Aspects, techniques and design of advanced interferometric gravitational wave detectors


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1 Aspects, techniques and design of advanced interferometric gravitational wave detectors by Pablo J. Barriga Campino Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy School of Physics The University of Western Australia 2009
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3 To: Sam and our future...
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5 Abstract The research described in this thesis investigates some of the key technologies required to improve the sensitivity of the next generation of interferometric gravitational wave detectors. A complete optical design of a high optical power, suspended modecleaner was undertaken in order to reduce any spatial or frequency instability of the input laser beam. This includes a study of thermal effects due to high circulating power, and a separate study of a vibration isolation system. In the last few years the Australian Consortium for Interferometric Gravitational Astronomy (ACIGA) has developed an advanced vibration isolation system, which is planned for use in the Australian International Gravitational Observatory (AIGO). A local control system originally developed for the modecleaner vibration isolator has evolved for application to the main vibration isolation system. Two vibration isolator systems have been assembled and installed at the Gingin Test Facility ( 80 km north of Perth in Western Australia) for performance testing, requiring installation of a Nd:YAG laser to measure the cavity longitudinal residual motion. Results demonstrate residual motion at nanometre level at 1 Hz. Increasing the circulating power in the main arm cavities of the interferometer can amplify the photonphonon interaction between the test mass and the circulating beam, enhancing the threemode parametric interaction and creating an optical spring effect. As part of a broader study of parametric instabilities, in collaboration with the California Institute of Technology, a simulation of the circulating beam in the main arms of Advanced LIGO was completed to determine the characteristics of the higher order optical modes. The simulation encompassed diffraction losses, optical gain, optical mode Qfactor and mode frequency separation. The results are presented for varying mode orders as a function of mirror diameter. The effect of test mass tilt on diffraction losses, and different coatings configurations are also presented. These simulations led to a study of the effect of power recycling cavities in higher order mode i
6 suppression. Current interferometric gravitational wave detectors in operation have marginally stable recycling cavities, where higher order modes are enhanced by the power recycling cavity, effectively increasing the parametric gain. A study of stable recycling cavities was undertaken in collaboration with the University of Florida, which has evolved into a proposed design for a 5 kilometre AIGO interferometer. This thesis concludes with an analysis of the work here presented and an outline of future work that will help to improve the design of advanced interferometric gravitational wave detectors. ii
7 Acknowledgements This thesis would have not been possible without the support, help and contribution of many people. First and foremost I would like to thank my supervisors Prof. David Blair, Dr. Li Ju (Juli) and Dr. Chunnong Zhao for their support and encouragement throughout this research. I am grateful to David for welcoming me into the gravitational waves group fresh from Chile, for finding the time to discuss and review different technical aspects of this research within his always busy schedule and for driving this project. I thank him for his confidence in me representing the group at international conferences and his understanding of Spanglish. I thank Juli for always being an open and approachable person, ready to help in different aspects of this research, including lending me her desktop computer to run simulations for several weeks. I also thank Zhao for his support and knowledge in the more technical aspects of this research, his guidance in the Gingin experiments were invaluable. It is unfortunate the busy work schedule worked against us having more football games. There are also a large number of people that have helped me in one way or another throughout this research. Thank you Riccardo DeSalvo and Phil Willems for being great hosts during my visit to the California Institute of Technology and P. Willems for his time in discussing the FFT code, which he helped to improve. Thanks also to Guido Mueller for his support and patience during the development of the optical design for recycling cavities and useful discussions during the Gingin workshop and afterwards. Also a big thanks to the people at LIGO Hanford for making those days and night shifts more enjoyable (especially during the owl shift) and to to the people at ANU and Adelaide University with whom I shared some good moments in various conferences. I would like to thank many people for their friendship and support; Jerome and iii
8 Sascha; Andrew and Jean Charles (long days and nights in Gingin making things work); the regular Gingin pilgrims Fan, Lucianne and Sunil; and David Coward and Eric. Special thanks to Slawek for sharing enormous amounts of coffee and invaluable discussions about gravitational waves and any other topics that came to mind. Of course none of these would have been possible without the love and support of my family (Francisco, Linda and Laura, Rodrigo, Daphne and Emilia) and my parents Pablo and Victoria. We went through difficult times when the distance between Chile and Australia did not help at all. My sister Alejandra is dearly missed. I would also like to thank Nigel and Christine for their support and interest in what their favourite son in law is doing. And last, but definitely not least, thank you to my wife Sam for her support, encouragement, love, faith, patience, for a few welldeserved kicks every nowandthen, and for reading this thesis (more than once) and for her interest in understanding what I have been doing all these years. iv
9 Preface This Preface presents an overview of the content of this thesis. The chapters predominantly correspond to published papers, submitted papers and a report to the LIGO Scientific Collaboration. The chapters in this thesis present advanced vibration isolation systems, optical design and simulation of optical subsystems for advanced interferometric gravitational wave detectors. The order of the chapters is not related to the time order of the work, nor to the order in which the papers were published. The Introduction (Chapter One) presents a review of interferometric gravitational wave detectors including the history behind the development of this field and its future direction. It also describes a number of other projects around the world that are aiming to directly detect gravitational waves and presents some of the noise sources that need to be overcome. The two following chapters are related to seismic noise and advanced vibration isolation systems. Chapter Two presents a novel design for a modecleaner vibration isolation system. The content of this chapter corresponds to early work from the author in vibration isolation and contains predictions on the performance of the proposed isolation system. Research undertaken on the control electronics and the local control system for the modecleaner vibration isolator was then upgraded for use with an advanced vibration isolation system. This work is presented in Chapter Three, which describes the assembly of two advanced vibration isolators at the High Optical Power Facility in Gingin, 80 km north of Perth in Western Australia. This chapter contains two recently submitted papers, which present the performance of the vibration isolation systems and the the local control system developed for their operation. The complete design of the control electronics for the advanced vibration isolator is presented in Appendix C. Chapter Four contains three papers which present the research undertaken for the optical design of a high power modecleaner. This work was started during developv
10 ment of the vibration isolation system for the modecleaner. The first paper presents the basic optical design of the triangular ring cavity used as an optical modecleaner, including a preliminary study of thermal effects in this cavity. The thermal effects and their consequences are then studied thoroughly in the following paper. In the final paper of this chapter a design for an astigmatism free modecleaner is proposed. Continuing with the study of optical modes suppression in advanced interferometers, Chapter Five simulates the behaviour of higher order optical modes in the main arms of an advanced interferometer. Although the simulations were made specifically for an Advanced LIGO type of cavity, they can be easily extended to any advanced interferometric gravitational wave detector. The calculations of the diffraction losses are an important component of the parametric instabilities studies carried out by the University of Western Australia. The studies by the author also included the effects of mirror tilt and how they affect diffraction losses. A study of the potential use of an apodising coating to reduce the possibility of parametric instabilities is also presented. This work was presented at the California Institute of Technology and later published as a technical report (T Z). Research forming the core of Chapter Five was undertaken in 2006 using the design parameters of Advanced LIGO at that time. Since then the design parameters for Advanced LIGO have changed and an actualisation of the main calculations is presented as a postscript in this chapter. Appendix B presents an overview of the theory behind parametric instabilities and strategies for the control of parametric instabilities in advanced gravitational wave detectors. The studies of optical modes are then extended in Chapter Six to include recycling cavities and the design of a dual recycled interferometer for gravitational wave detection, utilising stable recycling cavities instead of the marginally stable cavities used in current interferometric detectors. The publication at the core of this chapter was developed in collaboration with the University of Florida following a parametric instabilities workshop held in Gingin. This chapter presents a proposed design for the Australian International Gravitational Observatory (AIGO). This design is complemented in Appendix A with a description of the science benefits of a southern hemisphere advanced gravitational wave detector, work that was presented at the vi
11 Amaldi 7 conference on gravitational waves. The Conclusions present a discussion of the work presented in this thesis and future work. This includes the proposed new optical design for an astigmatismfree modecleaner using a novel and compact vibration isolation system. This design could also be used for auxiliary optics in advanced gravitational wave detectors. An advanced vibration isolator was developed through a combination of different preisolation techniques and tested at the AIGO test facility. This included the development of the control electronics and a local control system, and demonstration of a residual motion of 1 nm at 1 Hz. However this performance could be improved with the addition of actuators on the Roberts linkage stage. This addition will improve performance at low frequencies with the use of a superspring configuration. The analysis of recycling cavities for advanced gravitational wave detectors gives promising results, although further analysis including their effects in parametric instabilities is still necessary. vii
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13 Publications used for this thesis P. Barriga, A. Woolley, C. Zhao, D. G. Blair, Application of new preisolation techniques to modecleaner design, Class. Quantum Grav. 21 (2004) S951 S958 (Chapter 2, sections ). P. Barriga, J.C. Dumas, C. Zhao, L. Ju, D. G. Blair, Compact vibration isolation and suspension system for AIGO: Performance in a 72 m FabryPerot cavity, Rev. Sci. Instrum., 2009, submitted (Chapter 3, sections ). J.C. Dumas, P. Barriga, C. Zhao, L. Ju, and D. G. Blair, Compact suspension systems for AIGO: Local control system, Rev. Sci. Instrum. 2009, submitted (Chapter 3, sections ). P. J. Barriga, C. Zhao, D. G. Blair, Astigmatism compensation in modecleaner cavities for the next generation of gravitational wave interferometric detectors, Phys. Lett. A 340 (2005) 1 6 (Chapter 4, sections ). P. Barriga, C. Zhao, D. G. Blair, Optical design of a high power modecleaner for AIGO, Gen. Relat. Gravit. 37 (2005) (Chapter 4, sections ). P. Barriga, C. Zhao, L. Ju, D. G. Blair, SelfCompensation of Astigmatism in ModeCleaners for Advanced Interferometers, J. Phys. Conf. Ser. 32 (2006) (Chapter 4, sections ). P. Barriga, B. Bhawal, L. Ju, D. G. Blair, Numerical calculations of diffraction losses in advanced interferometric gravitational wave detectors, J. Opt. Soc. Am. A 24 (2007) (Chapter 5, sections ) P. Barriga and R. DeSalvo, Study of the possible reduction of parametric instability gain using apodizing coatings in test masses, Technical Report, ix
14 T Z, LIGO, (2006) (Chapter 5, section 5.7). P. Barriga, M. A. Arain, G. Mueller, C. Zhao, D. G. Blair, Optical design of the proposed Australian International Gravitational Observatory, Opt. Express 17 (2009) (Chapter 6, sections ) D. G. Blair, P. Barriga, A. F. Brooks, P. Charlton, D. Coward, JC. Dumas, Y. Fan, D. Galloway, S. Gras, D. J. Hosken, E. Howell, S. Hughes, L. Ju, D. E. McClelland, A. Melatos, H. Miao, J. Munch, S. M. Scott, B. J. J. Slagmolen, P. J. Veitch, L. Wen, J. K. Webb, A. Wolley, Z. Yan, C. Zhao, The Science benefits and Preliminary Design of the Southern hemisphere Gravitational Wave Detector AIGO, J. Phys. Conf. Ser. 122 (2008) (6pp) (Appendix A) L. Ju, D. G. Blair, C. Zhao, S. Gras, Z. Zhang, P. Barriga, H. Miao, Y. Fan and L. Merrill, Strategies for the control of parametric instability in advanced gravitational wave detectors, Class. Quantum Grav. 26 (2009) (15pp) (Appendix B) x
15 Contents Abstract Acknowledgements Preface Publications used for this thesis Contents List of Figures List of Tables i iii v ix xi xvii xxiii 1 Introduction A bit of history Gravitational wave detection Interferometric gravitational wave detectors Seismic noise Laser source noise Optical noise Thermal noise Readout scheme Output modecleaner Sensing and control A network of gravitational wave detectors Resonant detectors Interferometric detectors xi
16 1.5.3 Space interferometry Pulsar timing Conclusions References ModeCleaner Vibration Isolator Preface Introduction Isolation and suspension design Noise and locking predictions Conclusions Postscript References Advanced Vibration Isolator Preface Introduction Vibration isolation design Preisolation components Isolation stages Control mass and test mass suspension Integrated system Experimental setup Cavity parameters Suspension system transfer function Laser control system Measurements and results Conclusions and future work Local control Introduction Experimental setup Isolator components xii
17 3.9.2 Control hardware Control implementation and degrees of freedom Optical lever The digital controller Control scheme Preisolation feedback Control mass feedback Optimised feedback for preisolation Initial cavity measurements Conclusions Postscript References ModeCleaner Optical Design Preface Introduction The modecleaner Modecleaner thermal lensing Conclusions Introduction Modecleaner intrinsic astigmatism Astigmatism free modecleaner Conclusions Introduction Substrate deformation Thermal lensing Conclusions References Diffraction Losses and Parametric Instabilities Preface Introduction xiii
18 5.3 Diffraction losses FFT simulation Results Diffraction losses Optical gain Mode frequency Mode shape Conclusions Apodising coating Introduction Apodising coating Results Conclusions Mirror tilt Introduction Mirror geometry Mirror tilt Conclusions Postscript References Stable Recycling Cavities Preface Introduction Dual recycling interferometers Higher order modes suppression Possible solutions for stable recycling cavities Straight stable recycling cavity Folded stable recycling cavity Comparison between designs Sidebands and the stable recycling cavities Modulation frequencies calculations xiv
19 6.6.2 Signal recycling cavity Summary Discussion Beamsplitter thermal effects Introduction Astigmatism in folded design Thermal effects Discussion and conclusions References Summary and Conclusions 239 Appendices 243 A Science Benefits of AIGO 245 A.1 Introduction A.2 Scientific benefits of the AIGO observatory A.3 Preliminary conceptual design for AIGO A.4 Discussion and conclusions A.5 References B Control of Parametric Instabilities 257 B.1 Introduction B.2 Parametric instabilities theory and modelling B.2.1 Summary of theory B.2.2 Modelling approach B.2.3 Modelling results B.3 Possible approaches to PI control B.3.1 Power reduction and thermal radius of curvature control B.3.2 Ring dampers or resonant acoustic dampers B.3.3 Acoustic excitation sensing and feedback B.3.4 Global optical sensing and electrostatic actuation B.3.5 Global optical sensing and direct radiation pressure xv
20 B.3.6 Global optical sensing and optical feedback B.4 Conclusions B.5 References C Vibration Isolator Control Electronics 283 C.1 Introduction C.2 Control electronics C.3 Conclusions D Publications List 313 D.1 Principal author D.2 References D.3 ACIGA collaboration D.4 References D.5 LIGO Scientific Collaboration D.6 References xvi
21 List of Figures 1.1 An artist s representation of a gravitational wave The two polarisations of gravitational wave radiation Simple Michelson interferometer Interferometer with Fabry  Perot cavities as arms Dual recycled interferometer Classical modecleaner layout Dual recycled interferometer Distribution of the different sensing apparatus of a GW detector Dualrecycled advanced interferometric GW detector Measured sensitivity of different GW detectors Angular uncertainty maps An artist s representation of the LISA mission Modecleaner suspension design Frequency space representation of the isolator transfer function The AIGO modecleaner layout Modecleaner suppression factor Horizontal and vertical seismic noise and predicted system response Modecleaner residual motion Modecleaner suspension (top view) Comparison between theoretical and measured transfer function Modecleaner vibration isolator Full vibration isolator system for AIGO Inverse pendulum and LaCoste schematic The Roberts linkage xvii
22 3.4 Detail of the multistage pendulum Diagram of one selfdamped pendulum stage Euler Spring vertical stage diagram The combined 3D stage Control mass stage with test mass suspended ETM during the assembly of the second suspension system Average of the cavity decay time ITM horizontal mechanical transfer function Laser control system Semitheoretical transfer function Control loop frequency response Locked cavity frequency response Locked cavity residual motion Pitch and yaw angular residual motion for the ITM Isolation stages of the AIGO suspension chain The shadow sensor The magnetcoil actuator Schematic of the inverse pendulum control Schematic of the LaCoste control Schematic of the Roberts linkage control Schematic horizontal control of the control mass Schematic of the pitch control of the control mass The optical lever setup Block diagram of the isolation local control system Loop gain and closed loop transfer function of the inverse pendulum Test mass frequency response Test mass yaw angular motion Illustration of the proposed preisolator feedback control Measured integrated residual motion of the cavity Time distribution of the power inside the cavity Test mass angular motion xviii
23 3.35 Test mass angular oscillation histograms Simplified schematic showing the layout of the AIGO modecleaner Transmission of the higher order modes Detail of transmission of the higher order modes Triangular ring cavity layout used as a modecleaner Spots eccentricity M1/M2, M3 and waist for AIGO modecleaner Eccentricity variation with input power Modecleaner optical setup Modecleaner spot size simulation Waist and M3 spot eccentricity variation with input power Steady state solution for the bulk absorption case Steady state temperature distributions for coating absorption Advanced LIGO substrate dimensions Diffraction losses for different higher order modes Intensity profile at the ITM, mirror of diameter 34 cm Comparison of diffraction losses Cavity optical gain for some HG modes of different orders Optical gain variation for higher order modes Intensity profile variation of mode HG Modes of order 7 as they would appear in an infinite sized mirror Frequency variations from the theoretical value Optical Qfactor for the higher order modes in the proposed cavity Intensity profile of mode HG33 in a infinite sized mirrors Diffraction losses for an Advanced LIGO type cavity Proposed apodising coatings design for ITM and ETM Diffraction losses of higher order modes Ratio between the different coatings Diffraction losses comparison between homogeneous apodising coatings Different dielectric absorption for higher order modes Optical mode LG21 in a perfectly aligned cavity with circular mirrors. 181 xix
24 5.19 Optical mode LG21 in a perfectly aligned cavity with flat sides Optical mode LG03 showing the effect of the flat sides Influence of mirror geometry in diffraction losses Frequency change with mirror size Frequency change with mirror size for modes of order Change in the optical path for the laser beam due to mirror tilt Change in the path length of the laser beam Spot size tuning Frequency variation with mirror tilt for modes of order Transverse mode frequency variation with mirror tilt Comparison of diffraction losses between Advanced LIGO designs Comparison of higher order modes optical gain Qfactor comparison Frequency shift comparison Configuration of the proposed AIGO interferometer Transmission of HOM as function of the Gouy phase shift Comparison of the intensity suppression of HOM Proposed stable PRC design for AIGO advanced interferometer Modematching drop as a function of PR Proposed stable PRC design for AIGO advanced interferometer Modematching as a function of proposed radius of curvature Variation of the accumulated Gouy phase Two possible solutions for the AIGO stable recycling cavity Astigmatism comparison between AIGO and Adv. LIGO designs Temperature profile in the BS due to the PRC circulating power Comparison of the inline arm waist position and size Comparison of the perpendicular arm waist position and size A.1 Angular area maps for world array A.2 Expected average number of galaxies B.1 Parametric scattering xx
25 B.2 Threemode interactions B.3 Number of unstable modes as function of ETM radius of curvature B.4 Maximum parametric gain with different ETM radius of curvature B.5 The parametric gain distribution of unstable modes B.6 Example of unstable modes suppression using a ring damper B.7 Internal modes Qfactor of a test mass with small resonant damper B.8 Fields of the fundamental mode, high order mode and cavity feedback. 276 B.9 Schematic diagram of the PI optical feedback control C.1 Block diagram of the local control C.2 Control electronics diagram C.3 Control electronics board distribution C.4 Vibration isolator sensors and actuators distribution C.5 6way cross for wire feedthrough C.6 The intermediate board at the ITM vibration isolator C.7 Control board block diagram C.8 LED circuit diagram C.9 Photodetectors circuit diagram C.10 Control signal circuit diagram C.11 Power supply circuit diagram C.12 P connector signal distribution diagram C.13 Filters board and power supply circuit diagram C.14 High current signal filters circuit diagram C.15 Intermediate board block diagram C.16 Horizontal axis connections circuit diagram C.17 Vertical axis connections circuit diagram C.18 Control mass connections circuit diagram C.19 Electrostatic board connections circuit diagram C.20 Backplane board block diagram C.21 Backplane horizontal axis signal distribution circuit diagram C.22 Backplane vertical axis signal distribution circuit diagram C.23 Backplane Roberts linkage signal distribution circuit diagram xxi
26 C.24 Backplane filter signal distribution circuit diagram C.25 Backplane control mass horizontal signal distribution circuit diagram. 308 C.26 Backplane control mass tilt signal distribution circuit diagram C.27 Backplane DB 25 signal distribution diagram C.28 Backplane Sub D 100 pin signal distribution diagram xxii
27 List of Tables 2.1 Design parameters of different modecleaners Parameters affected by different radii of curvature Theoretical parameters of the AIGO 12 m modecleaner Parameters for the 72 m cavity Cavity parameters derived from our measurements I/O channel allocation usage of DSP Stages and degrees of freedom in the control scheme Modecleaner configurations used by other GW interferometers Difference between hot and cold parameters for AIGO modecleaner Comparison of waist and M3 spot sizes Output beam thermal lensing focal length Diffraction losses and cavity optical gain of mode HG Frequency shift of modes of order 7 for different sized mirrors Diffraction losses for the HG 00 mode using different coatings Comparison between LG21 modes in mirrors with different geometry Comparison of Advanced LIGO design parameters Parameters of the arm cavities for the AIGO interferometer Distance between the different optical components Focal length of the different optical components Development of the spot size radius on different optical components Optical length of the different cavities proposed for the AIGO Distance between the different mirrors for the proposed AIGO Astigmatism induced in the stable folded cavity designs xxiii
28 6.8 Comparison between thermal effects in both arms of the interferometer. 234
29 Chapter 1 Introduction 1.1 A bit of history In November 1915 Albert Einstein finishes his work on the General Theory of Relativity and presents it in a 4part speech at the Prussian Academy of Sciences. Einstein has found a way to present the laws of physics independent of the frame in which they were expressed. Consistent with Special Relativity he explained gravity as distortions in the fabric of space, with the associated wave phenomenon being a gravitational wave (GW). In this way he described the structure of spacetime and gravity as the simple manifestation of spacetime curvature. The more massive the object, the greater the curvature it causes, and hence the stronger the gravity. As massive objects move around in spacetime, this curvature will change. As a consequence, a moving object or system of objects will cause fluctuations in spacetime that spread outward like ripples in the surface of a pond. These ripples are gravitational waves and like any wave they carry energy (and therefore information) from a source. The curvature of spacetime is governed by the Einstein field equation: G µν = 8πG N c 4 T µν (1.1) Where G µν is the Einstein tensor, G N is Newton s gravitational constant, c is the speed of light and T µν is the stressenergy tensor. Put simply by John Wheeler the equation can be interpreted as matter tells space how to curve, in turn space tells matter how to move. The scalar constant linking the two tensors has a value of , an extremely small value that shows that the interaction between the distribution of matter and energy and the distortion of spacetime is very weak. 1
30 2 CHAPTER 1. INTRODUCTION Figure 1.1: An artist s representation of a GW caused by two massive objects that are orbiting each other. Gravitational waves are propagating gravitational fields, ripples in the curvature of spacetime, generated by the motion of massive particles, such as two stars or two black holes orbiting each other (figure courtesy of J. C. Dumas). The only Lorenzinvariant speed is the speed of light. As a consequence and similar to electromagnetic waves it is expected that gravitational waves propagate at the speed of light, they can not propagate with infinite speed. They also have different polarisations, but unlike electromagnetic waves that are generated by a dipole, gravitational waves are generated by a quadrupole and therefore they can have two types of polarisations, plus (+) or cross ( ). However they can also be linearly or circularly polarised, with any combination expressed in terms of the basic polarisations + or. Consider a ring of particles floating in space as shown in figure 1.2. As a GW passes through these particles it will perturb the distance between them. Due to its transverse nature, a GW propagating into the page will perturb the distances in the plane of the page. For the first quarter of the GW period the particles will be stretched apart in one axis (chosen here for convenience as the vertical), and be brought closer together in the perpendicular (horizontal) axis (h + at π/2 radians). During the second and third quarters of the GW period the distance between the test particles will contract in the vertical axis and expand in the horizontal direction, as illustrated in figure 1.2 (h + at 3π/2 radians), passing through the initial distribution at π radians. During the last quarter the particles are stretched in the vertical axis
31 1.1. A BIT OF HISTORY 3 h + h x 0 π/2 π 3π/2 2π Figure 1.2: The two polarisations of GW radiation. The major axis of the elliptical displacement is determined by the polarisation of the GW. The axes of h + and h differ by π/4 radians. and contract in the horizontal returning to the initial distribution and completing a full period (h + at 2π radians). This is an example of one polarisation, normally referred to as (+) or h + polarisation. An orthogonally polarised wave, denoted by ( ) or h, would have the axes of the distortions rotated by 45 o. The strength of a GW is measured by its strain, h, which gives an indication of the fractional length change induced by the passing wave. The amount of distortion shown in figure 1.2 is grossly exaggerated. The action of a GW will produce a relative deformation l/l h, this dimensionless parameter is a measure of the deviation of the metric from the Euclidean metric in the field of a GW, which is of the order of to If such a wave was to pass between the earth and the sun, their separation would change by less than the radius of a hydrogen atom. To detect the presence of a GW passing between two objects, we must be able to measure the changes in their separation with unprecedented accuracy. In 1974 Russell Hulse and Joseph Taylor found indirect evidence for gravitational waves in observations of a binary pulsar known as PSR , the HulseTaylor Binary Pulsar [1]. Their efforts were recognised with a Nobel Prize in 1993, but experiments designed to detect the waves directly have so far drawn a blank. However upper limits have been set after each science run of the existing GW detectors.
32 4 CHAPTER 1. INTRODUCTION 1.2 Gravitational wave detection The quest for the direct detection of gravitational waves started in the late 1960 s when Joseph Weber developed the first GW detector, the resonant bar detector [2, 3]. He built two bar detectors, massive cylinders of metal whose resonant modes can be excited by the passage of a GW. Weber claimed to have detected gravitational waves, which were later dismissed after other detectors built around the world failed to detect them. However he succeeded in starting a new field of research 50 years after Einstein presented his theory. Weber suspended a 1 tonne aluminium bar in a vacuum tank and bonded a ring of piezoelectric transducers around its centre. Weber built more than one of these bars, which were resonant at around 1600 Hz, a frequency where the energy spectrum of signals from collapsing stars was predicted to peak. These detectors exploit the narrow mechanical resonance of the bar to achieve high sensitivities over a bandwidth of a few hertz around the mechanical resonance (typically several hundred hertz). Modern bar detectors are currently about 1000 times more sensitive than Weber s original design [4, 5]. At about the same time, Soviet physicists Mikhail E. Gertsenshtein and Vladislav I. Pustovoit argued a case for the use of the MichelsonMorley interferometer [6] as an instrument of GW detection [7], but detailed studies of this technique were not completed until a decade later. Early researchers in this field included G. Moss [8] and R. Weiss [9]. Robert Forward and other scientists built the first interferometer for GW detection in the early 1970 s in California at the Hughes Research Laboratories [10]. For a number of years resonant bar detectors were the main observational instrument for GW detection. These have since been surpassed in sensitivity by laser interferometry based detectors.the concept behind the interferometers is to replace the bar with freefalling test masses, that will interact with an incoming GW. By placing a freefalling mass at the end of two very long, perpendicular arms the principle of a Michelson interferometer like the one in figure 1.3 could be used to measure the differential displacement of the end mirrors used as test masses. Such small displacements would be measured by interference of the laser light returning from each mirror along the arms.
