1+ iω ω φ) x(t), (2.9) and in the treatment given, φ has been taken to be a constant and it has been assumed implicitly that F (t) and x(t) are both sinusoidal with frequency ω, so that x may be written as x(t) =xe iωt. Therefore, the velocity ẋ may be written as ẋ = iωx and Equation 2.9 becomes, F (t) =mẍ + kẋ φ(ω)+kx, (2.10) ω

63 2.7 The internal loss factor 37 or, [ ] F (t) =m ẍ + ω2 0φ ω ẋ + ω2 0x, (2.11) where ω 0 is the angular resonant frequency, ω is the frequency of the applied applied oscillating force, k is the spring constant and m isthemassoftheoscillator. This can be compared with the commonly encountered case of viscous damping, where in place of the complex spring constant of Equation 2.9, there is a velocity-dependent damping term γẋ giving F (t) = mẍ + γẋ + kx which may be written as, [ ] F (t) =m ẍ + γ mẋ + ω2 0x, (2.12) or as, [ ] F (t) =m ẍ + ω 0 Q ẋ + ω2 0x, (2.13) where Q = mω 0. Comparing Equations 2.12 and 2.13 shows that close to the γ resonance where ω = ω 0, Q = 1 φ(ω 0 ). (2.14) It is difficult to measure the mechanical loss in a material far from the internal resonances and over a broad range of frequencies. Therefore, the mechanical loss is evaluated from the measured Q-value of the internal resonance(s) of the system under study, as seen from Equation Chapters 4, 5 and 6 are dedicated to such measurements with the aim of achieving a better understanding of the internal loss factors of various materials being considered for use in both current and future gravitational wave detectors. A more detailed account of the technique used for the measurement of the Q-value of a sample isgiveninsection4.3.

64 2.8 Thermal noise resulting from spacially inhomogeneous mechanical dissipation Thermal noise resulting from spacially inhomogeneous mechanical dissipation When considering the thermal noise models described in Equations 2.1 and 2.7, the laser beam reflected off the front face of the mirrors detects the spatial average of the sum of the thermal displacements from the low-frequency tails of all of the test mass resonant modes [91]. Early calculations assumed that the internal dissipation of the optics was homogeneously distributed and that the motion of each mode was uncorrelated, the noise contributions therefore adding in quadrature. However, Levin [92] and others [93] [94] introduced a more general approach to the problem that allowed the actual spatial distribution of loss and the detailed shape of the laser beam to be taken into account. Using Levin s approach, which directly applies the fluctuation dissipation theorem to the system, the power spectral density of the thermally driven motion of the front face of a test mass is, S x (f) = 2k BT W diss, (2.15) π 2 f 2 F 2 0 where W diss is the power dissipated when applying a notional oscillatory force of magnitude F 0 on the front face of a test mass mirror. The pressure associated with the laser intensity has a profile identical to that of the laser spot. W diss =2πf ε(x, y, z)φ(x, y, z, f)dv, (2.16) vol where ε is the energy density of the elastic deformation under the peak applied pressure F 0. In the case for homogeneous loss where the beam diameter is considerably smaller than the dimensions of the mass, it is possible to model the test mass as being half-infinite and the thermal displacement noise due to Brownianmotioncanbeshowntobe, S x (f) = 2k BT πf 1 σ 2 2πEr0 φ substrate (f), (2.17)

65 2.9 Interferometer suspension thermal noise sources 39 where φ substrate (f) is the substrate mechanical loss, E and σ are the Young s modulus and Poisson s ratio of the material and r 0 is the radius of the laser beam where the intensity has fallen to 1/e of the maximum intensity. Note that this model shows the power dissipated to be directly related to the elastic deformation caused by the notional pressure on the front face of the mirror. It is therefore clear that dissipation physically located close to the front surfaces of the mirrors will therefore contribute to more dissipated power and will thus contribute a greater level of thermal displacement than dissipation located far from the front face of the mirror. This is of significant importance as regards interferometric detectors, since it is necessary to maintain low optical losses, to use ion-beam-sputtered multilayer dielectric coatings to give high reflectivity on the front surfaces of the test masses. These coatings are known to have higher levels of mechanical loss (where φ coating a few 10 4 [97]) than that of the substrate material of the mirror (φ substrate 10 8 ). For this reason, possible ways to decrease the mechanical dissipation associated with the mirror coatings is currently a significant area of study. Measurements of the mechanical dissipation of test masses with mirror coatings are presented in Chapter 4 in Section Interferometer suspension thermal noise sources Pendulum modes The test mass mirrors in gravitational wave detectors are suspended as pendulums in order to attenuate seismic noise from ground motion in the detector s frequency band of operation, the suspension lengths being chosen such that the resonant frequency of the pendulum is below the detection band. Some further details and investigations on the dynamics of a simple pendulum constructed with a fused silica ribbon are detailed in Appendix A.

66 2.9 Interferometer suspension thermal noise sources 40 There is thermal noise associated with the pendulum mode, resulting from the mechanical loss of the suspension material from which the test mass is hung. The mechanical loss of a pendulum is related the mechanical loss of the suspension material by [63], φ pend (ω 0 ) φ mat (ω 0 ) ξn TEI, (2.18) 2mgl where there are n suspension elements, T is the tension per suspension element, E is the Young s modulus of the suspension material and I is the moment of cross-sectional area. Additionally, ξ takes the value of 1 or 2 depending on whether the wire positions constrain them to bend at the top, or the top and bottom, respectively [98]. The power spectrum of the displacement of the front surface of the mirror due to the pendulum mode of the suspension, above the pendulum resonant frequency, can be found using Equation 2.7, S x (ω) 4k BTφ pend (ω) ω0 2 m ω, (2.19) 5 where m is the mass of the test mass, ω is the angular frequency in the operating range of the detector and ω 0 is the angular resonant frequency of the pendulum mode. It is assumed that the laser spot is suitably positioned on the test mass so that the thermal noise from the rocking or rotational modes of the pendulums do not couple significantly. Violin modes It is also necessary to consider the effect of the violin modes of the suspension elements on the displacement noise of the front face of the mirrors. The violin modes unfortunately form a harmonic series that lie within the detector s frequency band. If the damping is homogeneous, then the loss associated with the suspension violin modes is related to the pendulum mode loss such that, φ violin (ω) =2φ pendulum (ω), (2.20)

67 2.10 Brownian and thermoelastic noise associated with dielectric mirror coatings 41 which is true in the case when the rocking mode of the pendulum has been constrained [98] [86]. The resultant off-resonance displacement introduced to the front face of the test mass from the violin modes of the suspension is much smaller than that resulting from the pendulum mode and the internal modes of the test mass. Since the suspension is constructed from a low loss material, most of the thermally induced motion lies within the narrow frequencies of the resonances. Since these peak are narrow, they can be easily removed from the signal from the detectors Brownian and thermoelastic noise associated with dielectric mirror coatings The dissipation resulting from the dielectric coatings applied to the test masses to form mirrors has been identified as being significant [152]. These coatings consist of multi-layers of typically silica (SiO 2 )andtantala(ta 2 O 5 ). The dissipation from these coatings will thus contribute strongly to the thermal displacement noise since this region is strongly sensed by the incident laser beam, as described by Levin, see Section 2.8. The power spectral density of the Brownian motion associated with mirror coatings can be written as [152] [153], S x (f) = 2k { BT 1 φ π 3/2 substrate + 1 ( d E f ωe π w E φ + E )} E φ, (2.21) where f is the frequency in Hz, T is the temperature in Kelvin, E and E are the Young s Modulus values for the substrate (bulk) and coatings respectively, φ and φ are the mechanical loss values for the coating for strains parallel and perpendicular to the coating surface, d is the coating thickness and w is the field amplitude radius. There also exists a form of thermoelastic dissipation associated with the different thermo-mechanical properties of the multi-layers [99] [100]. This can also set an important limit to detector sensitivity and the power spectral den-

68 2.11 Combined thermal noise in a detector 42 sity can be written as S x (f) 8 2k B T 2 π 2πf l 2 2r 2 0 (1 + σ s ) 2 C2 avg C 2 s α 2 s κs C sub 2, (2.22) where the subscript, s, refers to a substrate property and { ( 2 Cs α ( 1+σ +(1 2σ s ) E )) 1 } 2, (2.23) 2α s C avg 1 σ 1+σ s E s avg where the averaging takes into account the relative thicknesses of the alternating layers of two types of material. Experimental measurements are consistent with there being both Brownian and thermoelastic displacement noise [101]. Direct measurements of the thermal displacement noise from multi-layer coated optics have started [102] and appear to be of a level marginally lower than expected from the mechanical loss measurements [103] Combined thermal noise in a detector To evaluate the effect of total thermal noise in the interferometer to the achievable displacement sensitivity, it is necessary to combined the contribution of the thermally driven motion of the optics from the internal modes of the test mass (Equation 2.17), from the mirror coatings (Equation 2.21), in addition to the suspension contribution from the pendulum mode (Equation 2.19) and the violin resonances. A closer study of the thermal noise expected in the GEO600 detector, compared to the other sources of noise, is presented in Chapter 4 in Section 4.5. An illustration of the expected levels of thermal noise in a typical mirror planned in the upgrades to the first generation of detectors, taking into account all the previous considerations, has been calculated by Rowan and colleagues [104], as shown in Figure 2.2.

69 2.11 Combined thermal noise in a detector 43 linear spectral density of thermal displacement [m/ Hz] substrate brownian substrate thermoelastic coating total (substrate + coating) suspension Frequency [Hz] Figure 2.2: Calculated levels of expected thermal and thermoelastic noise in a single suspended fused silica mirror illuminated by a laser beam of radius r 0 = 3.9 cm, assuming a test mass loss of [105], and an ionbeam sputtered coating formed from alternating multi-layers of SiO 2 and Ta 2 O 5 [106].

70 Chapter 3 Influence of temperature and hydroxide concentration on the settling time of Hydroxy-Catalysis Bonds 3.1 Introduction The technique of hydroxy-catalysis bonding can be used to joint materials whose surfaces can be hydrated and dehydrated in the presence of a hydroxide catalyst. The technique was developed and patented by D.H. Gwo [107] [108] at Stanford University for bonding fused silica, for use in the Gravity Probe B mission [109] launched in 2004, see Figure 3.1. This mission uses fused quartz gyroscopes for measuring a frame-dragging effect predicted by Einstein s General Theory of Relativity. The gyroscopes sit in a fused quartz block that is silicate (hydroxy-catalysis) bonded to a fused quartz telescope such that the assembly of components does not distort or break when cooled to the cryogenic temperatures experienced. The hydroxy-catalysis bonding technique has been developed further by researchers in Glasgow and used in the construction of the ultra-low loss, quasi-monolithic fused silica suspensions installed in the GEO600 ground based gravitational wave detector [112] [113], with variants of 44

71 3.1 Introduction 45 the GEO suspension design planned for the Advanced LIGO system [114] in addition to possible use in future upgrades to VIRGO [115]. Gravity Probe B precision gyroscope Figure 3.1: Launch of the NASA Gravity Probe B Relativity Mission on April 20th The hydroxy-catalysis bonding technique was developed for the jointing of components for the telescope assembly to ensure the long-term stability of the components position. Astronomical events, such as massive coalescing black holes and resolved and unresolved galactic binary stars, create gravitational radiation at frequencies around 10 4 Hz to 1 Hz. Ground-based interferometers are expected to be limited by gravitational gradient noise in this frequency range [116]. For this reason, space-based interferometers are being planned and built. In addition, at these frequencies, to achieve the sensitivities required the arms of these interferometers are required to be very long, making space the only place where it is feasible is to build such a detector. The Laser Interferometer Space Antenna (LISA) mission [49] is proposed for launch in The LISA interferometer is formed between three spacecraft, each separated by 5 million kilometers, following Earth s orbit around the Sun, each of which contains freely floating test masses, see Figure 3.2. There are many technological challenges in launching a mission such as LISA. In order to verify some aspects of the expected performance of such a detector, a demonstrator mission called LISA Pathfinder [110] is planned

72 3.1 Introduction 46 Earth 6 5x10 km 20 o 60 o Venus Sun Mercury Figure 3.2: The planned orbit of the LISA space-based gravitational wave antenna. for launch in LISA Pathfinder will test some of the key components of precision interferometry in space, such as laser stabilisation, modulators, phase readout of interferometric signals, and drag-free control. On board the three spacecraft of LISA are located local interferometers which accurately measure the phase of the laser light received from the other two spacecraft. The LISA Pathfinder mission also requires a similar interferometric setup, although only being on a single spacecraft. The performance of both the missions depends greatly on the mechanical stability of the optical bench where the local interferometers are mounted [111]. The proposed design of optical bench incorporates silica optical components, attached to a Zerodur (low-expansion glass ceramic) slabs using silicate bonding to mount the silica optical pieces precisely and rigidly to the Zerodur bench, see Figure 3.3. One possible disadvantage of this technique is that the time taken for a bond to set at room temperature is in the region of a few tens of seconds. This only allows a short period of time in which to align the various components on the optical bench. Described here are aspects of research carried out to investigate ways to extend the bonding time to allow precise pre-alignment of the optical components.

73 3.2 The chemistry of hydroxy-catalysis bonding 47 bonded silica optical components Zerodur slab 212 mm Figure 3.3: The engineering model optical bench interferometer for the LISA Technology Package (LTP) which will fly on board LISA Pathfinder. 3.2 The chemistry of hydroxy-catalysis bonding Hydroxy-catalysis bonding has typically been used for jointing oxide materials. Two surfaces (flat to λ/10, where λ = 633 nm) in close proximity may be bonded using a very small quantity of aqueous hydroxide solution, such as sodium or potassium hydroxide. This solution is commonly referred to as the bonding solution. Through a series of chemical processes this solution forms a silicate gel that solidifies over time. A strong, rigid, yet very thin bond thus forms between the pieces being jointed. A short description of the stages of the bonding process is in the case of jointing fused silica is given below Hydration The surface of silica is hydrophilic and will attract OH ions to fill any open bonds from the silica. This process is known as hydration. When fully hydrated, as shown in Figure 3.4, there are 4 to 6 silanol (SiO 3 OH) groups per nm 2 [117]. It is clear that contaminants on the silica surface will inhibit hydration. The silica surfaces to be jointed are thus taken through a cleaning process, described in section 3.3.1, to remove any contaminants and to ensure

74 3.2 The chemistry of hydroxy-catalysis bonding 48 maximum hydration. - Attacted OH ions form silanol molecules on the surface of silica O O OH Si O OH Si Si O Si O O OH Si O Si bulk substrate (silica) Figure 3.4: A simplified, 2D schematic of the surface of silica when hydrated Etching Placing a solution with a high concentration of OH ions on the surface of silica causes etching to take place. This starts when additional OH ions form weak bonds with silicon atoms on the substrate surface, as shown in Figure 3.5(a). This increases the coordination number (number of bonds) of the silicon atom from four to five or six. As a result, the original lattice bonds will weaken with the possibility of the silicate molecule breaking away from the bulk structure, (b) (c). A quantity of liberated Si(OH) 5 molecules therefore become available in solution. OH ions form weak bonds with surface Si atoms Si bonds to the bulk can therefore become weaker Liberated Si from the bulk silica in the form of silicate Si Si O O H O OH Si O H O OH Si O Si Si O O H O OH Si O H O OH Si O Si Si Si(OH) 5 O OH OH OH OH Si bulk substrate (silica) bulk substrate (silica) bulk substrate (silica) (a) (b) (c) O - O O Figure 3.5: Schematic of the stages in the etching process due to an hydroxide solution applied to the surface of silica.