33 1.3. INTERFEROMETRIC GRAVITATIONAL WAVE DETECTORS 5 ETM Y BS ETM X Laser To Detector Bench Figure 1.3: Schematic diagram of a Michelson interferometer including a laser source, a beamsplitter (BS) and two end test masses, ETM X and ETM Y. 1.3 Interferometric gravitational wave detectors A GW interacting with the test masses at the end of the arms of an interferometer will induce a phase shift in the carrier light. The phase change (φ(t)) incurred is given by: φ (t) = ω o t r 2ω ol c ± ω o 2 t t 2l/c h + (t) dt. (1.2) Where t r is the round trip time of the light in the arm cavity, which is modulated by the incoming GW signal, assuming that h + 1 t r can be defined as t r 2l c ± 1 2 t t l/c h + (t) dt, (1.3) ω o is the angular frequency of the carrier light, 2l/c is the round trip light transit down the arm cavity of length l, and h + (t) corresponds to a sinusoidal GW with angular frequency ω g and peak amplitude h 0,
34 6 CHAPTER 1. INTRODUCTION This implies that: h + (t) = h o cos (ω g t). (1.4) δφ ω o 2 t t 2l/c h o cos (ω g t) dt (for h o 1) h o 2 ω o ω g {sin (ω g t) sin [ω g (t 2l/c)]} h o ω o sin (ω g l/c) ω g cos [ω g (t 2l/c)]. (1.5) As mentioned previously the amplitude of an incoming GW is of the order of 10 21, inducing a modulation of the order of 10 9 radians. Even though this value depends on the arm length the modulation index can be approximated to: δφ h o ω o sin(ω g l/c) ω g h oω o l. (1.6) c This is almost proportional to the initial distance l between the test masses, as long as it is short compared to half a wavelength of the GW, ω g l c π 2. (1.7) In other words, to obtain an optimum sensitivity to gravitational waves it is necessary that the light takes one half period of the GW to do a round trip in one arm of the interferometer. Therefore the required time for a GW with a frequency of 1 khz is 0.5 msec, which implies an arm length of approx 75 km, and even longer for sources of lower frequencies. These distances are impractical for a groundbased experiment. However the arms of the interferometer can be effectively folded by making the light resonate or by folding the light path. The effective number of bounces can then be traded off against the overall length to achieve a desired total path length or storage time. One possible solution is to replace the arms of the interferometer with delaylines, effectively folding the path into multiple reflections, as in a Herriott delay line [11, 12]. A different solution is to increase the light travel time by storing the laser light in a resonant cavity. This requires keeping the round trip length of the cavity
35 1.3. INTERFEROMETRIC GRAVITATIONAL WAVE DETECTORS 7 an integer number of times the wavelength of the input monochromatic laser, which can be achieved by using Fabry  Perot cavities as the arms of the interferometer as in figure 1.4. The main advantage of a resonant optical cavity over delay lines is the size of the mirrors. A delay line needs several reflections in different spots of a mirror (or several mirrors to accommodate each reflection), while in a Fabry  Perot cavity by superimposing the beam reflections the mirror size can be reduced, only to be limited by diffraction losses. A disadvantage is that the light enters a Fabry  Perot cavity through the substrate of an input test mass (ITM), which has to be of outstanding optical quality. Effectively this design corresponds to a Michelson interferometer with coupled Fabry  Perot cavities on the arms. ETM Y L Y FabryPerot Cavity ITM Y BS ITM X L X ETM X Laser FabryPerot Cavity To Detector Bench Figure 1.4: Interferometer with Fabry  Perot cavities as arms. The addition of two input test masses (ITM X and ITM Y ) form two Fabry  Perot cavities of lengths L X and L Y. The input laser light has to be resonant in the cavities in order to be able to detect gravitational waves. As in the Michelson interferometer case, a passing GW will induce changes in the length of the arm cavities by relative displacement of the end mirrors. These changes in length will induce changes in the phase of the reflected light. A passing
36 8 CHAPTER 1. INTRODUCTION GW modulates the phase of the light at its frequency, generating sidebands at angular frequencies from the laser frequency. The end test mass (ETM) has a much higher reflectivity than the ITM. Assuming that the losses in the cavity are much lower than the ITM transmission most of the light is transmitted back to the beamsplitter (BS) where it recombines with the light from the other, perpendicular arm. The differential changes on each arm are then transmitted to the output port (also known as dark port or asymmetric port) where they will be detected. By placing a mirror at the input port (also refer to as bright port or symmetric port) of the Michelson interferometer, just before the BS, the bright fringe will be reflected back into the interferometer rather than propagated towards the laser. The bright fringe corresponds to the common mode strains in the two interferometer arms. Therefore the presence of this additional mirror does not affect the frequency response of the interferometer, but does increase the sensitivity by increasing the stored optical power in the interferometer arms. This technique, known as power recycling and the resulting cavity, known as the power recycling cavity (PRC), were independently proposed in the early 80 s by R. Schilling and by R. Drever [13]. Another technique to further enhance the sensitivity of an interferometric GW detector is to add a mirror of specifically selected reflectivity at the output. The addition of this mirror adds an extra cavity to the interferometer known as the Signal Recycling Cavity (SRC). As described, a passing GW will generate sidebands on the carrier light of the arm cavities. Since these sidebands will not interfere destructively at the BS, they will appear at the output. This Signal Recycling Mirror (SRM) can be used to reflect the sidebands back into the interferometer. If the reflected sidebands are in phase with the sidebands in the arm cavities they will add coherently, enhancing the signal over a given bandwidth set by the mirror reflectivity. This technique is known as Signal Recycling (SR) [14, 15, 16]. The transmittance and reflectivity of the compound mirror formed by the SRM and the ITM is dependant on frequency. In signal recycling, this cavity is tuned so that the GW signal will see a lower transmittance (higher reflectivity) than that of the ITM alone. This controls the effective number of round trips over which the GW sidebands are summed, increasing their storage time, and determining the bandwidth of the
37 1.3. INTERFEROMETRIC GRAVITATIONAL WAVE DETECTORS 9 interferometer. The position of the SRM controls which GW sideband frequencies will add constructively and which will add destructively, thus determining the tuning of the interferometer and its frequency response. The bandwidth of an interferometric GW detector with FabryPerot arm cavities, without a SRM, is set by the bandwidth of the cavities. High finesse arm cavities are desirable in order to reduce photon shotnoise, simultaneously the storage time for the signal sidebands must be kept short enough to obtain the desire detection bandwidth. This determines what is referred to as storage time limit. Resonant Sideband Extraction (RSE) was proposed by Mizuno et al [17] as a new optical configuration to overcome this limitation. In this mode of operation the extra cavity at the output of the interferometer is usually referred to as the Signal Extraction Cavity (SEC). The purpose is to reduce the storage time for the GW signal, allowing for long storage times in the arm cavities without sacrificing the detector bandwidth [18, 19]. The tuning of this SEC in this case results in a bandwidth wider than that of the interferometer without a SRM. It is then possible to create a compound mirror with a transmissivity higher (lower reflectivity) than that of the ITM alone. This reduces the storage time of the signal frequencies of interest, resulting in an increased bandwidth for the interferometer. Since the interferometer bandwidth is not limited by the bandwidth of the arm cavities, FabryPerot cavities with high finesse, narrow bandwidth, can be used to maximise the stored energy. This has also the advantage of reducing the amount of light power which must be transmitted through the optical substrates of both ITMs and the BS in order to obtain the same amount of energy stored in the arm cavities. As a consequence the thermally induced distortion inside the substrates is effectively reduced. Both modes of operation (SR and RSE) also have the possibility of operating in a detuned mode. Conventionally the special case of maximum response at zero signal frequency is termed tuned; all other cases, with peak response at a finite frequency, are called detuned. This detuning is usually described by either the frequency of peak response or by the shift of the SRM away from the tuned point, often in terms of an optical phase shift. This positioning or detuning of the SRM within a wavelength of the carrier light allows for some narrowing of the detection bandwidth at the expense
38 10 CHAPTER 1. INTRODUCTION of loss of sensitivity outside the bandwidth. This could be valuable in searches for continuous wave sources of gravitational radiation, like rapidly rotating neutron stars and fast pulsars or even tracking of a changing periodic signal like a chirp [20]. The configuration of a Michelson interferometer with Fabry  Perot cavities in the arms and the addition of a PRC and a SRC is referred to as a dual recycled interferometer. The concept of dual recycling and the design of stable recycling cavities (PRC and SRC) is reviewed in Chapter 6. Figure 1.5 shows a simple diagram of this design, which is the one favoured for the next generation of interferometric GW detectors. However this is only part of the story behind the next generation of advanced GW detectors, which required the development of new techniques to overcome several noise sources. ETM Y L Y FabryPerot Cavity ITM Y PRM BS ITM X L X ETM X Laser FabryPerot Cavity SRM To Detector Bench Figure 1.5: Dual recycled interferometer. The addition of a power recycling mirror (PRM) increases the arms circulating power. The addition of a signal recycling mirror (SRM) enhances the signal response of the interferometer and could also be used for frequency tuning and narrow band operation.
39 1.3. INTERFEROMETRIC GRAVITATIONAL WAVE DETECTORS Seismic noise An important source of displacement noise in all groundbased detectors is seismic noise. Unfortunately it is not possible to control seismic noise, but it is possible to reduce its effects on the interferometer. Seismic vibration falls sharply with frequency (f), typically at a level of 10 7 /f 2 m/ Hz, depending on the location of the site in the world. This makes this source of noise a problem primarily below 100 Hz. At levels of hundreds of mhz the spectrum of the seismic noise is enhanced by what is known as microseismic peaks. Dominant microseismic signals from the oceans are linked to characteristic ocean swell periods, and occur approximately between 4 to 30 seconds [21]. Techniques for reducing the effects of seismic noise have long been known and yet new designs and approaches continue to improve vibration isolation. The most basic vibration isolation system is a simple pendulum which has evolved to the complex systems of today. The advanced vibration isolation systems planned for the next generation of interferometric GW detectors have an array of different techniques for preisolation in order to reduce the seismic vibration to levels where the detection of gravitational waves is feasible. This is only possible however down to a limit of about 10 Hz. This limit is due to random gravitational forces (or gravity gradient noise) caused by density fluctuations of the medium surrounding the GW detector; also including contributions from the motion of isolated bodies in the vicinity normally related to human activity [22]. Substantial effort has been put into mechanical vibration isolation systems. A review of the vibration isolation efforts is presented in Chapter 3, including the performance of an advanced vibration isolation system for AIGO Laser source noise Even though lasers are commonly thought to be stable sources of light, they have both frequency and spatial fluctuations. The spatial instability of a laser beam, known as beam jitter, is due to the mixing of higher order modes with the fundamental mode TEM00. Amplitude fluctuations are created by beam jitter whenever the beam interacts with a spatially sensitive element such as an optical cavity. The noise at
40 12 CHAPTER 1. INTRODUCTION the dark fringe of the interferometer output will be affected by such beam jitter effects. In addition, frequency fluctuations of the laser fundamental mode give rise to additional noise at the dark fringe. In the case of GW detection these variations can introduce substantial measurement noise. Filtering the laser light with an optical filter or modecleaner as illustrated in figure 1.6 can solve this problem. The input modecleaner (IMC) acts as a spatial filter. It provides passive stabilisation of time dependant higher order spatial modes, transmitting the fundamental mode TEM00 and attenuating the higher order modes. The concept was first suggested by Rüdiger et al in 1981 [23]. As a frequency stability element it can also suppress frequency fluctuations of the fundamental mode, but without DC stability. A complete optical design of a high power modecleaner is presented in Chapter 4. M1 M3 M2 Figure 1.6: A classical input modecleaner layout showing the laser light path and the relative position of the mirrors, where M1 (input coupler) and M2 (output coupler) are flat mirrors and M3 a convex curved mirror Optical noise Optical noise is traditionally divided into two sources of noise. The first is related to the interaction of light with the test masses, while the second is related to the counting of photons by a photodetector. The power that builds up inside the cavity exerts significant forces on the mirrors due to the momentum of the photons. The more power circulating in the cavity, the more photons, and a greater force acting on the test masses. The number of photons bouncingoff the test mass at any given time follows a Poisson distribution and the statistical distribution of the force on the test mass is known as radiation pressure noise. In interferometric GW detectors this is mainly
41 1.3. INTERFEROMETRIC GRAVITATIONAL WAVE DETECTORS 13 a problem at low frequencies (around a few Hz). However radiation pressure can generate other problems like parametric instabilities [24] and optical spring effects [25], which can be difficult to control. A discussion of parametric instabilities through the estimation of diffraction losses is presented in Chapter 5. Photon shot noise is the second source of optical noise. It is related to the statistical arrival time of photons and also follows a Poisson distribution for the counting of photons. At the interference fringes the fluctuations due to the random arrival of photons can look like a GW signal. The more photons we use, the smoother the interference fringe. Therefore very high power from a continuous wave laser is necessary to minimise the shot noise. The source of fluctuations for both shot and radiation pressure noise are the vacuum fluctuations, which enter the interferometer from the output port [26]. When laser light and vacuum fluctuations are injected into optical cavities with suspended mirrors, the vacuum fluctuations are ponderomotively squeezed by the back action of its radiation pressure on the suspended mirrors. The shot noise spectral density is flat, while the radiation pressure amplitude spectral density has a 1/f shape. The standard quantum limit is then defined by the quadrature sum of the shot and radiation pressure noise at a given frequency. However the standard quantum limit for the noise of an optical measurement scheme usually refers to the minimum level of quantum noise which can be obtained without the use of squeezed states of light [27, 28] Thermal noise Another source of fundamental noise that contributes to the displacement noise of a test mass is due to thermal fluctuations. Thermal noise is caused by mechanical loss in the system in accordance with the FluctuationDissipation Theorem [29]. In GW interferometers thermal noise is usually divided into two categories, suspension thermal noise and test mass thermal noise. In both cases the noise comes from the vibration of atoms which depend on Boltzmann s constant k B and is proportional to k B T, where T is the temperature. Suspension thermal noise corresponds to the effect of thermal fluctuations from the suspension system as seen at the mirror, thus the strongest effect is from the last stage of the suspension directly in contact with the
42 14 CHAPTER 1. INTRODUCTION test mass. The second component is due to the atoms on the test mass itself, causing the surface of the mirror to vibrate. However the test mass thermal noise can also be defined according to the source of the noise, substrate and coating. Coating thermal noises are defined by differences between the coating material and the substrate material, mechanical losses between coating layers contribute to the total Brownian noise of the test mass. Thermal fluctuations in the coating produce noise via thermoelastic and thermorefractive mechanisms. The study of multilayer dielectric coatings is an active area of research with new coatings under test for GW interferometers [30, 31] Readout scheme An incoming GW signal of frequency ω g will induce sidebands on the carrier light (frequency of Hz). These sidebands will be located at ±ω g from the carrier light. To be able to read such a high frequency signal, an optical oscillator is necessary in order to demodulate the output signal and read the gravitational wave signal. The first generation of interferometric GW detectors favoured a heterodyne readout scheme, where radiofrequency (RF) sidebands are modulated on the carrier light before entering the interferometer. A large mirror offset (centimetre scale) to create a macroscopic asymmetry in the Michelson arms of the interferometer (also known as Schnupp asymmetry [32]) is required to allow leakage of the RF sidebands through the output port of the interferometer whilst maintaining a dark fringe. This signal can be used as the local oscillator. The next generation of interferometric GW detectors however will see the addition of a SRC. The operation of this cavity in a detuned configuration will introduce an imbalance on the control sidebands, reducing the sensitivity of the interferometer [33]. The sensitivity reduction will depend on the relative level of imbalance between the sidebands. A homodyne readout scheme requires a small fraction of carrier light to be used as the local oscillator. One approach is to obtain a signal from the incoming beam before it enters the interferometer, and feed this signal as the local oscillator at the output port. Another approach that also allows a small amount of light to appear at the output port is to introduce a very small offset (picometre scale) at the Michelson arms of the interferometer. This technique, known as directconversion (DC) readout,
43 1.3. INTERFEROMETRIC GRAVITATIONAL WAVE DETECTORS 15 was proposed by Fritschel in 2000 [34] and tested at the 40 m interferometer at the California Institute of Technology [35]. It also has the advantage of increasing the signal to shot noise ratio by eliminating the vacuum fluctuations at twice the frequency of the modulation [36] Output modecleaner Despite best efforts to keep the arm cavities and recycling cavities perfectly aligned there will always be some degree of misalignment, for example due to radii of curvature mismatch and/or surface deformation of the mirrors. These factors can individually be very small, but they all contribute to light leaking to the dark port, which increases the noise and reduces the fringe contrast. With the addition of an output modecleaner (OMC) all but the fundamental mode component of the contrast defect will be rejected [37, 38, 39, 40]. Figure 1.7 shows a dual recycled interferometer including an IMC and an OMC. The OMC design depends on selection of either a RF (optical heterodyne detection) or DC (optical homodyne detection) readout configuration. In a RF readout scheme the control sidebands are used as local oscillator for signal demodulation, as a consequence a long OMC with individually isolated mirrors could be used in order to transmit the sidebands to the output port. However a short OMC with a broad bandwidth could also be used to reject higher order optical modes. A DC readout has several technical advantages over the traditional RF readout technique, including laser and oscillator noise coupling as well as reduced shot noise [41]. Current designs for the next generation of interferometric GW detectors favour a DC readout scheme (Advanced LIGO, Advanced VIRGO, and GEOHF). The DC readout scheme relies on a small offset (usually on the order of a few ten picometres) that allows a small amount of carrier light into the otherwise dark port. This carrier light is used as the local oscillator, and as a consequence the sidebands are not needed at the dark port. However they will still be present in the interferometer, since they are needed for autoalignment error signal. For a fixed OMC cavity length, the transmission of the sidebands decreases as the finesse increases. Therefore the OMC requirements include low finesse for reducing the sidebands, the transmission of the carrier funda
44 16 CHAPTER 1. INTRODUCTION ETM Y L Y FabryPerot Cavity Input Mode Cleaner M3 L IMC PRM BS ITM X L X ETM X Laser M1 M2 FabryPerot Cavity SRM Output Modecleaner To Detector Bench Figure 1.7: Diagram showing a dual recycled interferometer with input modecleaner and output modecleaner. Arm cavities, input modecleaner and output modecleaner are not at the same scale. mental mode and low backscatter light so as to not reduce the sensitivity with added phase noise. Taking this into account, the favoured configuration for an OMC is a four mirror cavity in a bowtie arrangement. This is small enough to be made from a monolithic piece of fused silica that is then suspended from a vibration isolation system. In future configurations the OMC could provide an excellent reference for aligning the injection of squeezed light [42]. 1.4 Sensing and control We have seen that interferometric GW detectors are a collection of optical cavities coupled together with such precision to allow for the detection of gravitational waves. However to obtain a working interferometer it is necessary to keep these cavities locked at all times. This requires a sophisticated sensing and control system, which
45 1.4. SENSING AND CONTROL 17 has to bring the interferometer from an unlocked state to a configuration appropriate for collecting science data. This can be divided into three steps. First is the lock acquisition, where all initially uncontrolled length degrees of freedom are globally controlled and brought to their operating point. Second is the transition from a locked interferometer with all degrees of freedom controlled to a configuration where science data can be collected. Third is to maintain the science mode and the data collection. All of the interferometer sensing and control ports (show in figure 1.8) will be equipped with wavefront sensors (WFS) which also give DC outputs allowing for fast beam stabilisation. Optical levers are considered for angular control of the core optics, as well as CCD cameras to locate the beam positions on the high reflectivity surfaces of all core optics. In addition, the transmitted beam of each optic will be monitored by a quadphotodetector (QPD). The main difference between QPD and WFS is that the QPD are sensitive to the carrier light, whilst the WFS are sensitive to the RF sidebands [43, 44]. The input signals for the sensing and control systems come from the different sensors installed around the interferometer. Tuned signals that are bandpass filtered before being synchronously demodulated are used for detection of the various degrees of freedom. The outputs of the length and sensing control system, and the alignment and sensing control system are processed by the global control system before being distributed to each of the core optics components, through the suspension local control system, and to the PreStabilised Laser (PSL) for actuation on the laser frequency. In principle two radio frequency (RF) sidebands are enough to control all 5 length degrees of freedom of an advanced dual recycled interferometer. These two RF modulation frequencies (f 1 and f 2 ) make the use of double and/or differential demodulation techniques possible. The two RF sidebands allow us to obtain length signals using single demodulation at f 1 and f 2 or differential demodulation at f 1 ± f 2, and double demodulation at a combination of f 1 f 2 and f 1 + f 2 where the signal is produced by the beat between the two RF sidebands, equivalent to (f 1 f 2 ). The signal of each WFS is processed in order to obtain a strong control signal from the sidebands. The RF signal is split and demodulated at 0 o (inphase) and
46 18 CHAPTER 1. INTRODUCTION Figure 1.8: Distribution of the different sensing apparatus of an advanced interferometric GW detector. These include WFS and QPD. It also includes the aid of CCD cameras and optical levers, which are auxiliary lasers and QPD to monitor the optics alignment. (Figure courtesy of the LIGO Scientific Collaboration (LSC) [43].) 90 o (quadraturephase) giving two signals per detector [45]. Each of these signals is fed to an input matrix where each length degree of freedom can be calculated. These are the differential arm length (DARM or L ), Michelson length (MICH or l ), common arm length (CARM or L + ), power recycling cavity length (PRCL or l + ), and signal recycling cavity length (SRCL or l src ). Figure 1.9 shows their definitions in an advanced dual recycled interferometer configuration. The signals are digitally filtered and converted through an output matrix into control signals for the core optics. This is done by feeding the control signals to the local control of the suspension system, where the signals are added to the appropriate degrees of freedom of each optic. In general the length degrees of freedom that need to be controlled and which are also shown in figure 1.9 are defined as: DARM Differential arm length: L = (L x L y )/2
47 1.4. SENSING AND CONTROL 19 CARM Common arm length: L + = (L x + L y )/2 MICH Michelson length: l = (l x l y )/2 PRCL Power recycling cavity length: l + = l pr + (l x + l y )/2 SRCL Signal recycling cavity length: l src = l sr + (l x + l y )/2 ETM y Degrees of freedom Input Mode Cleaner M3 L IMC L y FabryPerot Cavity ITM y L + =(L x +L y )/2 L=(L x L y )/2 l + =l pr +(l x +l y )/2 l=(l x l y )/2 l src =l sr +(l x +l y )/2 Laser M1 M2 PR1 l p1 l p2 PR2 l y BS ITM x L x ETM x PR3 l p3 SR2 l x FabryPerot Cavity l pr = l p1 +l p2 +l p3 l sr = l s1 +l s2 +l s3 l s1 l s2 l s3 SR1 L OMC SR3 Output Modecleaner To Detector Bench Figure 1.9: Dualrecycled advanced interferometric GW detector. The figure shows stable recycling cavities and length degrees of freedom, including input modecleaner and output modecleaner. In general the input modecleaner is part of the input optics subsystem; while the output modecleaner is part of the output optics and they are not part of the interferometer control system. Before the interferometer longitudinal degrees of freedom can be controlled it is necessary to initially align the cavity axes to one another. This includes centring the beams on the optics as well as aligning the interferometer input beam with the interferometer optical axes. This is performed with the aid of optical levers installed on each of the core optics of the interferometer. At this stage there is no need to operate the laser at full power, which could compromise lock acquisition by radiation
48 20 CHAPTER 1. INTRODUCTION pressure effects. The objective of the initial alignment is to attain a sufficiently high power buildup in the cavities of the interferometer for locking the longitudinal degrees of freedom [46]. The next step is to lock the central part of the interferometer comprising the ITMs, BS, PRM and SRM. This corresponds to three longitudinal degrees of freedom, the MICH, the PRCL, and the SRCL. With the central interferometer locked it is possible to lock the arms through the CARM and the DARM signals, locking the whole interferometer [43]. It is expected that the locked central part of the interferometer will remain stable during the locking of the arm cavities. If the carrier were used to lock the central part of the interferometer, the lock could be lost when locking the arm cavities. The extra phase shift added to the carrier when resonating in the arm cavities would change the polarity of the control signal, which could drive the control system unstable. For this reason single demodulation is only used for lock acquisition of the central part. Standard PoundDreverHall [47] signals generated by the beating between carrier and sidebands are strongly dependent on the behaviour of the carrier inside the arm cavities and on the interferometer losses. Control signals can be obtained from the beating of the first sideband (f 1 ) with the second sideband (f 2 ) by demodulating the signal twice. Since double demodulation scheme does not depend on the carrier, the amplitude and polarity of the control signals obtained from the sidebands are not affected by the lock or unlock status of the arm cavities. Consequently before the arms are brought into lock, the control is handed over to double demodulation. However since double demodulation signals do not work effectively far from the locking point so that they are not ideal for lock acquisition. Moreover, in the configuration for broadband detection, there is no detuning of the SRC. In such a configuration the two sidebands can no longer beat and there is no double demodulation signal at all. The TAMA group [48] proposed and studied the use of signals demodulated at three times the modulation frequencies (3f) in order to obtain signals that are independent from the CARM offset necessary to keep the arm cavities out of lock. These signals are produced by the beating between 2f and f sidebands and between 3f sidebands with the carrier. The second contribution is typically smaller than the first, so that the
49 1.5. A NETWORK OF GRAVITATIONAL WAVE DETECTORS 21 3f signal depends very little on the carrier behaviour. Signals detected in reflection and demodulated at 3f 1 and 3f 2 are in fact very good signals for PRCL, MICH and SRCL. A combination of the transmitted powers of the two arm cavities is used for the CARM degree of freedom, (more precisely the square root of the sum). For the DARM degree of freedom the difference between the two transmitted powers is used. The DARM degree of freedom is controlled by feedback to the end mirrors, while the CARM degree of freedom is expected to be controlled by feedback to the laser frequency. With the described locking scheme the interferometer is brought to the operating point by passing through stable states, which allow the activation of a frequency servo even during lock acquisition. 1.5 A network of gravitational wave detectors In principle a single GW detector would be enough to detect gravitational waves. An individual GW detector is almost omnidirectional with a wide antenna pattern and consequently has poor angular resolution. In order to undertake GW astronomy and astrophysics, a network of detectors is indispensable Resonant detectors In 1997 the Gravitational Wave International Committee (GWIC) was established [49]. Its main goal is to facilitate international collaboration and cooperation in the construction, operation and use of the major GW detection facilities worldwide. As such it is not limited to interferometric GW detectors and also includes the operating bar detectors (AURIGA [4], EXPLORER, and NAUTILUS [5]), which operate at the high end of the frequency band between 500 Hz and 5 khz. Bars are very sensitive to the direction of the incoming GW, which led to the development of omnidirectional resonant detectors shaped as a sphere such as the Mario Schenberg [50] and Mini GRAIL, with a resonant frequency of 2.9 khz and a bandwidth around 230 Hz [51].
50 22 CHAPTER 1. INTRODUCTION Interferometric detectors Operating at a slightly lower frequency band are the groundbased interferometric GW detectors. The largest of these detectors is LIGO, the Laser Interferometric Gravitationalwave Observatory, consisting of three kilometrescale detectors L1, H1 and H2 [52, 53]. Two of these have 4 km long arms, L1 located in Livingston, Louisiana, USA and H1 in Hanford, Washington, USA. H2 is a 2 km long interferometer, which shares the vacuum envelop with H1 at the Hanford facility. In 2007 these detectors completed a twoyear data taking run, known as S5 (the fifth science run for LIGO). At the time of writing only H2 has been left running, while L1 and H1 are going through an upgrade to increase their sensitivity before starting a new science run as Enhanced LIGO in the second half of 2009 [54]. Enhanced LIGO will be an intermediate step between Initial LIGO and Advanced LIGO, which will see H2 upgraded to a 4 km interferometer. On a similar scale with arms stretching 3 km, is VIRGO near Pisa in Italy [55], built by a consortium of institutions from France and Italy and operated by the European Gravitational Observatory (EGO) [56]. VIRGO is also going through an upgrade to become VIRGO + [57] and will join Enhanced LIGO for its science run. A new set of upgrades will then take place in order to improve its sensitivity to become Advanced VIRGO [58]. On a smaller scale there is the GermanBritish GEO600 [59]. Often called a second generation detector since it already uses some of the advanced technology. This project has a slightly different topology, being a dual recycled Michelson interferometer with folded arms for its 600 m arms. A series of upgrades are also planned for the GEO600 project, designed to improve its sensitivity in the high frequency region above 1 khz, to become GEO HF [60]. In Tokyo, Japan has built a 300 m interferometer, TAMA, which was the first large scale interferometer to achieve continuous operation and has participated in joint science runs with LIGO [61, 62]. The sensitivity of some of these different systems are compared in figure In addition to the operating interferometers, GWIC also includes future groundbased projects such as the Large Cryogenic Gravitational Telescope (LCGT) in Japan, an interferometer with 3 km arms located 1000 m underground at the Kamioka mine site [64]. The Japanese group has developed the Cryogenic Laser Interferometric Ob
51 1.5. A NETWORK OF GRAVITATIONAL WAVE DETECTORS 23 Figure 1.10: Measured sensitivity of different interferometric GW detectors. The figure includes the latest sensitivity curves of LIGO H1 (LHO 4 km), L1 (LLO 2 km), and H2 (LHO 2 km); GEO600 and VIRGO. It also shows the design sensitivity of the VIRGO interferometer. (Image courtesy of the LIGO Scientific Collaboration (LSC) [63]). servatory (CLIO) [65], a 100 m interferometer testing technology for the future LCGT. There are also plans for an advanced GW detector in Australia. The Australian Consortium for Interferometric Gravitational Astronomy (ACIGA) is developing the plans for a kilometrescale interferometric GW detector, the Australian International Gravitationalwave Observatory (AIGO) [66, 67]. Part of the research and development towards this advanced interferometric GW detector is presented throughout this thesis, including a proposed optical design for AIGO in Chapter 6. To determine the position of a GW event in the sky, a coherent network analysis will allow accurate measurement of the different phase fronts of the incident GW on different detectors, thus obtaining a good angular resolution for an allsky monitor. Providing that the orientation of the detectors is different (and therefore the orientation of their antenna patterns), a network of detectors will provide enough information about both polarisations components, and allow reconstruction of the polarisation of the incoming GW. This is partly due to the different geographical location of each detector over the globe, covering different portions of the sky.
52 24 CHAPTER 1. INTRODUCTION In a network of detectors, broadband stationary noise reduces as the square root of the number of detectors. While this factor is not large it has a much larger effect on the number of detectable sources. This depends on the volume accessible by the detector, which increases as the cube of the detector strain sensitivity. Multiple detectors at separated sites are essential for rejecting instrumental and local environmental effects in the data, reducing nonstationary noise by requiring coincident detections in the analysis. The probability of spurious signals reduces as the power of the number of detectors. Figure 1.11 shows the angular uncertainty on the sky when locating a GW event. With the current network of large detectors (L1, H1 and VIRGO) figure 1.11 (a) shows the angular resolution corresponding to the red areas, characterised by highly elongated ellipsoids. Figure 1.11 (b) shows the addition of the planned advanced GW detectors LCGT and AIGO dramatically reducing the uncertainty areas, and improving the angular resolution of the GW network. The fact that AIGO will be located in the southern hemisphere adds an outofplane detector (the rest of the interferometers are in the northern hemisphere), which increases the maximum baseline and improves the existing angular resolution. This highlights not only the importance of a southern hemisphere advanced GW detector but the necessity of operate as a cohesive network. A more complete analysis of the science benefits of an advanced interferometric GW detector in the southern hemisphere is presented in Appendix A. The proposed Einstein gravitational wave Telescope (ET) takes a slightly different approach. Proposed by eight European research institutes is officially a design study project supported by the European Commission under the Framework Programme 7. As a 3 rd generation detector in its design stages there are still a few open questions. Its topology is still under study and could include multiple interferometers colocated [69]. It is most likely that the interferometer (or interferometers) will include cryogenic technology, will be placed underground and will incorporate an important increase in laser power, which will require increasing the weight of the mirrors. The main goal of these efforts is to improve the sensitivity and expand the lower end of the detection band. With arms 10 km long, the differential length variation will allow the detection
53 1.5. A NETWORK OF GRAVITATIONAL WAVE DETECTORS 25 (a) L+H VIRGO GEO Dec ( 0 ) RA cos (Dec) ( 0 ) (b) L+H V G C A 20 Dec ( 0 ) RA cos (Dec) ( 0 ) Figure 1.11: Angular area maps for a world network of interferometric GW detectors. (a) shows the antenna pattern for an array including LIGO Livingstone (L), LIGO Hanford (H), VIRGO (V) and GEO600 (G). (b) shows the antenna pattern when LCGT (C) and AIGO (A) are included. The angular uncertainty is shown as red ellipsoids in the sky, with a clear reduction of the uncertainty by adding LCGT and AIGO to the GW network. (Figures courtesy of L. Wen [68]). band to be reduced to 1 Hz, only to be limited by gravity gradient noise. Higher power, requiring heavier test masses, will also expand the higher end of the detection band to about 10 khz [70].
54 26 CHAPTER 1. INTRODUCTION Space interferometry Joining the groundbased efforts are the space detectors, mainly the Laser Interferometer Space Antenna (LISA), a joint ESA and NASA effort [71]. Three identical spacecrafts act as an interferometer with an arm length of km. The spacecrafts are joined by a low power laser giving LISA its triangular shape, and creating three independent arms as shown in figure Gravitational waves from distant sources will warp spacetime, stretching and compressing the triangle, which constitutes a very large GW antenna. Due to its extremely large arms, LISA is expected to detect gravitational waves at low frequencies (0.1 mhz mhz). More ambitious plans to increase the sensitivity of space detectors have been proposed in order to detect the stochastic background of gravitational waves. The Big Bang Observer (BBO) will consist of four LISAlike spacecraft constellations [72]. There are plans for another spacebased interferometer, the Japanese DECIGO (DECIhertz interferometer Gravitational wave Observatory) [73], designed to cover the gap between LISA and the Advanced detectors in the frequency band between 100 mhz and 10 Hz. The first step for the space detectors is the launch of the LISA Pathfinder [74]. At the time of writing the LISA Pathfinder spacecraft is schedule to be launched in the first semester of It will test in flight the very concept of GW detection by putting two test masses in a nearperfect gravitational freefall. It will control and measure their motion with unprecedented accuracy providing invaluable data regarding the key technologies to be used in the LISA spacecrafts Pulsar timing In order to measure gravitational waves at even lower frequencies a different approached is required. By monitoring radio beams of distant pulsars for evidence of gravitational waves interacting with a beam of electromagnetic radiation. Radiotelescope groups are seeking to detect gravitational waves via precise timing of a large number of radio pulsars through programs like Parkes Pulsar Timing Array (PPTA) [75], the Nanohertz Observatory for Gravitational Waves (NANOGrav) [76], and the European Pulsar Timing Array (EPTA) [77]. These groups are in the process of organising themselves as the International Pulsar Timing Array or IPTA. Due to
55 1.6. CONCLUSIONS 27 Figure 1.12: An artist s representation of the LISA mission. Three spacecraft, each with a Yshaped payload, form an equilateral triangle with sides 5 million km in length. The two branches of the Y at one corner, together with one branch each from the spacecraft at the other two corners, form one of up to three Michelsontype interferometers, operated with infrared laser beams. ( c NASA images) the astronomical distances involved these observations are expected to detect gravitational waves at very low frequencies ( Hz). Sources in this regime will include coalescing supermassive black hole binary systems. At ultralow frequencies gravitational waves in the early universe may have left their imprint on the polarisation of the cosmic microwave background. 1.6 Conclusions These are exciting times for the GW community. We are getting closer to achieving direct detection of gravitational waves, which has encouraged continual improvement of the existing facilities. At the time of writing the improvements to LIGO L1 and H1 detectors are almost complete and soon a new science run will start (S6) using the most sensitive instrument built to date, Enhanced LIGO. They will be joined by VIRGO+ in this new attempt for direct detection of gravitational waves. The aim of S6 is to collect a year s worth of data after which the upgrades for Advanced
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64 36 CHAPTER 1. INTRODUCTION
65 Chapter 2 ModeCleaner Vibration Isolator 2.1 Preface With the design of the advanced vibration isolator for the test masses for an interferometric GW detector well under way, it became evident that a smaller and more compact version of the isolator would be an excellent system for use with the auxiliary optics. The first test for this system would be the modecleaner optics, which can be tested as a standalone system unlike the recycling cavities optics. For this design it was considered that in general a vibration isolator system for the auxiliary optics demands less in terms of seismic isolation than the full scale system. The design began with an inverse pendulum made of four legs supporting a rectangular structure used as a table. Weights were used to tune and lower the resonant frequency of the inverse pendulum, acting as a first stage of horizontal preisolation. A pyramidal Roberts linkage was attached to this structure, providing a second stage of horizontal preisolation. An Euler spring developed for vertical isolation was attached to this stage and a selfdamped pendulum with cantilever springs was in turn used to hang a breadboard where the optics will be placed. Since the vibration isolator system had not been completed at the time of writing the presented paper, only the predicted performance is shown. The system has since been completed and resulting measurements are presented in the postscript of this chapter. The vibration isolators were built by Andrew Woolley and the technical staff from the UWA School of Physics workshop. 37
66 38 CHAPTER 2. MODECLEANER VIBRATION ISOLATOR Application of New Preisolation Techniques to Modecleaner Design Pablo Barriga 1, Andrew Woolley 1, Chunnong Zhao 2,, and David G. Blair 1 1 School of Physics, University of Western Australia, Crawley, WA6009, Australia 2 Computer and Information Science, Edith Cowan University, Mount Lawley, WA 6050, Australia Two very low frequency preisolation stages can greatly reduce the residual motion of suspended optical components. In a mode cleaner this can reduce the control forces required on the mirrors, simplifying lock acquisition and reducing noise injection through control forces. This paper describes a 12 m triangular suspended mode cleaner under construction for AIGO high optical power interferometer. A novel and very compact multistage isolator supports the cavity mirrors. It combines an inverse pendulum in series with a low mass Roberts Linkage, both with pendulum frequencies below 0.1 Hz. The suspension chain is connected to the Roberts Linkage via an Euler spring stage and a cantilever spring assembly for vertical isolation. We present an analysis of the mode cleaner, emphasising the advantage of the improved mode cleaner suspension and its power handling capability. The effect of seismic noise on the residual velocity of the mirrors and the predicted frequency stability of the optical cavity are presented. 2.2 Introduction A high optical power interferometer is being built at Gingin near Perth to develop advanced interferometer technology required for advanced gravitational wave (GW) detectors. The project includes a 12 m triangular ring cavity modecleaner to minimise variations in laser beam geometry. The interferometer is designed to have parameters Now at School of Physics, The University of Western Australia, Crawley, WA6009, Australia
67 2.2. INTRODUCTION 39 as close as practicable to Advanced LIGO to enable the critical issues of thermal lensing and radiation pressure effects to be examined. All interferometer GW detectors include one or more modecleaners for laser spatial stabilisation before it is injected into the sensing cavities. All modecleaners have the same layout, a threemirror triangular ring cavity. Two flat mirrors define the short side of the triangle, and a concave mirror the acute side of it. Modecleaners generally use a simpler suspension system than those used for interferometer test masses. For advance interferometers we need modecleaners capable of transmitting at least 100 W of laser power. They all must solve several problems; seismic isolation, residual velocity, power handling and frequency discrimination. The mirror positioning control is based on the wellknown PoundDreverHall technique [1]. By reducing the vibration even at frequencies below the GW detection band (10 Hz 1 khz) we seek to facilitate the locking of high finesse cavities. This not only applies to the main FabryPerot long arms of the interferometer, but to the modecleaner and other optical components like the input optics and the beamsplitter among others. The reduction of the vibration will reduce the residual motion of the mirrors; as a consequence we will need less force on the actuators to control them. With lower servo forces required to maintain the locking noise injection by the actuators will be reduced. All of it simplifies the locking of the cavities and the control system design. Project Length (m) Finesse FSR (MHz) End mirror ROC (m) LIGO (4 km) GEO , , , , 6.72 VIRGO TAMA Advanced LIGO AIGO Table 2.1: Characteristics of modecleaners used on interferometric gravitational wave detectors around the world, including one proposition for Advanced LIGO (LIGOG D).
68 40 CHAPTER 2. MODECLEANER VIBRATION ISOLATOR Different configurations have been used for each long base interferometer as summarised in table 2.1. GEO600 has two modecleaners in series. LIGO uses a simple pendulum whilst Advanced LIGO is planning the use of a triple pendulum mirror suspension system based on the GEO600 suspensions [2]. TAMA300 and GEO600 use double pendulums for the modecleaners [3, 4, 5, 6]. At the same time LIGO and TAMA teams are involved in the implementation of a new suspension system called Seismic Attenuation System (SAS). VIRGO uses a simpler version of their test mass superattenuators with only two seismic filters mounted on an inverse pendulum [7, 8, 9]. In this paper we present the expected behaviour of a modecleaner which uses double preisolation stages to obtain very low residual motion of the mirrors. In order to model the performance we work with seismic data taken at AIGO in Gingin and with the preisolator stages transfer function. We show that very low residual velocity can be achieved simplifying locking and reducing noise injection. 2.3 Isolation and suspension design The suspension system for the test masses developed at UWA for the AIGO GW detector consists of different stages of preisolation in an effort to achieve the isolation requirements at low frequencies. The three dimensional (3D) preisolator consists of an inverse pendulum horizontal stage cascaded with a LaCoste vertical stage and a Roberts Linkage horizontal stage of preisolation [10, 11]. A four stage multipendulum system is then mounted, where Euler springs for vertical suspension are included [12]. Mounting each of the intermediate masses of the pendulum from gimbals allowed them to freely rock with respect to a short rigid section of the main pendulum chain, then viscously coupling these two together with magnetic eddy current coupling for selfdamping [13]. The modecleaner isolation design uses an improved and more compact suspension for a small optical table where the mirrors are mounted. One of the big differences between the test masses and the modecleaner systems is the available space to fit them. The modecleaner suspension is designed to fit inside a 12 m long and 1 m
69 2.3. ISOLATION AND SUSPENSION DESIGN 41 (a) (b) Inverse Pendulum Euler Spring Roberts Linkage Gimbal Cantilever Spring Self Damping Breadboard Figure 2.1: (a) Modecleaner isolator design, where it is possible to see the inverse pendulum with the Roberts Linkage and suspension chain. (b) Design of the Roberts Linkage and suspension chain. From the Euler spring follow the cantilever springs from where the breadboard with the optics will hang. 50 Horizontal and Vertical Isolator Transfer Function Magnitude [db] Horizontal Vertical frequency [Hz] Figure 2.2: Frequency space representation of the isolation system transfer function, where it is possible to identify the resonant frequencies of each of the horizontal and vertical stages of the preisolator. We expect some internal modes (not shown in the figure) of the order of a few tenths of hertz and Q 100 on the vertical transfer function. This representation was calculated assuming a low pass filter model for each stage.
70 42 CHAPTER 2. MODECLEANER VIBRATION ISOLATOR PreIsolator M1 Nd:YAG Laser PreIsolator M3 l 1 200mm M2 r 1 = r 2 = r l 12000mm 3 = L = 2l + l 2 1 Figure 2.3: The AIGO modecleaner layout formed by two flat mirrors (M1 and M2) in one end and one curved mirror (M3) at the other end, giving shape to the triangular ring cavity. diameter high vacuum pipe The first stage of horizontal preisolation is an inverse pendulum (figure 2.1(a)) with a period of 20 s. Suspended from it is a Roberts Linkage horizontal pendulum with a resonant frequency less than 0.05 Hz. The configuration of this Roberts Linkage differs from the one designed for the AIGO test masses in having very low mass and three attachment points (wires) instead of four, making it easier to tune [11]. Further isolation stages hang from the Roberts Linkage apex (figure 2.1(b)). These consist of an Euler spring stage and cantilever spring assembly for vertical isolation. A gimbal is situated in between for a high moment of inertia rocking mass to create a selfdamped pendulum. An array of magnets will provide the damping due to the induced eddy currents. This provides passive damping for the pendulum modes of the suspension chain and the vertical modes of the cantilever spring assembly. An aluminium optical breadboard is suspended from the cantilever spring assembly to mount the optical components. The modecleaner acts as a spatial filter, providing passive stabilisation of timedependant higherorder spatial modes and transmitting the fundamental mode (T EM 00 ). A key factor to reduce the transmission of the higherorder modes is the radius of curvature of the end mirror (M3 on figure 2.3). A simplified expression of the suppression factor is given in equation (2.1) [14]. The suppression depends directly on the ratio between the cavity length (L) and the radius of curvature (R), the reflectivity of each mirror (r 1, r 2, r 3 ) and the order of the mode expressed by the (m + n) parameter.
71 2.3. ISOLATION AND SUSPENSION DESIGN 43 Radius (m) Waist (mm) Stability (g) Power density for W (kw cm 2 ) Table 2.2: Some of the parameters affected by different radii of curvature and in particular the effect on the power density on the cavity flat mirrors. S mn = ( [ ] [ π 2 r 1 r 2 r 3 π 2 (1 r 1 r 2 r 3 ) 2 sin {(m 2 + n)cos 1 1 R]}) L 1/2. (2.1) The suppression of the higherorder modes is not the only consideration when choosing the curvature of the mirror. The builtup power inside the cavity depends on the finesse, but the area where this power will concentrate depends on the radius of curvature of the end mirror M3. This defines the size of the beam waist that defines the spot size on each mirror and hence the power density on them. Therefore the curvature of the mirror is limited by the damage threshold of the mirror coating, normally 1 MW cm 2 and by the stability factor g of the cavity. Due to the high laser power there is a thermal lensing effect that needs to be taken into account, which for simplicity and limited space has not been considered. Figure 2.4 shows the suppression factor for different higher modes. It is clear from the figure that the smaller the radius of curvature more higher modes will be transmitted. The larger the radius of curvature the wider is the window from where we can select a radius that will suppress most of the higher modes. At the same time the size of the waist will increase. As a consequence the power density on the flat mirrors will be reduced, but the price is that the cavity becomes less stable as summarised in table 2.2. Therefore we need to choose a radius of curvature that will give us a good compromise between power density and cavity stability. A radius of curvature of 31.8 m will give us a safety margin of 300% on the damage of the mirror coating with a stability factor of Other choices such as 38.9 m and 43.3 m give higher safety margins but with reduced cavity stability.
72 44 CHAPTER 2. MODECLEANER VIBRATION ISOLATOR Suppression Factor [db] M3 Radius of Curvature Figure 2.4: Simplified calculations of the suppression factor of the first 15 higher order modes for a 12 m triangular ring cavity. 2.4 Noise and locking predictions In order to predict the behaviour of the isolator we used the theoretical transfer function and seismic data measured at the AIGO site in Gingin. With this data it is possible to determine the residual motion of the mirrors as shown in figure 2.5. Here we see that at 10 Hz the residual motion is about m Hz 1/2 (Advanced LIGO expects a horizontal noise of m Hz 1/2 at 10 Hz; LIGOG D). This corresponds to a frequency stability of δl L = δf f, f Hz δf 7.9mHz Hz 1/2. (2.2) The lock acquisition of the cavity will be easier to achieve if the mirror velocities are below the critical level. This is the velocity at which the frequency shift due to the Doppler Effect becomes equal to the linewidth of the cavity [15]. The velocity is inversely proportional to the cavity storage time τ, but as can be seen in equation (2.4) this depends on the length (L) and the finesse (F) of the cavity: ν cr = λ 2τF where F = π r1 r 2 r 3 (1 r 1 r 2 r 3 ), (2.3)
73 2.4. NOISE AND LOCKING PREDICTIONS 45 ModeCleaner Preisolator Horizontal Response to Seismic Noise at Gingin Magnitude m/ Hz Horizontal Seismic Noise Vertical Seismic Noise Horizontal System Response Coupled Vertical System Response frequency [Hz] Figure 2.5: Gingin horizontal and vertical seismic noise and predicted system response. The vertical response is reduced in three orders of magnitude to simulate the vertical coupling into the horizontal stages. τ = 2(L/c) ln(r 1 r 2 r 3 ) 2L F c π. (2.4) The lower the velocity the easier it will be to lock the cavity. Table 2.3 summarises some of the parameters calculated for AIGO modecleaner assuming mirror losses are only due to transmission. Using the isolator theoretical transfer function, we obtain the rms residual motion (indefinite integral) of the modecleaner mirrors for the measured seismic noise at Gingin, as shown in figure 2.6 (a). At 1 Hz the residual motion is about m. Free spectral range MHz Cavity optical bandwidth khz Finesse 1495 Storage time µs Critical velocity 9.26 µm s 1 Table 2.3: Theoretical parameters of the AIGO 12 m modecleaner.
74 46 CHAPTER 2. MODECLEANER VIBRATION ISOLATOR Figure 2.6 (b) shows the rms mirror velocity where at 1 Hz the mirror velocity is about m sec 1, well below v cr calculated above as 9.26 m sec 1. This means that in order to lock and control the mirrors we will need to apply forces of about N at 1 Hz. It can also be seen that above 0.4 Hz the noise is dominated by the vertical noise coupled into the horizontal stages. The analysis here has ignored the thermal noise of the test masses which becomes a significant noise source above 10 Hz. 2.5 Conclusions Using a theoretical transfer function and onsite measured seismic noise we characterise the behaviour of a compact and novel isolation system that will be used for the modecleaner of AIGO high power facility. We have shown that two steps of preisolation allow frequency stability, residual motion and mirror velocities to be kept well below the critical value. From the PoundDrever technique only low frequency corrections are needed to control the mirror position, therefore from these results we expect to be able to use low servo forces for cavity locking, and hence to minimise noise injection from the control system. As part of the design of the modecleaner we also calculate the radius of curvature of the end mirror M3. It is possible to achieve more than 40 db suppression of the first 15 modes and to allow the transmission of 100 W of laser power into the main arms of the detector with a power density of 305 kw cm 2. Acknowledgements The authors want to thank Li Ju, John Winterflood, Jérôme Degallaix and John Jacob for helpful discussions. This work was supported by the Australian Research Council, and is part of the research program of the Australian Consortium for Interferometric Gravitational Astronomy.
75 2.5. CONCLUSIONS 47 (a) 10 5 Mirror Residual Motion 10 0 Mirror Residual Motion (m) Horizontal Coupled Vertical Frequency (Hz) (b) 10 0 Mirror Velocity 105 Mirror Velocity (m/sec) Horizontal Coupled Vertical Frequency (Hz) Figure 2.6: (a) Mirror rms residual motion as a function of frequency for the modecleaner mirror suspension. (b) Mirror rms velocity as a function of frequency for the modecleaner mirror suspension.
76 48 CHAPTER 2. MODECLEANER VIBRATION ISOLATOR 2.6 Postscript In order to test the modecleaner vibration isolator, a full system was assembled on top of a table. The purpose was to shake the whole system in order to measure the mechanical transfer function. The driving system comprised a large loudspeaker specially modified in order to drive the suspension system at low frequencies. The driving signal was generated by the builtin source signal generator of a spectrum analyser. This signal was connected to the modified loudspeaker through a high voltage amplifier. The spectrum analyser was then used in swept sine mode to perform the measurements at different frequency ranges. Two geophones were mounted on the suspension system; one on top of the inverse pendulum stage where the vibration isolator was attached to the driving system and the other on top of the breadboard. Their differential signal gives the mechanical transfer function of the isolation system. Measurements were performed for the two horizontal axes X and Y as shown on figure 2.7. Following several tests and revision of the theoretical model, a new set of measurements was performed using the described experimental setup. Originally the inverse pendulum stage was modelled as a low pass filter with a corner frequency at 0.1 Hz and a 20 db/dec rolloff above this corner frequency. Since this rolloff does not continue indefinitely, a floor at around 30 db was added to the inverse pendulum Y  Axis X  Axis Figure 2.7: Top view of the modecleaner vibration isolation system defining X and Y axes used for the measurements of the horizontal transfer function.
77 2.6. POSTSCRIPT 49 theoretical model. The measurements agreed well with the new theoretical model. Due to operational limitations of the geophones it was not possible to obtain an accurate measurement of the inverse pendulum frequency response for low frequencies. Therefore the measured curves presented in figure 2.8 show the measurements of the isolation chain from the Roberts Linkage stage down to the breadboard, with the theoretical inverse pendulum curve added for comparison. (a) 50 Horizontal Transfer Function 0 Magnitude (db) Theoretical Horizontal Complete Isolator Transfer Function (YAxis) Complete Isolator Transfer Function (XAxis) Frequency (Hz) (b) 50 Vertical Transfer Function 0 Magnitude (db) 50 Theoretic Vertical Measured Vertical Frequency (Hz) Figure 2.8: Comparison between theoretical and measured transfer function for both (a) horizontal axes X and Y and (b) for the vertical axis. Figure 2.8 (a) shows the preliminary measurement of the horizontal transfer func
78 50 CHAPTER 2. MODECLEANER VIBRATION ISOLATOR tion. At low frequencies the measurements show a double peak at about 0.1 Hz which corresponds to the inverse pendulum frequency. The double peak is generated by the coupling of the X and Y axes of the inverse pendulum, each having slightly different frequencies. Unfortunately limitations of the geophones do not allow for accurate measurement at these low frequencies. The inverse pendulum peak is followed by the Roberts Linkage normal mode at 0.2 Hz, which is higher than the design value. The normal modes of the preisolation stages are followed by two pendulum modes at 1.7 Hz and 3.6 Hz, which correspond to the pendulum that holds the cantilever spring stage from the Euler spring and the wires that attach the breadboard to the cantilever stage. Higher order modes at 7.2 Hz (in the X direction only), 10 Hz (10.2 Hz in the Y axis and 10.7 Hz in the X axis), and around 17 Hz (16.5 Hz in the X axis and 18.1 Hz in the Y axis) correspond to resonance modes of the breadboard stage. These higher order modes were viscously damped using eddy current coupling. This damping was achieved by attaching a short leg with a neodymium boride permanent magnet at the end to the tip of each cantilever spring and adding a small copper plate to each corner of the breadboard just underneath each legmagnet pair. Figure 2.8 (b) shows the preliminary measurement of the vertical axis of the vibration isolator. The first peak at 1.4 Hz corresponds to the Euler spring resonant mode and the second peak at 3.5 Hz to the cantilever spring stage. In comparison with the measured results, the theoretical model shows lower Qfactor and a lower resonant frequency of 0.8 Hz for the Euler spring stage and a lower frequency close to 2 Hz for the cantilever spring stage. This difference can be partly attributed to the higher frequency of the Euler spring resonant mode. The measurement also shows a third resonant mode close to 10 Hz that corresponds to the coupling of a high frequency mode from the breadboard. After several sets of measurements, recommendations for system improvements were made. Including the improvement of the inverse pendulum legflexure alignment control to achieve more consistent tuning at low frequencies. A more robust Roberts linkage frame with additional tune damping at the top. Replacement and tuning of the Euler spring blades in order to lower the resonant frequency. A more radical improvement will be the addition of a LaCoste stage, but utilising an Euler spring
79 2.7. REFERENCES 51 Mode Cleaner Measurements Figure 2.9: Modecleaner vibration isolator system mounted inside a pipe for testing. At the far right hand side of the suspension it is possible to see the ports in the pipe that were intended for use as the main laser port and cable feedthrough. design approach for the spring components. This will allow for a sensing and actuation scheme similar to the main advanced vibration isolators. Unfortunately the GW research group suffered a funding shortage. This affected several projects including the development of the seismic isolation system for the modecleaner. This resulted in postponement of both the planned vibration isolation system modifications and the completion of measurements using more sensitive devices such as shadow sensors. This forced us to focus on a related system, a suspended optical cavity utilising an advanced isolation system, which is presented in the next chapter. The electronics developed for the modecleaner isolation system were then used for the local control of the advanced vibration isolation system. 2.7 References [1] R. W. P. Drever, J. L. Hall, F. V. Kowalski, et al, Laser Phase and Frequency Stabilization Using an Optical Resonator, Appl. Phys. B 31 (1983) [2] N. A. Robertson, G. Cagnoli, D. R. M. Crooks, et al, Quadruple suspension design for Advanced LIGO, Class. Quantum Grav. 19 (2002)
80 52 CHAPTER 2. MODECLEANER VIBRATION ISOLATOR [3] S. Goßler, M. M. Casey, A. Freise, et al, The modecleaner system and suspension aspects of GEO 600, Class. Quantum Grav. 19 (2002) [4] M. V. Plissi, K. A. Strain, C. I. Torrie, et al, Aspects of the suspension system for GEO 600, Rev. Sci. Instrum. 69 (1998) [5] M. Ando, K. Tsubono for the TAMA collaboration, TAMA project: Design and current status, AIP Conf. Proc. 523 (2000) [6] A. Takamori, M. Ando, A. Bertolini, et al, Mirror suspension system for the TAMA SAS, Class. Quantum Grav. 19 (2002) [7] F. Bondu, A. Brillet, F. Cleva, et al, The VIRGO injection system, Class. Quantum Grav. 19 (2002) [8] G. Losurdo, M. Bernardini, S. Braccini, et al, An inverted pendulum preisolator stage for the VIRGO suspension system, Rev. Sci. Instrum. 70 (1999) [9] A. Bernardini, E. Majorana, P. Puppo, et al, Suspension last stages for the mirrors of the VIRGO interferometric gravitational wave antenna, Rev. Sci. Instrum. 70 (1999) [10] J. Winterflood, High performance vibration isolation for gravitational wave detection, PhD Thesis, School of Physics, The University of Western Australia, Chapter 7, [11] F. Garoi, J. Winterflood, L. Ju, et al, Passive vibration isolation using a Roberts linkage, Rev. Sci. Instrum. 74 (2003) [12] J. Winterflood, Z. B. Zhou, L. Ju, D. G. Blair, Tilt suppression for ultralow residual motion vibration isolation in gravitational wave detection, Phys. Lett. A 277 (2000) [13] J. Winterflood, High performance vibration isolation for gravitational wave detection, PhD Thesis, School of Physics, The University of Western Australia, Chapter 3, 2001.