75 3.2 The chemistry of hydroxy-catalysis bonding Polymerisation - a definition for the settling time The settling time of a bond between two pieces of silica is typically defined as the time for the bonding solution to become rigid. That is, the point at which the silica pieces can no longer move relative to one another without causing damage to the sample surfaces. To understand where in the chemical reaction this takes place, we must look closer at what happens during the formation of Si(OH) 5 molecules. Si(OH) - 5 Si(OH) 4 +OH- (3.1) Above ph 11 the available OH (hydroxyl) ions convert any Si(OH) 4 to easily dissolved silicate ions (Si(OH) 5 ), thus allowing the silica to remain in solution. However, below ph 11, the silicate ion hydrolyzes to soluble Si(OH) 4 and OH allowing the etching process to continue. When the concentration of Si(OH) 4 molecules reaches 1 2%, the bonding solution becomes rigid [118]. This occurs because Si(OH) 4 is a monomer which likes to form a polymer arrangement as shown in Figure 3.6. This particular example illustrates how the polymerisation process can eventually bond together two nearby silica surfaces. Let us assume, for the bonding solution considered (at least 0.1 mol bulk substrate (silica) Si O OH Si OH OH H 2 O OH OH Si OH O OH Si OH O Si bulk substrate (silica) Figure 3.6: Mechanism for polymerisation for the formation of siloxane chains from a solution containing silicate. l 1, that is 5.6g KOH dissolved per litre of pure, de-ionised water) that around 1 2% Si(OH) 5 is liberated by etching and present in solution before the ph drops below 11. If the rate at which the conversion of Si(OH) 5 Si(OH) 4

76 3.2 The chemistry of hydroxy-catalysis bonding 50 and the resultant polymerisation is fast compared to the rate of etching (liberation of silicate from the substrate surfaces), then the settling time can be defined as the time taken for the ph of the bonding solution to drop to ph=11. Therefore, perhaps counterintuitively without prior knowledge of the processes involved, the settling time will increase as the concentration of OH ions increases. This is because the greater the concentration of OH ions in solution, the longer the etching process continues. This results in deeper etching and consequently a thicker bond layer for greater OH ion availability. A thicker bond layer may significantly affect the properties of the bond, such as the mechanical strength. A graph can therefore be shown to illustrate the changing molarity of [OH] (where [X] denotes the concentration of X in solution) against time (Figure 3.7). This definition of settling time is not exact. For example, - molarity of OH (b) (a) representation of a typical (exponential) chemical reaction molarity corresponding to a ph of 11 Bonding time for conc n (a) t Bonding time for conc n (b) time Figure 3.7: Graph of the rate of change of [OH] over time for two solution with initial OH concentrations labelled (a) and (b). the cation in solution with the hydroxide, typically a Group-I element from the periodic table, are known to dissolve silica in neutral solutions [118]. This most likely occurs since these elements combine with any any Si(OH) 5 as it is

77 3.3 Catalysis bonding procedure 51 liberated from the silica surface by the catalytic OH ion, helping to hold the reversible process in Equation 3.1 to the left. This helps prevent the solution from saturating. There are also other speculative processes existing that may affect the rate of bonding. For example, sodium ions (Na + ) have a stronger coulomb force than potassium (K + ) and may therefore help the reaction to propagate deeper into the bulk substrate [107]. Impurities and contaminants have also been observed to affect etch rates of glasses. Van Lier et al. discovered that NaCl (salt) increased the rate of etching of quartz in water by factors of 4, 14 and 67 in 10 3,10 2 and 10 1 mol l 1 solutions [119]. Silica is known not to etch in very pure water, free from impurities [120]. Clearly any study of the dependencies in the settling time for the hydroxy-catalysis process must be carried out in a controlled and clean environment with specially prepared samples. 3.3 Catalysis bonding procedure Silica surface preparation Surface preparation is of significant importance since surface hydration (section 3.2.1) is required for the initial stages of the hydroxy-catalysis process. For this reason, a thorough cleaning process must be implemented so that the silica surfaces are left free from contaminants and significantly hydrated. Also for successful, repeatable bonding of samples we require a typical global surface flatness of λ/10, where λ = 633 nm. Surfaces that are less flat may still be bonded, however bonding solutions already containing silicate may be required in order to fill in the larger spaces between the samples. The steps undertaken for the necessary surface preparation for silica samples are as follows: silica samples are rinsed in a stream of de-ionised water. the surfaces to be bonded are then rubbed with a paste of cerium oxide for seconds, i.e. a light abrasive cleaning.

78 3.3 Catalysis bonding procedure 52 silica samples again rinsed with de-ionised water to remove excess cerium oxide from the surfaces. the bond surfaces are then rubbed with a paste of sodium bicarbonate. Sodium bicarbonate is required to clean off microscopic particles of cerium oxide which are electrostatically bonded to the silica surface. silica samples are given a final, thorough rinse in de-ionised water. surface cleanliness can be verified if a film of water will sit across the entire sample with no coagulation. the bond surfaces are dried by giving a single wipe across the sample with a clean, non-abrasive cloth lightly dampened with methanol. water coagulates on unclean silica surface 1 2 surface clean with cerium oxide paste cerium removed with sodium bicarbonate 3 4 water settles evenly on clean silica surface Figure 3.8: Images of the major steps involved in preparing/cleaning of the silica sample surface prior to bonding. The purpose of this cleaning process is primarily to remove surface contaminants and allow the surface silicon atoms to become hydrated. Additionally,

79 3.3 Catalysis bonding procedure 53 such an abrasive process will also increase the local surface roughness, increasing the relative surface area of the surface of the silica in contact with the bonding solution and will increase the initial etching rate Bonding procedure The prepared silica samples can then be easily bonded using the hydroxycatalysis bonding technique. Here we carefully apply around 0.4 µl cm 2 of bonding solution to one of the surfaces to be bonded using a pipette. Then the second sample is carefully lowered onto the first sample where the bonding solution was applied. If the surfaces are of the required flatness and cleanliness then this small quantity of solution will evenly spread out across the entire contact region. If not physically held, the samples will at first be very free to move with respect to one another in the plane parallel to the surfaces to be bonded, although being very hard to lift off each other due to the surface tension of the solution. The solution will gradually become more viscous (as etching takes place and the concentration of silicate in solution increases) making the samples harder to move, until at a certain point the bond appears to take. At this point, the bond will be difficult to break without causing some level of damage to the sample surfaces. However, placing the bonded samples in an ultrasonic bath of very weak etching solution, even a number of hours after bonding, will often allow the samples to lift off each other. This would imply that the observed grab is likely due to a large degree of polymerisation within the bond layer yet with very few siloxane networks having propagated across the entire bond layer, adjoining the adjacent surfaces.

80 3.4 Catalysis bonding dependence on hydroxide concentration Catalysis bonding dependence on hydroxide concentration The rate of most chemical reactions, depending only on the concentration of one reactant, can be described by: rate = k[x] α (3.2) This is known as the Rate Law for chemical reactions, where k is the rate constant, [X] is the concentration of substance X and α is known as the order of the reaction. The hydroxy-catalysis reaction can be approximated a first order reaction (α=1), the reaction being etching of the silica substrate dependent on the OH ion availability/concentration. Therefore, we can say: [OH] = k[oh] (3.3) or for [OH] as a function of time, [OH](t) =Ae kt (3.4) Let us define the initial concentration of OH ions in the bonding solution such that, [OH] i = A (3.5) and define the final concentration of OH ions in the bonding solution (i.e. at a ph=11 when coagulation takes place) as, [OH] f = Ae kt (3.6) Now, taking the natural logarithm of each equation will give us, ln [OH] i = lna (3.7) ln [OH] f = lna kt (3.8)

81 3.4 Catalysis bonding dependence on hydroxide concentration 55 Therefore, ln [OH] i ln [OH] f = k t (3.9) where t is the settling time. This result means that we can define settling time as a function of [OH] or even ph. t = ln [OH] i ln [OH] f k t = (ph i ph f )ln10 k (3.10) (3.11) A graph of initial ph verses settling time (t) should give a straight line, with gradient = k/ ln 10 and a y-intercept =ph i, since, ph(t) = k ln 10 t +ph i (3.12) 15 settling time data points fit where: y = x calculated initial ph Settling time (s) Figure 3.9: Graph of ph (calculated ph =14 log 10 [OH]) against settling time. Figure 3.9 shows that some non-linearity occurs at large concentrations greater than around ph=14. However, at lower ph levels there appears a linear relationship that may give us more information regarding hydroxy-catalysis

82 3.5 Temperature dependence of settling time 56 bonding using the presented analysis. A result of k = s -1 is therefore obtained and a characteristic time for the reaction (for concentration to decrease by 1/e) of τ = 308 seconds, approximately 5 minutes. The results obtained here clearly show that increasing the concentration of hydroxide in the bonding solution will increase the settling time for the bond to set or grab. Over the ph range around there appears a trend that closely agrees with the analysis presented, see Figure 3.9. This method for increasing the settling time gives the option in applications such as SMART-2 for increasing the time available for precise optical alignment of various optical components. Two factors still require investigation. Firstly, the relative thickness of the bonding layer may increase when increasing the concentration of the hydroxide bonding solution since the etch time increase. This may have an affect on the mechanical strength of the bonds. Secondly, the mechanical loss associated with the bond will increase as the bond thickness increases. In the case of the bonded optics of ground based detectors, this may have implications for the levels of thermal noise [121] [122] and therefore would need to be considered when altering hydroxide concentrations. 3.5 Temperature dependence of settling time Experimental setup - temperature controlled environment A different way to increase the settling time is to lower temperature at which bonding occurs. Reducing the temperature will result in the reactants (whether molecules or atoms) having a lower level of kinetic energy and therefore decrease the rate of collisions. In general, chemical reaction will follow the Arrhenius equation [123] which describes an exponential dependence on temperature. A temperature controlled environment would allow investigations into the dependence of the settling time of the hydroxy-catalysis bonding on temperature. Therefore an enclosure was designed and built to enable a user to

83 3.5 Temperature dependence of settling time 57 carry out bonds at given temperature in the range 0 20 o C [124]. Two internal cross-blower fans are used to circulate the air in a box of approximately 0.35m 3 over heat-sinks cooled by using Peltier devices, as seen in Figure A capillary thermostat was used to sense the internal temperature and allow control by means of regulating the power to the Peltier devices. The inside oftheboxwasmadefromlexan R polycarbonate (dimensions 1.0 m 0.7 m 0.5 m) and was insulated with polystyrene and covered with aluminium foil to reflect away radiated heat. The front of the box was double-glazed with polycarbonate with a 16 mm air gap for optimal thermal insulation [125]. Two glove holes on the front window allowed access for bonding to be carried out within the environment. These could be blocked off by aluminium foil covered polystyrene stoppers when internal handling was not required. capillary thermostat circulated air cross-blower fan heat sink peltier heat extraction glove holes work area Figure 3.10: Schematic diagram of temperature controlled environment constructed for carrying out hydroxy-catalysis bonding. Two sets of results were taken using this temperature controlled environment. A preliminary set of measurements of the settling time for bonding with help from E. Elliffe and were measured by continually moving the bonding silica samples manually. Two bonds were made at each temperature. The results for the measured settling time as a function of temperature are shown in

84 3.5 Temperature dependence of settling time 58 temperature readout display thermostat control fan and heatsink (heat extraction) glove box internal air ciculation Figure 3.11: Picture of temperature controlled environment constructed for carrying out hydroxy-catalysis bonding.

85 3.5 Temperature dependence of settling time 59 Figure It is clear that, for the bonds studied here, reducing the temperature at which bonding took place significantly increased the settling time of a bond. 300 Average settling time results 250 (0.23 mol l -1 NaOH) Fit: y = 347.9*exp(x/7.3)+36.3 settling time (s) o T ( C) Figure 3.12: Plot of settling time as a function of temperature for the hydroxycatalysis bonding technique when using 0.23 mol l -1 NaOH solution (9.2g NaOH per litre of water, a typical concentration used by Gwo et. al [107]) with exponential fit Experimental setup - precise measurement of bond settling time The repeatability of measuring the settling time of bonds manually is clearly a problem. Firstly, there may be discrepancies as to when the end-point, or grab, of the bond occurs. Secondly, the more the silica samples are moved around with respect to each other, the more the bonding solution will be spread across a larger surface. It would seem likely that the observed bonding time will therefore shorten as the silica samples are increasingly moved around. This occurs since the bonding solution is spread more thinly, causing the relative proportion of surface area of silica to volume of hydroxide solution to increase. For this reason, a more repeatable method for measuring the bonding time was sought.

86 3.5 Temperature dependence of settling time 60 Initially, it was thought this could be achieved by physically shaking the lower silica sample at a given frequency as the upper sample was placed on with hydroxide solution between [126]. The phase difference was monitored between the force applied to the lower silica sample and the photodiode readout monitoring the upper silica sample. At first, when little silicate is dissolved by etching, the viscosity of the solution will be low. The phase difference would therefore be large. Over time, as the quantity of dissolved silicate increases, the solution viscosity will increase causing the phase difference to decrease. When the bond finally grabs, the phase difference would be zero. The simplicity of this method of measurement would likely imply a large degree of repeatability. However, two significant problems arose with this method. The upper silica sample would wander across the surface of the lower silica sample due to the shaking of the lower sample and the cushion of solution between them. This then would spread the bonding solution out into a thinner layer and also cause the upper sample to move outwith the view of the readout sensor. Also, the upper piece of silica would often only move for a short time (few seconds) and then become stuck, giving a phase readout of zero. This was likely due to either two hills on the silica surfaces coming into contact and sticking, or some small particle in the bond layer linking the two surfaces. A repeatable readout of the settling time was impossible to achieved. Therefore a similar scheme was developed to separate more efficiently the relative motions of the two silica samples being bonded [127]. The upper piece of silica was held and an oscillating force applied to it by means of a loudspeaker and elastic band. This upper sample was then brought into contact with the lower silica sample, which was rigidly clamped. The bonding solution was carefully placed between the surfaces as they were brought into contact. During bonding, the amplitude of the upper silica sample s motion decreased due to the friction in the bonding solution increasing. Again a simple photodiode readout system was implemented for recording the amplitude over the duration of the

87 3.5 Temperature dependence of settling time 61 bonding process up to the grab point. The typical magnitude of the motion of the upper silica sample was found to lie in the range of ±100 to 500 µm. The readout setup can be seen in Figures 3.13 and 3.14, with the readout circuit used shown in Figure shadow sensor flag loudspeaker holder rubber band silica flat holder silica ear LED Figure 3.13: Picture of shaking and readout scheme used to measure the settling time for hydroxy-catalysis bonds. rigid clamp mechanical oscillation of holder holder for silica ear loudspeaker spring or rubber band surfaces to be bonded silica flat silica ear Figure 3.14: Schematic of the sample and the mechanical oscillation used for measuring the settling time for hydroxy-catalysis bonds. Note that no vertical downward load was applied to the silica ear and the samples were held together during testing only by the surface tension of the bonding solution.

88 3.6 Temperature dependent bonding results Ω 300Ω flag (shadow) shadow cast between photodiodes loudspeaker d 100kΩ 30V 10,000µF (electrolytic 50V) - OPA227 + GND silica flat silica ear Readout signal filtered LED circuit silica samples photodiode readout circuit Figure 3.15: Readout circuit diagram for measuring the amplitude of shaking of the upper silica sample as it is hydroxy-catalysis bonded to a lower clamped silica sample. The optical sensing was carried out using an infrared LED (810nm) with a beam angle of 20 used to illuminate two mm single-element silicon planar photodiodes separated by 500µm where a parallel shadow was cast by the 4 mm wide flag mounted on top of the silica ear sample holder. 3.6 Temperature dependent bonding results A constant amplitude of shaking was used for all the results taken. The initial amplitude of shaking was observed to be around ±500 µm. This amplitude would decrease by around a factor of 5 during the bonding and would come to a sudden stop as the bond grabbed, as likewise observed when shaking the bond by hand. A typical plot of sample amplitude as a function of time can be seen in Figure The frequency of oscillation was set at 14 Hz and was chosen because this frequency coupled very strongly and cleanly to the sample holder. On several occasions the bonding samples appeared to grab in this setup when in fact the bonding solution was still fluid. This again may be an example of parts of the silica surfaces coming into contact, possibly due to contaminant particles on the bonding solution. In these cases the silica ear holder was gently tapped by hand and the shaking would freely continue.

89 3.7 Temperature dependent bonding analysis 63 silica ear amplitude readout (V) settling time result taken at 0 o C time (s) Figure 3.16: Example of silica ear shaking amplitude as a function of time during bonding using readout scheme described. A set of results for the bonding time was carried out using 0.1 mol l 1 KOH solution at 0, 5, 10, 15 and 20 o C. Potassium hydroxide, instead of sodium hydroxide, was chosen since other results suggest that potassium hydroxide bonds are stronger [108]. The results can be seen in Figure Each point shows the average of between 8 and 10 measurements. A total of 69 bonds were fabricated for these results. However, sometimes the bonding solution did not apply correctly to the silica surfaces (due to confined space and restricted visibility of samples within the setup used) and the results had to be excluded. This was observed by eye where the solution occasionally missed the target area between the samples or some solution continued to remain in the tip of the pipette. 3.7 Temperature dependent bonding analysis The reaction rate of the hydroxy-catalysis reaction should follow the Arrhenius equation [123], with a reaction constant, k. The rate of any reaction can be

90 3.7 Temperature dependent bonding analysis Average settling time results (0.1 mol l -1 KOH) Fit: y = 612.1*exp(x/34.2) settling time (s) o T(C) Figure 3.17: Plot of settling time as a function of temperature for the hydroxycatalysis bonding technique when using 0.1 mol l -1 KOH solution with exponential fit. defined by: rate = k[a] α [B] β (3.13) = Ae Ea k BT [A] α [B] β where [A] is the concentration of the first reactant, [B] is the concentration of the second reactant, E a is the activation energy for the reaction, k B is Boltzmann s constant and T is the temperature in Kelvin. The factors α and β describe the order of the reaction. Assuming hydroxy-catalysis is a first order reaction dependent only on the concentration of available OH ions, then we can say, rate = Ae Ea k BT [OH], (3.14) where [OH] is the concentration of OH ions in the bonding solution. Another reasonable approximation we can make is to say that the settling time is proportional to the inverse of reaction rate: settling time 1 rate. (3.15)

91 3.7 Temperature dependent bonding analysis 65 This now gives, settling time = constant e Ea k B T [OH] (3.16) Taking the natural logarithm of equation 3.16 will therefore yield, ( ) 1 ln (settling time) = ln (constant) + ln + E a [OH] k B T, (3.17) Now, for each bond, the concentration of OH can be assumed to be the same at the start (since the hydroxide solution s concentration was always 0.1 mol l 1 ) and likewise the same at the end (see definition of settling time in Section 3.2.3). Therefore the average concentration of OH ions in the bonding solution is likely to be constant at all temperatures. Assuming this to be true, a plot of ln(s), where s is the settling time, against 1/T will have a gradient of E a /k B, as seen in Figures 3.18 and The gradient of the best fit slope for the preliminary results, using 0.23 mol l 1 NaOH, is Multiplying this by Boltzmann s constant (k B = JK 1 ) we calculate E a =(9.6 ± 1.7) J/molecule, or (5.8 ± 1.1) 10 4 Jmol 1. This is equivalent to 0.60 ± 0.11 ev for the activation energy per molecule of OH (at room temperature k B T 1 ). Note that this value does not correspond to one specific reaction. It instead describes the activation 40 energy for a compound reaction, that is, the etching and polymerisation processes in the bonding. However, since the etching is likely to be the principal reaction over the time taken to bond silica samples together, the activation energy may largely correspond to the energy given per mole of hydroxide in the etching of silica. The gradient of the best fit slope for the results using the readout scheme described earlier and 0.1 mol l 1 KOH, is This would correspond to an activation energy of 0.54 ± 0.06 ev per molecule of OH. Let s look more closely at the results taken with the readout scheme shown in Figure There appears a very strong linear relationship here with the

92 3.7 Temperature dependent bonding analysis 66 exception of the data point at room temperature (20 o C). There were however two differences in the method in which this result was taken to the others results presented. The results at 20 o C were not taken in the temperature controlled environment. Perhaps the readout system was (unintentionally) altered when the readout was moved inside of the unit. Another difference is that the potassium hydroxide solution was filtered for all the temperature points except for the 20 o Cpoint. It must also be noted that the settling times were comparable when using the two different hydroxide solutions (sodium hydroxide and potassium hydroxide) despite the fact the concentrations were different (0.23 mol l 1 NaOH and 0.1 mol l 1 KOH). This suggests the different Group-I hydroxides may have quite different corresponding settling times when used in hydroxycatalysis bonding. However, as yet we do not know whether the action of hand-movements on the silica samples significantly affected the settling times for the experiment using NaOH. 5.5 Preliminary temperature dependence results (0.23 mol l -1 NaOH) Ln[settling time (s)] x x x10-3 1/T (K -1 ) Figure 3.18: The natural log of the settling time plotted as a function of 1/temperature for the hydroxy-catalysis bonding technique when using 0.23 mol l -1 NaOH solution (using the preliminary results - measured by hand) with linear fit.