81 2.7. REFERENCES 53 [14] M. Barsuglia, Stabilisation en fréquence du laser et contrôle de cavités optiques à miroirs suspendus pour le détecteur interféromètrique d ondes gravitationnelles VIRGO, PhD Thesis, Université de ParisSud, Orsay, Chapter 7, [15] M. Rakhmanov, Dopplerinduced dynamics of fields in FabryPerot cavities with suspended mirrors, Appl. Opt. 40 (2001)
82 54 CHAPTER 2. MODECLEANER VIBRATION ISOLATOR
83 Chapter 3 Advanced Vibration Isolator 3.1 Preface This chapter comprises two papers describing the advanced vibration isolation system developed at UWA. Both papers have been submitted to the Review of Scientific Instruments journal. The first paper presents a review of the vibration isolator and includes the measurements of the frequency response of the cavity, which led to the characterisation of vibration isolator performance. The second paper introduces the local control system which allows for feedback control and normal mode damping. The control electronics and its diagrams are presented in Appendix C. Development of the modecleaner vibration isolation system included design and development of control electronics. These were developed to amplify and filter the input signal from the shadow sensors and to amplify the signal that drives the coils of the magnetic actuators. Since the shadow sensors and actuators installed in the modecleaner are very similar to the ones installed in the full scale vibration isolation system, the same electronics (with the exception of gain tuning) were used for the advanced vibration isolator local control system. In addition to development of the electronic boards, a backplane mounted on the back of a 6U chassis was necessary in order to distribute the I/O signals between the vibration isolator and the DSP (Digital Signal Processor) at the core of the local control system. This required the design of signal distribution and cabling to connect the control electronics to the different sensors and actuators distributed along the vibration isolator structure. In order to facilitate these connections, a vacuumcompatible intermediate board was designed and installed on the vibration isolators. The author 55
84 56 CHAPTER 3. ADVANCED VIBRATION ISOLATOR was not involved in the wiring of the advanced vibration isolators. The intermediate board for the signal distribution was designed in consultancy with J. C. Dumas. The builtin libraries provided by the DSP manufacturer allow for the control loops to run on the DSP board whilst isolated from the local operating system. The user control and interface was written in LabView R predominantly by J. C. Dumas. The second vibration isolator was being assembled and tuned at the AIGO test facility in Gingin by J. C. Dumas, A. Woolley and technicians. With the first vibration isolator fully operational in the main lab, several tests were carried out to determine some of the loop gains to be used in the PID control loops (even though not all the loops are PID, some only use integral gain and some only derivative gain for damping). Tests of the mechanical transfer functions were also completed. At the same time the author prepared an optical table and installed a 300 mw Nd:YAG laser with the necessary optics for cavity modematching, phasemodulation and optical isolation of the system. With support from C. Zhao, the author tested the possibility of using a reference cavity for laser stabilisation and an acoustooptic modulator for locking the main cavity. After several tests the dynamic range of the acoustooptic modulator (used in double pass mode) was found to be insufficient for following the longitudinal variations of the cavity. Therefore laser locking was achieved directly to the main cavity, as presented in the first paper of this chapter.
85 3.2. INTRODUCTION 57 Compact vibration isolation and suspension for AIGO: Performance in a 72 m Fabry Perot cavity P. Barriga, J. C. Dumas, A. A. Woolley, C. Zhao, D. G. Blair School of Physics, The University of Western Australia, Crawley WA 6009, Australia This paper describes the first demonstration of vibration isolation and suspension systems which have been developed with view to application in the proposed Australian International Gravitational Observatory (AIGO). In order to achieve optimal performance at low frequencies new components and techniques have been combined to create a compact advanced vibration isolator structure. The design includes two stages of horizontal preisolation, and one stage of vertical preisolation with resonant frequencies 100 mhz. The nested structure facilitates a compact design and enables horizontal preisolation stages to be configured to create a superspring configuration, where active feedback can enable performance close to the limit set by seismic tilt coupling. The preisolation stages are combined with multistage 3D pendulums. Two isolators suspending mirror test masses have been developed to form a 72 m optical cavity with finesse 700 in order to test their performance. The suitability of the isolators for use in suspended optical cavities is demonstrated through their ease of locking, long term stability and low residual motion. An accompanying paper presents the local control system and shows how simple upgrades can substantially improve residual motion performance. 3.2 Introduction The first vibration isolation systems used multiple stages of massspring elements based on rubber and steel stacks followed by a pendulum stage where individual mode frequencies were of a few Hertz. This technology was inherited from resonant bar gravitational wave detectors. In the early 1990 s the University of Western Australia (UWA) group introduced cantilever blade springs, both for use on cryogenic
86 58 CHAPTER 3. ADVANCED VIBRATION ISOLATOR resonant bar detector Niobe [1], and then on an 8 m prototype interferometer [2]. All metal vibration isolation components are required for use in high vacuum environments (while rubber is not permissible). However multiple stages of massspring elements suffer from relatively high normal mode Qfactors which comes as strong resonant enhancements in the transfer function. To avoid this problem the UWA group proposed the use of preisolators [3]. The idea was to precede a multistage isolator with a single very low frequency stage, which would reduce the seismic excitation of the higher frequency normal modes [4]. The VIRGO project was the first to use this concept in a full scale detector [5], by suspending vibration isolation stages consisting of long pendulums and cantilever springs by a 6 m inverse pendulum stage with very low resonant frequency [6]. In the 1980 s a group at JILA developed active vibration isolator based on the superspring concept. In this case a two element spring with a sensing transducer and actuator allows a synthetic spring to be created, with resonant frequency determined by a control system [7, 8]. During the 1990 s Winterflood et al [9, 10, 11] developed a range of preisolation techniques based on geometric cancellation of elastic spring constants so called geometric antisprings. Of these, greatest attention has been given to the Roberts Linkage [12] which is described in section 3.3. The VIRGO group developed magnetic antisprings to lower the vertical frequency of their cantilever spring systems [13]. At the top of the inverse pendulum is the first vertical filter. It contains a set of maraging steel triangular blades from where a five stage pendulum chain is suspended. Each intermediate mass is a 100 kg drumshaped vertical filter. Each one of them also includes magnetic antispring systems in order to reduce resonance frequencies [14]. From the last stage, known as marionetta, the test mass and a reaction mass are suspended providing three degrees of freedom (translation, pitch and yaw) [15]. This design will also be used in Advanced VIRGO [16]. A new approach to vertical vibration isolation was developed by Winterflood et al [17] the Euler spring. This involves loading a vertical elastic column just beyond the Euler buckling instability; where it becomes a well behave spring with frequency equal to 1/ 2 of the frequency of a pendulum of the same length. Methods were found
87 3.2. INTRODUCTION 59 of adding geometric antispring elements to this system to create even lower resonant frequencies [18]. The Euler spring approach is advantageous because the stored elastic energy is reduced by a factor of (working range/effective length) 2, enabling the spring to be of much lower mass, and with much higher internal mode frequencies. A problem with multiple pendulum stages is the difficulty of damping their normal modes without noise injection. Horizontal isolation at the GEO600 project is provided by a triple pendulum system. This assembly is suspended from a two layer isolation stack consisting of an active and a passive stage. A separate reaction pendulum is included so global control forces can be applied to the test mass from a seismically isolated platform [19]. At UWA we developed the concept of selfdamped pendulum which provides a passive solution to the problem, through dissipative coupling to an angular degree of freedom within the pendulum mass [20]. The TAMA project in collaboration with the LIGO laboratory is also testing an advanced vibration isolation system called Seismic Attenuator System or SAS [21]. TAMASAS is conceptually similar to the VIRGO design, with a shorter inverse pendulum (2.5 m compared to VIRGO 6 m), a similar top vertical filter and a suspended triple pendulum stage [22]. Meanwhile the Advanced LIGO project favoured a stiff preisolation technique based on hydraulic systems distributed on each corner of the vacuum chamber [23]. These provide vertical and horizontal preisolation for the whole chamber combining geophones and seismometers with high resonant frequencies in a high bandwidth control loop. Based on the GEO600 design Advanced LIGO will use a quadruple pendulum suspension for horizontal isolation [24]. Therefore two stages of active isolation are followed by a pair of parallel quadruple pendulums, the last being the interferometer test mass, which is opposed by a reaction mass used for actuation All of the above concepts have been integrated into a single advanced vibration isolation system. The system we have created uses double preisolation, to enable the superspring concept to be used to suppress normal peaks. It is nested so that the entire structure is less than 3 m in height. Including a multipendulum stage with selfdamping. Two systems have been installed at the East arm of the AIGO research facility in Gingin, Western Australia. Test masses are suspended from each
88 60 CHAPTER 3. ADVANCED VIBRATION ISOLATOR system to form a 72 m optical cavity. A 300 mw Nd:YAG laser locked to the cavity through the PoundDreverHall technique [25] provides an error signal to measure the performance of the suspended cavity at low frequencies. While the vibration isolation system is mainly a passive design, some feedback is required for low frequency control, such as alignment and drift corrections, and damping of normal modes. Each suspension system is integrated with a digital local control system with feedback to the preisolation stages and the penultimate pendulum stage. Details of the local control strategy are discussed in an accompanying article by Dumas et al [26]. In this article we present the first results of the noise performance in the suspended cavity. In section 3.3 we review the overall isolator design, introducing individual isolation concepts and components. The experimental setup is presented in section 3.4 including cavity parameters and the laser control system. In section 3.5 we discuss the results obtained during the operation of the locked cavity. Finally in section 3.6 we discuss the performance of these isolators and future developments. 3.3 Vibration isolation design The Australian Consortium for Interferometric Gravitational Astronomy (ACIGA) has developed a high performance compact vibration isolation system for the proposed AIGO interferometer. Novel isolation elements for all vertical and horizontal stages were individually developed and tested. First they were designed to include multiple passive low frequency preisolation to minimise the seismic excitation of isolator normal modes. This was achieved by applying simple geometric antispring techniques to achieve very low resonant frequencies. Second, they were designed to attain passive damping of pendulum modes through the concept of selfdamped pendulum. Third, Euler springs were used to obtain vertical normal modes frequencies well matched to the pendulum frequencies. Fourth, special attention was given to the materials used for the design and construction of the suspension in order to be vacuum compatible. Finally centre of percussion tuning was used when possible to optimise the transfer function of individual stages [27].
89 3.3. VIBRATION ISOLATION DESIGN 61 Rigid links Inverse Pendulum Roberts Linkage LaCoste vertical stage Flexure Top of preisoltator stand Euler springs vertical stages Three vibration Isolation stages Selfdamping system Control Mass Niobium Suspension Test Mass Mirror Figure 3.1: Full vibration isolator system and schematic that show the different stages of preisolation and the multipendulum stage with a test mass at the bottom of the chain Preisolation components The vibration isolation system consists of multiple cascaded stages as shown in figure 3.1 applying several different techniques to attenuate seismic noise. At the top of the suspension four short inverse pendulums legs provide the first stage of preisolation. This first preisolation stage is effectively a square table mounted on four inverse pendulums for legs. This allows for horizontal translation, while being rigid to tilt. The resonant frequency of these pendulums can be tuned to very low frequency providing very effective preisolation. Its cube shape allows for the integration of a spring system that provides vertical preisolation. The LaCoste linkage consists of diagonally attached springs between each of the four legs on two structures (the inverse pendulum and the LaCoste supporting frame). The four lower pivot arms were fitted with counterweights in order to provide centre of percussion tuning for the eight of them. Pretensed springs were used in order to obtain the zerolength requirement of the LaCoste geometry [28]. Horizontal springs were added and stretched
90 62 CHAPTER 3. ADVANCED VIBRATION ISOLATOR more than the separation between the pivoting points creating an inverse pendulum effect. This effect produces a negative spring constant that counteracts the significant spring constant of the flexure pivots. By making the width adjustable one can tune the spring constant in order to obtain very low frequencies and a large dynamic range. Therefore both preisolation stages have resonant frequencies below 100 mhz. Both the inverse pendulum and the LaCoste stages have a cubic geometry which allows for their combination into a single 3D structure as illustrated in figure 3.2. Figure 3.2: First stage horizontal and vertical preisolation. The preisolator combines two very low frequency stages: (a) The horizontal Inverse Pendulum and (b) the vertical LaCoste linkage. The concept of the antispring for flexure spring constant cancellation is shown. The rigidity of this structure allows for the suspension of a Roberts linkage stage nested within the two preisolators. It is a relatively simple design consisting of a cube frame suspended by four wires hung off the LaCoste stage as illustrated in figure 3.3. Its geometry is tuned to restrict the suspension point of the load to an almost flat horizontal plane, thus making the gravitational potential energy almost independent of displacement and minimising the restoring force resulting in a low resonance frequency [12, 27]. At only 1 m height the whole top section is very compact, including also the topmost stage of the multistage pendulum in the same volume. The preisolation stages are mounted on top of a rigid frame so as to have enough height to suspend the isolation chain from the top of the Roberts linkage as seen in
91 3.3. VIBRATION ISOLATION DESIGN 63 figure 3.1. The interweaving of the preisolation stages together with the use of large dynamic range sensors and actuators allows for an active feedback control of the preisolator stages enabling performance close to the limit set by seismic tilt coupling [26]. Figure 3.3: The Roberts linkage. (a) shows a onedimensional diagram of a Roberts Linkage with a suspended load from point P, which stays in the same plane for variations in the position of C and D. (b) shows a diagram of the cube shaped design used in the AIGO suspension Isolation stages Intermediate masses of 40 kg are suspended to form a selfdamped pendulum arrangement as illustrated in figure 3.4. The selfdamping concept consists of viscously coupling different degrees of freedom of the pendulum mass as shown in figure 3.5. Each intermediate mass is pivoted at its centre of mass, with a light shaped frame fixed to the pendulum link to provide a reference against tilt. Neodymium boride magnets in a comblike distribution are mounted to the frame. These are paired with intermeshing copper plates attached to each corner of the square rocker mass to create a viscous damping through eddy current coupling reducing the Qfactor of the pendulum normal modes [20]. An aluminium arm is attached on each side of the top rocker mass of the pendulum chain. The purpose is to increase the moment of inertia to further reduce the Qfactor of the lower resonant mode of the multistage pendulum.
92 64 CHAPTER 3. ADVANCED VIBRATION ISOLATOR Rotational arm rests on Euler springs and is suspension point for next stage Al web rigidly clamped to the suspension tube Copper plates attached to Al web Euler springs Magnets, attached to rocker mass Bottom of springs is clamped to suspension tube Rocker mass Control mass Niobium flexures Test mass Figure 3.4: The multistage pendulum including three intermediate masses showing the rigid section, rocker mass, eddy current damping and Euler springs for vertical isolation. At the bottom of the chain the control mass stage provides sensing and control for a test mass suspended with niobium flexures. Each intermediate mass in the multistage pendulum is attached to the next using Euler springs tuned for low frequency vertical isolation effectively attenuating the vertical component of the seismic noise. Euler spring stages can be tuned with antispring geometries to achieve good low frequency performance within a very compact design [17, 18, 29]. Figure 3.6 shows a diagram of an intermediate mass with Euler spring attachment, while figure 3.7 illustrate in detail one of the intermediate masses Control mass and test mass suspension At the bottom of the multistage pendulum chain shown in figure 3.4 a 30 kg control mass stage provides the interface to the test mass. The test mass consists of a 50 mm, 30 mm thick fused silica mirror supported in an aluminium and stainless steel cylinder.
93 3.3. VIBRATION ISOLATION DESIGN 65 Double wire suspended on a pivot, free to swing. Rocker mass, high moment of inertia. Viscous damping Pivot Figure 3.5: Diagram of one selfdamped pendulum stage. Magnets that generate eddy currents on copper plates create the viscous damping for a high moment of inertia rocker mass. Figure 3.6: Schematic of the intermediate mass showing the Euler Spring vertical stage and the attachment to the rocker mass [29]. The purpose is to closely replicate a real test mass and allows for the characterisation of the high performance vibration isolation chain developed at UWA using an optical cavity. Four niobium ribbons each of 25 µm thick, 3 mm wide and 300 mm long are used to suspend the test mass from the control mass stage. These are design to minimise internal modes providing at the same time a high Qfactor [30]. For the initial suspension we have used temporary brass pins clamped to the suspension ribbons through a high pressure contact tooth instead of the high pressure contact pins bonded to the end of the ribbons [31]. The ribbon clamping mechanism, as opposed to a permanent bond, is not ideal. However, the current design provides high enough contact pressure (approaching the yield strength of niobium) to minimise slip stick friction, whilst not weakening the suspension ribbon at the point of clamping.
94 66 CHAPTER 3. ADVANCED VIBRATION ISOLATOR Al web rigidly clamped to the suspension tube Magnets attached to rocker mass Euler springs Rocker mass Copper plates attached to Al web Rotational arm rests on Euler springs and is suspension point for next stage Bottom of springs is clamped to suspension tube Integrated 3D isolator stage Figure 3.7: Intermediate mass showing the integration of the high moment of inertia rocker mass with the Euler spring vertical stage. The intermediate mass has a hollow tube in the center to allow for the suspension wire to go through all the stages. The figure also shows the shaped frame that hold the copper plates on top of the rocker mass. Attached to the rocker mass are the magnets that create the damping through eddy current generation on the copper plates [20]. Even though the same pin design has been used for the initial brass pins extra thermal noise may be induced. Currently, priority lies in testing the performance of the vibration isolation system. Therefore a temporary brass/niobium suspension at the expense of a possible increase in thermal noise is acceptable. Figure 3.8 illustrates the control mass stage. A cage attached to the control mass provides both mechanical safety stops for the test mass, and a low noise reference from which actuation and local sensing of the test mass can be performed. The control mass is suspended from the vibration isolation system using a single wire. It also contains actuators and sensors from which 5 degrees of freedom (translation in all 3 dimensions, yaw and pitch) can be accessed [26] Integrated system The isolator structures for AIGO were designed to solve a range of problems in vibration isolation. This system relies on passive damping design, three stages of preisolation and the use of several novel isolation techniques to achieve nanometer
95 3.3. VIBRATION ISOLATION DESIGN 67 Actuation arms Control mass Suspension cage Test mass Niobium ribbons Figure 3.8: Control mass stage with test mass suspended. Actuation arms holding permanent magnets are attached to the control mass. From the control mass a suspension cage is attached and a test mass is suspended by four niobium ribbons. residual motion at low frequencies. The three preisolation stages incorporate an inverse pendulum as a first horizontal preisolation and a LaCoste linkage for vertical preisolation. Nested inside is a second stage of horizontal preisolation based on a Roberts linkage. This compact triple preisolation structure supports an isolation stack which consists of three low frequency threedimensional isolator stages combining selfdamped pendulums and Euler springs. At the bottom of the chain a control mass provides the interface with the test mass through niobium flexures. The local control system is used to compensate for movements induced by the seismic noise in the local reference system of a single suspension, in particular at the resonant frequencies of the suspension system. Position readout is done through large dynamic range shadow sensors. Feedback forces maintain position and alignment of the preisolation stages and the control mass through magnetic actuators attached to an inertial frame and referred to the ground. An optical lever was added outside of the vacuum system for test mass readout. The control system is also used to damp some normal modes through velocity feedback. Details of the local control system
96 68 CHAPTER 3. ADVANCED VIBRATION ISOLATOR and its performance can be found in an accompanying paper by Dumas et al [26]. 3.4 Experimental setup The experimental setup consists of two vibration isolation systems. Each one with a mirror as a test mass. A second isolator is necessary in order to have an inertial reference. In this way we create a 72 m optical cavity. Each system is inside a tank and under vacuum (10 6 mbar) interconnected through a 70 m long 400 mm diameter pipe. Both systems run their own local control system as described by Dumas et al [26]. The light source is a continuous wave 300 mw single frequency, nonplanar ring oscillator, Nd:YAG laser (λ = µm) modematched to the suspended cavity. The laser beam centring onto the input test mass (ITM) and the end test mass (ETM) is done with the aid of a CCD camera outside each station. The locking of the cavity is done using a standard PoundDreverHall technique. Neither frequency nor power stabilisation nor spatial modecleaner are used in this experiment. Figure 3.9: The picture shows the ETM during the assembly of the second suspension system. A 2 inches mirror is mounted in a stainless steel support that gives the test mass the same size of a real test mass. At the back of the ETM we can see the electrostatic board.
97 3.4. EXPERIMENTAL SETUP Cavity parameters At the bottom of the suspension system a test mass hangs from the control mass stage using niobium ribbons. An extra opticmodulator was added to the laser control loop in order to measure the cavity free spectral range (FSR) using the sideband locking method [32]. The FSR was measured at MHz which corresponds to a cavity length of m. Each of these test masses is made of a 50 mm fused silica mirror mounted in an aluminium and stainless steel mass. The purpose is to facilitate the mounting of the smaller mirror to the suspension system and give the payload the size of a real test mass. The ITM is flat and the ETM has a radius of curvature of 720 m. The parameters of the cavity formed by these two test masses are summarised in table 3.1. A picture of the ETM during assembly of the second suspension system can be seen in figure 3.9. ITM radius of curvature ETM radius of curvature 720 m Cavity g factor Waist size radius mm ITM spot size radius mm ETM spot size radius mm Waist position from ITM 0 m Free spectral range MHz Table 3.1: Parameters for the 72 m cavity In order to characterise the cavity we measured the intensity decay time of the transmitted beam. Figure 3.10 shows an average of the measurements of the cavity decay time with a value of τ c = 56.3 ± 0.2 µs. This value allows us to derive more parameters for this cavity as shown in table 3.2 [33] Suspension system transfer function The mechanical transfer function of the ITM suspension system was measured. The driving signal was injected at one of the actuators of the inverse pendulum. The
98 70 CHAPTER 3. ADVANCED VIBRATION ISOLATOR 2 Cavity Decay Amplitude (V) ( t) e time (ms) Figure 3.10: Average of the cavity decay time. Several measurements were done in order to determine an average decay time of 56.3 ± 0.2 µs. Cavity Parameters Formula Value Finesse 2π (c/l) τ c 732 Cavity bandwidth (2πτ c ) Hz Cavity Q factor 2πf o τ c Total losses L/ (cτ c ) 8585 ppm Reflectivity product (1 L/ (2cτ c )) Cavity pole 1/ (4πτ c ) 1413 Hz Table 3.2: Cavity parameters derived from our measurements. Here L corresponds to the round trip optical path length, f o the optical frequency, c the speed of light in vacuum, and τ c corresponds to the measured characteristic decay time of the intensity defined as I(t) = I 0 exp ( t/τ c ). response was measured at the control mass stage on one of the shadow sensors on the same axis. As a consequence the higher frequency pendulum modes are too small to be detected at the control mass shadow sensor. During the measurements there was no control of any of the stages of the vibration isolator. Figure 3.11 shows the measured mechanical transfer function of a complete vi
99 3.4. EXPERIMENTAL SETUP ITM frequency response 20 0 Magnitude (db) frequency (Hz) Figure 3.11: ITM horizontal mechanical transfer function. The transfer function was measured at the control mass level using the shadow sensor for sensing. The driving signal was injected at the magnetic actuator mounted on the inverse pendulum. bration isolator under vacuum. Two peaks at 58 mhz and 69 mhz correspond to the X and Y axis respectively of the first preisolation stage. This is followed by the Roberts linkage, which also has different frequencies for X and Y axis. The last peak before the rolloff corresponds to the main pendulum mode at around 500 mhz [20]. This was confirmed with separate measurements on each axis [34]. The mechanical transfer function has the characteristic features of a soft system, with high peaks at frequencies below 100 mhz. The laser control system will need high gain at low frequencies in order to follow the cavity length variations due to the low frequency displacement of the test masses Laser control system The isolation system can greatly reduce residual motion at high frequencies. However a laser control system is necessary for the low frequencies. Based on a couple of low noise amplifiers (Stanford Research Systems SR560) we design a simple control system for the laser as shown in figure A phasemodulator with a local oscillator of 10 MHz is used to generate the required sideband. The error signal is then obtained
100 72 CHAPTER 3. ADVANCED VIBRATION ISOLATOR by a photodetector reading the reflected signal which is synchronously demodulated at 10 MHz. This error signal is send to a low noise voltage preamplifier with a cutoff frequency of 3 Hz, a gain of 5, and a rolloff of 20 db/dec, which compensates for the pendulum modes above 1 Hz. The resonant frequencies of the preisolation stages are around 0.1 Hz. Therefore large displacements at low frequencies are generated by microseismic noise which could drive the system unstable. To control these low frequency oscillations a second SR560 is installed. This instrument is set as a second order (40 db/dec rolloff) low pass filter with a corner frequency of 0.3 Hz and a gain of 5 so as to avoid instabilities due to the phase lag induced by the filter around the corner frequency. Ideally matching the rolloff of both filters is necessary in order to obtain a smooth slope between the two filters and the optical cavity frequency response. However the gains at the SR560 instruments were limited to a maximum of 5 otherwise the second filter (cutoff 0.3 Hz) saturates. The combined signal is added in parallel to the second signal from the power splitter. The resulting signal goes through a high voltage amplifier with a gain of 40 before going into the PZT port of the laser. The voltage applied to the PZT (bond to the laser gain medium) stresses the laser medium to adjust the laser frequency. The PZT has a frequency response flat to about 100 khz, at this frequency the phase is already diverging from zero. The East Arm Cavity ~80m PM FI Laser Current Temp PZT Laser Controller Power Supply + LO IF HV Amp Gain MHz RF A LNA 3Hz 6 db/oct Gain 5 Inv Output LNA 0.3 Hz 12 db/oct Gain 5 (10) Figure 3.12: Diagram that shows the laser control system. A signal generator is used to generate the 10 MHz sideband used for cavity locking. PM corresponds to a phase modulator and FI a faraday isolator. The diagram does not include the optical components necessary to steer and modematch the beam into the main cavity.
101 3.4. EXPERIMENTAL SETUP Semitheoretical transfer function Magnitude (db) 50 0 Unity Gain: khz frequency (Hz) Phase (deg) Phase: 37.5 o frequency (Hz) Figure 3.13: The semitheoretical transfer function is a combination of the measurements of the electronics in the control loop and the optical cavity theoretical frequency response and the PZT frequency response. optical cavity can be modelled as a first order low pass filter with a cutoff frequency given by the cavity pole at 1.4 khz and a rolloff 20 db/dec. The resulting semitheoretical transfer function of this system can be seen in figure As expected the magnitude curve shows a 20 db/dec slope between 3 Hz and the cavity pole. However above the cavity pole the slope increases to a measured value of db/dec around unity gain; lower than the expected 40 db/dec, which could have led to an unstable system. This difference comes from the PZT frequency response and the Qfactor around its corner frequency effectively lifting the slope of the frequency response. The phase also diverge from the slope around 100 khz due to the phaselag introduced by the PZT response. Figure 3.14 shows the measurements of the laser control loop around the unity gain. The measurements were made using a spectrum analyser connected to point A in the control loop as shown in figure Using the swept sine mode we inject the source signal and add it to the error signal then measure the frequency response between the resulting signal and the input one (source + error signal). The figure shows a good agreement between the semitheoretical and the measurements of the
102 74 CHAPTER 3. ADVANCED VIBRATION ISOLATOR Measured frequency response 40 Magnitude (db) Unity Gain: khz Unity Gain: khz frequency (Hz) Phase (deg) Phase: 37.5 o Phase: 37.0 o frequency (Hz) Figure 3.14: Comparison between the semitheoretical curve and an average of a few measurements of the loop frequency response at high frequency. frequency response. The semitheoretical curve shows a unity gain around 14.7 khz and the average of the measurements a unity gain at 14.9 khz. Correspondingly the measured phase is in good agreement with the semitheoretical curve at the unity gain frequency with an average of 37.0 o with a semitheoretical value of 37.5 o. This phase value also shows the stability of the control loop. A gain of 150 was calculated in order to match the semitheoretical curve with the measured closed loop transfer function. This gain includes the cavity gain, the mixer and the PZT. These components were not included when measuring the electronics. 3.5 Measurements and results The experimental setup presented in the previous section was used to measure the performance of the advanced vibration isolation system. The main measurements were the residual motion of the test masses, including longitudinal motion as well as pitch and yaw measurements. Figure 3.15 shows the frequency response of the PZT driving signal. This signal corresponds to the contribution of the two vibration isolator systems to the longitudinal displacement. Around 70 mhz the contributions of the main preisolator frequency
103 3.5. MEASUREMENTS AND RESULTS 75 East arm displacement 10 0 Cavity displacement Laser noise 10 Magnitude (dbv rms / Hz) frequency (Hz) Figure 3.15: Measurement of the frequency response of the laser PZT signal. The top (blue) line shows the cavity frequency response and the bottom (red) line the laser noise. We notice that above 1 Hz the main contribution to the cavity displacement frequency response comes mainly from the laser noise. East arm residual motion (PZT) 106 rms displacement (m) PZT: 5.8e Hz frequency (Hz) Figure 3.16: Residual motion of the east arm cavity derived from the frequency response measurements. A residual motion of m can be seen at 1 Hz which is reduced to m just below 5 Hz.
104 76 CHAPTER 3. ADVANCED VIBRATION ISOLATOR can be seen in the figure. The following peaks correspond to the Roberts linkage resonant frequency (265 mhz) and the main pendulum frequencies at 640 mhz, and 950 mhz respectively. Figure 3.15 also shows the measured laser noise, which shows that above 1.2 Hz the main component of the measured cavity displacement signal corresponds to laser noise. This measurement allows us to calculate the cavity residual motion shown in figure As expected there is an increase in residual motion at the resonant frequencies. The residual motion of this cavity up to 1 Hz is m which is then reduced to m at just below 5 Hz. However this mainly corresponds to the laser noise contribution Pitch and Yaw angular residual motion Yaw: 4.62e Hz Yaw: 4.79e Hz Pitch: 6.74e Hz Pitch: 4.14e Hz (rad) frequency (Hz) Figure 3.17: Pitch and yaw angular residual motion for the ITM. The dotted lines show the angular residual motion for the control mass when controlled only with the shadow sensor. The continuous line shows the angular residual motion using the optical lever control loop. One of the main problems during the assembly and operation of the isolators and the main cavity was the pendulum mode of the niobium test mass suspension. Without the electrostatic control there were no sensing of the test mass it self. Therefore only the shadow sensor signal at the control mass stage was available for the signal readout. The signal to noise ratio at the shadow sensor turned out to be too poor to be able to detect pitch and yaw pendulum modes of the niobium ribbons at 3.3 Hz
105 3.6. CONCLUSIONS AND FUTURE WORK 77 and 1.75 Hz respectively, with particularly high Q in the yaw mode. The addition of an optical lever allowed for the measurement of these modes. This was installed outside the vacuum tanks in both isolators using a quad photodetector to feedback the signal to the control electronics and into the DSP. Due to the limited area of the quad photodetector the first stage of control of the test mass was done using the shadow sensor readout, which allowed for a larger range for the control and positioning of the test mass. Once the control signal was within the quad photodetector range the control of this last stage was handover from the shadow sensor control loop to the optical lever control loop with a much better signal to noise ratio. This allowed for the use of bandpass filters and damping loops for each pendulum mode as part of the optical lever PID control loop. More details are presented in the accompanying paper by Dumas et al [26]. Figure 3.17 shows a comparison between the residual motion of the pitch and yaw modes. The dotted lines show the optical lever control loop off whilst the continuous lines show the optical lever control loop on. Control loop off means that only the shadow sensor readout and its corresponding PID loop are being use for controlling the test mass position (pitch and yaw). The optical lever control loop on means that the shadow sensor control loop is turned off, but the level of the control signal at the time of switching from shadow sensor to optical lever control is used as an offset for the optical lever control loop. Figure 3.17 shows at 3.3 Hz a reduction of the pitch mode residual motion from 3.5 µrad to 1.9 µrad at the same frequency when the optical lever control loop is on. The yaw resonant mode at 1.75 Hz is reduced from 26 µrad to 2.5 µrad when the optical lever control loop is on. The angular residual motion obtained from these measurements is radians rms up to 100 mhz in yaw and radians rms up to 100 mhz in pitch. 3.6 Conclusions and future work We have shown for the first time that vibration isolators that combine multiple preisolation stages, selfdamping pendulums, Euler springs and niobium ribbons suspension have no unforeseen difficulties. Using a pair of advanced vibration isolators we
106 78 CHAPTER 3. ADVANCED VIBRATION ISOLATOR have shown that these systems have long term stability, and are responsive to the controls required to operate long optical cavities. Two vibration isolator systems were assembled 72 m apart as to form a suspended optical cavity. This was assembled and tested on the east arm of the AIGO research facility in Gingin. At the same time a local control system and its control electronics were developed for each of the vibration isolators. This allowed us to operate and control the suspended optical cavity. We have also demonstrated that without direct actuation at the test mass, it is possible to achieve a residual motion of 1 nm at 5 Hz for the suspended cavity. This includes the addition of an optical lever for improvement of the angular residual motion by actuation at the control mass and therefore no direct actuation on the test mass was required. Future work includes the development of an autoalignment system. Even though we were able to lock the cavity and operate for long periods of time, an autoalignment system will dramatically improve the duty cycle. This is part of the current development of a hierarchical global control scheme, which will allow for the operation of longer cavities with higher finesse. This will be assembled and tested at the AIGO research facility. Improvements to the local control system are also planned including the addition of an electrostatic control for the test mass. The addition of a reference cavity and/or a premodecleaner will reduce the laser noise at high frequencies, allowing isolator performance to be characterised above 10 Hz. However without a full interferometer it will be difficult to characterise the isolator response in the khz range. Acknowledgements The authors would like to thanks the technical staff at The University of Western Australia and Gingin for building each of the thousands of pieces that form the isolators, in particular, Steve Pople, Peter Wilkinson, Peter Hay and Daniel Stone. We would also like to thanks EuJeen Chin and Ben Lee for their collaboration. This work was supported by the Australian Research Council, and is part of the research program of the Australian Consortium for Interferometric Gravitational Astronomy.
107 3.7. LOCAL CONTROL Local control Compact vibration isolation and suspension for AIGO: Local control system J. C. Dumas, P. Barriga, C. Zhao, L. Ju, D. G. Blair School of Physics, The University of Western Australia, Crawley WA 6009, Australia High performance vibration isolators are required for ground based gravitational wave detectors. To attain very high performance at low frequencies we have developed multistage isolators for the proposed AIGO detector in Australia. New concepts in vibration isolation including self damping, Euler springs, LaCoste springs, Roberts Linkages, and double preisolation require novel sensors and actuators. Double preisolation enables internal feedback to be used to suppress low frequency seismic noise. Multidegree of freedom control systems are required to attain high performance. Here we describe the control components and control systems used to control all degrees of freedom. Feedback forces are injected at the preisolation stages and at the penultimate suspension stage. There is no direct actuation on test masses. A digital local control system hosted on a DSP (digital signal processor) maintains alignment and position, corrects drifts, and damps the low frequency linear and torsional modes without exciting the very high Qfactor test mass suspension. The control system maintains an optical cavity locked to a laser with a high duty cycle even in the absence of an autoalignment system. An accompanying paper presents the mechanics of the system, and the optical cavity used to determine isolation performance. A feedback method is presented which is expected to improve the residual motion at 1 Hz by more than one order of magnitude.
108 80 CHAPTER 3. ADVANCED VIBRATION ISOLATOR 3.8 Introduction In a companion article [35] we describe test mass vibration isolation and suspension systems developed for the proposed Australian International Gravitational Observatory (AIGO). The performance of an individual isolator system cannot be measured due to the lack of an inertial reference. For this reason a pair of isolators were configured to suspend mirrors for a 72 m optical cavity. The isolators were developed to satisfy the sensitivity requirements of advanced interferometric gravitational wave detectors which require test masses to be isolated from seismic noise at frequencies down to a few hertz. For a target sensitivity of m this typically requires a seismic attenuation of more than 10 orders of magnitude. In addition to high performance isolation within the detection bandwidth, it is critical to interferometer operation that the isolator provides minimal residual motion at low frequencies. This facilitates cavity locking and minimises noise injection through actuation forces. Pendulum systems inherently have large Qfactors, therefore it is often necessary to damp the normal modes of the suspension system. Such requirements can be addressed by a local control system. A feedback loop injects control forces at various actuation points on the isolator. These forces are derived by applying appropriate filters to error signals from local transducers such as position sensors. Typically the local control system is responsible for several tasks, each requiring different bandwidths and different filters. For example, in the frequency band DC to 10 mhz the control system is responsible for drift correction, positioning and alignment. In the frequency band up to 1 Hz the control system is mainly needed for damping normal mode peaks. For higher frequencies (Hz khz) the control system may be required to suppress high frequency noise, in the form of active vibration isolation. Despite being conceptually simple, the operation of local control systems is complicated by resonant modes and mode interaction between isolation components and different degrees of freedom. As a result, the feedback scheme often requires complex filter designs to avoid noise injection at critical eigenmodes that would interfere with the noise budget of the test mass. In the gravitational wave community there have been two broad approaches to the vibration isolation problem. The first, and most widely adopted, including this work,
109 3.8. INTRODUCTION 81 has been to create mostly passive vibration isolators based on massspring systems. To stabilise them a local control system is used to control or damp certain normal modes. The second approach is to invest heavily in very sensitive seismometers to measure the seismic noise, and then to use active feedback in a rather stiff system to actively suppress the measured motion. In the first case, one allows the system itself to provide an inertial reference. In the second, the inertial reference is provided by the test mass of the seismometer. In the design presented here we extend the idea of the system itself being the inertial sensor, by using a pair of very low frequency stages that are designed specifically to allow relative sensing and feedback, to provide an additional means of active suppression of very low frequency seismic noise. The VIRGO project [5, 6] and the TAMA project [36, 21] have used the first approach using multiple passive isolation stages, and relying on the control system mainly for damping and alignment. The GEO600 project [19] uses a combination of an active layer and several passive stages. The LIGO project [37, 38] first implemented an active preisolation stage to overcome problems of excess seismic noise. For the Advanced LIGO project they have constructed a system which combines stages of stiff active isolation with multiple pendulums [23]. The AIGO vibration isolator is conceptually similar to the VIRGO Superattenuator [39], but is more compact and has an extra stage of preisolation. The compactness is made possible through the use of novel isolation techniques with multiple preisolation stages [27]. Pendulum normal modes are passively damped through selfdamping [20]. Feedback is applied to the preisolation stages and the penultimate suspension stage to steer the mirrors and maintain alignment and positioning, while correcting for various sources of drift such as temperature fluctuations. Low frequency control loops also damp fundamental modes of the preisolation stages and angular modes (yaw) of the suspension chain to minimise low frequency residual noise dominated by the eigenmodes of the isolation chain. These low frequency resonances can otherwise have large amplitudes as they are located close to the microseismic peak of the seismic background. Therefore careful consideration has to be taken when designing the feedback loop in order to avoid exciting the test mass suspension fundamental mode, which by design has extremely low loss and hence low Brownian motion. To
110 82 CHAPTER 3. ADVANCED VIBRATION ISOLATOR facilitate this, the pitch and yaw of the test mass are monitored directly through an optical lever. Control forces are applied indirectly to through a control mass from which the test mass is suspended, in similar fashion to the VIRGO marionetta [40]. In order to provide a flexible platform capable of satisfying the various control requirements, a digital control system was implemented on a Sheldon Instrument DSP board [41], running on a PCI bus. Each vibration isolator is integrated with an independent digital control system. The two only differ in minor adjustments of gains and corner frequencies to match small differences in the mechanical modes of the isolation stack. Two vibration isolators were installed at the AIGO test facility in Gingin, Western Australia. The systems were installed in the East arm of the vacuum envelope to form a 72 m suspended cavity. A Nd:YAG laser was locked to the cavity using the PoundDreverHall [42] technique. In a companion article [35] we discuss the vibration isolator design and the low frequency performance as determined from the error signal of the optical cavity. This article presents the control architecture and systems to enable the potential performance of the isolators to be realised. In section 3.9 we describe the integration of the isolator components with specially developed sensor and actuator components including high dynamic range shadow sensors, high force magnetic actuators and ohmic position control systems. The control scheme is presented in section 3.10, where we also show how the novel dual preisolation approach is expected to allow major improvements in performance through use of so called superspring techniques. Finally in section 3.11 we review cavity locking results which confirm the performance of the control system, and demonstrate the capability of the system for use in interferometric gravitational wave detectors. 3.9 Experimental setup Isolator components The AIGO vibration isolation design is discussed in detail in the companion article [35], and only a brief review is presented here. It consists of 9 cascaded stages
111 3.9. EXPERIMENTAL SETUP 83 as illustrated in figure 3.18, including 3 stages of preisolation in a compact nested structure from which is suspended a triple selfdamped pendulum. A vertical stage based on Euler springs is colocated with each of the 3 pendulums. AntiSpring geometries are implemented into various stages to reduce fundamental mode frequencies [29, 43, 27]. Inverse pendulum Roberts Linkage a d c b e LaCoste stage Euler stage SelfDamped pendulums Control mass Test mass Figure 3.18: Isolation stages of the AIGO suspension chain. The preisolation stages include a, b and c. The isolation stack is defined as the three identical stages of selfdamped pendulums with Euler stages; (a) Inverse pendulum preisolator [4] (b) LaCoste Linkage [4] (c) Roberts Linkage [27] (d) Euler springs [29] (e) Selfdamped pendulums [20]. The preisolator consists of several Ultra Low Frequency (ULF) stages; the Inverse Pendulum, the LaCoste linkage [4] and the Roberts linkage [12], each with their resonant frequencies in the order of 100 mhz. The inverse pendulum stage can be tuned close to 0.05 Hz to provide low frequency horizontal preisolation. It has a large dynamic range with ±10 mm in all directions, which can be used to buffer
112 84 CHAPTER 3. ADVANCED VIBRATION ISOLATOR temperature drifts and tidal effects. The inverse pendulum requires minimal force to displace it and is therefore well adapted to actuation [44]. Vertical preisolation is provided by the LaCoste linkage which is a combination of negative springs to null out flexure stiffness and zero length coil springs to support a 250 kg load. The LaCoste linkage, like the inverse pendulum, has a large dynamic range and can be tuned below 0.05 Hz. The Roberts linkage provides the second horizontal preisolation stage [27]. The combination of coil and magnet provides the actuation for the positioning and damping for both the inverse pendulum and the LaCoste stages. Additional heating of the coil springs at the LaCoste stage provides a means of compensating for slow temperature drifts in the vertical direction as well as correcting for creep in the isolation chain. Position control at the Roberts linkage stage is done through heating of the individual wires. A low frequency isolation stack is suspended from the Roberts linkage consisting of three almost identical stages (see figure 3.18). A 40 kg mass load is suspended from each stage in a selfdamped pendulum arrangement [20] and each is combined with an Euler spring for vertical isolation [29]. The test mass is suspended from a control mass which can be actuated in pitch, yaw, and horizontal translation. The suspension design uses four Niobium ribbons [31] to form a low loss suspension with pendulum Qfactor The control mass itself is suspended from the isolation system by a single suspension wire. All optical cavity testing has been done without direct sensing or actuation on the final test mass of the system, beside the use of optical levers. An integrated electrostatic actuator/rf sensor has been developed [45] as a final stage of low level control. This will be implemented when the vibration isolators are used in a full interferometer configuration Control hardware Shadow sensors The position of several stages of the isolation stack is monitored by the local control system through optical shadow sensors. A shadow sensor as illustrated in figure 3.19 is a simple device consisting of an LED shining an infrared beam onto two photodiodes 40 mm away, each photodiode is 10 mm in length. A long and thin flag of the same
113 3.9. EXPERIMENTAL SETUP 85 width (10 mm), is attached to the stage to be monitored. It is positioned perpendicularly to the two sensors, such that the flag forms a shadow falling roughly equally on both photodiodes. The resulting current from each photodiode is approximately proportional to the area illuminated, and as the flag is displaced across the sensor the difference of the two photodiode signals forms a linear response. Each photodiode signal is amplified and converted to a signal in the ±10 V range for the ADC module of the DSP board to be read by the digital control system. The shadow sensor has a relatively large dynamic range of ±5 mm, and typical sensitivity of m/ Hz [4]. Infrared LED Shadow card Photodiodes Figure 3.19: The shadow sensor is a simple device, where an LED shines a beam onto two photodiodes, and an intermediate shadow mask is attached to the part to be measured. Magnetcoil actuator Magneticcoil actuators are used to control several stages as described in section Each actuator consists of a pair of coils assembled together as illustrated in figure Two designs of magnetic actuators are used in the isolator. Large actuators are used on the preisolation stage, for position control, drift correction, and damping ULF normal modes. This design uses coils with 1600 turns of 0.25 mm wire to form a coil diameter mm, with a resistance of 115 Ω. A mm permanent neodymium boride magnet is used to result in a force of 160 mn with a current of 100 ma driving the actuator (50 ma each coil, connected in parallel). A smaller actuator design is used at the control mass, with a coil diameter of mm, made of 600 turns of 0.25 mm wire. These coils have a resistance of 37 Ω, and are paired
114 86 CHAPTER 3. ADVANCED VIBRATION ISOLATOR with a mm magnet. The magnetic field within the actuator coils is nearly uniform within 1% in the central 10 mm of its range, allowing a large dynamic range in the control system. Coil Coil Magnet Figure 3.20: The magnetcoil actuator. A magnet mounted on an isolation stage is placed in the centner of two coils that are mounted on the support frame. Wire heating In addition to the magnetic actuators, some stages are controlled by passing a current through particular suspension elements. The elements warm up and lengthen through thermal expansion. This control method is relatively effective when the system is under vacuum, as heat does not dissipate through convection, but only by the relatively slow processes of radiation and conduction. The advantage of this method is that it removes the complication of added parts, and in some cases provides much greater dynamic range. This is used to correct for large drifts caused by daily and seasonal ambient temperature changes. This actuation method responds with a quadratic relationship, but it is linearised in the digital feedback loop. The horizontal control of the Roberts linkage, and the vertical control of the LaCoste linkage, both employ this strategy. The four suspension wires of the Roberts linkage are individually wired to current power supplies, allowing the length control of each of them by thermal expansion as they warm up. Since the Roberts linkage is very sensitive to any change of tension in any of the wires due to it s carefully tuned folded configuration, a relatively small
115 3.9. EXPERIMENTAL SETUP 87 change of length is enough to control the stage through it s entire dynamic range 10 mm. The position of the Roberts linkage is controlled via the circulating current which in turn is controlled by the local control system through integral feedback to the current power supplies. The LaCoste linkage has a large dynamic range and can be controlled through its entirety by magnetic actuators at a fixed ambient temperature. However daily and seasonal temperature fluctuation cause drifts that would far exceed the capacity of the actuators. A 1 o C change will offset the balance point of the LaCoste linkage by it s entire range of 10 mm. For this reason, wire heating is an essential part of the LaCoste control loop. The coil springs of the LaCoste linkage are electrically connected in series, and can be heated to change the spring constant of the springs, which greatly affects the vertical position or balance point of the stage. Low frequency control (DC 10mHz) of this stage is achieved by regulating this current, while the magnetic actuators are used at higher frequencies ( 100s mhz) Control implementation and degrees of freedom The inverse pendulum can be monitored and actuated in 3 Degrees of Freedom (DoF), two in the horizontal plane (X and Y ), and one angular (yaw φ), i.e. the rotation about the vertical axis. These are sensed and actuated through 4 shadow sensors and 4 actuators that are colocated on the inverse pendulum frame as shown in figure The four signals are diagonalised into the 3 DoF X,Y and φ, and each is controlled independently as a Single Input Single Output (SISO) feedback loop. The LaCoste stage is a purely 1dimensional vertical stage (DoF: Z). Two actuators are mounted on opposing sides as illustrated in figure 3.22 and figure They are used to damp the ULF normal mode of the stage. In addition, the LaCoste linkage can be controlled by heating the coil springs on all four sides of the stage, which are all connected to a high current supply. A shadow sensor mounted on the side of the preisolation structure monitors the vertical position of the LaCoste linkage. The Roberts linkage in figure 3.23 is controlled in 2 DoF, X and Y, by passing a current through its suspension wires. Each of the four suspension wires are electrically isolated from the rest of the vibration isolator structure and independently connected
116 88 CHAPTER 3. ADVANCED VIBRATION ISOLATOR Horizontal actuator Y X Shadow sensor LaCoste frame Inverse pendulum Vertical actuator (LaCoste frame) Figure 3.21: The inverse pendulum is controlled through shadow sensors and magnetic actuators. Shadow sensor Z X Vertical actuator Heated suspension coil spring Inverse Pendulum Figure 3.22: The LaCoste stage is controlled through a shadow sensor and magnetic actuator as well as the heating of the suspension coil spring. to a high current power supply. By controlling the circulating current on each wire it is possible to control its length, and therefore the position. Since the suspension system is under vacuum the heat loss by convection is minimal. The X and Y signals from the control mass shadow sensors are used to feedback to this Roberts linkage actuation method. This control system provides a low frequency correction of any drift in the Roberts linkage and ultimately of the multistage pendulum and the test mass. The control mass can be controlled in 5 DoF, three orthogonal translations X, Y, Z, the rotation about the vertical axis, yaw (φ) as shown in figure 3.24, and the rotation about the horizontal axis perpendicular to the laser axis, pitch (θ) shown
117 3.9. EXPERIMENTAL SETUP 89 Y Heated suspension wire X LaCoste Frame Figure 3.23: The Roberts linkage is controlled through shadow sensors and the heating of the four suspension wires. in figure Three horizontal actuators and shadow sensors are colocated in a 120 o arrangement on the horizontal plane as seen in figure 3.24, while two vertical shadow sensors and actuators are colocated on opposing sides of control mass along the laser axis as in figure The signals are digitalised by a sensing matrix into five orthogonal DoF, X, Y, Z, φ and θ. These are treated by five separate control loops as independent SISO systems, before the signals are recombined by a driving matrix into the appropriate actuator signals. 120 Shadow sensor Actuator Y X Figure 3.24: The control mass has three actuators and shadow sensors collocated on the horizontal plane, in a 120 o arrangement. These three signals are converted to an orthogonal reference frame X, Y, Z and φ by a sensing matrix.
118 90 CHAPTER 3. ADVANCED VIBRATION ISOLATOR Shadow sensor Z Actuator Y Figure 3.25: The pitch of the control mass is actuated by two vertical magnetic actuators Optical lever Due to poor coupling of the mirror suspension angular modes to the control mass, it is necessary to have a direct readout of the mirror angular orientation. This was achieved by a simple optical lever as illustrated in figure A laser outside the vacuum envelope is reflected off the test mass and is measured by a quadrant photodiode, also outside the vacuum envelope. In addition to being a direct measurement from the mirror surface the optical lever provides better sensitivity to angular motion as it is placed further away from the centre of rotation of the mirror, such that the same angular rotation corresponds to a much larger arclength measured by the quadrant photodiode 5 m away from the mirror, than the shadow sensors which is only 200 mm away. The drawback is the limited dynamic range, it provides 1 mrad which is greatly exceeded by the test mass suspension oscillation when it is excited. Therefore the shadow sensor feedback is used for initial damping of the angular modes, before the optical lever signal can be used for feedback, as discussed in section The digital controller The control system is hosted by a Sheldon Instrument DSP board, forming a flexible multidimensional digital control platform. The board is a SIC33DSP on a PCI bus, based on a 150 MHz Texas Instruments TMS320VC33 DSP using a mezzanine board SIMOD6800 to provide 32 input channels (16 bit ADC), 16 output channels (16 bit
119 3.9. EXPERIMENTAL SETUP 91 Vacuum envelope Shadow sensors Quadrant photodetector 200 mm Test mass 5 m Laser Figure 3.26: vacuum envelope. The optical lever setup, using a quadrant photodiode placed outside the DAC), and digital input/outputs [41]. The input channels are used as described in table 3.3. Most input channels are used for the shadow sensors which require two inputs each, one per photodiode. Two more inputs are used for the vertical and horizontal axis readout of the quadrant photodiode used with the optical lever, and two more are wired to auxiliary connectors to inject any arbitrary analogue signal. The output channels are used for the control signals of actuators and current power supplies. Stage inputs outputs Inverse Pendulum 8 4 LaCoste Stage 2 2 Roberts Linkage 4 4 Control Mass 10 5 Optical Lever 2 0 Auxiliary 2 1 Total: Table 3.3: I/O channel allocation usage of DSP An intermediate analogue system amplifies and filters the input and output chan
120 92 CHAPTER 3. ADVANCED VIBRATION ISOLATOR nels between the DSP and the control components (shadow sensors, wire heating and actuators) with the exception of the quadrant photodetector used in the optical lever which is integrated on a board with preamplification. The analogue electronics consists of 13 boards in a standard 6U rack, each board contains a dual photodetector circuit for the pair of photodiodes in one shadow sensor, and a control signal circuit to drive an actuator. The dual photodetector circuit contains a transimpedance amplifier, antialiasing filters and an amplifier. The signal of both photodiodes is then distributed to two inputs on the DSP board. The control signal is distributed from one DSP output to the corresponding channel on the control circuit, which contains antialiasing filters and a high speed current amplifier before distribution to the actuator coils. An additional board in the 6U rack contains five filter circuits for the wire heating control signals. These five signals are then distributed to five external voltage controlled current supplies. The control scheme, algorithms, and the user interface, are written and operated in LabView R. The builtin libraries provided by the DSP manufacturer allows for the control loops to run on the DSP board in real time including ADC and DAC at a 100 Hz sampling rate. The user interface is ran on a host PC to monitor every stage of the isolation chain and adjust control parameters, such as filters and loop gains as necessary Control scheme The control scheme has three main purposes. One is to maintain alignment and positioning for all stages, against drifts such as caused by ambient temperature changes, or tidal effects. These effects are extremely low frequency, with timescales from tens of minutes to days. Therefore the control strategy consists of low gain integration feedback. The control system has also to maintain the test mass alignment to obtain a resonant cavity. The alignment of the test mass is controlled indirectly via pitch and yaw of the control mass stage. However sensing is done with the control mass shadow sensors and directly from the test mass trough the optical lever. The control is achieved by both proportional and integration control while aligning the cavity,
121 3.10. CONTROL SCHEME 93 and integration only when maintaining the cavity locked. While the isolation design relies on passive isolation, some active damping is required for some ULF resonant modes of the preisolation, as well as low frequency torsional modes (yaw) of the entire chain. Additionally, the two normal modes of the test mass suspension (pitch and yaw) must be damped at least initially after any alignment or positioning offset. The extremely high Qfactor of the Niobium suspension would otherwise result in several days of oscillations after any large perturbation. While controlling the alignment the control system must also avoid driving the suspension resonant modes, this is done through carefully placed filters in the feedback gain. Figure 3.27: Block diagram of the isolation local control system. The signals from shadow sensors and a quadrant photodiode are used to feedback to several stages using magnetic actuators or high current heating. At each stage the sensor signals are converted into orthogonal DoF by a sensing matrix, such that the control system consists of independent SISO systems, which simplifies the control strategy and requirements over a MIMO system (Multiple Input Multiple Output). The various DoF of each stage relevant to the control system, as
122 94 CHAPTER 3. ADVANCED VIBRATION ISOLATOR discussed in section 3.9.3, can be summarised in table 3.4. Figure 3.27 illustrate the physical location for sensing and actuation of the feedback loops. Stage Inverse Pendulum LaCoste Stage Roberts Linkage Control Mass Optical Lever DoF X, Y, φ Z X, Y X, Y, Z, φ, θ φ, θ Table 3.4: Stages and relevant degrees of freedom in the control scheme Preisolation feedback Damping of the yaw (φ) mode of the inverse pendulum is done via velocity feedback (F (s) = (φ t IP φ ground )) in a bandpass filter 0.3 Hz<f<0.7 Hz. But the two horizontal modes require some consideration to maintain stability. Figure 3.28 shows the loop gain with feedback loop gain using a relatively high damping gain and a 2 nd order low pass filter at 0.7 Hz. Note that lowering the corner frequency of the low pass filter would lead to instability as can be seen from the small gain margin. In order to damp the large resonance at 70 mhz we apply a damping gain using the inverse pendulum shadow sensor output (which is a measurement of x 1 x 0, as illustrated in figure 3.31). This control method is being replaced by the scheme described in section but it was used for initial testing of the cavity. In principle it should also be possible to damp the second resonant mode (cause by the Roberts linkage) with phase compensation at the appropriate frequency, but this is difficult to implement effectively while maintaining stability due to the small frequency separation of the two preisolation stages. However damping of the Roberts linkage can be done with the method described in section The LaCoste Stage feedback method in the vertical direction Z is divided between the actuator and the coil heating. The shadow sensor signal is fedback to the actuator with a damping gain and a low pass filter at 0.7 Hz. The vertical signal from the
123 3.10. CONTROL SCHEME 95 (a) Magnitude (db) Loop gain Closed loop TF Frequency 100 Phase Frequency (b) Figure 3.28: H(s) = Loop gain G(s) = IP (s)s(s)c(s)a(s) and closed loop transfer function A(s)IP (s)s(s) 1+A(s)IP (s)s(s)c(s) of Inverse Pendulum horizontal DoF. The diagonalised signal from the inverse pendulum shadow sensors is fedback at the inverse pendulum actuators (x 1 and F 1 in figure 3.31 respectively). The digital compensator C(s) t (X ip X ground )LP F with a low pass filter at 0.7 Hz. control mass is also fedback to the coil heating supply with a low integration gain, after being linearised by taking the square root of the resulting control signal. The coil heating method allows to correct for a large range of ambient temperature which would be impossible with the actuators alone, while the fast response of the magnetic coils is used to damp the ULF normal mode. It is possible to also use the control mass vertical signal for damping feedback to the LaCoste actuators, but this is not currently implemented to avoid coupling to the control mass pitch, pending a thorough investigation of the sensing matrix digitalisation and coupling of the DOFs.