93 3.8 Conclusions Temperature dependence results (0.1 mol l -1 KOH) 5.5 Ln[settling time (s)] /T (K -1 ) Figure 3.19: The natural log of the settling time plotted as a function of 1/temperature for the hydroxy-catalysis bonding technique when using 0.1 mol l -1 KOH solution, using the readout scheme described, with linear fit. 3.8 Conclusions It has been shown that varying the temperature and concentration of hydroxide solutions can vary the settling time for the hydroxy-catalysis bonding process. Using such techniques may allow high-rigidity bonding of multi-component optical benches whilst allowing sufficient time for ulta-precision alignment. As discussed earlier in this chapter, this technique may meet the tight requirements demanded by the SMART-2 test mission and subsequent LISA space gravitational wave detector when considering the construction of their optical benches. However, this technique may find other applications where optical high precision measurements are required.

94 Chapter 4 Quality Factors of Selected Test Mass Mirror Substrates and Coatings 4.1 Introduction Much of the work presented within this thesis focuses on the measurement of the quality factor, or Q factor of various materials currently used or being considered for use as mirror substrate materials in interferometric gravitational wave detectors. As discussed in detail in Chapter 2, one important limit to the displacement sensitivity of these interferometers is set by off-resonance thermal noise in the interferometer mirrors and suspensions. The magnitude of this offresonance thermal noise is directly related to the mechanical dissipation (loss) of the substrate material. Materials with a low mechanical loss exhibit lower levels of off-resonance thermal noise. For this reason, the materials used for the substrates for the interferometer s mirrors and suspensions have to be chosen carefully in order that their mechanical losses and thus their thermal noise is minimised. In addition, for a material to be suitable for use as a mirror substrate it must satisfy a number of other requirements, as discussed below. 68

95 4.2 Requirements for test mass mirrors Requirements for test mass mirrors The required sensitivities for long-baseline gravitational wave detectors impose a number of constraints on the test masses being used as mirrors. The material should be available in large enough sizes (10 s of kg), be capable of being suitably figured and polished, and for transmissive optics, exhibit low optical absorption at the wavelength of the laser light to be used to minimise the distortion due to heat deposited as the light passes through [128]. The combination of these requirements has led to fused silica being chosen as the test mass material for all currently operating interferometric gravitational wave detectors in addition to being the most likely choice for the test mass substrates in the planned upgrades to the LIGO and VIRGO detector systems (Advanced LIGO and Advanced VIRGO). Recent research by Penn et al [129] suggests that the mechanical loss in fused silica may be lower than previously expected and vary for different grades of fused silica. For this reason, several different grades/types of fused silica cylinders have been studied and in this chapter their mechanical losses are compared to the expected loss values from Penn s semi-empirical formula [129]. However, there remain limitations to the achievable performance of future interferometric detectors operating with fused silica optics despite its very low levels of mechanical loss. That is, the levels of laser power needed to achieve the desired shot noise limited sensitivity in these upgraded detector systems are such that that thermally induced deformations of the fused silica mirrors resulting from residual absorption in the mirror substrates and coatings are expected to be at a level that requires active thermal compensation. In particular Advanced LIGO will operate with approximately 830kW of laser power in the interferometer s arms and compensation techniques, involving a combination of CO 2 laser based and radiative heating are proposed to ameliorate the effect of thermal distortion [130] [131]. Any further significant increase in the laser power incident on silica mirrors would cause them to thermally distort to an extent difficult to compensate.

96 4.3 Experimental set-up and procedure 70 Since the magnitude of the thermally induced deformation is proportional to α/κ, whereα is the coefficient of linear thermal expansion and κ is the thermal conductivity of the substrate material [128], moving to an alternative substrate material in which this ratio is minimised could significantly improve the tolerance to thermally induced deformations and thus enable higher standing laser powers and potentially improved shot-noise-limited performance. Therefore the study of other materials which may have a combination of low mechanical loss and low α/κ is of interest. A proposed substrate material for this is silicon [74] [132]. Silicon s high thermal conductivity and moderate expansion coefficient means that localised expansion effects can be more easily tolerated however it is opaque to Nd:YAG laser light at 1064nm. Constructing detectors in which an all-reflective topology using diffractive optics is used will bypass the problem of thermal distortion of light passing through an input cavity optic [133] [134] [135]. Silicon is widely available in large sizes and has low mechanical loss as well as possessing some other interesting thermo-mechanical properties when cooled, thus silicon is being considered as an optic for operation at cryogenic temperatures. Further details of the possible advantages of silicon as an optic and suspension element along with investigations of it s mechanical loss are presented in Chapters 5 and 6. Sapphire has been the material of choice for the LCGT interferometer in Japan [47] currently being built and the planned AIGO detector in Western Australia [136]. This Chapter will focus on silica and sapphire substrate test masses along with some coated (for high-reflectivity) silica samples. 4.3 Experimental set-up and procedure In order to measure the internal loss φ(ω), or inversely the Q value, of a test material, the sample must be free to move and isolated from the outside world. A very good way of achieving this is to suspend a sample as a pendulum

97 4.3 Experimental set-up and procedure 71 and carry out measurements under vacuum. A simple pendulum suspension consisting of a loop of twisted silk thread was used. Silk is known to enable low mechanical loss to be measured in such a system [137]. The loop holds the cylindrical mass around the centre and is held above by a metal clamp, see Figure 4.1. An internal resonance of the test mass is then excited by a signal applied to an electrostatic pusher [139] [141]. silk thread clamp to high volts 3" (76mm) electrostatic drive test mass Figure 4.1: Schematic diagram and picture of a typical test mass suspension. The electrostatic drive is switched off and the resonance allowed to decay. A Michelson interferometer is used to sense the amplitude of the freely decaying oscillation. The envelope of the amplitude of the free oscillations can be expressed as, A(t) = A o e ωot 2Q (4.1) = A o e 1 2 φ total(ω o)ω ot, (4.2) which can be rearranged as, ln[a(t)/a o ]= ω otφ total (ω o ) 2 (4.3) Here A o is the initial amplitude (at time t =0),ω o is the resonant angular frequency and φ total (ω o ) is the total loss. Therefore, from Equation 4.3, if ln[a(t)/a o ] is plotted against time t the gradient of the linear decay will be ωo φ total(ω o), from which the loss angle φ 2 total (ω o ) can be calculated, since the natural (angular) frequency of the resonance is know.

98 4.3 Experimental set-up and procedure 72 electrostatic drive test mass sample in silk thread loop suspension low-pass filter and amplifier mirror (mounted on loudspeaker) filtering beamsplitter photodiode LASER mirror 2 (mounted on piezo) signal generator lock-in amplifier spectrum analyser to data aquisition Figure 4.2: Schematic diagram of the interferometer readout used for measuring the amplitude ring-down of resonant modes of a test mass. Note that the majority of test masses studied are considerably smaller than the size of the mirrors used in current interferometric gravitational wave detectors. For handling and cost reasons it is more practical to test smaller samples. Using this setup it was found that the Q values measured tended to vary considerably between different suspension conditions. The following points outline techniques that were found to produce the highest Q values. Silk thread produced the best Q values. The only other suspension material that produced high Q values was polished tungsten wire. The state of the polish of such wires was seen to affect significantly the final Q value. These wires were polished by hand using diamond paste. Perhaps a better polishing technique may allow tungsten to achieve higher Q values than silk thread. Note: the highest measured Q value on polished tungsten wire was

99 4.3 Experimental set-up and procedure for silicon (100) sample (length 100 mm, diameter 98 mm) where was measured for the resonant mode at 43.2 khz using silk thread. The thickness of silk thread used also was seen to affect the Q value. The thinner the thread, without breaking, produced the highest Q values. The breaking strengths of various thicknesses of thread were measured in order to choose the minimum thread thickness. The application of a small quantity of grease (pig fat) to the test mass or the suspension thread was often seen to improve the measured Q. It was observed that the best Q values were generally obtained by very light greasing of the silk thread with no grease applied directly to the mass. This effect could be due to the following two reasons. Firstly, the grease adds friction between the thread and the test mass. This may decrease what is commonly known as slip-and-stick loss associated with the thread rubbing against the test mass surface. Also, the grease was observed to glue down small frayed ends along the thread. These small hair-like structures may also be a source of slip-and-stick loss in the suspension or indeed even be extracting energy from the resonance of the test mass by themselves being resonantly excited. A final method of decreasing the suspension loss was found by pretensioning the thread before being clamped with a teflon spacer between the test mass and clamp as shown in Figure 4.3. A good method for doing this was to rigidly hold one end of the thread as a Newton balance held the other end. The Newton balance could then be pulled until the force pulling the thread was near its breaking strength. This kept the suspension length to a minimum (the suspension sling did not relax to a significantly lower position when finally receiving the full weight of the test mass) and generally allowed the highest Q values to be obtained.

100 4.3 Experimental set-up and procedure 74 force applied clamped Newton balance teflon spacer Figure 4.3: Schematic diagram of pre-tensioning technique used for a suspension using silk thread. Figure 4.4 shows a typical plot generated from the readout of the laser interferometer sensing the amplitude ringdown of a resonant mode of a test mass. This particular sample was a sapphire rod, 10 cm long and of diameter 3cm. 10 resonant mode amplitude readout exponetial fit, φ(ω)=4.6x10-9 amplitde readout (V) 1 thin aluminium mirror 10cm time (mins) Figure 4.4: Typical ringdown for Hemex sapphire sample of length 10 cm and diameter 3 cm.

101 4.4 Bulk silica results Bulk silica results There are only two vendors of fused silica samples of suitable size and quality, Corning (who supplied the silica currently used in some of the LIGO optics) and Heraeus (whose silica is used in the remaining LIGO optics and used for the VIRGO and GEO optics). Also each vendor makes a number of different optical grades of silica, i.e. with varying specifications for absorption and homogeneity. However, as noted, both the optical and mechanical qualities are of significant importance to the sensitivity of interferometric gravitational wave detectors. Empirical measurements [142] suggest that Heraeus fused silica has a lower mechanical loss than that of Corning fused silica and that the various Heraeus Suprasil grades have different losses from each other. Specifically, Suprasil 311 and 312 are the grades of silica found to have the lowest mechanical loss [142]. However, we do not yet understand in detail what processing optimises the mechanical performance of these grades of silica, such as annealing temperature, cooling rate and geometry. As will be discussed, understanding these factors could perhaps allow further improvements to reduce the level of mechanical loss. Considerable studies of silica and its mechanical properties in relation to its use for the optics of gravitational wave detectors have been carried out by a number of authors [140][141][91][144][145]. An empirical model has been developed for the mechanical loss in fused silica [129]. This model includes dependencies on both surface-to-volume ratio and frequency, such that the mechanical loss in a samples of fused silica, φ(f, V ), may be expressed as, S ( φ f, V ) ( / ) V = C 1 + C 2 f C 3 + C 4 φ th, (4.4) S S where V is the sample volume in mm, S is the surface area in mm, f is frequency in Hz and φ th is the thermoelastic loss. The coefficients C 1, C 2, C 3 and C 4 are obtained empirically depending on the particular type of fused silica. The surface loss term, (C 1 / V ), is associated with the resultant damage S

102 4.4 Bulk silica results 76 from abrasive polishing, or in the the case of flame polished or flame drawn fused silica samples this is due to adsorbed contaminants and/or micro surface cracks. The bulk loss term, C 2 f C 3, (for high quality pure fused silica) is associated with the strained Si-O-Si bonds, where there exists a minima of energy of the bond at two different bond angles, forming an asymmetric double-well potential [146]. When a strain is applied, there is a redistribution of bond angles, resulting in a source of mechanical dissipation. The thermoelastic loss, φ th, has been discussed previously in Section 6.8 and can be calculated from the material properties of the fused silica. It can be shown that, for the above samples, at the frequencies under consideration, we are well away from the thermoelastic peak (see Appendix B) and the thermoelastic loss may be assumed negligible. The coefficients that have been found to best describe the loss mechanism in Heraeus Suprasil 312 fused silica are [129]: C 1 = C 2 = (4.5) C 3 = In the lab in Glasgow, Heraeus 311 type fused silica was tested. Heraeus 311 is manufactured identically to 312 type except that an additional annealing cycle is performed in order to increase the homogeneity of the refractive index. Work carried out by Numata et al studied the mechanical losses of samples fabricated from these two types of fused silica and found them to be identical within the precision of the experiment [142]. Figure 4.5 shows the experimental loss values measured for the Suprasil 311 mass compared to the predictions of the model using the coefficients for Suprasil 312. Each point corresponds to at least three ringdowns. The sample was re-suspended around ten times and the lowest average loss values of each mode in a particular suspension were accepted. The standard error in the measured loss was observed to be between 5and10%.

103 4.4 Bulk silica results 77 It can be seen in this plot that the lowest measured loss factors have values lying close to the empirical model. The loss values that lie significantly above the empirical model are likely to be due to the measurements being limited by suspension losses, i.e. we are measuring the loss associated with the suspension holding the test mass and not the internal loss of the test mass itself. Lowest measured loss Empirical model for loss φ(ω) 10-7 (a) (b) frequency (khz) Figure 4.5: Plot of (a) measured and (b) predicted loss factors for the resonant modes of a Suprasil 311 fused silica cylinder of 65 mm diameter and 70 mm length, where the standard error of each point was calculated to be between 5-10%. In order to determine the detailed mode shapes of the modes measured here a Finite Element Analysis program, Ansys, was used to model the mass. The following two plots show an example of two of the modes that had loss values that lay above the values predicted by the empirical model. These plots show the summed displacements, U, in all directions (when the modes are excited) such that U = x 2 + y 2 + z 2. The maximum displacements within each mode are coloured in red and the minimum in blue. For modes that have measured loss values above the empirical model, e.g. the 70,357 Hz mode, there is considerable motion observed at points where the thread forming the suspension lay. This could lead to slip-and-stick loss as

104 4.4 Bulk silica results 78 f = Hz f = Hz Figure 4.6: Illustration of 32,151 Hz and 70,357 Hz mode shapes for a Suprasil 311 fused silica cylinder of 65 mm diameter and 70 mm length where the relative displacements U = x 2 + y 2 + z 2 are plotted in dimensionless units. f = Hz f = Hz Figure 4.7: Illustration of 39,645 Hz and 53,990 Hz mode shapes for a Suprasil 311 fused silica cylinder of 65 mm diameter and 70 mm length where the relative displacements U = x 2 + y 2 + z 2 are plotted in dimensionless units. the thread rubs against the silica surface. Consider the equivalent plots for modes with loss values that were very close to those predicted by the empirical model. The relative displacements around the centre of the barrel in Figure 4.7 are significantly lower than those for the plot in Figure This is consistent with the hypothesis that the measured losses for the modes of this sample are limited by suspension losses for modes where there is significant motion of points where the suspension thread lies.