124 96 CHAPTER 3. ADVANCED VIBRATION ISOLATOR The Roberts linkage control merely consists of position feedback to the heating current supplies, with a low gain integral loop, to the two axes, X and Y, at frequencies below 10 mhz. Position sensing of the control mass is fedback to the position control of the Roberts linkage, to minimise forces injected at the control mass Control mass feedback The control mass horizontal translation X and Y, is fedback to the preisolation stages with a low integration gain, to minimise noise injection at the control mass. Pitch (θ) and yaw (φ) are controlled to maintain alignment of the optical cavity, as well as to damp the pitch and yaw resonant modes of the control mass. The control mass is an order of magnitude heavier than the test mass, hence the suspension resonances are weakly coupled to the control mass. This results in a poor signal to noise ratio at the suspension resonances. An optical lever is therefore used to monitor the pitch and yaw directly from the mirror. The yaw resonance is damped with the shadow sensor control otherwise its amplitude is too large for the range of the optical lever 1 mrad. Although the oscillation couples weakly to the control mass, there is sufficient signal to noise ratio to damp the test mass into a range where the optical lever becomes operable, then a much better level of control can be achieved ( µrad). Low frequency resonances caused by the torsion of the suspension wires dominate the residual angular motion with either control method. The optical lever achieves better performance due to higher angular sensitivity, and decoupling from translation motion (X, Y, Z). If the optical lever goes out of range, either due to a large offset, or large amplitude in the suspension normal mode, the control system will damp the pitch and yaw modes with a strong velocity feedback gain. Integral and proportional feedback are used in the loop for cavity alignment, it also includes bandpass filters to damp the suspension modes. In particular, the phase of the control signal at the suspension resonances must be reversed, since the control mass coupled oscillation is out of phase to the test mass oscillation. Once the optical lever is in range, the damping feedback automatically changes the source of its error signal to the optical lever signal (quadrant photodiode). This process is automated by the control system though a set of boolean operations
125 3.10. CONTROL SCHEME 97 to determine the state of the system according to threshold values on the range of the optical lever and shadow sensors. The transition from one loop to the other is smooth in the sense that the DC force resulting from integral and proportional setpoints are passed on from one loop to the other. Since the pitch and yaw damping feedback operate at the highest bandwidth of the control scheme, they are the most affected by the phase lag due to the 100 Hz sampling rate. At the pitch resonance of 3.3 Hz, the minimum possible phase lag due to the 100 Hz operation of the ADC/DAC, is approximately 12 o, at the yaw resonance of 1.75 Hz it is 6 o. Figure 3.29 shows a comparison between the frequency responses of the pitch and yaw modes with the optical lever control loop off (dotted line) and optical lever control loop on (continuous line). Optical lever control loop off means that only the shadow sensor readout and its corresponding control loop are being use for controlling the test mass position. The optical lever control loop on means that the shadow sensors control loop is turned off, but the level of the control signal at the time of switching the loop is used as an offset for the optical lever control loop. Figure 3.29 shows a pitch mode reduction of 20 db at 3.3 Hz when using the optical lever control loop. The reduction on the yaw mode is more significant with about 40 db at 1.75 Hz. Figure 3.30 shows the yaw angular motion of the test mass in time using either shadow sensors or optical lever feedback. The setpoint has been removed so as to centre the curves on zero removing the offset introduce by the control loop setpoint. The total yaw displacement is dominated by the torsional mode of the suspension wires, causing very low frequency yaw oscillations at 20 mhz Optimised feedback for preisolation A feedback scheme was devised based on the superspring concept which takes advantage of the dual stage of horizontal preisolation in our design. This is an active control method consisting of feeding back the position of the loaded end of a massspring system to the spring suspension, and keep their relative distance constant in order to synthesise a verylow frequency system [8, 46, 47, 48]. The concept can be equally applied to pendulum systems. Shadow sensors mounted between the inverse
126 98 CHAPTER 3. ADVANCED VIBRATION ISOLATOR (a) 10 3 Control mass Pitch frequency response SS OL Magnitude (rad/ö Hz) frequency (Hz) (b) 10 3 Control Mass Yaw frequency response SS OL Magnitude (rad/ö Hz) Frequency (Hz) Figure 3.29: Measurement of the frequency response of the test mass in it s two angular DoF (the resonant modes are highlighted). (a) pitch θ with test mass suspension mode at 3.3 Hz. Note that the broad peak at 280 mhz is due to the rocking mode of the control mass. (b) yaw φ with suspension mode at 1.75 Hz. The dotted line shows the measurements with the optical lever feedback off, using only the shadow sensor signal for feedback. The solid line shows the measurements with the optical lever control loop on. The optical lever was used for the measurement of both curves. pendulum frame and the Roberts linkage as shown in section could be used for such a purpose. These sensors provide a signal (x 2 x 1 ) as illustrated in figure Feedback to the inverse pendulum actuators would permit to lower the effective in
127 3.10. CONTROL SCHEME Yaw µ rad SS OL time (sec) Figure 3.30: Test mass yaw angular motion using different feedback loop. The optical lever feedback greatly improves the limit imposed by the low signal to noise ration of the shadow sensor when sensing the suspension normal modes. verse pendulum resonance. In addition, velocity feedback (x t 2 x 1 ) would also allow effective damping of the Roberts linkage resonant mode and simplify stability considerations compared to feedback of the inverse pendulum position. Modelling of this control method shows that the low frequency isolation below a few Hertz can be improved by an order of magnitude compared to that described in section A comparison of the theoretical performance with the initial setup is shown in section Figure 3.31: A simple 2 pendulum system illustrating the preisolation feedback using shadow sensors. The inverse pendulum position is referenced to the ground, for low frequency position control ( (x 1 x 0 )dt. The Roberts linkage is referenced to the inverse pendulum, and can be feedback to lower the first resonant mode, and damp the second ( (x 2 x 1 ) + t (x 2 x 1 ).
128 100 CHAPTER 3. ADVANCED VIBRATION ISOLATOR 3.11 Initial cavity measurements Preliminary performance results of the cavity displacement noise have been obtained from initial trials of cavity locking. The locking scheme employed in these runs where as described in section 3.10, with simple damping feedback at the inverse pendulum stage, and did not yet implement the superspring feedback concept in the preisolation stage. A standard PoundDreverHall feedback system was implemented to keep the laser locked to the 72 m cavity. The companion paper [35] describes the optical scheme and laser feedback in more details. The integrated residual motion per test mass can be calculated from the control signal and it is shown in figure Here we see that above 1 Hz, the residual motion is 3 nm per test mass. The measured curves compares closely with the predicted performance for the simple feedback at the inverse pendulum. Note that the theoretical performance without any control is an order of magnitude lower at the same frequency, however the large resonance of the preisolation mode at 70 mhz is too large to practically lock the optical cavity. The inverse pendulum feedback loop permits to damp the normal mode, at the sacrifice of high frequency noise injection. This simple feedback is also ineffective in damping the second resonant mode of the two body system formed by the inverse pendulum and Roberts linkage. An optimised feedback system was devised using the superspring concept as discussed in section , which could achieve an acceptable residual motion at low frequencies without compromising the isolation performance. This scheme which requires additional sensors on the Roberts linkage stage, has been tested on a single isolator, and will be implemented and tested in the cavity in the near future. The stability of the cavity, and in particular the angular noise of the test mass, can be demonstrated by a long term record of the power inside the cavity. This was achieved by measuring the transmitted light at the end test mass (ETM). In figure 3.33 we plot a histogram of measured power over two hours. As an autoalignment scheme has not been implemented at this initial stage, continuous lock could not be achieved over long periods (24 hours).
129 3.12. CONCLUSIONS 101 Residual motion (m) (d) (c) Cavity residual motion Measured performance Model with IP feedback Model without control Model with RL feedback (a) (b) Frequency (Hz) Figure 3.32: (a) The measured integrated residual motion of the cavity (x r ms = f xf 2 df). It is at the nanometre level above 1 Hz. Note that the measurement is limited by laser noise above 2 Hz [35], due to the free running laser. (b) A model with the same feedback scheme as used for the measurement. (c) The modelled performance with an optimised preisolation feedback scheme, using the superspring concept. (d) A modelled performance if no feedback was implemented Conclusions A local control system was implemented in a novel isolation and suspension design for laser interferometer gravitational wave detectors. The system provides feedback for position control, cavity alignment, and damping of normal modes. Three DoF are controlled by ohmic thermal tuning of the length of pendulum wires in the Roberts linkage and by thermal spring constant of the LaCoste linkage. Large dynamic range shadow sensors and actuators allow more than ±5 mm three dimensional position control of the test mass. Two suspensions systems and their associated control systems were installed to form a 72 m optical cavity. Without using direct test mass control, it was possible to lock the cavity and maintain lock, with a residual motion of 3 nm per test mass above 1 Hz. Low frequency residual motion at microseismic frequencies is expected to improve by over an order of magnitude once the second horizontal preisolation stage is used for feedback. Angular control of the test mass is
130 102 CHAPTER 3. ADVANCED VIBRATION ISOLATOR 3 x 104 Histogram of transmitted power count Amplitude (V) Figure 3.33: The power inside the cavity is represented in a histogram over a period of two hours. Note that the Xaxis is of arbitrary unit: the voltage of the photo detector measuring transmitted light. aided by an optical lever, with an automatic transition between local sensors and the optical lever. Additionally both sensing and direct actuation of the test mass will be possible by a capacitive plate mounted on the control cage. The cavity demonstrates long term stability, indicating that there are no unexpected noise sources or drifts in the system. Acknowledgments We would like to acknowledge the Australian Research Council for their support of this work. We also thank David Ottaway for his helpful discussions. This project is part of the research program of the Australian Consortium for Interferometric Gravitational Astronomy.
131 3.13. POSTSCRIPT Postscript A thorough study of the optical lever was undertaken during measurements of vibration isolator performance described in the presented paper. This section includes additional information removed from the paper due to page limits recommended by Review of Scientific Instruments journal. Figure 3.34 shows the difference between the test mass control using only the shadow sensors and the reduction on the test mass angular displacement when using the optical lever. For these graphs the setpoint on each axis has been removed to centre the curves on zero, removing the offset introduced by the control loop setpoint. The distribution of the test mass position in time shows a Gaussian distribution from which we can determine the standard deviation for each case. Therefore the yaw pendulum mode when using the shadow sensor has a standard deviation of µrad, which is reduced to 35.3 µrad when using the optical lever. This also improves the median value and improves the precision and the accuracy when reaching the setpoint. Figure 3.35 (b) shows the reduction in pitch mode to be much smaller, with a stan µ rad Yaw 400 SS 600 OL time (sec) Pitch µ rad 050 SS 100 OL time (sec) Figure 3.34: Test mass angular motion when controlled using only the shadow sensors; and when controlled using the optical lever. We can clearly see the difference in the yaw mode, however there is a much smaller difference in the pitch mode.
132 104 CHAPTER 3. ADVANCED VIBRATION ISOLATOR dard deviation of 60.1 µrad when using the shadow sensor and reduced to 57.8 µrad with the optical lever. However there is an improvement in the median value. (a) 2500 Shadow sensor angular motion distribution: Yaw Shadow Sensor Optical Lever Count Angle (µ rad) (b) Shadow sensor angular motion distribution: Pitch Shadow Sensor Optical Lever 4000 Count Angle (µ rad) Figure 3.35: Histograms showing the gaussian distribution of the angular motion for the ITM in pitch and yaw. (a) shows a broad distribution when driving the control mass with the signal from the shadow sensors and a much narrow distribution when using the optical lever signal. (b) also shows the difference between controlling the control mass with the shadow sensor signal and the optical lever signal, however the difference is much smaller.
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136 108 CHAPTER 3. ADVANCED VIBRATION ISOLATOR [32] A. Araya, S. Telada, K. Tochikubo, et al, Absolutelength determination of a longbaseline FabryPerot cavity by means of resonating modulation sidebands, Appl. Opt. 38 (1999) [33] D. Z. Anderson, J. C. Frisch, and C. Masser, Mirror reflectometer based on optical cavity decay time, Appl. Opt. 23 (1983) [34] J. C. Dumas, K. T. Lee, J. Winterflood, et al, Selfdamping pendulums in vibration isolators, Phys. Lett. A 2009, submitted. [35] P. Barriga, J. C. Dumas, A. Woolley, et al, Compact vibration isolation and suspension for AIGO: Performance in a 72 m cavity, Rev. Sci. Instrum. 2009, submitted. [36] R. Takahashi, K. Arai, and the TAMA Collaboration, Improvement of the vibration isolation system for TAMA300, Class. Quantum Grav. 19 (2002) [37] R. Abbott, R. Adhikari, G. Allen, et al, Seismic isolation for Advanced LIGO, Class. Quantum Grav. 19 (2002) [38] R. Abbott, R. Adhikari, G. Allen, et al, Seismic isolation enhancements for initial and Advanced LIGO, Class. Quantum Grav. 21 (2004) S915 S921. [39] S. Braccini, L. Barsotti, C. Bradaschia, et al, Measurement of the seismic attenuation performance of the VIRGO superattenuator, Astropart. Phys. 23 (2005) [40] G. Ballardin, S. Braccini, C. Bradaschia, et al, Measurement of the transfer function of the steering filter of the VIRGO super attenuator suspension, Rev. Sci. Instrum. 72 (2001) [41] [42] E. D. Black, An introduction to Pound Drever Hall laser frequency stabilization, Am. J. Phys. 69 (2001)
137 3.14. REFERENCES 109 [43] J. Winterflood, Z. B. Zhou, and D. G. Blair, Reducing lowfrequency residual motion in vibration isolation to the nanometre level, AIP 523 (2000) [44] C. Y. Lee, C. Zhao, E. J. Chin, et al, Control of preisolators for gravitational wave detection, Class. Quantum Grav. 21 (2004) S1015 S1022. [45] B. H. Lee, Advanced Test Mass Suspensions and Electrostatic Control for AIGO, PhD Thesis, School of Physics, University of Western Australia, Chapter 6, [46] N. A. Robertson, R. W. P. Drever, I. Kerr, and J. Hough, Passive and active seismic isolation for gravitational radiation detectors and other instruments, J. Phys. E: Sci. Instrum. 15 (1982) [47] J. P. Schwarz, D. S. Robertson, T. M. Niebauer, and J. E. Faller, A freefall determination of the newtonian constant of gravity., Science 282 (1998) [48] J. M. Hensley, A. Peters, and S. Chu, Active low frequency vertical vibration isolation, Rev. Sci. Instrum. 70 (1999)
138 110 CHAPTER 3. ADVANCED VIBRATION ISOLATOR
139 Chapter 4 ModeCleaner Optical Design 4.1 Preface Following the work completed with the vibration isolation system for the modecleaner, it was natural to progress to the optical design. A preliminary design was started during the vibration isolator design presented in the previous chapter. The initial design assumes that the suppression of the higher order modes is similar to a two mirror cavity, which is not necessarily correct since the input modecleaner design consists of three mirrors in a triangular distribution. The symmetry of the suppression of higher order transverse modes is broken due to the odd number of mirrors in the modecleaner ring cavity. Therefore a new approach for the optical design was necessary. The optical design is divided into three parts corresponding to three different publications. The first paper shows a complete analysis of the higher order mode suppression properties of the triangular ring cavity used as an input modecleaner. This first paper also includes a preliminary analysis of the thermal lensing problem induced by the high circulating power in the modecleaner. This work was followed by a more complete study of the effects of high circulating power in the input modecleaner. In the third publication a complete simulation of thermal effects are presented, including the thermal gradients induced in the modecleaner optics. The effects of astigmatism induced by the high circulating power are calculated and a new optical design for an astigmatismfree modecleaner for advanced interferometric gravitational wave detectors is presented. 111
140 112 CHAPTER 4. MODECLEANER OPTICAL DESIGN Optical Design of a High Power ModeCleaner for AIGO Pablo Barriga 1, Andrew Woolley 1, Chunnong Zhao 2,, and David G. Blair 1 1 School of Physics, University of Western Australia, Crawley, WA6009, Australia 2 Computer and Information Science, Edith Cowan University, Mount Lawley, WA 6050, Australia Laser beam geometry variations such as beam jitter and frequency fluctuations are a critical source of noise in the output signal of a laser interferometer gravitational wave detector. In order to minimise this noise a resonant vibration isolated optical filter or modecleaner is required. For advanced gravitational wave detectors such a modecleaner is required to be able to handle transmitted power 100 W, and an internal circulating power of 45 kw. This paper addresses the design requirements of such a modecleaner. We characterise the modecleaner requirements and the effects of high laser power on the optics and its consequence on the suppression of higher order modes. 4.2 Introduction The Australian Consortium for Gravitational Astronomy (ACIGA) has built a high optical power test facility at the site of the proposed Australian International Gravitational Observatory (AIGO), north of Perth in Western Australia. This facility will play three vital roles in gravitational wave research. In the short term it will be used to collaborate in the development of high optical power technologies required for the next generation of Advanced Gravitational Wave (GW) detectors. The second is to demonstrate the operation of a very low noise 80 m base line advanced interferometer. Third will be the development of the southern hemisphere long baseline detector, AIGO. Now at School of Physics, The University of Western Australia, Crawley, WA6009, Australia
141 4.2. INTRODUCTION 113 The 80 m interferometer has been designed to have parameters as close as practicable to Advanced LIGO to enable the critical issues of thermal lensing, radiation pressure and optical spring effects to be examined. An essential part of the facility will be a triangular ring cavity modecleaner to minimise variations in laser beam geometry. The spatial instability of a laser beam, known as beam jitter, is due to the mixing of higher order modes with the fundamental mode (TEM 00 ). Amplitude fluctuations are created by beam jitter whenever the beam interacts with a spatially sensitive element such as an optical cavity. The noise at the dark fringe at the interferometer output will be affected by such beam jitter effects. In addition frequency fluctuations of the laser fundamental mode give rise to additional noise at the dark fringe. All these noise sources can be minimised by using a modecleaner as illustrated in figure 4.1. l 1 Vibration Isolator M1 M2 Vibration Isolator M3 l 2 2L = 2l + l 2 1 Figure 4.1: Simplified schematic showing the layout of the AIGO modecleaner. The modecleaner acts as a spatial filter. It provides passive stabilisation of time dependant higher order spatial modes, transmitting the fundamental mode (TEM 00 ) and attenuating the higher order modes. The concept was first suggested by Rüdiger et al in 1981 [1]. As a frequency stability element it can also suppress frequency fluctuations of the fundamental mode, but without DC stability. The designer is able to vary cavity length, mirror radius of curvature (ROC) and number of modecleaners in series. The choice of parameters can allow optimisation of frequency stability, power handling capacity, and choice of free spectral range (FSR). However these factors are not independent and there is no unique optimum solution,
142 114 CHAPTER 4. MODECLEANER OPTICAL DESIGN Project Length [m] FSR [MHz] ROC [m] Stability GEO600 [2] VIRGO [3] TAMA [4] LIGO Caltech [5] Advanced LIGO [6] Table 4.1: Summary of the modecleaner configurations used by other gravitational wave interferometers. as can be seen in table 4.1 where the various modecleaner solutions adopted by the other GW interferometric projects are presented. The FSR needs to match desired interferometer modulation frequencies (to allow sideband transmission). Large ROC leads to increased spot size and higher power handling capacity, but with loss of angular stability. In the following sections, we first present a summary of the key design aspects and theoretical background to a modecleaner and use this to derive suitable parameters for the AIGO modecleaner. Then we address the issue of thermal lensing for the AIGO high power modecleaner. We show that design choices depend on the highest higher order mode for which suppression is desired, and that issues remain concerning astigmatism due to thermal lensing in the 45 o mirrors. 4.3 The modecleaner A modecleaner is a cavity used in transmission, with a geometry chosen such that the fundamental mode is nondegenerate. This is a cavity where the fundamental mode is resonant and the higher order modes, having different cavity eigenfrequencies, are attenuated or suppressed. How effective the cavity will be for filtering the higher order modes is then given by the suppression factor S mn of the cavity [7], given by:
143 4.3. THE MODECLEANER 115 where S mn = [1 + 4F ( 2 1/2 2π νmn L)] π 2 sin2, (4.1) c r1 r 2 r 3 F = π (4.2) (1 r 1 r 2 r 3 ) ( ) ν mn = c 2L (m + n) 1 π arccos 1 L. (4.3) R Equation (4.1) shows the suppression factor of any higher order mode TEM mn. Here F corresponds to the finesse of the cavity as shown in equation (4.2), where r 1, r 2, and r 3 are the reflectivity of each mirror. L is the length of the cavity and c is the speed of light in vacuum. Equation (4.3) corresponds to the difference in frequency between any higher order mode TEM mn and the fundamental mode TEM 00. This frequency difference not only depends of the order values m and n, but also depends on the length of the cavity and the ROC of the end mirror R. The relation between the length of the cavity and the ROC is commonly known as the stability gfactor of the cavity as shown in equation (4.4). In the case of the modecleaner, and in order to have a stable cavity this value has to be 0 < g < 1. g = 1 L R. (4.4) Clearly the ROC of the end mirror M3 is one of the essential variables when designing a modecleaner. By equation (4.4), the ROC and the length of the cavity define the stability gfactor. Through equations (4.1) and (4.3), the gfactor defines the suppression factor of the higher order modes. The power that builds up inside the cavity depends on its finesse, but the area of the mirrors where this power will concentrate depends on the ROC of the end mirror M3. Therefore the ROC defines the size of the beam waist that defines the spot size on each mirror and hence the power density. Consequently the curvature of the mirror is limited by the damage threshold of the mirror coating and by the stability gfactor of the cavity. Typical damage threshold is 1 MW cm 2 [8]. The transmission factor of higher order modes, T mn is given by the ratio of the transmission factor of the fundamental mode, T 00, and the suppression factor of the
144 116 CHAPTER 4. MODECLEANER OPTICAL DESIGN higher order modes as shown in equation (4.5). The transmission of the fundamental mode will depend on the transmission of the input and output couplers M1 and M2 (t 1 and t 2 ), and the mirrors reflectivity as shown in equation (4.6), where 1 T mn = T 00 [ 1 + ( 2 Fsin ( 2πL ν )) ] 2 1/2 (4.5) π c mn T 00 = t 1t 2 (r 1 r 2 r 3 ). (4.6) The general rule for the resonant condition of a cavity says that this one way phase shift must be an integer number of half cycles [9]. Since the total round trip phase shift must be an integer multiple of 2, it must satisfy: 2kL 2(m + n + 1) arccos ( g) = 2πq. (4.7) Where k = ω/c corresponds to the wave number, g is the cavity stability factor and q an integer that represents the axial mode number. All of the above is true for a two mirror cavity and in general for any cavity with an even number of mirrors. It has been shown that symmetry with respect to the vertical axis implies that the spatial dependence of the field with the plane of incidence is an even function. For this analysis the vertical axis is defined perpendicular to the plane of incidence. In this case a ring cavity formed by three or four mirrors is completely equivalent. If the field distribution is antisymmetric with respect to the vertical axis, then there is a difference of half a wavelength between an even and an odd number of mirrors [10]. As a consequence a ring cavity with an odd number of mirrors like the modecleaner of figure 4.1 will have the same frequency shift for the higher order modes with similar (m + n) value, depending on the symmetry of the horizontal modes m with the vertical axis. We can formalise this by writing the following general expression: 2kL 2(m + n + 1) arccos ( g) π (1 ( 1)m ) 2 = 2πq. (4.8)
145 4.3. THE MODECLEANER 117 The FSR is defined by equation (4.9). The frequency for any higher order mode at any axial mode q is then given by equation (4.10): ν mnq = qν 0 + ν 0 π (m + n + 1) arccos ( g) + ν 0 2 ν 0 = c 2L, (4.9) (1 ( 1) m ). (4.10) 2 The frequency difference between the fundamental mode and any higher order mode associated with the same q axial mode is then given by: ν mn = ν 0 π (m + n) arccos ( g) + ν 0 2 (1 ( 1) m ). (4.11) 2 In this frequency difference expression two factors can be distinguished: 1 2 arccos ( g ), (4.12) π (1 ( 1) m ). (4.13) 2 The first shown in (4.12) corresponds to the Gouy phase shift factor [9]. However the limiting values for this factor will also depend on the geometrical design of the optical cavity. In our case there are two possible limiting values for the Gouy phase shift factor: g close to 1 and g = 0 a. When g is close to 1 the Gouy phase shift factor becomes close to 0. Therefore the frequency of all transverse modes associated with a given axial mode q are clustered on the high frequency side. This is the case where the end mirror M3 corresponds to a flat mirror which gives rise to a minimum ν mn. This implies that most of the higher order modes will be transmitted together with the fundamental mode. Hence a modecleaner will not be effective if g is very close to 1. Such a cavity also has extreme angular sensitivity and is almost unstable. If g is equal to 0 the Gouy phase shift factor becomes 0.5 and the TEM 01 associated with the qth axial mode will fall exactly half way between the q and q+1 axial modes, although the TEM 10 associated with the same qth axial mode will fall exactly at the a A third possible limit is g close to 1 but this is not applicable to a stable modecleaner, where 0 < g < 1.