105 4.4 Bulk silica results 79 Two other types of fused silica were studied, Heraeus Type 311SV (manufactured differently to reduce water content) and Corning 7980 (manufactured in a different way by another vendor). These are of interest since Heraeus Type 311SV is used for the beamsplitter in GEO600 and Corning 7980 is used for some of the LIGO optics. The loss values obtained for these types of silica lay above the level of the loss that the empirical model would predict for Suprasil 311 and 312. It is therefore necessary to fit the coefficients in the empirical model to these types of fused silica. The coefficient C 3 corresponds to the power-law of relaxation in silica due to thermally activated transitions [146]. This property is inherent in silica glass and therefore unlikely to differ between different types of silica. This leaves the coefficients C 1 (the magnitude of surface loss) and C 2 (the magnitude of the frequency dependent bulk loss) to be considered. It is likely that the surface losses will be dependent on the surface treatment (polish) of the samples. Since the final polishing steps for each sample were similar, it would seem likely that the surface loss would also be similar. Assuming this to be true, the only component of loss that could be expected to significantly change will be that of the bulk loss of the different types of silica. Further confidence in this assumption came when the experimental results for the modes where there was the least motion at points around the centre of the cylinders fitted the empirical model with a standard deviation several times lower when changing C 2 than when changing C 1. In Figure 4.8 shows measured loss values for the sample of suprasil 311SV studied. The two resonant modes with least motion around the suspension point were fitted to the empirical model. This was achieved by setting C 2 = which corresponds to an increase of approximately a factor of two in the bulk loss compared to Suprasils 311 and 312. A similar analysis was carried out for the results from the measurements of loss of the Corning 7980 silica sample, see Figure 4.9. The value for C 2 was found to be which corresponds to an increase of approximately

106 4.4 Bulk silica results 80 Lowest measured losses for Suprasil 311SV Empirical model for loss using coefficients for 311/312 Scaled empirical model for loss found to best fit the 311SV results 10-7 (a) (c) φ(ω) (b) frequency (khz) Figure 4.8: Plot of (a) measured loss factors of the resonant modes of a Suprasil 311SV fused silica sample 76.2 mm diameter and 25.4 mm length, (b) predicted loss factors for Suprasil 311 and 312 of the same dimensions and (c) predicted loss factors allowing an increase in the bulk loss term of a factor of two. The standard error of each point was calculated to be between 5-10% Lowest measured losses for Corning 7980 Empirical model for loss using coefficients for 311/312 Scaled empirical model for loss found to best fit the Corning 7980 results (a) (c) φ(ω) (b) frequency (khz) Figure 4.9: Plot of (a) measured loss factors of the resonant modes of a Corning 7980 fused silica sample 76.2 mm diameter and 25.4 mm length, (b) predicted loss factors for Suprasil 311 and 312 of the same dimensions and (c) predicted loss factors allowing an increase in the bulk loss term of a factor of 1.2. The standard error of each point was calculated to be between 5-10%.

107 4.5 Calculation of the substrate thermal noise in the GEO600 detector 81 a factor of 1.2 in the bulk loss compared to Suprasils 311 and 312. These postulated increases in bulk loss for the Suprasil 311SV and Corning 7980 samples were not considered to be due to suspension losses since it was possible to measure lower loss factors from the Heraeus 311 with the same suspension type, as shown in Figure Calculation of the substrate thermal noise in the GEO600 detector The expected level of thermal noise in gravitational wave interferometers due to the internal mechanical dissipation of the mirrors can be calculated using analytical or finite element techniques. Typically this substrate thermal noise has been calculated assuming the mechanical loss of fused silica is frequency independent. However, as discussed earlier, recent research is consistent with the bulk dissipation in fused silica samples having a distinct frequency dependence, see Equation 4.4. It is therefore of interest to recalculate the levels of substrate thermal noise expected for current interferometers. Until recently the displacement noise due to thermal fluctuations within the test masses in gravitational wave detectors was carried out using a normalmode expansion technique as described by Saulson in 1990 [63] and later Gillespie and Raab in 1995 [143]. In that case, the displacement of the front face of a test mass, sensed by a Gaussian beam, is computed for a large number of individual resonant modes of the test mass and the calculated displacements summed incoherently after appropriate weighting. However, the calculation is only accurate when the spatial distribution of the mechanical dissipation is homogeneous. This is therefore not exact since in practice the surface dissipation (due to surface damage and contamination) in test mass samples is known to be of a significant level compared to the mechanical dissipation associated with the bulk material. More recently an alternative method applying the

108 4.5 Calculation of the substrate thermal noise in the GEO600 detector 82 fluctuation-dissipation theorem was developed by Levin [92]. Levin calculated the thermal noise for a half-infinite test mass, where this can be assumed to be a reasonable approximation for the cases where the beam radius is small compared to the radius of the mirror. Later in 2000, Liu and Thorne formulated a correction to his calculation which could be used to estimate the noise for a finite mirror, with known radius and thickness [94]. They found a correction factor (C ftm ) for the calculated displacement thermal noise (using an AdLIGO type mirror) between the finite and infinite cases to vary from for beam spot radii (r o ) ranging from 1 6 cm respectively. The power spectral density of the thermal noise of a test mass, approximated as a half-infinite slab, S ITM x (f), may be expressed as [92], Sx ITM (f) = 4k BT ω 1 σ 2 2πEro φ(ω), (4.6) and the power spectral density of the thermal noise of a finite sized test mass, S FTM x (f), as [94], Sx FTM (f) = 8k BT ω φ(ω)(u o + U), (4.7) where k B is Boltzmann s constant, T is the temperature, ω the frequency, φ(ω) the mechanical loss or dissipation, σ is the Poisson ratio and (U o + U) incorporates the required numerical correction from a half-infinite to finite sized test mass. The level of this thermal noise is expected to be the limiting noise source in GEO600 s most sensitive frequency band [148]. As mentioned before, the level of substrate thermal noise was previously calculated using a constant value for the mechanical dissipation, for GEO600 this was assumed to be for the test mass mirrors [149]. Using the more complete description of bulk loss where there exists a frequency dependence, as described by Penn, the level of thermal noise may be reevaluated. The optics that contribute significantly to the thermal noise in the readout scheme of GEO600 are the mirrors (NMs and FMs) and beamsplitter (BS), as shown in Figure The types of silica

109 4.5 Calculation of the substrate thermal noise in the GEO600 detector 83 material used for the mirrors and beamsplitter are not identical. The GEO mirrors are fabricated from Heraeus Suprasil 1 type fused silica whereas the beamsplitter is fabricated from Heraeus 311SV silica. This material is used for the beamsplitter in order to reduce the optical loss and the associated change in the refractive index caused when light is absorbed and heats up local areas of the optic. This effect is then observed as thermorefractive noise within the readout scheme [151]. The mechanical loss of Suprasil 1 type fused silica has been studied by other authors. Taking the mechanical loss values for Suprasil 1 type fused silica obtained by Numata et al [150] and fitting the empirical model for loss by the same method illustrated in Figures 4.8 and 4.9 gives a bulk loss term scaled by a factor of approximated 2.4 above that for 311 and 312 type. Using the semi-empirical model for loss and these results for Suprasil 1 and 311SV brands of silica, the level of thermal noise associated with the bulk substrate material of GEO s optics may be estimated. A development in this analysis was also to split the calculation into two parts: one focusing on calculating the thermal noise resulting from the frequency dependent bulk loss and and the second focusing on the surface loss (assumed frequency independent). Since the surface has some characteristic surface depth, the thermal noise here must be modeled differently than described by Equation 4.7 only. The lossy surface layer is considered to be equivalent to a lossy coating surrounding the entire surface of the test mass. An expression for the power spectral density, S x (f), of the thermal noise associated with a lossy coating applied to the front face of a test mass was derived by Nakagawa [152] and can be expressed as [153], S x (f) = 4k ( BT d E π 2 fe w 2 E φ + E ) E φ, (4.8) where f is the frequency, w is the field amplitude radius, E and E are the Young s Modulus values for the substrate (bulk) and coatings respectively, φ and φ are the mechanical loss values for the coating for strains parallel and

110 4.5 Calculation of the substrate thermal noise in the GEO600 detector 84 perpendicular to the coating surface, d is the coating thickness and w is the field amplitude radius. In the analysis presented here, the coating describes the lossy surface layer of the silica samples in question. Here we assume that E = E and φ = φ in this situation, since both the coating and bulk are fused silica. FM pathlength = 1200m light from modecleaners PR NM BS NM SR FM output photodetector Figure 4.10: Simplified schematic diagram of the optical layout of GEO600 comprising the near and far mirrors (NM and FM), beamsplitter (BS) and power recycling mirror (PR). Note, however, that the surface loss value arising from Penn s empirical model is the effective loss of a fused silica substrate arising from surface effects, whereas, φ and φ describe the actual loss of the surface layer. Therefore scaling is required to estimate the pure surface loss used in Equation 4.8. We already know from Equation 4.4 that the loss arising from the surface layer can be described by, ( ) S φ s = C 1 (4.9) V where φ s is the expected loss of a fused silica sample associated with the surface loss only, which in turn can be calculated from the actual loss of the surface layer (as if one strips off the surface and measures it independently), φ surf,

111 4.5 Calculation of the substrate thermal noise in the GEO600 detector 85 scaled by the proportional strain energy residing in the surface layer such that, E Bulk φ surf = φ s (4.10) E Surface where we may approximate the ratio of energy stored in the surface layer to the energy stored in the bulk to be equivalent to the volume of the surface layer to the volume of the bulk, therefore, φ surf = φ s E Bulk E Surface (4.11) = C 1 S V E Bulk E Surface (4.12) C 1 S V V Bulk C 1 V Surface d (4.13) where d is the thickness of the damaged surface layer and V Bulk and V Surface are the volumes associated with the bulk and surface of the sample. This estimated value of the surface loss, φ surf, was used to evaluated the thermal noise associated with the damaged surface layer using Equation 4.8 such that φ = φ = φ surf. The thickness of the damaged surface layer, d, did not need to be accurately determined since φ surf 1 which in turn cancels the d thickness term shown from Equation 4.8 where S x (f) d before Equation 4.13 is substituted in for φ surf. The estimated substrate thermal noise using this analysis was calculated by combining the thermal noise associated with the bulk substrate (using Equation 4.7) and associated with the surface effects (using Equations 4.8 with the surface loss estimated from Equation 4.13) in quadrature, shown in Figure Note that the thermal noise associated with the mirrors will be different to the beamsplitter since the bulk mechanical losses are not of the same magnitude. The combined thermal noise in the GEO600 detector can be shown to be a contribution of the various optics such that [151] X2 NM +2X2 FM X2 BS h(f) =, (4.14) L

112 4.5 Calculation of the substrate thermal noise in the GEO600 detector NM and FM total φ BS total φ φ(ω) BS bulk φ NM and FM bulk φ NM and FM surface φ BS surface φ frequency (Hz) Figure 4.11: Plot of the expected contributions to the effective (or measured) mechanical losses in the relevant GEO600 optics. The bulk mechanical losses have been estimated from the empirical formula for mechanical loss in fused silica [129], where the model has been verified experimentally in the case of the interferometer mirrors (see Figure 4.8) and modified in the case of the beamsplitter (see Figure 4.5) as detailed earlier. The surface loss was extrapolated from the empirical model for loss in fused silica.

113 4.5 Calculation of the substrate thermal noise in the GEO600 detector 87 where X NM, X FM and X BS are the thermal displacement noise as sensed by the Gaussian laser beam of the near mass, far mass and beamsplitter optics of GEO600, shown in Figure The estimated thermal noise calculated using this analysis is plotted alongside the currently accepted values for the other relevant noise sources [154] [155], shown in Figure h(hz) -1/2 1E-19 1E-20 1E-21 1E-22 seismic suspension TN substrate TN coating TN thermorefractive shot noise, detuned at ~350Hz recalculated substrate TN including surface loss existing total noise recalculated total noise 1E-23 1E frequency (Hz) Figure 4.12: Noise contributions in the GEO600 detector with the recalculated substrate thermal noise and updated total noise. The results show that, when considering the frequency dependence of the mechanical loss in fused silica and when considering a concentrated level of loss in a surface around the test mass mirrors, the estimated level of substrate thermal displacement noise is around a factor of ten lower than previously estimated. This highlights the significance of the other sources of noise, particularly at the lower frequency band of operation of GEO, where previously the substrate thermal noise was expected to be dominant. In particular, thermal noise associated with the mirror coatings remains of significant importance as one of the limiting factors to current detector sensitivities. Investigations into reducing the level of mechanical dissipation associated with the mirror coatings is therefore likely to be one of the most significant challenges in the

114 4.6 Bulk silica with HR coatings 88 upgrades to the current interferometers, such as AdvLIGO. Investigations of the mechanical losses associated with mirror coatings are detailed in the following section. 4.6 Bulk silica with HR coatings It is necessary for the test masses in interferometric gravitational wave detectors to be coated so they can act as optically highly reflecting mirrors. This is achieved by coating the front surface of the mass with alternate layers of different dielectric materials (typically λ/4 in optical thickness, where λ is the wavelength of laser light used, 1.064µm for gravitational wave detectors) to achieve a highly reflective surface. Such a coating will introduce another source of loss causing another source of thermal noise in the detector. The losses of such mirror coatings must therefore be characterised to allow the limits to detector sensitivity set by thermal noise from the mirror coatings to be calculated. All current mirror coatings in use in interferometric gravitational wave detectors are formed from alternate layers of ion-beam sputtered silica (SiO 2 ) and tantalum pentoxide (Ta 2 O 5 ), or tantala, for short. Previous work on dielectric coatings suggested that the residual loss of the coating was dominated by the tantala component [106] [73]. Preliminary experiments also suggest that doping the tantala coating with titania (TiO 2 ) can reduce the mechanical loss [99]. As mentioned before in Section 4.5, understanding and reducing the residual mechanical loss of these optical coatings is of significant importance for reducing the associated coating thermal noise in future interferometric gravitational wave detectors. The mechanical loss of a coated mass can be represented by [153], φ coated φ bulk + E coating, total φ coating, (4.15) E bulk where, φ coating = φ residual + E coating, volume change E coating, total φ thermoelastic, (4.16)

115 4.6 Bulk silica with HR coatings 89 where E coating, total is the total energy stored in the coating layer, E bulk is the total energy in the substrate and E coating, volume change is the energy in the coating where there is a change in volume [147]. Energy in the coating that does not cause a change in volume (from shear distortion) is not subject to thermoelastic loss. The loss must be calculated for each individual mode since the strainenergy distributions vary between resonant modes. A set of three 30-layer silica-tantala (SiO 2 -Ta 2 O 5 ) coatings with various levels of titania (TiO 2 ) doping were produced by LMA in France [160] on fused silica substrates of length 76.2 mm and diameter 25.4 mm. In addition to this, two silica substrates of the same dimensions had single layers of tantala (equivalent thickness to 37 layers of λ/4) and silica (equivalent thickness to 25 layers of λ/4). For comparison, an equivalent undoped 30-layer silica-tantala coating was produced by a separate vendor, CSIRO in Australia [161]. The measured loss values for the coated samples mentioned are presented in Figure It can be seen that, for these particular samples studied, the greater the percentage of titania doping, the lower the mechanical loss. The 1E-6 A - control (uncoated) for C & D B - 30 layer Si/Ta nominally 3% TiO2 C - 30 layer Si/Ta nominally 6% TiO2 φ(ω) 1E-7 D - 30 layer Si/Ta nominally 50% TiO2 E - single layer silica (equivalent to 25 layers) F - single layer tantala (equivalent to 37 layers) G - 30 layers Si/Ta (CSIRO) frequency (Hz) Figure 4.13: Measured mechanical loss values for a variety of coated silica samples where the standard error of each point was calculated to be between 5-10%. The frequency dependence of the measured losses are not seen to be dependent only on the mode shapes, as shown in Figure 4.14.