146 118 CHAPTER 4. MODECLEANER OPTICAL DESIGN q + 1 axial mode. In our case this situation occurs if the end mirror M3 has a ROC equal to the length of the cavity. Therefore if m is even and (m + n) is even the mode will be transmitted. Moreover if m is odd and (m + n) is odd the mode will also be transmitted. This case can be extended to every M3 ROC that is a multiple of the length of the cavity L, where different multiples of the modes will be transmitted. The second factor shown in (4.13) corresponds to the frequency shift between modes of the same order, but present only if the horizontal mode m is odd. This introduces an extra shift of half the FSR. Combining equations (4.5) and (4.11) the transmission factor is given by: 1 T mn = T 00 [ ( 1 + 4F 2 sin π 2 (m + n) arccos ( ) g + π 2 2 )] (1/2). (4.14) (1 ( 1) m ) 2 The phase shift introduced by the fact that the modecleaner is formed by an odd number of mirrors is normally not taken into consideration. If this is not considered there will be no difference between m odd or even for any TEM mn mode. For example TEM 31 is not the same as TEM 22, even though in both cases (m + n) = 4. As can be deduced from equation (4.11) each of these modes will have a different frequency. In figure 4.2 it is possible to see that the transmission factor does depend on this value, as suggested in equation (4.14). Using expression (4.14) it is possible to simulate the transmission factor of the higher order modes for the modecleaner as a function of the gfactor. Figure 4.2 shows the distribution of the higher order modes. Here for example, even though TEM 13 and TEM 22 have the same value of (m + n), because they will have different eigenfrequencies they need different gfactor to resonate in the modecleaner. For the same cavity length a higher gfactor (closer to 1) means a larger ROC, therefore less power density and less thermal effects on the mirrors. Using the first 20 modes (corresponding to a total of 231 modes with (m+n) 20) we calculate a gfactor that minimise the transmission of the higher order modes as shown in figure 4.3. Choosing an end mirror with a ROC close to the length of the cavity many higher order modes will be transmitted. Also we have to consider the damage threshold of the mirror coatings. In our case with an input power of 100 W
147 4.3. THE MODECLEANER 119 Transmission of Higher Order Modes 1 TEM02 TEM04 TEM10 TEM12 TEM20 TEM30 TEM40 TEM13 TEM31 TEM03 TEM21 TEM04 TEM11 TEM22 TEM40 TEM12 TEM30 TEM13 TEM31 TEM01 TEM20 TEM02 TEM03 TEM04 TEM21 TEM40 Transmission factor TEM01 TEM02 TEM03 TEM04 TEM10 TEM11 TEM12 TEM13 TEM20 TEM21 TEM22 TEM30 TEM31 TEM g factor Figure 4.2: Transmission of the higher order modes as function of the stability gfactor of a three mirror ring cavity to be used as a modecleaner for the AIGO interferometer. by choosing a ROC larger than 20 m (gfactor higher than 0.5) we assume a safety margin higher than 50% with just 0.43 MW cm 2 on the mirrors M1 and M2. It is important to notice that by including more higher order modes to the calculations we are reducing the suppression of the lower higher order modes by 6 db to 10 db. This due to the fact that we need to move away from what could be a minimum of a lower mode in order to avoid the transmission of a higher one. Also important is to give higher priority to the lower modes when choosing a gfactor or a radius of curvature, since these modes are more likely to appear in the incident beam. This is the main reason not to choose a radius of curvature around 21.5 m where the highest transmission (or lowest suppression) corresponds to TEM 11 mode. All of the above lead us to choose a ROC of 22.5 m, which means a gfactor of This results in a waist radius of mm and power density of 378 kw cm 2 M1 and M2. For this simulation a circular Gaussian beam was assumed inside the modecleaner.
148 120 CHAPTER 4. MODECLEANER OPTICAL DESIGN The purpose is to select the best ROC for M3 that maximises the modecleaner rejection of higher order modes. Therefore the elliptical profile that appears due to the fact that the input and output couplers are at 45 o in relation to the input beam has not being considered here Higher Order Modes Sum 10 1 Magnitude M3 Radius of Curvature [m] Figure 4.3: Detail of transmission of the higher order modes as function of the radius of curvature of the end mirror M3. The graph shows the transmission factor for modes with (m + n) 20 for radius of curvature between 20 m and 25 m. 4.4 Modecleaner thermal lensing In Advanced GW Interferometers high power lasers will be used [11]. This will introduce thermal lensing effects in the main cavities of the interferometer where the power builds up to the order of hundreds of kilowatts [12]. This modecleaner is designed with a finesse of For an input laser power of 100 W, the expected power inside the cavity will be 45 kw. Considering fused silica substrates for the mirrors that form the modecleaner cavity it is possible to calculate the thermal effect produced in them. Due to the high
149 4.4. MODECLEANER THERMAL LENSING 121 laser power circulating inside the cavity the mechanical parameters of the mirrors will suffer a small but significant change. This change in the curvature of the mirror will alter some of the cavity parameters, like the gfactor and the waist size. Among other effects these changes will affect the suppression (transmission) factor of the cavity [13]. The change in the ROC of the mirrors is really a change in sagitta, which corresponds to the curvature depth of the mirror measured across the beam diameter [14]. The sagitta s o of the mirror surface with a radius of curvature R over a spot size ω is given by: s o = R R 2 ω 2. (4.15) The change in sagitta caused by heating can be expressed as: δs αp a 4πk. (4.16) Here P a corresponds to the absorbed light power, k the heat conductivity of the substrate, and α the thermal expansion. Coating absorption losses of 1 ppm, a thermal expansion of K 1, and heat conductivity of 1.38 W m 1 K 1 have been assumed for the fused silica substrates. Due to the geometric distribution of the mirrors to form the modecleaner the laser beam will impinge upon the flat mirrors at an angle of 45 o. In our case with a distance of 20 cm between the input and output couplers the angle is o. As a result the spot size on the mirrors can not be considered as circular anymore but elliptical. The elliptical spot at the mirrors means different sagitta values for the horizontal and vertical planes causing an elliptical deformation of the mirror. An elliptical profile will lead to differences in wavefront curvature between the two transverse directions or astigmatism. After the high power laser has been switched on and the cavity enters a steady state, some of the parameters of the cavity will change due to thermal effects in the mirrors. This steady state situation is what is known as hot cavity. Table 4.2 summarise some of these changes, where for comparison the cold cavity status still
150 122 CHAPTER 4. MODECLEANER OPTICAL DESIGN assumes a circular beam profile. The end mirror M3 is expected to increase its ROC and consequently M1 and M2 will become slightly convex. Hence the minus sign in the hot cavity ROC of M1 and M2. Due to the elliptical profile of the beam we now see that the deformation of the mirrors due to thermal effects it is not even in both axes but elliptical as well. In fact on the flat mirrors the effect will be stronger on the Yaxis (tangential plane) than on the Xaxis (sagittal plane) producing differences in wavefront curvature between the two transverse directions. As seen before the frequency of the higher order modes depends on the gfactor of the cavity. The change in ROC due to thermal effects alters the cavity gfactor, changing the frequency spacing between the higher order modes and the fundamental mode. Within addition to this frequency shift there is a change in the suppression (transmission) factor of each mode. The increase in the ROC of the mirrors makes the cavity gfactor also increase, therefore the Gouy phase shift decreases. The frequency Definition unit Cold Hot Cavity Cavity Xaxis Yaxis Modecleaner length m 10 M1 radius of curvature m flat M2 radius of curvature m flat M3 radius of curvature m Cavity gfactor Modecleaner FSR MHz Finesse 1495 Modecleaner waist mm Rayleigh range m Input power W 100 Stored MC power kw Table 4.2: Shows the difference between hot and cold parameters for the AIGO modecleaner (MC). Considering that all mirrors substrates are made of fused silica.
151 4.4. MODECLEANER THERMAL LENSING 123 difference between the fundamental mode and the higher order modes also decreases, ultimately shifting the higher order modes closer to the fundamental one. By doing separate analysis for tangential and sagittal planes it is possible to determine the level of astigmatism introduced by the mirror deformation due to thermal absorption. We calculate the difference in the frequency shift of the fundamental mode for each axis as shown in equation (4.10). The difference between X and Y axis for the fundamental mode is about 65 khz corresponding to a 0.17%. Also there is a difference in the frequency shift of the higher order modes suggesting that the coupling of higher order modes will be stronger on the horizontal axis than in the vertical one. The worst case is in the Xaxis, where the mode TEM (12)(1) will have a transmission of the 4.3% due to the astigmatism present inside the cavity. This effect could be compensated using radiant heat to vary the temperature profile on the mirror using compensating methods already developed for thermal lensing in test masses. This problem is an area of actual study by LIGO [15] as well as ACIGA. However further study is required to address this issue. The level of the higher order modes transmitted by the modecleaner will strongly depend on the quality of the incident beam. Moreover the excellence of this beam will not only depend on the quality of the laser, but also on the performance of the Premodecleaner. In order to achieve these levels of suppression of the higher order modes it is necessary to specify the ROC of the end mirror quite carefully for a fixed cavity. More realistic a careful control of the cavity length is needed, in order to match the length of the cavity to the mirror ROC to obtain the adequate gfactor. Having different values for the transmission of the higher order modes for the horizontal and vertical axes shows the effect of the astigmatism inside the cavity. As a result the beam inside the cavity and therefore the transmitted one will not be a pure HermiteGaussian beam. It will be the fundamental mode plus some small components from the higher order modes coupled to the fundamental one.
152 124 CHAPTER 4. MODECLEANER OPTICAL DESIGN 4.5 Conclusions We have seen some of the effects caused by the high power stored in the modecleaner, including mechanical changes in the optics and how they alter some of the cavity parameters such as the gfactor and cavity waist. The frequency shift on the higher order modes and the change produced in the suppression (transmission) factor were also presented. The high power stored in the cavity creates a thermal lens effect in the modecleaner optics, which will introduce some astigmatism in the cavity. This effect should be taken into account when performing the mode matching for the power recycling cavity and main arms of the future AIGO interferometer. It will also be necessary to estimate the effect of the thermal lensing in the sidebands that need to be transmitted through the modecleaner in order to control the other cavities in the interferometer. High power modecleaner design presents a challenge due to astigmatic thermal lensing. Design performance represents a trade off between low order and high order normal modes. If the injection beam quality is such that mode numbers of order 20 need to be strongly suppressed, either the cavity length or the mirror radii of curvature need to be tuneable to a precision 0.1%. Astigmatic thermal compensation is also required to achieve sufficient rejection of certain modes with mode number > 10. However, we note that the fine tuning required can be achieved through the suspension system design previously reported [16], which has an ultralow frequency stage capable of fine translation over 0.1% of the cavity length. Astigmatic thermal compensation can be provided using astigmatic heating, based on small modifications of existing thermal compensation techniques [12]. We have shown that a modecleaner with a gfactor 0.55 and 10 m length with a ROC of 22.5 m allows the transmission of the fundamental mode reducing the coupling of the higher order modes. In addition, this design has good immunity to thermal lensing effects, with the high order modes frequency offset from the carrier changing by less than 350 khz from cavity switch on to high power operation in the worse case (Yaxis). The modecleaner is designed to support 100 W of input power with a safety margin of more than 60% on the coating power threshold for standard high power mirrors. Future work include a more detailed study of the effect of the astigmatism
153 4.5. CONCLUSIONS 125 due to high power stored in the cavity and the coupling of higher order modes into the fundamental mode. Acknowledgements The authors would like to thank Jérôme Degallaix and Bram Slagmolen for helpful discussions and Andrew Woolley for his construction of the isolator stage for the modecleaner. We thank David Reitze, Amber L. Bullington and Ken Y. Frazen from the LIGO group for useful discussions. This work was supported by the Australian Research Council, and is part of the research program of the Australian Consortium for Interferometric Gravitational Astronomy.
154 126 CHAPTER 4. MODECLEANER OPTICAL DESIGN Astigmatism compensation in modecleaner cavities for the next generation of gravitational wave interferometric detectors Pablo Barriga 1, Andrew Woolley 1, Chunnong Zhao 2,, and David G. Blair 1 1 School of Physics, University of Western Australia, Crawley, WA6009, Australia 2 Computer and Information Science, Edith Cowan University, Mount Lawley, WA 6050, Australia Interferometric gravitational wave detectors use triangular ring cavities to filter spatial and frequency instabilities from the input laser beam. The next generation of interferometric detectors will use high laser power and greatly increased circulating power inside the cavities. The increased power inside the cavities increases thermal effects in their mirrors. The triangular configuration of conventional modecleaners creates an intrinsic astigmatism that can be corrected by using the thermal effects to advantage. In this paper we show that an astigmatism free output beam can be created if the design parameters are correctly chosen. 4.6 Introduction The configuration of interferometric Gravitational Wave (GW) detectors includes at least one input modecleaner [2, 3, 17, 18]. Current detectors have all similar configuration, consisting of two flat mirrors (M1, M2) used as input and output couplers and a concave end mirror (M3) as show in figure 4.4. This configuration is preferred since the reflected light at the input mirror will not be reflected back to the laser increasing the noise, but used to control the cavity locking. The triangular configuration presents an intrinsic astigmatism inside the cavity which increases when the input power is increased. Now at School of Physics, The University of Western Australia, Crawley, WA6009, Australia
155 4.6. INTRODUCTION 127 l 1 Vibration Isolator M1 M2 Vibration Isolator M3 l 2 2L = 2l + l 2 1 Figure 4.4: Triangular ring cavity layout used as modecleaner for the interferometric gravitational wave detector. This configuration has the advantage of low optical feedback. The next generation of interferometric GW detectors will also include input modecleaners. Designs have been proposed for Advanced Laser Interferometer Gravitationalwave Observatory (Advanced LIGO) [6], the Largescale Cryogenic Gravitationalwave Telescope (LCGT) [19], and for the Australian International Gravitational Observatory (AIGO) being developed in Gingin 90 km north of Perth in Western Australia. The results presented below are applied to the modecleaner design being developed for AIGO. The proposed design consists of a 10 m long triangular ring cavity. Two flat mirrors are used as input and output couplers. At 9.9 m a concave end mirror with a radius of curvature of 22.5 m is installed. This means cavity stability or gfactor of The design is chosen to minimise transmission of higher order modes, and to enable spot size large enough to keep the power density below the coating damage threshold. The details of this design are given in ref [20]. With a finesse of 1495 a circulating power of 45 kw for 100 W of input power is expected. Some of this power will be absorbed by the mirrors causing thermal effects in the substrate and their deformation as a consequence [14]. The amount of power absorbed by the mirrors and the substrate greatly depends on the quality of the coating. The magnitude of the thermal deformation depends on the thermomechanical properties of the substrate, usually fused silica. It has been previously shown that triangular ring cavities present an intrinsic astigmatism [21, 22, 23]. In this paper we show that modecleaners can use the thermal effects due to the high circulating power to advantage to compensate and greatly
156 128 CHAPTER 4. MODECLEANER OPTICAL DESIGN reduce the astigmatism, producing a modecleaner with near ideal performance and astigmatism free output. We show that there are various possible solutions. We present different alternatives varying radius of curvature, substrate materials and coating absorptions in order to obtain an output beam free of astigmatism. 4.7 Modecleaner intrinsic astigmatism The actual input modecleaners used by different GW detectors are based on a design proposed by Rüdiger et al in 1981 [1]. The design basically consists of vibration isolation suspension and suspended mirrors as shown in figure 4.4. Since the beam impinges onto the flat mirrors at an angle 45 o the spot on the surface of these mirrors is strongly elliptical with an eccentricity 0.7, which as a consequence produces an elliptical cavity waist. It has been shown that a cavity with these characteristics will always have an astigmatic output [23]. An elliptical spot means different sagitta values for the xaxis (parallel to the plane of incidence) and yaxis (perpendicular to the plane of incidence) causing an elliptical deformation of the mirror. In fact on the flat mirrors the effect will be stronger on the yaxis (tangential plane) than on the xaxis (sagittal plane) producing differences in wavefront curvature between the two transverse directions or astigmatism. Previously we have shown the astigmatism level of a high power modecleaner for AIGO [20]. In principle the modecleaner design consists of a 10 m modecleaner with two flat mirrors and a 22.5 m radius of curvature end mirror. In cold cavity conditions (very low or no input power) the waist will have an eccentricity of Cold Cavity Hot Cavity Parameter Xaxis Yaxis Xaxis Yaxis Waist [mm] M3 Spot [mm] Table 4.3: Comparison of waist and M3 spot sizes in sagittal and tangential planes between cold and hot cavity.
157 4.7. MODECLEANER INTRINSIC ASTIGMATISM Eccentricity variation with Power M1 Spot Eccentricity Eccentricity M3 Spot Waist Size Input Power [W] Figure 4.5: Eccentricity of the spots M1/M2, M3 and the waist for the AIGO high power modecleaner. The upper graph shows the reduction of the spot eccentricity of M1 and M2 when the input power increase. The lower graph shows the eccentricity variation of the M3 spot and cavity waist for different input power values. compare to an eccentricity of under the hot cavity conditions ( 100 W input power). Increasing an order of magnitude the waist eccentricity, which implies stronger astigmatism, to levels in which mode matching into the interferometer is significantly degraded. The intrinsic astigmatism also produces higher order modes that will degrade the interferometer operation. Table 4.3 present the values for the xaxis and the yaxis under cold and hot cavity conditions. The calculations were made using a Matlab R code written by the authors based on matrix simulations [24, 25]. Together with the enlargement of the waist and the spot size at M3 due to the thermal effects we note that both eccentricities also increase. Note that not only the eccentricity increases, but the major axis of the ellipse changes from the xaxis to the yaxis. This implies that the waist must cross a circular profile for a certain level of input power. Fused silica substrate was assumed for the thermal simulation presented in figure 4.5 with thermal expansion of K 1 and heat conductivity of 1.38 W m 1 K 1. Coating absorption losses of 1 ppm has also been assumed.
158 130 CHAPTER 4. MODECLEANER OPTICAL DESIGN So far we have just considered the thermal effects on the mechanical properties of the substrate, but we also need to consider the change in curvature of the wave front of the transmitted beam. This change mainly depends on the effect of a heated hemisphere due to the absorption at the dielectric coating. There is also a smaller contribution from the absorption that occurs inside the substrate along the transmitted path. Contributions to the path changes due to the variation of the refractive index to stress (photoelastic effect) can be neglected [26]. The following equations have been proposed as good approximations to calculate the path difference between the centre and the outer parts of the beam when transmitted through a substrate [14, 27]. Equation (4.17) refers to the effect due to the power absorbed by the dielectric coating, and equation (4.18) to the change due to the power absorbed by the substrate. δs n T P a 4πk (4.17) δs = 1.32 n T p a d, (4.18) 4πk where n/ T corresponds to the temperature dependence of the refractive index, P a is the absorbed light power, k is the thermal conductivity, p a is the power absorbed per unit length and d is the substrate thickness. With a value of K 1 for n/ T, we assume an absorption of 2 ppm/cm for fused silica substrate and that 90.7% of the input power will be transmitted into the power recycling cavity. Therefore the light path difference for the transmitted beam is 22.8 nm due to the coating absorption and 8.1 nm due to the substrate absorption for a 25 mm thick substrate, which corresponds to an error of λ/35. Reduction of the coating absorption losses can be sufficient to correct the astigmatism at low power levels. For high circulating power this needs to be combined with changes to the mirrors radius of curvature in order to reduce the astigmatism.
159 4.8. ASTIGMATISM FREE MODECLEANER Eccentricity variation with Power Eccentricity M1 Spot M3 Spot Waist Size Eccentricity Input Power [W] Figure 4.6: Eccentricity variation with input power for fused silica substrate with 0.6 ppm coating absorption losses. The upper graph shows the reduction of the M1 and M2 spot eccentricity for different levels of input power. The lower graph shows the variation of the M3 spot and waist eccentricity variation for different levels of input power. 4.8 Astigmatism free modecleaner The results presented suggest that with the right combination of parameters we can obtain a modecleaner with an astigmatism free output. One possible solution is to create a circular waist inside the cavity. In order to obtain a circular waist the following configuration is proposed: Mirrors substrate will be made of the same type of fused silica, but instead of flat input/output couplers we will have 500 m radius of curvature ones. In order to keep the transmission properties of the modecleaner similar to the original design we change the radius of curvature of the end mirror M3 to 24 m. In the previous design it was suggested a coating with 1 ppm absorption losses that needs to be improved down to 0.6 ppm. Figure 4.6 shows that by choosing the parameters mentioned before we will create a waist with a radius of mm across both axes. This means a circular cavity waist for W of input power. The beam will also have a Rayleigh range of m in both axes and a beam radius of curvature of m in both axes. We still have the problem of the thermal lensing when the beam is transmitted
160 132 CHAPTER 4. MODECLEANER OPTICAL DESIGN through the substrate. In this new configuration the values for the path difference are of 13.6 nm due to coating absorption and 8.1 nm due to transmission through the substrate for a 25 mm thick substrate, which corresponds to an error of λ/50. This result shows a 57% improvement when compared to the previous configuration, mainly due to less absorbed power in the coating. Good results with a circular waist of mm can also be obtained using input and output couplers made of fused silica with a radius of curvature of 750 m. The best results for this case are obtained for W of input power. In order to keep the higher order modes suppression levels an end mirror with a radius of curvature of 23.7 m is chosen. For this configuration it is necessary to reduce the absorption losses of the coating down to 0.4 ppm, reducing also the path difference for the transmitted light to λ/64. This result can be improved just by using M3 as the output coupler. Under this configuration not only the astigmatism is reduced also the light path difference is improved to λ/74. It is possible to obtain similar results using sapphire as substrate. In this case the radius of curvature of the input and output couplers needs to be 1000 m and 22.8 m for the end mirror. In such case 0.75 ppm coating absorption losses are needed. The circular waist obtained with an input power of W which has a radius of mm in both axes. Rayleigh range of m and a beam radius of curvature of m are also achieved. Given that mirror coating parameters are difficult to precisely specify, it will in practice be necessary to adjust the input power to tune the modecleaner near to the zero astigmatism condition. Due to its homogeneity it is assume that in fused silica there is no problem in transmitting the light through the substrate at 45 o, and that the difference between a normal incidence of the beam and a 45 o one it is only at the amount of substrate that the light will have to go through. In the case of sapphire this assumption is not valid anymore, and we will have to consider the substrate axis orientation. Therefore if we want to use sapphire as a substrate for the modecleaner mirrors we will need to consider the use of the end mirror (M3 in figure 4.4) as the output coupler in order to minimise the incident angle of the beam.
161 4.9. CONCLUSIONS Conclusions Triangular modecleaners have a mild astigmatism when operating under low power conditions. High power lasers under development for the next generation of interferometric GW detectors will introduce significant thermal effects in the optical cavities. In the modecleaners thermal deformation worsens the astigmatism of the output beam. We have shown that by carefully choosing the parameters for the modecleaner it is possible to use those thermal effects to our advantage and dramatically reduce the astigmatism within the modecleaner. Even if we remove the astigmatism inside the modecleaner the use of M2 as an output coupler has the disadvantage of crossing the substrate at nearly 45 o. This means longer lightpath through the substrate, therefore more power absorbed and a larger change in the light path between the centre of the beam and its edge. Moreover the spot at this mirror is elliptical and as a consequence the wavefront path changes are different for the xaxis and for the yaxis. In order to avoid this situation we can use the end mirror M3 as an output coupler. Due to the long distance between M1, M2 and M3 the laser beam crosses the substrate nearly perpendicular minimising the astigmatism due to thermal lensing. Future work on thermal effects due to high circulating power in modecleaners will address this problem. We have shown that by improving the coating absorption losses for fused silica mirrors it is possible to considerably reduce the astigmatism for 100 W of input power. The right combination between the mirrors radius of curvature and coating reduces the astigmatism while keeping the cavity stability and the higher order modes suppression levels. A solution using sapphire substrates was also presented. Similar solutions can be obtained for any high power modecleaner. Acknowledgements The authors would like to thank Jérôme Degallaix, Li Ju and Bram Slagmolen for helpful discussions. This work was supported by the Australian Research Council, and is part of the research program of the Australian Consortium for Interferometric Gravitational Astronomy.
162 134 CHAPTER 4. MODECLEANER OPTICAL DESIGN SelfCompensation of Astigmatism in ModeCleaners for Advanced Interferometers Pablo Barriga, Chunnong Zhao, Li Ju, David G. Blair School of Physics, University of Western Australia, Crawley, WA6009, Australia Using a conventional modecleaner with the output beam taken through a diagonal mirror it is impossible to achieve a nonastigmatic output. The geometrical astigmatism of triangular modecleaners for gravitational wave detectors can be selfcompensated by thermally induced astigmatism in the mirrors substrates. We present results from finite element modelling of the temperature distribution of the suspended modecleaner mirrors and the associated beam profiles. We use these results to demonstrate and present a selfcompensated modecleaner design. We show that the total astigmatism of the output beam can be reduced to for ±10% variation of input power about a nominal value when using the end mirror of the cavity as output coupler Introduction Input modecleaners are used in gravitational wave interferometers presently in operation. One on each LIGO detector in the USA [17], one in the French Italian VIRGO [7], two in series at the British German GEO600 [2] and one in TAMA300 in Japan [18]. As an important part of the input optics system modecleaners are used to reduce any spatial or frequency instability of the laser beam. In addition frequency fluctuations of the laser fundamental mode give rise to additional noise at the dark fringe. It provides passive stabilisation of time dependant higher order spatial modes, transmitting the fundamental mode TEM 00 and attenuating the higher order modes. Plans for the next generation of advanced interferometers (Advanced LIGO [6], LCGT [19] and AIGO) will also include at least one input modecleaner. In order to overcome photon shot noise high power (>100 W) single frequency, continuous wave Nd:YAG lasers are needed [28]. A small portion of the circulating power will remain
163 4.11. SUBSTRATE DEFORMATION 135 in the mirrors due to substrate and coating absorptions. This energy will increase the temperature on the mirror causing a thermal expansion and a change in its radius of curvature [14]. Therefore the higher the circulating power the stronger the thermal effects. It has been shown that triangular ring cavities like the modecleaner will always have a certain level of astigmatism due to the angles at which the beam impinges on the mirrors [23]. This creates a mild astigmatism in the beam that circulates inside the cavity, which is worsen when the input (or circulating) power is increased. This geometrical astigmatism is a consequence of the geometrical distribution of the mirrors in the triangular ring cavity. Whenever the beam crosses a piece of optics there is a change in path length between the centre and the outer beam [14]. This thermal lensing effect is induced by the power absorbed by the coating and the substrate of the optics. If the beam crosses the output coupler at a relatively large angle then this effect will be stronger and even worse at high power. Similar effects were reported at the beamsplitter of the Phase Noise Interferometer at the Massachusetts Institute of Technology (MIT), where the beam crosses at 45 o as well [29]. In this paper we first analyse the substrate deformation due to the thermal effects inside the modecleaner. This deformation introduces a thermally induced astigmatism inside the cavity. We propose a solution to this problem using the thermally induced astigmatism to advantage in order to compensate the geometrical astigmatism present in the modecleaner output beam. We also show the temperature distribution in the substrate when the mirror is used as an output coupler. This will be used to determine the thermal lensing effect of the output beam due to the different path between the centre and the outer parts of the beam. This will help us to quantify the astigmatism of the output beam and to select the best solution that minimise this effect Substrate deformation Input modecleaners actually in use consist mainly of two flat mirrors that define the short side of the triangle and a concave mirror that forms the acute angle of an isosceles triangle as seen in figure 4.7. Due to this configuration and depending on
164 136 CHAPTER 4. MODECLEANER OPTICAL DESIGN l 1 Vibration Isolator M1 M2 Vibration Isolator M3 l 2 2L = 2l + l 2 1 Figure 4.7: Due to its low optical feedback a triangular ring cavity using three suspended mirrors is preferred as the input modecleaner for interferometric gravitational wave detectors. the distance that separates the two flat mirrors from the concave end mirror (usually several metres) the laser spot at the flat mirrors will be strongly elliptical while at the end mirror will be closer to a circumference as can be see in figure 4.8. With an elliptical spot on the surface of the flat mirrors the thermal effects will be different for each axis. (a) (b) Figure 4.8: (a) Elliptical spot at the M2 mirror inside the cavity. The elliptical spot is the result of the 45 o of the laser beam incident angle. (b) Due to the long distance between M1/M2 to M3 the spot at the end mirror is nearly circular under the cold cavity conditions.