116 4.6 Bulk silica with HR coatings khz khz khz khz U= ¼ ½ ¾ 1 Figure 4.14: Illustration of khz, khz, khz and khz mode shapes for a fused silica test mass of 25.4 mm diameter and 76.2 mm length where the relative displacements U = x 2 + y 2 + z 2 are plotted in normalised, dimensionless units. results suggest that that the residual loss of the single layer of silica is also significantly lower than that of the tantala coating, which is consistent with previous analysis of multi-layer coatings [73]. However, it should be noted that the single layer of undoped tantala had small cracks across the coating which formed during the annealing process. These small cracks could have contributed to the increased levels of mechanical loss. The residual coating loss is found by fitting the loss measurements to equations 4.15 and In order to do this, the value of φ bulk must be known. This is achieved by measuring a control sample, that is, an uncoated test mass manufactured identically and having undergone any of the same annealing stages. Additional to this, the energy ratios of the coating layers and the bulk material must be evaluated. This is carried out using a finite element model to eval-

117 4.6 Bulk silica with HR coatings 91 uate the relative displacement of the test masses and coatings for each mode studied. The relative strain-energy values can therefore be found for the coating layers and the bulk as the mass distorts when resonating. The calculated residual loss of these coatings is modeled as having frequency independent and dependent components, such that, φ residual = φ frequency independent + f φ frequency dependent (4.17) Using these relationships, the coating losses in Table 4.1 were calculated by this method by David Crooks for the measured test masses: Sample Coating type φ freq. independent φ freq. dependent /Hz B 30 layer SiO 2 /Ta 2 O ± ± 1.8 (Vendor: LMA) nominally 3% TiO 2 doped C 30 layer SiO 2 /Ta 2 O ± ± 0.6 (Vendor: LMA) nominally 6% TiO 2 doped D 30 layer SiO 2 /Ta 2 O ± ± 0.8 (Vendor: LMA) nominally 50% TiO 2 doped E single layer SiO ± ± 1.0 (Vendor: LMA) equivalent to 25 layer F single layer undoped Ta 2 O ± ± 5.3 (Vendor: LMA) equivalent to 37 layer G 30 layer SiO 2 /Ta 2 O ± ± 1.0 (Vendor: CSIRO) undoped Table 4.1: Calculated residual losses for various coated test masses. Note that doping is only applied to the tantala component of the multi-layer. A 30 layer undoped silica-tantala coating was later measured by Peter Murray where the residual loss of the coating was calculated to be (2.4 ± 0.5) f(1.2 ± 1.0) 10 9 [162]. Comparing the results for the LMA coatings therefore suggests that doping with tantala reduces the mechanical loss by approximately 30% to 50%. The residual loss for the undoped CSIRO sample is also slightly higher than the undoped LMA sample, although the reason for

118 4.7 Bulk sapphire results 92 this is yet unknown. However investigations into these factor are continuing. The level of doping indicated by the vendor for each sample was only approximate. MacLaren et al, from the solid-state physics group in Glasgow, used the Electron Energy Loss Spectroscopy technique (EELS) [163] to precisely study the material composition of such multi-layer coatings. Preliminary results for the coating stated to have 3% titania doping were found to contain 8.5 ± 1.2% by cation of titania when studied using EELS. This clearly showed that the stated quantities of doping from the vendor were not precise and therefore further investigations using the EELS technique are continuing in Glasgow. Further studies of ion-beam sputtered coatings from other vendors, such as from CSIRO in Australia, are also continuing. Additionally, studies into why titania helps reduced the mechanical loss for such coatings and the effect of different annealing temperatures and cool-down times are of significant interest. Annealing at high temperatures allows internal stresses in the multi-layers, a known source of excess mechanical loss, to relax. Using the values for residual coating loss found here, the level of thermal noise expected in an interferometer using such coated optics can be calculated using Equation 4.8. As discussed in Section 4.5, this level of thermal noise sets an important limit to detector sensitivity. Arguably, this particular source of thermal noise is perhaps the only limiting noise source in the current interferometers, such as GEO600, that doesn t already have a clear method by which it may be reduced (for example, shot noise can be reduced by increasing the laser power). Therefore, when considering future upgrades to the detectors, finding ways in which high-reflectivity coatings may have their residual mechanical losses reduced is of great importance. 4.7 Bulk sapphire results Fused silica (and originally fused quartz) has been the material of choice for constructing test masses suspended as mirrors in the interferometers for grav-

119 4.7 Bulk sapphire results 93 itational wave detection. The low optical losses, low thermal noise (resulting from silica s low mechanical dissipation) and availability in large pieces made it ideal for the purpose. Sapphire was previously proposed as a high Q/low loss material, and its properties studied by the Braginsky group, see [137]. In the late 1990 s, the lab in Glasgow measured very low loss factors associated with a HEMEX sapphire sample [156]. This sample was fabricated by Crystal Systems [164] using a heat exchanger method, where the highest quality grade was called HEMEX. The international gravitational wave community then began to consider sapphire as a serious candidate for a mirror substrate material. The pendulum (of favour and not the detectors) has swung back-and-forth over the years as to which material was superior. A long down-select process between fused silica and sapphire for the substrate material for the Advanced LIGO project (see Chapter 1 Section 1.6) was eventually concluded early in The decision was not easy since both materials had advantages over the other and both met the design sensitivity goal for Advanced LIGO of 200 Mpc range for binary neutron star systems. Silica has a lower level of thermal noise at low frequencies ( 100 Hz), making a detector using it more sensitive to low frequency sources such as high-mass black hole binaries. There was better confidence in the performance of fused silica, especially with regard to its suitability to support suitable optical coatings and the jointing of the lowerstage optics using silicate bonding. Sapphire instead has lower thermal noise at higher frequencies which may lead to sensitivities being improved in the frequency band where LMXBs (Low Mass X-ray Binaries) are candidates for detection [157]. Sapphire also has a higher thermal conductivity which would allow increased levels of laser power to circulate in the interferometers without the mirrors distorting due to their thermal load. Unfortunately sapphire also exhibits higher levels of homogeneous and inhomogeneous optical loss and issues of optical scattering have been observed. The down-select working group for Advanced LIGO decided on silica as the material of choice, largely based on

120 4.7 Bulk sapphire results 94 better understanding of silica s properties and of its required coatings. Despite this, sapphire remains of considerable interest for other planned gravitational wave detectors [158] [159], such as LCGT being built in Japan [47]. In Figure 4.15 the loss factors for three sapphire samples of length 76.2 mm and diameter 25.4 mm are plotted. All the samples are A-axis grown sapphire manufactured in the same way. The only difference is that two of the samples have their faces commercially polished whereas the other was super-polished. Super-polished denotes a sub-angstrom micro-roughness whereas commercial polish may be a factor of ten above this level. The lowest loss factor here for sapphire was measured to be φ min = , which is approximately half that of the silica Corning 7980 sample where φ min = Therefore, these results also appear to confirm that sapphire exhibits a lower bulk loss at these frequencies than fused silica. It is unclear to what extent the different surface finish from the two polishing techniques may be affecting the observed mechanical losses. The typical surface roughnesses of the sapphire samples were not observed to be greatly different and both sub-nanometer, as seen in Figure The commercial and super-polished surfaces of our sapphire samples appeared comparable despite the differences in the polishing techniques. The super-polish technique has been observed to create local surface buildups, especially in softer glassy materials, from the polishing compound and substrate particles during polishing [165] [166] [167] [168]. This local buildup of material on the substrate surface, although helping to fill in valleys to result in a higher optical figure, will create an amorphous, contaminated surface layer. Therefore, it is a possibility that the associated surface loss of a super-polished substrate is greater than that of a commercially polished substrate. To determine whether excess surface loss exists for super-polished samples, the estimated energy stored in the surface layer must be considered. Two finite element models were considered using Ansys, as can be seen in Figure Ideally a multi-component F.E. model should be constructed (model (a)

121 4.7 Bulk sapphire results 95 1x10-7 8x10-8 6x10-8 Commercial polish sample 1 Commercial polish sample 2 Super polish sample 3 4x10-8 φ(ω) 2x frequency (Hz) Figure 4.15: Loss measurements for one super polished and two commercially polished sapphire samples of length 76.2 mm and diameter 25.4 mm. super-polish commercial polish Ra = nm RP-V = nm Ra = nm RP-V =0.615nm Figure 4.16: Surface profiles for one super polished and one commercially polished sapphire sample, where in 1-dimension Ra = 1 l l 0 h(x) dx where l is the length of sample studied and h(x) is the height above or below the mean and where R P V is the average height between the ten largest peaks and the ten lowest valleys on the sample surface.

122 4.7 Bulk sapphire results 96 (a) multi-component F.E. model (b) single component F.E. model Figure 4.17: Finite element models for measuring the stored strain energy in a damaged surface layer. in Figure 4.17) in order to simulate a very thin surface layer to a comparably large test mass. However, the attachment methods implemented for gluing the respective sections of this model together resulted in 5 10% of the nodes having errors. Errors are highlighted in this case when the F.E. package finds that the equations defining the adjoining surface and bulk regions in the model are observed to yield inconsistent (or false) results. This would likely distort the solution in many of the nodes in the model and can create singularities, i.e. where the strain energy at a specific node (or point) tends to infinity. It was therefore decided, for the sake of accuracy and simplicity, to implement a series of models similar to (b) in Figure Here, the analysis only meshes (divides) the single bulk test mass, using the outermost elements to define the surface layer. The limit to the minimum surface thickness is then set by the maximum number of elements licensed to run in a particular version of Ansys. In our case, the maximum node number is 128, 000. This sets the minimum surface thickness in this model to be 1.4 mm. Typically the damaged layer resulting from the polishing process is in the region of 1 µm. Therefore, a variety of models with thicknesses ranging from mm which were then extrapolated down to zero thickness and zero energy (since at zero thickness the surface layer contains zero energy). The results for the fractional surface

123 4.7 Bulk sapphire results 97 energies in the modes measured in the sapphire samples are shown in Figure The plot gives an indication on the relative amount of energy stored in a surface layer between the different resonant modes. Appendix C shows that the trend in Figure 4.18 is not due to a convergence of accuracy as the mesh density of the model is increased). ES EB ratio of energy stored in the surface to energy stored in the bulk khz mode 36.2 khz mode 64.2 khz mode 86.7 khz mode 86.9 khz mode x x x x x10-3 surface thickness (m) Figure 4.18: Energy ratios in sapphire samples studied with varying thickness of surface layers for five of the resonant modes. The total measured loss can be divided into two components, bulk loss and surface loss, such that, φ total = E bulk E total φ bulk + E surface E total φ surface, (4.18) φ bulk + E surface φ surface, (4.19) E bulk where φ bulk is the bulk loss of the substrate, φ surface is the loss of the damaged surface layer, E total is the total energy stored in the sample, E bulk is the energy stored in the bulk substrate and E surface is the energy stored in the surface layer. The approximation given can be used when E total E bulk, i.e. the surface layer is very small, which is true for large samples such as discussed here.

124 4.8 Conclusions 98 There appears to be a small link between the estimated energy stored in a surface layer and the measured loss factors in Figure For the two modes around 87 khz, the super-polished loss factors were significantly below the commercially polished samples. These two modes contained the lower stored energy in their surfaces. Therefore, if the super-polishing technique has caused increased surface loss then we would expect these loss measurements to be lower. However, if this was the case then the loss measured should be closer to the bulk loss of the substrate and therefore the measured losses should be similar between all the samples, since the substrate materials are identical. Since there appears no clear trend in the data, there is a possibility that most of the measurements are being limited in some way by suspension losses. 4.8 Conclusions The results presented here for measurements of the mechanical loss of bulk silica indicate that the expected level of substrate thermal noise within the GEO600 detector is likely to be significantly lower than previously estimated. This is of importance as it shows that the substrate thermal noise is considerably lower than some of the other noise sources, most notably coating thermal noise and thermorefractive noise. Efforts to reducing these other noises is one therefore of great importance, particularly with regard to future interferometer upgrades. The results for coated silica samples show that the majority of the excess loss from the coatings arises from the tantala component and that this loss can be reduced by doping with titania. Reducing coating loss is will therefore play a significant role in the research towards future second and third generation detectors. The results for bulk sapphire samples strengthen that case that sapphire is a possible candidate substrate due to its low level of mechanical loss. However, further samples must be tested in order to understand better the loss mechanisms present and to verify that the loss factors presented are

125 4.8 Conclusions 99 not limited by external loss mechanism, such as suspension loss.

126 Chapter 5 Silicon as a Mirror Substrate for Third Generation Detectors 5.1 Introduction As previously described in chapter 1, long baseline gravitational wave detectors operate using laser interferometry to sense the differential strain, caused by the passage of gravitational waves, between mirrors suspended as pendulums. These detectors operate over a frequency range above the pendulum modes of the suspensions (typically few Hz) and well below the lowest internal resonances of the mirrors (few 10 s of khz). One important limit to the displacement sensitivity of current and planned detectors in the frequency range of operation is off-resonance thermal noise in the mirrors and suspensions driven by thermal fluctuations. Thus low mechanical loss materials, such as silica, sapphire and silicon are currently used or proposed for detectors at the forefront of this research. Improved sensitivity at low frequencies (few Hz to few 100Hz) will require further reduction in the level of thermal noise from the test masses and their suspensions. A possible route for achieving this is through cooling. Fused silica, the most commonly used test mass material, exhibits a broad dissipation peak at around 40K and therefore is not a promising candidate for cooling [137]. 100

127 5.1 Introduction 101 Sapphire and silicon however are good candidates. Work is currently being carried out in Japan on developing cooled sapphire test masses and suspension fibers for use in a transmissive Fabry-Perot based interferometer [159] [47], and in Europe and the US research is underway on the use of silicon at low temperatures [74] [132]. At higher frequencies (greater than a few 100Hz) the performance of current interferometers is not limited by thermal noise from the optics but by photoelectron shot noise, whose significance can be reduced by circulating higher optical powers in the interferometer. However, power absorbed by the test masses and mirror coatings can cause excessive thermally induced deformations of the optics, causing the interferometer to become unstable. The extent of this deformation is proportional to α/κ [128] where α is the linear coefficient of thermal expansion and κ the thermal conductivity of the test mass material. Changing from a transmissive to a reflective topology could eliminate thermal loading from substrate absorption, provided coatings of suitably low transmission are available. Used in such a topology, the high thermal conductivity of a silicon mirror substrate would allow circulating powers approximately seven times higher than could be supported by sapphire and approximately 22 times higher for fused silica for the same induced surface deformation making silicon of significant interest as a test mass substrate from a thermal loading standpoint [74]. At room temperature the thermal noise resulting from thermo-elastic effects in interferometers using crystalline optics has been predicted to be a significant noise source in the frequency band of gravitational wave detection [90]. The spectral density for the thermoelastic noise resulting from thermodynamical fluctuations of temperature in a substrate material was calculated by Braginsky et al. and shown to be [90], S TD (ω) 8 α 2 (1 + σ) 2 κt 2 a 2 1 (2π) ρc r0 3 ω, (5.1) 3 where a 2 = κ/ρc, κ is the thermal conductivity, ρ is the density, σ is the

128 5.1 Introduction 102 Poisson coefficient, r 0 is the mirror radius, T is the temperature and ω is the angular frequency under consideration. Another other main source of dissipation to be considered is the intrinsic dissipation. This dissipation arises from the presence of point defects or line dislocations in the crystalline structure. The level of intrinsic and expected thermo-elastic dissipation in silicon is broadly comparable to sapphire at room temperature [171]. However on cooling the linear thermal expansion coefficient of silicon becomes zero at two temperatures, 125 K and 18 K [169], and thus around these two temperatures thermoelastic dissipation could be expected to be negligible, see Figure 5.1. It is thus of interest to study the temperature dependence of mechanical dissipation in silicon samples for potential use as suspension elements and mirror blanks. Figure 5.2 estimates the magnitude, calculated by S. Rowan et al. [170], for the intrinsic thermal noise (from experimental measurements of loss in silicon [138]) and theoretical thermoelastic thermal noise [90],[169] thermal coefficient of expansion for single crystal silicon 2.0 α (10-6 K -1 ) Temperature (K) Figure 5.1: Plot of the thermal expansion coefficient of single crystal silicon over the temperature range 5 K 300 K where α 0at 125 K, with inset, magnified plot showing where α 0at 18 K.

129 5.2 Experimental setup Temperature (K) displacement (m / Hz 1/2 ) intrinsic thermal noise thermoelastic noise Figure 5.2: Plot of intrinsic amd thermoelastic noise of silicon as a function of temperature. (calculated intrinsic thermal and thermoelastic noise at 100 Hz in a single silicon test mass, sensed with a laser beam of radius 6 cm.) The intrinsic thermal noise exhibits two peaks at temperatures similar to those where the thermoelastic noise tends to zero (where α tending to zero). It is not known if the intrinsic loss peaks are related to these zeros in α and further studies are required. As a start to such an investigation, this chapter details measurements of the mechanical loss of six single-crystal silicon test masses at room temperature. In the following chapter further studies of silicon as a possible suspension element at cryogenic temperatures are presented. 5.2 Experimental setup The experimental setup used for measuring the loss factors of the silicon test masses presented here is identical to that described in chapter 4. Each sample was suspended using silk thread, very lightly greased. Again, as in chapter 4, the minimum loss measurements were taken for each resonant mode of the silicon test masses over a range of suspension runs. An average of at least three separate ring-downs were taken for each loss measurement quoted. A picture of one of the silicon suspensions used is shown in Figure 5.3.

130 5.3 Results and conclusions 104 Figure 5.3: Setup of one of the suspensions used to measure the mechanical loss of one of the large silicon test masses. 5.3 Results and conclusions The results presented in Figure 5.4 are for three small bulk silicon samples. The mechanical losses obtained lie slightly above the typical values for high grades of fused silica, see Figures 4.5, 4.8 and 4.9. The results in Figure 5.6 shows the mechanical losses for two larger silicon samples. Interestingly the (111) orientation gave lower loss values, down to (9.6 ± 0.3) This is comparable to the lowest loss factors measured for other materials. Unfortunately these two samples also differ in their levels of doping and therefore it is not possible to determine which factor is causing this difference in the mechanical loss. However, the loss values for the smaller and doped (100) samples in Figure 5.4 are more comparable to the larger (100) undoped test mass seen in Figure 5.6. This suggests that the crystal cut is the definitive factor, rather than the level or type of doping. Perhaps it is the case that varying the doping to some given level will help

131 5.3 Results and conclusions 105 6x10-7 5x10-7 Silicon coin (3"x1") Silicon rod 1 Silicon rod 2 4x10-7 φ(ω) 3x10-7 2x10-7 1x Frequency (khz) Figure 5.4: Plot of the room temperature mechanical loss for a silicon coin (length 25.4 mm, diameter 76.2 mm) and two silicon rods of length 150 mm and diameter 53 mm, all (100) orientation, boron doped with resistivity 1 10 Ohm cm, where the standard error of each point was calculated to be between 5-10%. Figure 5.5: Picture of the larger silicon cylinders tested.

132 5.3 Results and conclusions x10-7 Silicon (111) doped Silicon (100) undoped 1.2x x10-7 φ(ω) 8.0x x x x Frequency (khz) Figure 5.6: Plot of the room temperature mechanical loss for two larger silicon cylinders (length 100 mm, diameter 98 mm), 1 (111) with resistivity 20 Ohm cm, 1 (100) undoped with resistivity 7000 Ohm cm, where the standard error of each point was calculated to be between 5-10%. decrease the mechanical losses in silicon. Doping can impact both the levels of thermoelastic dissipation and the mirror figure distortion under thermal loads by changing the rate at which heat flows in the material caused by altering silicon s thermal conductivity [169]. Further study would be required here to increase our understanding of these parameters and how they would affect the final sensitivity noise of an interferometer operating with silicon optics.

133 Chapter 6 Silicon as a Suspension Element for Third Generation Detectors 6.1 Introduction As discussed in chapter 2, the level of thermal noise observed in the mirrors and suspensions in our interferometers is related to the level of mechanical loss in the substrate material. Future substrate materials will therefore be required to have lower levels of mechanical loss, and therefore lower levels of thermal noise, in order to improve sensitivities further. Studies of dissipation in silicon samples of a variety of geometries and types have been carried out by other authors. In particular, dissipation in silicon flexures has been studied in samples of the type used in Atomic Force Microscopes [174] [175]. However these cantilevers have dimensions considerably smaller than would be suitable for use in the test mass suspensions of gravitational wave interferometers and thus are in a regime where measured dissipation may be dominated by different sources of dissipation than the dimensions that have been studied here [176]. 107

134 6.2 Requirements for cooling pendulum systems Requirements for cooling pendulum systems Silicon suspension elements may be used in cryogenically cooled monolithic silicon optics for future gravitational wave detectors. These suspension elements must meet the following demands: the mechanical loss must be very low in order to minimise off-resonance thermal displacement noise. have the capability to extract heat dissipated by the laser beam from the mirror substrate This chapter investigates the level of mechanical loss associated with samples of a comparable geometry to suspension elements likely to be used in our detectors. Studies on the heat flow through silicon suspension elements have been carried out by Martin et al. [172]. Preliminary calculations show that an all-silicon test mass suspension, intended to operate at 125K, withthe upper test mass held at 77 K, supporting 830 kw of 1064 nm laser light (assuming 0.1 ppm coating loss) would require a minimum cross-sectional area of ribbon to be m 2, when using four ribbons in the suspension. With each ribbon being 1 mm wide, this would result in a minimum thickness for the sufficient rate of heat flow (extraction) to be 123 µm. Given a yield strength of 7 GPa for silicon [173], a maximum load before failure for such dimensions would be approximately 87 kg per ribbon, where the maximum load is calculated from the product of the cross-sectional area and the yield strength. Although a safety margin would be necessary, the expected strength and high thermal conductivity of such ribbons make silicon a promising material for fabricating the suspension elements supporting test mass mirrors of many 10 s of kg s in future cryogenically cooled interferometers.

135 6.3 Proposed sample design Proposed sample design Three basic requirements were considered for the design of the silicon samples tested. Firstly, the dimensions as discussed had to be comparable to what future detectors may be expected to use, i.e. 10 s to 100 s of µm. Secondly, a thicker clamping block would be required so the effects of slip-stick losses were minimised [178]. Thirdly, since thermoelastic dissipation is likely to give some limit to the sensitivity expected from using silicon optics (as discussed in Chapter 5 [90]), it may be useful to design the silicon samples with dimensions such that this dissipation mechanism is optimised. This can be done by matching the characteristic frequency of the thermoelastic dissipation ( 1 t where t is the sample thickness) to the bending mode frequencies ( t ). Figure 6.1 L 2 shows the proposed design for a silicon cantilever to be machined from bulk silicon by a commercial vendor. 15mm 15mm 15mm 100mm 500 µm thickness Figure 6.1: Schematic diagram of proposed silicon cantilever to be manufactured. Such a design of cantilever would have several bending mode resonances situated around the thermoelastic dissipation peak, over a wide range of temperatures, as shown in Figure 6.2. However, problems arose during the production of these silicon cantilevers by machining and grinding. The harsh machining process cause such samples to break at the transition between the thick clamping block and the thin cantilever. Another process was found for the production of samples meeting the necessary requirements.

136 6.4 Silicon flexure fabrication 110 thermoelastic loss φthermoelastic(ω) 1E-4 1E-5 1E-6 1E-7 1E-8 1E-9 f2 f1 f3 f4 f5 f6 fn = room temperature bending mode resonant frequencies for machined cantilever T = 293K T=160K T=120K T=100K Frequency (Hz) Figure 6.2: Plot of the calculated thermoelastic loss for the proposed silicon cantilever to be manufactured with the calculated resonant frequencies of the bending modes marked with arrows. 6.4 Silicon flexure fabrication A non-abrasive and non-contact method was found for fabricating singlecrystal cantilevers, this time from silicon wafers, by using a hydroxide chemical etch. The fabrication of these samples was carried out by Stefan Zappe at Stanford University. The anisotropic nature of such etching allows the reduction of thickness whilst a masked, thick end can remain as a clamping block to reduce any slip-stick losses as the cantilever flexes [178]. The geometry of the cantilevers obtained is shown in Figure 6.3. The thickness of the silicon wafers used for producing these cantilevers was 550µm, allowing a relatively thick clamping block to be maintained. The cantilever thickness was etched down to a thinner 95±5 µm. This resulted in the peak in the thermoelastic dissipation at room temperature to be pushed up to 16 khz, as seen in Figure 6.4. The resonant modes measured and presented within this chapter all lie below this thermoelastic peak. The silicon was boron doped with a resistivity of 10-20

137 6.5 Experimental Procedure 111 Ωcm with the <110> crystal axis along the length of the sample. clamping block 550µm thick cantilever surface on (100) plane cantilever 92µm thick cantilever length along <110> axis width 10mm Figure 6.3: Schematic diagram of silicon cantilever tested. 1E-4 fn = room temperature bending mode resonant frequencies for cantilever of length 57mm thermoelastic loss φthermoelastic(ω) 1E-5 1E-6 1E-7 1E-8 1E-9 fn = room temperature bending mode resonant frequencies for cantilever of length 34mm T = 293K T = 160 K T = 120 K T = 100 K f1 f2 f3 f4 f5 f1 f2 f3 f4 f5 f6 room temperature thermoelastic peak at ~16kHz Frequency (Hz) Figure 6.4: Plot of the calculated thermoelastic loss for the two tested silicon cantilevers, fabricated by wet chemical etching, with the calculated resonant frequencies of the bending modes marked with arrows. 6.5 Experimental Procedure Two readout setups were under consideration for taking cryogenic measurements of the silicon samples. Firstly, a shadow sensor very similar to that used in Chapter 3, could be adapted for monitoring the cantilever amplitude. The drawback to such a configuration is that two or three mirrors would have to be placed within the experimental chamber of the cryostat (Figure 6.5).

138 6.5 Experimental Procedure 112 For example, laser light would be directed down the long optical feed-through pipe into the experimental chamber, reflected using a mirror across the silicon sample and then reflected back out from the cryostat by a second mirror. The position of the shadow cast from the silicon cantilever would then be measured using a split photodiode. Two potential drawbacks appear with implementing this experimental setup: the placement of the mirrors would be difficult due to the confined space and the alignment of the mirrors may change significantly during cooling to cryogenic temperatures caused by the differential thermal expansion coefficients of the various materials. Therefore, a second configuration was proposed where a reflection sensor was implemented. In this setup laser light was directed down the optical feed-through pipe of the cryostat, reflected off the silicon cantilever s surface back out of the cryostat. The advantage here is that no mirrors are required within the experimental chamber. The reflected laser beam was then directed onto a split photodiode. A prototype setup was arranged on a bench can be seen in Figure 6.6 The prototype setup illustrated using a steel cantilever and excited by an electromagnetic actuator was installed into the cryostat system for taking measurements. However, a problem arose that made measurements of the cantilever amplitude impossible. The large distance between the silicon cantilever and the split photodiode readout resulted in the reflected laser spot s motion being large even for small cantilever amplitudes. Even the background excitation of the fundamental mode of the two cantilevers studied remained considerably above the saturation level of such a readout setup. The dynamic range of a split photodiode readout scheme is ±( 1 D), where D is the diameter of the laser spot, as shown in Figure 6.7. In fact, to maintain a 2 linear position sensing this should be kept to ±( 1 D). Therefore a variation of 10 this readout scheme was implemented where the split photodiode was replaced by a single photodiode. This single, longer photodiode was masked as to only keep a narrow triangular strip exposed at one side with the laser spot directed

139 6.5 Experimental Procedure 113 cable feed-through vacuum pumping optical feed-through temperature sensor mounted within clamp between heater and sample. silicon stainless steel clamp ceramic spacer resistive heater 1.4 m outer cryostat vacuum jacket experimental chamber liquid nitrogen bath silicon Figure 6.5: Schematic diagram of cryostat. steel blade reflected laser spot coil exciter reflected laser spot He-Ne laser laser reflected onto split photodiode sensor Figure 6.6: Illustration of the prototype setup of the readout system for the cryostat using a split photodiode arrangement as shown in Figure 6.7 consisting of two 5 5 mm single-element silicon planar photodiodes separated by 500µm.

140 6.5 Experimental Procedure 114 zero readout min max readout readout laser spot photodiode D D/2 Figure 6.7: Schematic diagram showing the dynamic range of a split photodiode readout scheme where the differential signal from both photodiodes gives the readout signal. midway along the triangle, see Figure 6.8. As the cantilever bends back-andforth, the angle of the reflected laser beam changes and the laser spot moves across the photodiode. Due to the triangular geometry of the sensing part of the photodiode, this results in the photodiode readout current increasing and decreasing. Using this modified readout scheme significantly increases the dynamic range from the split photodiode arrangement, up to ±10 mm which is at least a factor of ten larger than the split-photodiode arrangement. The calibration plots (or response plots) for both readout schemes are shown in Figures 6.9. The resultant sensitivity curves are shown in Figures The thick end of each cantilever was then held in a stainless steel clamp and placed within the cryostat, shown in Figure 6.5, evacuated to approximately mb. The resonant modes of each cantilever were excited in turn using an electrostatic drive plate. Laser light reflected from the silicon surface and directed onto the photodiode external to the cryostat allowed the angular motion of the end of the cantilever to be detected. The length of the lever arm due to the optical pipe leading to the inner experimental chamber of the cryostat made the readout system very sensitive to the cantilever motion. As a consequence, loss measurements on the first bending mode were not possible since the readout system saturated before the mode was excited to a level

141 6.5 Experimental Procedure V 10pF masked off area of photodiode 1kΩ +15V -15V laser spot 25.4mm D (to reduce non-linearity due to circular laser spot need h~d/5) h narrow triangular strip of exposed photodiode to photodiode 10µF OPA227 or LF357-15V 10µF Figure 6.8: Circuit diagram for the readout system used for reading out the cantilever amplitude in the cryostat. One mm single-element silicon planar photodiode is used where the active length is 25.4 mm and the maximum active width after masking, labeled h, is 1 mm. The photodiode mask was made from standard black electrical insulating tape and the laser spot diameter was measured to be 5.0 ± 0.5 mm. Photodiode readout (V RMS ) split photodiode calibration data -15-4x x x10-4 4x10-4 Angle (rad) Photodiode readout ( V RMS ) 6x10-4 single photodiode calibration data 4x10-4 2x x x x Angle (rad) Figure 6.9: Responses of split photodiode readout (see Figure 6.6) and single photodiode readout (see Figure 6.8) systems for measuring silicon cantilever displacement angle.

142 6.5 Experimental Procedure 116 noise readout - rad/(hz) -1/ actual (single photodiode) readout system with lights off actual (single photodiode) readout system with lights on prototype (split photodiode) readout system with lights off Frequency (Hz) Figure 6.10: Noise curves for single photodiode readout system with room lights on and off compared with noise curve for prototype split photodiode readout system with room lights off. significantly above the background excitation due to ground vibrations. A solution to this would be to use a lens to reduce this range of movement, however the available range to the laser beam was also observed to be restricted by the diameter of the window of the cryostat s optical feed-through and this would also have needed to be changed. It was possible to measure the frequency of the first resonance, and this is used later in section 6.6. The mechanical quality factor Q of a resonance of angular frequency ω 0 can be calculated from measurements of the amplitude A of freely decaying resonant motion. It can be shown that the time dependence of the amplitude decay is given by A(t) =A 0 e ω0t/2q, (6.1) where A 0 is the initial amplitude of the motion. The mechanical loss φ(ω 0 ) is the inverse of the quality factor [179] [139]. The mechanical losses of the several modes of each cantilever were measured at temperatures from 85 K

143 6.6 Temperature dependence of mode frequencies g mass catcher piezo transducer clamp silicon cantilever electrostatic pusher resistor heater base-plate temperature sensor mounted within clamp total clamp height 60mm ceramic spacer Figure 6.11: Picture of silicon cantilever mounted in the stainless steel clamp. to 300 K. Presented within this chapter are loss measurements for the second (f 240 Hz), third (f 670 Hz), fourth (f 1320 Hz), fifth (f 2185 Hz) and sixth (f 3265 Hz) bending modes of a 57 mm long cantilever and the third (f 1935 Hz), fourth (f 3788 Hz) and fifth (f 6265 Hz) modes of a 34 mm long cantilever. 6.6 Temperature dependence of mode frequencies At a given frequency the expected thermoelastic dissipation depends on the sample thickness. The thickness of the silicon samples was measured to be (92 ± 2) µm using a Wyko NT1100 Optical Profiler. The frequency of each bending mode changes as the silicon is cooled due to the temperature dependence of Young s modulus, E(T ). E(T ) canbecal- culated using the semi-empirical formula [180], E(T )=E 0 BTe T0/T, (6.2) where E 0 is the Young s Modulus at 0 K, B is a temperature independent

144 6.6 Temperature dependence of mode frequencies 118 constant related to the bulk modulus, T is the temperature in Kelvin and T 0 is related to the Debye Temperature. The angular resonant frequency ω of the third bending mode of a homogeneous beam of thickness t and length L is given by [181], ω =(7.583) 2 t E L 2 12ρ, (6.3) where ρ is the material density. As noted by Gysin et al. [182] any change in ω resulting from a temperature dependant variation in T, L or ρ is smaller than that from the variation in E and may be ignored. Using values of E 0 =1.69GPaalongthe< 110 > axis and T 0 = 317 K from Gysin et al. [182], Equations 6.2 and 6.3 were used to find a best-fit curve to the observed temperature dependence of the frequency and thus a value for E 0 obtained. It was possible to measure the first cantilever length to be 57.0 mm to an accuracy of ±0.5 mm without contacting the cantilever surface, and the second cantilever length to be (34.0 ± 0.5) mm. For the first cantilever, the third mode saw the best agreement between the predicted and experimental frequencies with an E 0 value of (161.7 ± 0.8) GPa. The temperature dependence of the calculated and measured frequencies for this mode are shown in Figure Applying the same model to the first and fifth resonant modes of this sample (approximately 39 Hz and 2185 Hz) gave very similar values for E 0. The average value of E 0 with associated standard error is (163 ± 4) GPa. Likewise, the third and fifth bending modes for the cantilever of length 34 mm matched the predicted frequencies when E 0 =(165± 6) GPa. Combining the calculated E 0 values yields (164 ± 3) GPa which appears close to the literature value E 0 = GPa [187].

145 6.7 Loss as a function of temperature 119 Figure 6.12: Temperature dependence of the frequency of the third resonant mode of the cantilever of length 57 mm at 670 Hz. Measured data (o) and modeled (-). The curve is a best fit where E o is calculated from the resonant frequency when B (a constant related to the bulk modulus) and T o (related to the Debye Temperature) are known. 6.7 Loss as a function of temperature The measured loss, φ measured (ω), is the sum of dissipation arising from a number of sources, φ measured (ω) =φ thermoelastic (ω)+φ bulk (ω)+φ surface (ω)+φ clamp (ω)+φ gas (ω)+φ other (ω), (6.4) where φ thermoelastic (ω) is loss resulting from thermoelastic damping, φ bulk (ω) is the bulk (or volume) loss of the material, φ surface (ω) is the loss associated with the surface layer, φ clamp (ω) is the loss associated with the clamping structure, φ gas (ω) is the loss due to damping from residual gas molecules and φ other (ω) is loss from any other possible dissipation process. In order to estimate the level of thermal noise expected from using silicon in gravitational wave detectors test masses and suspensions, φ thermoelastic (ω), φ bulk (ω) andφ surface (ω) mustbe quantified. Therefore in our experiment all the other sources of loss must be minimised.

146 6.8 Thermoelastic loss Thermoelastic loss Thermoelastic loss is associated with the flexing of a thin suspension element where the cyclical stretching and compression of alternate sides of a flexing sample results in heat flow between the compressed and expanded regions [81] [183]. The flow of heat is a source of loss. In the simple case of a bending bar of rectangular cross section, the thermoelastic loss can be expressed as [184], φ thermoelastic (ω) = ωτ 1+ω 2 τ, (6.5) 2 where = Eα2 T ρc, (6.6) and τ = 1 ρct 2 π 2 κ, (6.7) with τ the characteristic time for heat transfer across the bar, C is the specific heat capacity of the material and other parameters are as defined earlier. Equations 6.5 to 6.7 may be used to calculate the temperature dependent thermoelastic loss for our sample using the relevant material parameters. Table 6.8 shows the room temperature parameters used. The temperature dependent parameters were taken from Thermophysical Properties of Matter (Touloukian) [169], except for the value of Young s Modulus (E) which was worked out from the frequency measurements of the resonant modes (see Equation 6.2). Data for the coefficient of thermal expansion comes from the recommended curve, Touloukian Vol. 13 p.155, and the specific heat from curve 2, Touloukian Vol. 5 p.204. The thermal conductivity data is taken from the curves presented in Touloukian Vol. 2 p.326. Here minimum, median and maximum values are taken to represent the spread of data for single-crystal silicon at each temperature.

147 6.9 Mechanical loss in silicon flexures as a function of temperature 121 Parameter Young s Modulus (E) Coefficient of linear thermal expansion (α) Density (ρ) Specific heat capacity (C) Thermal conductivity (κ) Magnitude GPa K Kg m J Kg 1 K 1 min: 130 W m 1 K 1 median: 145 W m 1 K 1 max: 160 W m 1 K 1 Table 6.1: Room temperature parameters for silicon, the variations in thermal conducitivity dependent on the doping [169]. The uncertainty in the calculated magnitude of the thermoelastic loss comes predominantly from this variation in thermal conductivity (κ) between silicon samples. 6.9 Mechanical loss in silicon flexures as a function of temperature The measured mechanical losses of the fifth and third bending modes of the silicon cantilever of length 57 mm are shown in Figures 6.14 and The results obtained from the second, fourth and sixth bending modes showed similar trends across the temperature range. The measured mechanical loss of the third, fourth and fifth bending modes of the silicon cantilever of length 34 mm is shown in Figures 6.18, 6.19 and Plotted alongside are the predicted levels of thermoelastic dissipation for each mode. Each data point in Figures 6.14 to 6.20 represents the average of at least three consecutive loss measurements. To investigate the reproducibility of the measurements, the sample was repeatedly cooled to an initial temperature of 85K and loss measurements made as temperature was increased.

148 6.9 Mechanical loss in silicon flexures as a function of temperature (a) 10-6 φ(ω) (b) calculated thermoelastic loss 24 Jan Jan Mar Mar Mar Mar Mar Temperature (K) Figure 6.13: Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the second bending mode at 240 Hz, for cantilever of length 57mm (a) 10-6 φ(ω) (b) calculated thermoelastic loss 21 Jan Jan Feb Feb Feb Feb Feb Temperature (K) Figure 6.14: Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the third bending mode at 670 Hz, for cantilever of length 57mm.

149 6.9 Mechanical loss in silicon flexures as a function of temperature (a) 10-5 φ(ω) 10-6 (b) calculated thermoelastic loss 17 Jan Mar Mar Mar Temperature (K) Figure 6.15: Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the fourth bending mode at 1320 Hz, for cantilever of length 57mm (a) φ(ω) (b) calculated thermoelastic loss 10 Jan Jan Feb Feb Feb Feb Temperature (K) Figure 6.16: Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the fifth bending mode at 2185 Hz, for cantilever of length 57mm.

150 6.9 Mechanical loss in silicon flexures as a function of temperature (a) φ(ω) (b) calculated thermoelastic loss 17 Jan Jan Feb Feb Feb Feb Temperature (K) Figure 6.17: Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the sixth bending mode at 3265 Hz, for cantilever of length 57mm (a) 10-6 φ(ω) (b) calculated thermoelastic loss 15 Apr Apr Apr Temperature (K) Figure 6.18: Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the third bending mode at 1935 Hz, for cantilever of length 34mm.

151 6.9 Mechanical loss in silicon flexures as a function of temperature (a) φ(ω) (b) calculated thermoelastic loss 18 Apr Apr Temperature (K) Figure 6.19: Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the fourth bending mode at 3788 Hz, for cantilever of length 34mm. (a) φ(ω) (b) calculated thermoelastic loss 18 Apr Apr Temperature (K) Figure 6.20: Temperature dependence of (a) measured loss, and (b) calculated thermoelastic loss for the fifth bending mode at 6265 Hz, for cantilever of length 34mm.

152 6.10 Analysis of low temperature silicon results 126 The data shows a number of interesting features. First consider the measured mechanical loss of the fifth bending mode as shown in Figure Measurements of the mechanical loss at temperatures between 85K and 150K when made on different days could differ by up to a factor of 2.4. Also, during two measurement runs a broad dissipation peak was observed at around 200 K. However this is unlikely to result from an intrinsic loss mechanism in the sample since the peak is not observed in the results from 1st February. We believe that these effects are due to energy coupling into the clamping structure. Similar dissipation peaks are also observed in the higher frequency modes of the short cantilever in Figures 6.19 and This evidence of a coupling that is dependent on the temperature distribution inside the system, which may differ from run to run, is studied in section In contrast, the temperature dependence of the loss factors of the third bending modes for both cantilevers, shown in Figures 6.14 and 6.18, showed no sign of any dissipation peaks and appeared to have a smaller variation between experimental runs. It can be seen that the dominant loss mechanism at temperatures above approximately 160K is broadly consistent with thermoelastic effects. Possible candidates for the additional sources of loss observed at lower temperatures are discussed in the following section Analysis of low temperature silicon results It is useful to consider each possible loss mechanism in turn. This helps to estimate the magnitude of the various components of the measured loss measurements results presented and thus gain a better understanding of the intrinsic mechanical properties of silicon and the possible benefits of using silicon optics in third generation gravitational wave detectors.

153 6.11 Surface loss Surface loss Mechanical loss measurements carried out on silicon samples of sub-micron thickness suggest that the measured loss is dominated by surface losses [182] [185]. These may be due to a combination of the following: a thin layer of oxidized silicon on the surface [185]. shallow damage to the crystal structure (atomic lattice) from surface treatment. contaminants absorbed on or into the surface from the surroundings or from polishing. general (or local) surface roughness [186]. Yasumura et al measured the loss factors of single-crystal cantilevers with thickness in the range 0.06 µm to0.24µm and found they could be represented by [176], φ surface = 6δ t E S 1 E 1 φ s, (6.8) where φ surface is the limit to the measurable loss of a cantilever of thickness t and Young s modulus E 1 set by the presence of a surface layer of thickness δ, Young s modulus E S 1, and loss φ s. If for simplicity we assume E 1 E S 1, then Equation 6.8 may be used to estimate the limit to measurable dissipation for our sample, set by surface loss, by scaling with thickness the results of Yasumura et al. The magnitude of this scaled loss, φ surface (ω), summed with the upper limit to thermoelastic loss, φ thermoelastic (ω), for the third bending mode of the cantilever of length 57 mm is shown in Figure Recall that the measured loss varies from run to run and is most likely due to changes in the system during different cycles of cooling and heating. Since this spread is not intrinsic to the

154 6.11 Surface loss (a) φ(ω) (b) φ measured (ω) φ surface (ω) + φ thermoleastic (ω) Temperature (K) Figure 6.21: Plot of (a) the minimum measured loss of the third bending mode at f 670 Hz for the cantilever of length 57 mm compared with (b) the sum of the estimated surface loss and calculated thermoelastic loss. sample, the minimum measured losses are presented at each temperature point for comparison. It can be seen that below 160 K the sum of the estimated surface and thermoelastic loss is still lower than the experimental loss by up to a factor of six, thus other loss mechanisms are of a significant level. A possibility is that the depth of the damaged surface layer or the value of φ s is larger than in Yasamura s case. To estimate the depth of the damaged layer, the surface roughness of the polished and ground-etched surfaces of the silicon samples were measured using a Wyko NT1100 Optical Profiler, see Figures 6.22 and For the sake of argument, we approximate the thickness of a damaged surface layer to be approximately twice the typical surface roughness. Equation 6.8 can be rearranged for the case of a single surface such that, φ surface φ s = 3δ t E S 1 E 1 = energy stored in the damaged surface layer. (6.9) energy stored in bulk substrate Estimating the damaged surface layer, δ, resulting from the polishing and chemical etching technique to be approximately 8 nm and 1 µm respectively

155 6.11 Surface loss 129 will therefore yield ratios of the energy stored in the damaged surface layer to the energy stored in the bulk substrate of and respectively. The mechanical loss associated with a damaged surface layer, φ s,canthenbe calculated from the measured loss, φ surface, of a sample dominated by surface effects using the ratio of stored energies in Equation 6.9. Therefore, a cantilever of the dimensions tested here, with a mechanical loss dominated by surface effects and of the order of φ surface (ω) would have a surface loss of φ s (polished) if resulting from contamination and damaged from the polishing and φ s (etched) if resulting from contamination and damage due to the the chemical etching. Further investigations into such surface effects are required, and if found to be significant, techniques may be investigated for reducing this excess loss associated with the silicon cantilever surface, for example annealing [188] [189] or optimised forms of electro- or chemo-mechanical polishing. (a) (b) Figure 6.22: Plot of the (a) 2-D surface profile and (b) 3-D surface profile for the upper (ground and etched) surface of the cantilever of length 57 mm. RMS surface roughness = nm

156 6.12 Gas damping 130 (a) (b) Figure 6.23: Plot of the (a) 2-D surface profile and (b) 3-D surface profile for the lower (polished) surface of the cantilever of length 57 mm. RMS surface roughness = 3.99 nm 6.12 Gas damping Suitable vacuum pressures must be reached in order to avoid the measured loss being limited by the result of damping from residual gas molecules in the system. At room temperature the recorded range of gas pressures was mb. However, an accurate measure of the gas pressure within the experimental chamber was not possible as the sensor was some distance from the experimental chamber. A residual gas analyser sensitive to molecular weights up to 200 indicated that residual molecules were mainly nitrogen (N 2 )andwater(h 2 O). The level of loss due to gas damping of an oscillator can be expressed as [139], φ gas AP M mω RT MRT, (6.10) where A is the surface area, P is the pressure, m is the mass of the oscillator, ω is the angular frequency of the resonant mode, M is the mass of one mole

157 6.13 Bulk loss 131 of the gas, R is the gas constant and T the temperature. For gas damping to be the dominant loss mechanism at room temperature the gas pressure in the experimental chamber would be approximately mb, assuming the gas to be N 2, which is much higher than could be reasonably expected, particularly as the residual gas pressure is expected to decrease with temperature. In Figures 6.14 and 6.18, below 160 K the level of dissipation for both modes at temperatures at best approaches levels around 10 6 and thus there is no apparent 1/ω 0 dependence on frequency which would be expected in the case of gas damping Bulk loss Results presented in chapter 5 and measurements by other authors have shown that the intrinsic bulk dissipation of single-crystal silicon cylinders at room temperature can be as low as and in general is found to decrease as temperature decreases [190] to [191]. This suggests that the bulk loss of silicon is significantly lower than the measured losses of the cantilever studied here. However the measurements of McGuigan et al. [138] revealed dissipation peaks near the temperatures where the coefficient of thermal expansion goes to zero. Other experimenters have also observed dissipation peaks at these, and other temperatures, see for example [176]. There are a variety of explanations postulated in the literature for the existences of each of the peaks observed - however there appears no reason that the peaks should be related to the zeros in the coefficient of expansion. Therefore it is of interest to investigate whether such peaks in the loss are observed in the sample being studied here. Over the temperature range from K there is a plateau in the loss in all the modes presented in Figures 6.14 to 6.18 and there is no sign of a clear dissipation peak within the temperature range of these measurements, at the levels of dissipation observed.

158 6.14 Modeled elastic loss in the clamping structure from a resonating flexure Modeled elastic loss in the clamping structure from a resonating flexure. Another source of loss to be considered is the elastic loss in the stainless steel clamp as the silicon cantilever flexes. As the cantilever bends the clamping structure experiences an elastic distortion proportional to the force exerted by the cantilever divided by the elastic modulus of the clamp. Subsequently a fraction of this elastic energy will be lost over each cycle of the cantilever s bending resulting from the mechanical loss (or dissipation) of the clamping structure s material. To estimate this level of energy loss a finite element simulation was run using Ansys. The model consisted of a stainless steel structure to simulate the upper clamping block, lower clamping block and small connecting pads (comparable in dimension to the stainless steel screws). Within this clamping structure was placed a silicon block to simulate the thicker (550 µm) clamping block with an attached thinner (92 µm) flexing part to simulate the cantilever. Figure 6.24 shows this Ansys model. upper clamp cantilever thick end lower clamp cantilever 3mm "pads" (approx. 3mm screws) elements of F.E. model Figure 6.24: Illustration of Ansys finite element simulation to calculate the elastic energy residing in the clamp as the cantilever flexes. The model was run using the parameters shown in Table The silicon and stainless steel surfaces that were in contact were rigidly bonded using the CEINTF command in Ansys for generating constraint equations between

159 6.14 Modeled elastic loss in the clamping structure from a resonating flexure. 133 Parameter Silicon Stainless steel Young s Modulus (E) GPa GPa Density (ρ) 2330 Kg m Kg m 3 Poison s Ratio (v) Table 6.2: Room temperature parameters for silicon and stainless steel. the nodes of the silicon clamping block and the elements of the stainless steel clamp in contact. The elastic energy was found by summing the strain-energy components for each element in the model. Figures 6.25 and 6.26 show the displacement of the clamp and cantilever for two of the cantilever s resonances. The proportion of elastic energy stored within the clamping structure as each of the cantilevers flex as they resonant at their bending modes has been modeled from 0 Hz 13 khz and is shown in figure (a) clamp elastic displacement (b) cantilever elastic displacement Figure 6.25: Finite element simulation of the elastic displacement in the stainless steel clamp and the silicon cantilever for the third bending mode of the cantilever of length 57mm at f 670 Hz. The results presented predict the level of elastic loss due the clamping structure to be somewhere below 10 4 for the cantilever of length 57 mm and below for the cantilever of length 34 mm. A reasonable upper limit to the loss of stainless steel would be around Therefore, the minimum expected loss measurable for the longer cantilever would be 10 7 at 6kHz

160 6.14 Modeled elastic loss in the clamping structure from a resonating flexure. 134 (a) clamp elastic displacement (b) cantilever elastic displacement Figure 6.26: Finite element simulation of the elastic displacement in the stainless steel clamp and the silicon cantilever for the sixth bending mode of the cantilever of length 57mm at f 6094 Hz. proportional elastic energy (clamp/total) 1x10-4 5x10-5 cantilever of length 34 mm cantilever of length 57 mm frequency (Hz) Figure 6.27: Plot of the proportional elastic energy residing in the stainless steel clamp structure as the silicon cantilever bends.

161 6.14 Modeled elastic loss in the clamping structure from a resonating flexure. 135 and several 10 8 at 3.2 khz and below where measurements were taken. The mechanical loss of the stainless steel is also likely to decrease with temperature and make this figure for the limiting dissipation due to elastic loss in the clamp lower still. The results also show an interesting feature. The level of elastic energy residing in the clamping structure broadly follows a linear relationship with an increased level between 5 and 10 khz for both cantilever lengths. In the linear portion, the majority of the energy appeared to reside in the clamping parts close to the silicon sample, as can be seen in Figure The results where an increased level of elastic energy was observed above this showed the small, upper clamping block apparently rocking back-and-forth, again seen in Figure 6.26, with elastic energy being driven throughout the whole clamping structure. This is likely to be due to a resonance of the upper part of the clamping structure close in frequency (6984 Hz) as shown in Figure This particular feature is perhaps only applicable to this simplified model of the clamping structure and not the actual experimental setup used. However, this result indicates that elastic loss may become a problem if the resonating cantilever was very close in frequency to a given internal resonance of the clamping structure, which is in the upper part of the clamp. clamp-cantilever resonance at 6984Hz Figure 6.28: Finite element simulation of the elastic displacement for the clampcantilever resonance at f 6984 Hz.

162 6.15 Clamping Loss Clamping Loss For the case of a two-dimensional system radiating into a semi-infinite silicon substrate the structural loss can be estimated by the following expression [192], ( ) 3 t φ support = β, (6.11) L where t and L are the thickness and length of the flexing beam and where the typical values for the constant β lie in the range 2 3 [192][193]. This would give a limiting loss factor of between and This is significantly below the measured losses for this particular cantilever. However stick-slip losses may also exist associated with friction at the clamped end of the oscillating sample [178] Other losses As previously discussed, for both cantilevers measured there is clear evidence of excess dissipation above that estimated for the sources detailed through Sections 6.11 to Intermittent dissipation peaks were seen in several of the measured modes between the two cantilevers. These peaks were most likely due to energy loss into the clamping structure. To investigate this, a piezo transducer was attached to the upper part of the clamp to sense displacements of the clamp which could result from energy coupling to the clamp from an excitedmodeofthecantilever.thiswascarriedoutforthefifthmodeofthe cantilever of length 57 mm. A comparison can then be made with the estimated excess loss measured in this particular cantilever mode. The excess loss was found by subtracting the calculated thermoelastic loss from the measured total loss. In the following plots, each mode was excited to a similar amplitude and the magnitude of the peak at the relevant modal frequency was found using a spectrum analyser. The magnitude of the signal sensed by the piezo was then divided by the amplitude of the signal from the oscillating cantilever to normalise the piezo data.

163 6.16 Other losses 137 φ (ω) -φ (ω) measured thermoelastic 1x10-5 9x10-6 8x10-6 7x10-6 6x10-6 5x10-6 4x10-6 3x kHz bending mode φ (ω)- φ (ω) measured thermoelastic Piezo response Temperature (K) 7x10-5 6x10-5 5x10-5 4x10-5 3x10-5 2x10-5 Normalised piezo response (arb. units) Figure 6.29: Plot of the average excess loss of the fourth bending mode at f 1.3 khz compared with the normalised magnitude of the signal from the piezo sensor. Sample length 57 mm. φ (ω) - φ (ω) measured thermoelastic 6x10-5 5x10-5 4x10-5 3x10-5 2x10-5 1x kHz bending mode φ (ω)-φ (ω) measured thermoelastic Piezo response Temperature (K) 6x10-4 5x10-4 4x10-4 3x10-4 2x10-4 1x10-4 Normalised piezo response (arb. units) Figure 6.30: Plot of the average excess loss of the fourth bending mode at f 2.2 khz compared with the normalised magnitude of the signal from the piezo sensor. Sample length 57 mm.

164 6.16 Other losses 138 φ (ω) - φ (ω) measured thermoelastic 1x10-5 8x10-6 6x10-6 4x10-6 2x kHz bending mode φ (ω)-φ (ω) measured thermoelastic Piezo response Temperature (K) 1x10-3 8x10-4 6x10-4 4x10-4 2x Normalised piezo response (arb. units) Figure 6.31: Plot of the average excess loss of the fifth bending mode at f 3.1 khz compared with the normalised magnitude of the signal from the piezo sensor. Sample length 57 mm. In general the above plots indicate a clear relationship between the excess loss factors measured in this sample and the energy coupling to the clamp at the same frequency. Further work must therefore be carried out to attempt to damp the resonances in the clamping structure or to shift the resonances by adding mass. The response of the clamping structure was also studied across the frequency range of measurements. In Figure 6.32, the response of the clamp from 0 Hz 10 khz is presented. Despite the significant detail in this plot, clear resonances are seen to reside in the clamp at many frequencies. The first clear resonances occur between 2.1 khz and 2.2 khz. This gives a likely explanation for the intermittent dissipation peak observed in Figure 6.16 in the 2185 Hz mode of the cantilever of length 57 mm. A mine-field of clamp resonances is seen at frequencies above 2 khz that may couple energy away from a clamped, resonating sample. This gives a likely explanation for the significantly higher mechanical losses measured in the higher frequency modes (see Figures 6.19 and 6.20) of the cantilever of length 34 mm.

165 6.16 Other losses Cantilever amplitude readout (V) (b) (a) piezo response readout (V) cantilever amplitude readout (V) stainless steel clamp transfer function piezo driven with 100mV pk-pk signal temperature of system = 295 K Piezo response (V) Hz fundemental 690 Hz 2nd bending 1927 Hz 3rd bending 2147 Hz 2nd torsional 3775 Hz 4th bending Frequency (Hz) 6243 Hz 5th bending 8166 Hz 8550 Hz 9037 Hz 9334 Hz Figure 6.32: Plot of (a) the response of the clamping structure to a constant driving signal and (b) the subsequent excitation of the modes of the cantilever of length 34 mm from this driving signal.

166 6.17 Conclusions 140 8x10-5 7x10-5 6x K 270K 285K 5x10-5 Piezo response (V) 4x10-5 3x10-5 2x Frequency (Hz) Figure 6.33: Plot of the response of the clamping structure to a constant driving signal at 240 K, 270 K and 280 K. Figure 6.33 shows that the frequencies of these clamp resonances are likely to shift as the temperature changes. This reveals that the clamp resonances observed depend on the temperature distribution within the clamping structure. Therefore it is likely that dissipation peaks observed in our samples, caused by a coupling to the clamping structure, may appear at different frequencies from run to run as seen in Figure Conclusions The measurements presented here of the mechanical dissipation of single crystal silicon cantilevers as a function of temperature in general show the dissipation decreasing as temperature decreases. At room temperature the measured dissipation is strongly dependent on the level of thermoelastic dissipation in the sample, however at lower temperatures other loss mechanisms become dominant. Losses associated with the surface of the samples are expected to be significant, but at a level lower than the measured losses. The level of surface loss will be investigated further by measuring the dissipation in samples of dif-

167 6.17 Conclusions 141 ferent thicknesses. A possible source of loss requiring further investigation is that of frictional losses associated with the end of the sample moving inside the clamp. To reduce this effect, samples with a greater ratio of thickness of clamp end to thickness of cantilever will be studied. In particular we do not see any distinct peaks in the dissipation close to 125K [169], at the level of dissipation found here. In common with other researchers, intermittent dissipation peaks have been observed at other temperatures in several of our experimental runs. However, it is very likely that these are not intrinsic to our silicon samples but rather here are due to couplings to the clamping structure used.

168 Chapter 7 Conclusions As current interferometric gravitational wave detectors reach their design sensitivities, we enter an era where the possibility of direct detection of gravitational waves from astrophysical sources moves from being a theory to a reality. Throughout the interferometer network, targeted R&D continues with the aim of further reducing the various sources of noise limiting the sensitivities of these current detectors. The monolithic suspension technology of GEO600 will be one such technique allowing the longer baseline detectors to be more sensitive, by reducing the thermal displacement noise. In addition to this, various other technical improvements and the implementation of higher power lasers with the necessary thermal compensation schemes should allow these second generation detectors to reach sensitivities around 10 times higher in their detection band. At this stage many signals, particularly those from coalescing compact binary systems, are expected to be at a detectable level. With this in mind, further R&D is required for developing the technology for third generation detectors operating with sensitivities allowing large-scale observational astronomy to be carried out. With regards to mirror substrate materials, results presented of the mechanical loss of various fused silica and sapphire samples show that these materials exhibit very low levels of internal dissipation, a requirement for good thermal noise performance in optics constructed using these materials. In par- 142

169 143 ticular, calculations of the substrate thermal noise in the GEO600 detector indicate that the substrate thermal noise levels are likely to be around a factor of 10 lower than previously predicted. Thermal noise associated with the mirror dielectric multi-layer coatings is thus likely to be the dominant source of thermal noise in the GEO600 detector. Reducing the mechanical loss associated with such dielectric coatings will then be crucial for reducing the coating thermal noise in future detectors. Results presented of the mechanical loss associated with dielectric coatings reveal that the dissipation arises predominantly from the tantala component of the silica-tantala coatings and that doping the tantalum pentoxide with titania can reduce the mechanical loss by a factor of approximately two. Further investigations into optimising the level of doping in order to reduce associated the mechanical losses will be one required step for performance improvements for future detectors. Additionally, the work carried out here highlights the benefits of moving to a reflective interferometry regime for third generation detectors, using silicon optics. Silicon s low mechanical loss and high thermal conductivity make this a very interesting material for the future. Measurements of bulk samples of silicon have shown the mechanical loss at room temperature to be as low as φ(ω) = for the [111] orientation, and experiments are ongoing to study losses at cryogenic temperatures. Measurements of 92 µm thick flexures for use as suspension elements have shown the mechanical loss to follow thermal elastic damping above 160 K and reach levels of φ(ω) = at 85 K and measurements at lower temperatures are envisaged. The high thermal conductivity of silicon will allow significantly higher levels of thermal loading to be tolerated over the currently used fused silica optics, allowing further increases in laser power and thus improved shot noise performance in such future detector designs. Investigations into hydroxy-catalysis bonding has quantified two techniques for extending the settling time for ultra-stable jointing of optical components;

170 144 by increasing the hydroxide concentration of the bonding solution and by lowering the temperature. This is important for any optical or mechanical system where extended times are required for the precision alignment of various components, and will find application in various space-based projects operating with precision optical sensing. The experimental data and the analysis using the Rate Law and Arrhenius formulae for chemical reactions appear to be in very close agreement. This will benefit any aspect of future work using this technique, whatever the application, in more precisely quantifying the various processes taking place during hydroxy-catalysis bonding. In regard to future gravitational wave detectors, using this knowledge to investigate methods for reducing the mechanical loss of the joints may be important, for example by optimising the mechanical strength in order to reduced the bond geometry and thus reduce the associated mechanical losses. Applying this technology to silicon optics will be required for the construction of monolithic silicon optics required for the third generation of interferometric detectors. In summary, the experimental investigations carried out here build on previous work carried out within the gravitational wave field, particularly in Glasgow, and contributes to designs and topologies for future detectors. With the possibility of direct detection of gravitational waves approaching and the maturing of the technology for second and third generation detectors, the ground is being set for the beginning of gravitational wave astronomy, with all the new knowledge that this will bring to modern astronomy.

171 Appendix A Aspects of Pendulum Dynamics A.1 Introduction Significant work has previously been carried out to study the mechanical losses of these pendulum systems [194] [195] [196] and questions have arisen as to the beat frequencies observed during the measurements and the possibilities of there existing chaotic effects due to non-linearities. One particular experiment carried out in Glasgow by Heptonstall et al [197] used a pendulum system under vacuum where the mass was suspended by a silica ribbon. The aim was to measure the mechanical loss factor of the fused silica ribbon suspension element. Here a silica mass was flame welded to a silica ribbon, suspended, then gently kicked. Care was taken to give the system kinetic energy only in the direction where the ribbon was thinnest (or least stiff). The pendulum was setup in a vacuum system, to eliminate the effects of gas damping. A 1D motion sensor was then used to monitor the amplitude of the pendulum s swing over time. Figure A.1 illustrates the motion applied to this pendulum. Once the pendulum was set swinging the air was evacuated from the vacuum tank. To reduce the air damping effects to a negligible level, approximately one week passed before a sufficient vacuum level (< 10 6 mb) could be reached to allow experimental measurements of the pendulum s Q to begin. The pendulum was expected to remain swinging back-and-forth in the direction it was originally 145

172 A.1 Introduction 146 Figure A.1: Motion applied to suspended mass in Heptonstall s pendulum setup. given motion. However, when the data acquisition started the pendulum motion appeared to have changed. The amplitude of the pendulum was instead increasing and decreasing with a period of approximately 3 minutes (182 s). If the period was much longer (many hours) then this apparent change in amplitude might be attributed to the Coriolis Effect, due to the rotation of the earth exerting a small torque on the clamped part of the pendulum. This phenomenon raises the question of what is causing the pendulum motion to change? Simple pendulums are understood to have two specific mode directions. The total energy of the pendulum is divided between these two modes. The energy in each mode would be expected to remain constant (in a lossless, linear system) and the displacement of the pendulum mass would be defined as the superposition resulting from each mode. If the pendulum was then observed along a direction which was not the direction of one of these modes, then the apparent magnitude may be seen to change periodically. This happens when the frequencies of each mode are different and the varying amplitude would in fact be the observed beating between these two modes. Another possible explanation is that there may exist a mechanism in the pendulum system for coupling energy between the two modes of swinging. This would be observed by the amplitude of the pendulum slowly decreasing in one direction, while the amplitude in another direction slowly increases.

173 A.2 Silica Pendulum Setup and Results 147 A simple pendulum system was constructed in an attempt to better understand the dynamics observed by Heptonstall et. al. An XY displacement sensor was constructed to more completely record the position of the pendulum over time. There are however two differences between the experiment carried out here and the one by Heptonstall. Firstly, this experiment was carried out in air and not under vacuum. Short data runs were therefore taken to minimise the observed amplitude decay resulting from gas damping. Secondly, the ribbon in this setup was clamped to the rigid supporting structure and likewise clamped to the suspended mass. In the experiment carried out by Heptonstall, the ribbon was instead welded to both the support and the mass. A.2 Silica Pendulum Setup and Results shadow from fibre/ribbon detectable region photodiode x y Figure A.2: XY Optical sensor comprising of two pairs of photodiodes each detecting orthogonal directions of motion. Two orthogonal sensors were mounted together using infra-red laser diodes to illuminate, and cast a shadow from, the ribbon on to the photodiode arrangement as shown in Figure A.2. The differential signal of the two photodiodes was amplified and the data sent to a computer data logger. Looking at Figure A.2 one can see that the differential signal of the photodiodes when the pendulum is at rest, or zero displacement, will be zero when the shadow

174 A.2 Silica Pendulum Setup and Results 148 is symmetrically positioned. During the experiment it was observed that, if the pendulum was only kicked in the direction where the ribbon is thinnest, it remained swinging primarily in that one direction. In order to observe the pendulum dynamics when both pendulum modes are excited, the suspended mass must be kicked at a different angle, here 45 o, as shown in Figure A.3. The shadow sensor was mounted near the bottom of the suspension at 45 o to the dimensions of the silica ribbon (the two orthogonal position sensors were looking diagonally through the dimensions of the ribbon). This was done to help prevent any transmitted light passing through the ribbon and hitting the photodiodes. A plot of the raw output signals from the photodidoes over a 2 minute period is shown in Figure A.4, where clearly there is an observed beating. The readout scheme had also been calibrated in order to calculate the underlying X and Y displacements of the suspended mass from the photodiode readout. Due to small amounts of crosstalk between the orthogonal sensors, a 2D calibration had to be implemented. That is, the shadow sensor designed to detect displacement in only the X- direction had some sensitivity to displacement in the Y -direction. This is most likely due to light being reflected or transmitted through the silica and then directed onto the orthogonal photodiodes. The results presented in Figures A.5 and A.6 show the displacement of pendulum motion over both a 20 and 40 second period. Energy conservation rules dictate that the modes of a simple pendulum must be orthogonal, as studied in the theoretical model in section A.3. This means the trace of the pendulum motion should map out a rectangle whose boundaries are set by the total energy of the system. The fact that the rectangular plot in Figures A.5 and A.6 is not a perfect rectangle may suggest the calibration was also not perfect. It would be desirable to design the orthogonal shadow sensors to be completely independent, thus having no degree of cross-talk. Also, the pendulum motion shows a degree of oscillation at a higher

175 A.2 Silica Pendulum Setup and Results 149 ~45 o Figure A.3: Motion applied to suspended mass of the pendulum first photodiode-pair readout (V) second photodiode-pair readout (V) time (s) Figure A.4: Output signals from orthogonal photodiodes showing observed beating in the pendulum motion.

176 A.2 Silica Pendulum Setup and Results trace of pendulum motion over 20s y-displacement (mm) x-displacement (mm) Figure A.5: XY plot of silica ribbon pendulum motion, over 20 seconds in air. 6 4 trace of pendulum motion over 40s y-displacement (mm) x-displacement (mm) Figure A.6: XY plot of silica ribbon pendulum motion, over 40 seconds in air.

177 A.3 Analysis of Observed Pendulum Dynamics 151 frequency than the pendulum mode and this effect appears more pronounced in some directions over others. Since this artifact appears to have an intrinsic frequency, it is likely due to another resonant dynamic of the pendulum, e.g. from the suspended mass rocking about the point where the ribbon is clamped. A.3 Analysis of Observed Pendulum Dynamics To analyse these experimental results, a theoretical model can be simulated to study the pendulum motion over time [198]. Let the displacements as a function of time of the two modes of such a pendulum, ξ and η respectively, be; ξ = A cos(ωt), (A.1) η = B cos((ω + ω)t), (A.2) where ω 1 = ω is the frequency associated with the first mode, and ω 2 = y η ξ θ 2 θ 1 x Figure A.7: Pendulum mode directions for the theoretical model. ω + ω is the frequency associated with the second mode, where ω is small.

178 A.3 Analysis of Observed Pendulum Dynamics 152 The x and y displacements and ẋ and ẏ velocities of the pendulum are therefore: x = ξ cos θ 1 + η cos θ 2, (A.3) y = ξ sin θ 1 + η sin θ 2, (A.4) ẋ = Aω 1 sin(ω 1 t)cosθ 1 Bω 2 sin(ω 2 t)cosθ 2, (A.5) ẏ = Aω 1 sin(ω 1 t)sinθ 1 Bω 2 sin(ω 2 t)sinθ 2, (A.6) Energy conservation in a lossless isolated system tell us that θ 1 and θ 2 must be orthogonal, although proof of this is not given here. A computer model was run using these equations to simulate pendulum motion. A plot of pendulum position over a time of 100 seconds is given in Figure A.8. The oscillation of the two modes over this time is shown in Figure A.9. Both plots are very comparable to the experimental data in Figures A.4, A.5 and A.6. Note that the beating modes are only seen in the experimental and the theory since we are observing the pendulum displacements at a different angle to the pendulum mode directions. If one was to look in the mode directions, ξ and η in the theory, there would be no observed beat Figure A.8: Theoretical X-Y plot of pendulum motion, A=1, B=0.7, θ 1 = π 4, θ 2 = π 4, ω 1 =2π, ω 2 = π

179 A.4 Further pendulum analysis 153 Figure A.9: Plot showing a simulation of amplitude of the X and Y displacements due to the oscillation of a pendulum as a function of time. The pendulum was modeled as being suspended by a single ribbon fibre. A.4 Further pendulum analysis It is also possible to consider the observed motion of a pendulum in terms of its chaotic behaviour, or conversely the degree of confinement and predictability. It would be interesting to study such effects in pendulum systems similar to those used for suspending the mirrors in the interferometers. For example, a chaotic system can be mathematically characterised by how it deviates from its expected path when it is perturbed by an external force. For example, if the deviation grows in an exponential fashion then it is characterised as chaotic. Whereas, in a non-chaotic system, the deviation will only grow in a linear fashion. The simple example of a butterfly flapping its wings causing a subsequent tornado on the other side of the globe is commonly used. One method for identifying chaotic behaviour is to plot the phase-space of the particle or system. Here, the position of the pendulum system at incremental units of time are plotted against each other, x n+1 versus x n and likewise y n+1 versus y n, shown in Figures A.10 and A.11. The larger degree of chaos in a given system, the more phase-space will be occupied, filling a larger area in phase-space plots. The limited phasespace occupied in Figures A.10 and A.10 is consistent with the large degree of