165 4.11. SUBSTRATE DEFORMATION 137 We define a coordinate system where the xaxis is parallel to the plane of incidence on the cavity mirrors and the yaxis is perpendicular to it, leaving zaxis along the direction of propagation. The thermal effects due to the absorption at the dielectric coating and the deformation of the substrate as a consequence are stronger in the yaxis (tangential plane) than in the xaxis (sagittal plane). This causes an elliptical deformation of the mirror substrate producing differences in wavefront curvature between the two transverse directions or astigmatism. This effect needs to be carefully calculated since it will change the gfactor of the cavity, changing the transmission (or suppression) factor of the higher order modes. We have shown that the astigmatism inside a triangular modecleaner is strongly dependant on the circulating power, which is defined by the input power and the cavity finesse [30]. Figure 4.9 shows the eccentricity variation of the M3 spot and cavity waist with input power. The eccentricity values depend on the radius of curvature of the mirrors that form the modecleaner and the cavity finesse. They also depend on the mirrors substrate and coating absorption. In our case we assume that all substrates are of fused silica with substrate absorption of 2 ppm/cm, heat conductivity k = 1.38 Wm 1 K 1, thermal expansion coefficient α = K 1 and refractive index temperature dependence β = K 1. Input modecleaners at the operational gravitational wave interferometers are all designed with two flat mirrors and one end concave mirror at the acute end of it. In this case when operating at low power the cavity waist will have an almost circular profile. As soon as the circulating power is increased the waist gets strongly elliptical and therefore with high eccentricity, as shown in figure 4.9 (a). This figure also shows that operating a standard modecleaner at a 100 W of input power we have a relatively high eccentricity of 0.06 at the waist, even if we use M3 as an output coupler we still get an eccentricity of When using the design proposed in figure 4.9 (b), we obtain a beam free of astigmatism at mirror M3. This result is very difficult to obtain in reality due to power variations and the fact that coating and substrates are never exactly the ones predicted. However we notice that a variation of ±10% in the input power only increases the eccentricity from nearly 0 to when using M3 as an output coupler. A similar
166 138 CHAPTER 4. MODECLEANER OPTICAL DESIGN (a) Eccentricity M1/M2 = Flat M3 = 22.5 m Eccentricity variation with Power M3 Spot Waist Size (b) 0.06 Eccentricity M1/M2 = 1000m M3 = 23.3 m (c) Eccentricity M1/M2 = 470 m M3 = 23.3 m Input Power [W] Figure 4.9: Waist and M3 spot eccentricity variation with input power. The upper graph shows two flat mirrors and an end concave mirror assuming coating absorption of 1 ppm (best result for M3 at 28.5 W). Figure 4.9 (b) shows the cavity tuned for 100 W input power, while 4.9 (c) shows an Advanced LIGO type of modecleaner tuned for 160 W of input power. For the last two cases we assumed coating absorption of 0.5 ppm. case is presented in figure 4.9 (c). With a different design, this time for 160 W of input power. In this case a variation of ±10% in the input the input power increases the eccentricity up to when using M3 as the output coupler. By choosing different radius of curvature for the mirrors and combining them with lower absorption coatings it is possible to design a mode cleaner free of astigmatism. When doing the design it is important to maintain the intended cavity gfactor, in order to keep the higher order modes nondegenerated in the cavity. Graphs at figure 4.9 only consider the thermal deformation of the substrate. However we note that with the right combination it is possible to design a modecleaner free of astigmatism for different input power levels. It is interesting to see that at the waist and at M3 there is also a change in the ellipse major axis. The thermal effects inside the cavity will stretch the yaxis to the point that crosses a circular profile. It even gets larger than
167 4.12. THERMAL LENSING 139 the xaxis when the input power (and therefore the circulating power) is increased. The geometrical astigmatism has the opposite sign from the thermally induced astigmatism from the mirror absorption. Using the parameters for the mirrors we show this thermally induced astigmatism can be used to correct the intrinsic geometrical astigmatism of an isosceles triangular ring cavity modecleaner. The balance of these two effects can lead to a self compensated modecleaner. For the simulations here presented it was assumed a perfectly mode matched gaussian beam, however the effects of the substrate in the input beam when entering the cavity were considered. In reality the input beam quality is such that the input performance of the modecleaner is dominated by the poor quality of the input beam. Therefore mode matching losses are generally significantly smaller than the mode cleaning losses, and as a consequence negligible. Figure 4.10: Steady state solution for the bulk absorption case. If M2 is used as output coupler the diagonally transmitted beam produces strong astigmatic thermal lensing. (90.5 W transmitted power, 0.6 ppm coating absorption, fused silica substrate absorption of 2 ppm/cm).
168 140 CHAPTER 4. MODECLEANER OPTICAL DESIGN 4.12 Thermal lensing The thermal lensing effects as described by Hello and Vinet will cause a deformation in the curvature of the wavefront of the transmitted beam [31]. This change mainly depends on the effect produced by the power absorbed by the dielectric coating due to the high power circulating inside the cavity. In our case with a finesse of 1500 and 100 W of input power there will be 45 kw of circulating power. There is also a small contribution from the power absorbed by the substrate during the transmission of the output beam. In most cases the two flat mirrors are used as input and output couplers for the modecleaner cavity. This can be a problem in high optical power modecleaners, not only due to the deformation of the mirror s substrate, but mainly due to the thermal lensing effects when transmitting the beam through the output coupler. Figure 4.10 shows the steady state temperature distribution of a flat mirror when Figure 4.11: Steady state temperature distributions for coating absorption due to the circulating power inside the modecleaner. M2 high reflectivity coating absorption produces astigmatic thermal lensing. The spot ellipticity produces different distribution between X and Y axis. (45 kw of circulating power, 0.6 ppm coating absorption, fused silica substrate absorption of 2 ppm/cm).
169 4.12. THERMAL LENSING 141 used as the output coupler. By crossing the substrate at almost 45 o the laser beam will cross a larger section of substrate, which will increase the thermal lensing effect produced by the temperature distribution compared to a perpendicular transmission. The main contribution to the thermal lensing from the power absorbed by the coating, is even larger as can be inferred from the higher temperatures in figure The wavefront change in the light path can be quantified as a change in the focal length of the beam. In this case there is higher temperature along the yaxis (vertical crosssection) due to a smaller spot size along the yaxis. This will produce a stronger deformation of the wavefront that will in consequence produce a stronger astigmatism in the output beam. Since the distortion is different for each axis we will study both separately. By calculating the wavefront deformation including bulk and coating absorptions it is possible to determine the best lens fit for it. Using a Matlab R code written by the authors it was possible to calculate the new focal length for the output beam. This method was later compared to FFT simulations obtaining very close agreement. The main contributors to the thermally induced optical path change are the dependence of the refractive index on temperature, the strain and the thermal expansion. Different authors [27, 26] have already examined these effects, which are summarised in equation (4.19). δs = 1.32 p ( a n 4πk T n3 2 ρ 12α + 2αn ω ) d. (4.19) d Here δs corresponds to the path change, p a to the power absorbed per unit length, k to the heat conductivity, n/ T = β to the refractive index change with temperature, n the fused silica refractive index, ρ 12 the fused silica photo elastic coefficient, α the thermoelastic coefficient, ω the beam radius of the intensity profile and d is the substrate thickness. If we use M2 as the output coupler the output beam will be strongly astigmatic. This astigmatism according to our simulations presented in table 4.4 will lead to a focal length difference between the xaxis and the yaxis of 133 m. If instead of using M2 we use M3 as the output coupler the astigmatism will be much smaller due to the small ellipticity of the spot, producing similar thermal effects
170 142 CHAPTER 4. MODECLEANER OPTICAL DESIGN in both axes. Even though at table 4.4 the results for M3 look the same they have a difference in focal length of about 1 µm. X  axis (m) Y  axis (m) M2 output coupler Strong astigmatism M3 output coupler Mild astigmatism Table 4.4: Output beam thermal lensing focal length. Best result is obtained when using M3 as output coupler. Input power of 99.6 W gives a focal length difference between X and Y axis of 10 6 m Conclusions The current modecleaner designs in use for interferometric gravitational wave detectors have mild astigmatism at low input power due only to the geometrical distribution of the mirrors. The same design used with higher laser power leads to significant astigmatism of the output beam due to thermal effects in the modecleaner mirrors. We have shown that the balance of the two effects can lead to a selfcompensated high optical power modecleaner. Our proposed solution for the AIGO modecleaner is to use fused silica substrates, M1 and M2 with a radius of curvature of 1000 m and 23.3 m for the end mirror, assuming coating absorption losses of 0.5 ppm. The need for an adaptive optic element for astigmatism control and modematching can be avoided, since using M3 as output coupler contributes negligible additional astigmatism in the output beam. The design presented keeps the astigmatism below 0.5% over a ±10% power range for 100 W of input power. Self compensation can be adjusted to any input power level using an additional single fixed astigmatic corrector at the output. Acknowledgements The authors wish to thanks Bram Slagmolen and Stefan Goßler for useful discussions and Jérôme Degallaix for helpful hints in writing the code. This work is supported by the Australian Research Council, the Department of Education, Science and Training
171 4.14. REFERENCES 143 (DEST), and is part of the research programme of the Australian Consortium for Interferometric Gravitational Astronomy References [1] A. Rüdiger, R. Schilling, L. Schnupp, et al, A mode selector to suppress fluctuations in laser beam geometry, J. Mod. Opt. 28 (1981) [2] S. Goßler, M. M. Casey, A. Freise, et al, Modecleaning and injection optics of the gravitationalwave detector GEO600, Rev. Sci. Instrum. 74 (2003) [3] F. Bondu, A. Brillet, F. Cleva, et al, The VIRGO injection system, Class. Quantum Grav. 19 (2002) [4] G. Heinzel, K. Arai and N. Mitaka, TAMA Modecleaner alignment, Technical Report, Version 1.6, TAMA, (2001). [5] B. Bonfield, B. Abbott, O. Miyakawa, et al, Chracterizing the length sensing and control system of the mode cleaner in the LIGO 40 m lab, Technical Report, T R, LIGO, (2002). [6] G. Mueller, D. Reitze, D. Tanner and J. Camp, Reference Design for the LIGO II Input Optics, LSC Presentation, G D, LIGO, (2000). [7] M. Barsuglia, Stabilisation en fréquence du laser et contrôle de cavités optiques à miroirs suspendus pour le détecteur interféromètrique d ondes gravitationnelles VIRGO, PhD Thesis, Université de ParisSud, Orsay, Chapter 7, [8] Melles Griot, The Practical Application of Light, (1999). [9] A. E. Siegman, Lasers, University Science Books, Sausalito, California, Chapter 19, (1986). [10] C. Mathis, C. Taggiasco, L. Bertarelli, et al, Resonances and instabilities in a bidirectional ring laser, Physica D 96 (1996)
172 144 CHAPTER 4. MODECLEANER OPTICAL DESIGN [11] LIGO II Conceptual Project Book, M AM, LIGO, (1999). [12] J. Degallaix, B. Slagmolen, C. Zhao, et al, Thermal lensing compensation principle for the ACIGA s High Optical Power Test Facility Test 1, Gen. Relat. Gravit. 37 (2005) [13] R. Amin, G. Mueller, M. Rakhmanov, et al, Input Optics Subsystem Conceptual Design Document, Technical Report, T D, LIGO, (2002). [14] W. Winkler, K. Danzmann, A. Rüdiger and R. Schilling, Heating by optical absorption and performance of interferometric gravitationalwave detectors, Phys. Rev. A 44 (1991) [15] A. L. Bullington, D. Reitze, K. Y. Franzen, Internal communication (2004). [16] P. Barriga, A. Woolley, C. Zhao and D. G. Blair, Application of new preisolation techniques to modecleaner design, Class. Quantum Grav. 21 (2004) S951 S958. [17] R. Adhikari, A. Bengston, Y. Buchler, et al, Input optics final design, Technical Report, T D, LIGO, (1998). [18] S. Nagano, M. A. Barton, H. Ishizuka, et al, Development of a light source with an injectionlocked Nd:YAG laser and a ring mode cleaner for the TAMA 300 gravitationalwave detector, Rev. Sci. Instrum. 73 (2002) [19] K. Kuroda, M. Ohashi, S. Miyoki, et al, Largescale cryogenic gravitational wave telescope, Int. J. Mod. Phys. D 8 (1999) [20] P. Barriga, C. Zhao, and D. G. Blair, Optical design of a high power modecleaner for AIGO, Gral. Relat. Gravit. 37 (2005) [21] F. J. Raab and S. E. Whitcomb, Estimation of special optical properties of a triangular ring cavity, Technical Report, T R, LIGO, (1992). [22] J. Betzwieser, K. Kawabe, and L. Matone, Study of the output mode cleaner prototype using the phasecamera, Technical Report, T D, LIGO, (2004).
173 4.14. REFERENCES 145 [23] T. Skettrup, T. Meelby, K. Færch, et al, Triangular laser resonators with astigmatic compensation, Appl. Opt. 39 (2000) [24] A. E. Siegman, Lasers, University Science Books, Sausalito, California, (1986). [25] H. Kogelnik and T. Li, Laser Beams and Resonators, Appl. Opt. 5 (1966) [26] J. D. Mansell, J. Hennawi, E. K. Gustafson,et al, Evaluating the effect of transmissive optic thermal lensing on laser beam quality with SchackHartmann wavefront sensor, Appl. Opt. 40 (2001) [27] K. A. Strain, K. Danzmann, J. Mizuno, et al, Thermal lensing in recycling interferometric gravitational wave detectors, Phys. Lett. A 194 (1994) [28] D. Mudge, M. Ostermeyer, D. J. Ottaway, et al, Highpower Nd:YAG lasers using stableunstable resonators, Class. Quantum Grav. 19 (2000) [29] B. T. Lantz, Quantum limited optical phase detection in a high power suspended interferometer, PhD Thesis, Department of Physics, Massachusetts Institute of Technology, Chapter 4, (1999). [30] P. J. Barriga, C. Zhao, and D. G. Blair, Astigmatism compensation in modecleaner cavities for the next generation of gravitational wave interferometric detectors, Phys. Lett. A 340 (2005) 1 6. [31] P. Hello and J. Y. Vinet, Analytical models of thermal aberrations in massive mirrors heated by high power laser beams, J. Phys. I France 51 (1990)
174 146 CHAPTER 4. MODECLEANER OPTICAL DESIGN
175 Chapter 5 Diffraction Losses and Parametric Instabilities 5.1 Preface The previous chapter presented the optical design of an input modecleaner for advanced interferometric GW detectors. The improved understanding of higher order optical modes obtained through this work led to the study of their behaviour in the arm cavities of the interferometer. Due to the high circulating power, higher order optical modes interact with the test mass acoustic modes in the arm cavities. If this threemode optoacoustic parametric interaction has enough gain, the arm cavities could potentially be driven out of lock, leaving the interferometer inoperable. It was therefore very important to understand this phenomenon and determine the parametric gain as accurately as possible. However there are several parameters that contribute to the parametric gain and in most cases they are not independent from each other. One is diffraction losses, which depend on the design parameters of the arm cavities. Normally they are treated with simple approximations that, as presented in this chapter, do not give a realistic result. Diffraction losses affect not only the power buildup of each mode, but also the frequency shift, the mode gain and its Qfactor, all of which contribute to the possibility of parametric instabilities. This research was started in collaboration with Biplab Bhawal from the California Institute of Technology in Pasadena, USA, and also coincided with the author s visit to the LIGO Hanford site as part of the science monitors program during the LIGO S5 science run. This visit allowed the author to further discuss parametric instabilities with the wider scientific community. In particular Phil Willems, who hosted the author s visit to the Caltech laboratory, gave the author several ideas on how to 147
176 148CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES improve the code developed for the simulations. Riccardo DeSalvo in Pasadena also encouraged the author to continue this research, suggesting study of the apodising coating and the possible benefits in reducing the parametric gain. At the core of this chapter is the paper that condenses these studies. This is followed by an extract of a technical report published for the LSC as a LIGO technical report T Z and an extract of an internal technical report which was presented at the International Parametric Instabilities Workshop held in Gingin in This presentation can be found at The introductions of both reports have been reduced due to similarities with the main introduction of the published paper. A paper published by the UWA group is presented in Appendix B for better understanding of parametric instabilities and related control strategies.
177 5.2. INTRODUCTION 149 Numerical calculations of diffraction losses and their influence in parametric instabilities in advanced interferometric gravitational wave detectors Pablo Barriga 1, Biplab Bhawal 2,, Li Ju 1, David G. Blair 1 1 School of Physics, The University of Western Australia, Crawley, WA 6009, Australia. 2 LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA. Knowledge of the diffraction losses in higher order modes of large optical cavities is essential for predicting threemode parametric photonphonon scattering, which can lead to mechanical instabilities in long baseline gravitational wave detectors. In this paper we explore different numerical methods in order to determine the diffraction losses of the higher order optical modes. Diffraction losses not only affect the power build up inside the cavity but also influence the shape and frequency of the mode, which ultimately affect the parametric instability gain. Results depend on both the optical mode shape (order) and mirror diameter. We also present a physical interpretation of these results. 5.2 Introduction In order to detect gravitational waves large laser interferometers have been built with arms formed by FabryPerot cavities stretching up to 4 km. The actual interferometers are very close to their design sensitivity, but this may still not be enough to detect gravitational waves. In order to increase their sensitivity, advanced laser interferometer gravitational wave detectors will require much higher circulating optical power. The high power increases the high frequency (>100 Hz) sensitivity, but also enhances undesired effects including the possibility of parametric instability, which Now at Google Inc., CA 94043, USA
178 150CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES were first predicted by Braginsky et al [1] with further study by Kells et al [2] and Zhao et al [3]. Parametric instabilities in advanced gravitational wave interferometers are predicted to arise due to a 3mode optoacoustic resonant scattering process in which the cavity fundamental mode w 0 scatters with test mass acoustic modes w m and optical cavity modes w 1, which satisfy the condition w 0 w m + w 1. The parametric gain R 0 for this process determines whether the system is stable (R 0 < 1) or unstable (R 0 1). Three mode optoacoustic parametric processes have not yet been observed. However 2mode processes which also couple optical and mechanical degrees of freedom via radiation pressure have been observed in resonant bar gravitational wave detectors with microwave resonators readouts [4]. More recently they have been observed in optical microcavities with very high Q [5], and low frequencies in short ( 1 m) suspended optical cavity [6]. In all these cases the mechanical mode frequency is within the electromagnetic mode linewidth. It has been shown that for an Advanced LIGO type of interferometer with fused silica test masses the parametric gain R 0 will typically have a value of 10 [7]. The parametric gain scales directly as the mechanical Q factor of the test masses and the optical Q factor of higher order modes. Hence errors in the Q factor of higher order modes directly affect the estimation of R 0. Thus it is very important to have an accurate estimation of the diffraction losses of the modes. Equation (5.1) shows the parametric gain in a power recycled interferometer. R 0 2P Q m ( Q 1 Λ 1 McLwm w1/δ Q 1a Λ 1a 1 + w1a/δ 2 1a 2 ). (5.1) Here P is the total power inside the cavity, M is the mass of the test mass, Q 1(a) are the quality factors of the Stokes (antistokes) modes, Q m is the quality factor of the acoustic mode, δ 1(a) = w 1(a) /2Q 1(a) corresponds to the relaxation rate, L is the cavity length, w 1(a) = w 0 w 1(a) w m is the possible detuning from the ideal resonance case, and Λ 1 and Λ 1a are the overlap factors between optical and acoustic modes. The overlap factor is defined as [1]: Λ 1(a) = V ( f 0 ( r )f 1(a) ( r )u z d r ) 2 f0 2 d r f1(a) 2 d r u 2 dv. (5.2)
179 5.2. INTRODUCTION 151 Here f 0 and f 1(a) describe the optical field distribution over the mirror surface for the fundamental and Stokes (antistokes) modes, respectively, u is the spatial displacement vector for the mechanical mode, u z is the component normal to the mirror surface. The integrals d r and dv correspond to integration over the mirror surface and mirror volume respectively. Traditionally it has been assumed that diffraction losses can be estimated by the clipping approximation. In this approximation it is assumed that the mode shape is not altered by the finite mirror geometry, and that the diffraction loss is simply determined by the fraction of the mode that overlaps the mirror. It has been demonstrated, for the fundamental mode of a two mirror symmetric cavity, that the geometry of the system determines whether the clipping approximation overestimates or underestimates the diffraction losses [8, 9]. Preliminary work by D Ambrosio et al [10] indicates that the clipping approximation is not valid for this particular case. For this reason we have undertaken careful numerical modelling of the mode losses based on the free propagation of the beams inside a long baseline interferometer. Using Fast Fourier Transform (FFT) simulations we determine the diffraction losses inside the main arms of a proposed advanced interferometer configuration. We present results for typical arm cavity design for advanced interferometers, based on the proposed design of Advanced LIGO (Laser Interferometer Gravitationalwave Observatory). We show that for these very large cavities the mode frequencies are shifted by a significant amount; and that the size of the mirror not only affects the diffraction losses but also the cavity gain, mode frequency, Qfactor and the mode shape. All these parameters are necessary to determine the overlap factor Λ and greatly affect the predicted parametric gain R 0. First we introduce the FFT simulation method to simulate the behaviour of a gaussian beam inside a cavity. In the next section this is applied to calculate the diffraction losses from HermiteGaussian (HG) and LaguerreGaussian (LG) modes in an advanced gravitational wave detector arm cavity, analysing also the limits of the method. In section 4 these results are compared with results from eigenvalues simulations. We also analyse the change in diffraction losses when using finite mirrors
180 152CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES of different sizes, including their impact on the optical gain and mode frequency, the optical Qfactor and mode shape. 5.3 Diffraction losses Diffraction losses occur in any optical cavity with finite size optics, even if the mirrors are large compare to the Gaussian spot size. The larger the Fresnel number, the weaker is the field intensity at the edge of the mirror, and smaller the power loss due to diffraction loss [11, 12]. If we consider a FabryPerot cavity (like the proposed Advanced LIGO arm cavities) with mirrors of diameter 2a and spot size radius we can say that only those modes of order less than the ratio of the two areas will oscillate inside the cavity with relatively low losses [13]. ( ) πa 2 N max. (5.3) πw 2 We use the proposed design of Advanced LIGO as a case in point. It has mirrors of 34 cm diameter with a spot size of approx 6 cm, which gives us N max = Therefore we will analyse modes up the 8 th order. We can expect that the cavity losses for higher order modes will rapidly increase with mode number. The clipping approximation determines the part of the higher order mode spot size that will fall outside the mirror s surface when the mode shape itself is due to that of an infinite diameter mirror. The diffraction loss in each reflection of a cavity mode off a mirror is given by: D clip = a U(r) 2 2πdr. (5.4) Here U(r) is the normalised field of a HG or LG mode with infinite size mirrors integrated outside a mirror of diameter 2a. It is already known that the clipping approximation yields a smaller loss than the calculations based in FFT methods for the T EM 00 mode in a long optical cavity. The FFT method enables the mode shape changes due to the finite mirror sizes to be estimated and hence enables a much better approximation of the diffraction losses.
181 5.3. DIFFRACTION LOSSES 153 A good explanation of this method can be found in ref [14], and a more general explanation in ref [15]. The following calculation relates the internal power with the diffraction losses due to finite size mirrors. Let T i, D i and L i be the transmission, diffraction and dielectric losses respectively for the Input Test Mass (ITM) and T e, D e and L e be the corresponding values for the End Test Mass (ETM). The finesse of such cavity can then be calculated as [16]: F = 2π T i + D i + L i + T e + D e + L e. (5.5) Since we are interested in the effects of the diffraction losses over the circulating power inside the cavity in steady state, we assume perfect mode matching for the input beam. Therefore the peak value for the circulating intensity in a purely passive cavity at resonance can be written as [17]: I circ 4(T i + L i ) (T i + D i + L i + T e + D e + L e ) 2 I inc. (5.6) In the case of infinite size mirrors there are no diffraction losses, then D i = D e = 0. For our simulations we use the following values, which coincide with proposed parameters for Advanced LIGO interferometers, where T i = 5000 ppm, T e = 1 ppm and L i = Le = 15 ppm [18]. In this case it is clear that the major loss contribution comes from the transmission losses of the ITM. Note that equation (5.6) does not contain a mode matching parameter because of the assumption of perfect mode matching. This assumption allows us to refer the incident beam to the cavity waist, this simplifies the analysis. In order to calculate the diffraction losses we use a lossless cavity in parallel with the cavity under study. After each round trip the resulting beam is normalised and propagated in to a lossless cavity, this cavity has the same characteristics of the cavity under study but with lossless mirrors. In such cavity diffraction losses are the only cause of power loss. Therefore we can calculate the diffraction losses per round trip using equation (5.7), where P Norm j corresponds to the normalised total circulating power per round trip. In the case of infinite sized mirror P Norm j diffraction losses. is always 1, hence no
182 154CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES D clip = 1 P Norm j. (5.7) This method proved to be more accurate than measuring the diffraction losses on every round trip using the incremental power of every round trip as shown in equation (5.8). This method is more susceptible to numerical errors due to the digitisation of the beam. D = ( 1 p ) j (T i + L i + T e + L e ). (5.8) p j 1 Here p j corresponds to the power contribution of round trip j. Therefore p j 1 corresponds to the power contribution of the previous round trip. A third way to estimate the diffraction losses is to calculate the eigenvalues for the cavity. At the end of section we comment on the verification of our results based on comparison with eigenvalues calculations. This eigenvalues γ mn are such that after one round trip the eigenmodes will satisfy the simplified round trip propagation expression [19]: γ mn U mn (x, y) = K(x, y, x 0, y 0 )U mn (x 0, y 0 )dx 0 dy 0. (5.9) The magnitude of the eigenvalues due to the round trip losses will be less than unity. Therefore we can calculate the power loss per round trip as: Power loss per round trip = 1 γ mn 2. (5.10) We now go on to present the details of our FFT simulation methods. 5.4 FFT simulation Since each mode has a different resonant frequency depending on the order of the mode we must analyse each mode separately. The first step is to generate the mode we are interested in. Part of the study also includes the comparison between HG and LG modes. For simplicity the mode is generated at the waist of the cavity. HG modes are given by:
183 5.4. FFT SIMULATION 155 U m,n (x, y, z) = ( ) exp {j(2m + 2n + 1)ψ(z)} 2π 2 m 2 n m!n!w(z) 2 ( ) ( ) 2x 2y H m H n exp w(z) { j2kz jk w(z) ( ) x 2 + y 2 x2 + y 2 2R(z) w(z) 2 }. (5.11) Here m and n correspond to the order of the transverse modes, w(z) is the spot size radius, R(z) is the beam radius of curvature, ψ(z) is the Gouy phase shift, k is the wave number and H m () is the m th order Hermite polynomial. LG modes are also a valid representation of the higher order modes, this time in cylindrical coordinates rather than rectangular, and given by: U l,m (r, φ, z) = ( 4l! exp {j(2l + m + 1)ψ(z)} (1 + δ 0,m )π(l + m)! w(z) ( ) m ( ) { 2r 2r 2 r 2 L l,m exp jkz jz w(z) w(z) 2 2R(z) r2 w(z) 2 ) cos(mφ) }.(5.12) Here l corresponds to the radial index and m to the azimuthal mode index, L l,m () are the generalised Laguerre polynomials, δ 0,m = 1 if m = 0 and δ 0,m = 0 if m > 0. The rest of the variables are the same as in the HG modes. The Fourier transform corresponds to the transformation of the beam profile into a spatial frequency domain. We can create a propagation matrix based on an expansion of the optical beam in a set of infinite plane waves travelling in slightly different directions [20] given by A(p, q, z L ) = exp { } jkz L + jπλ(p 2 + q 2 )z L. (5.13) Where z L corresponds to the distance which we will propagate the beam, p and q are the coordinates in the Fourier space or the spatial frequencies. To apply this propagation matrix it is also necessary to transform the field of the input beam by using a two dimensional FFT. The Fourier transform of a gaussian function is always another gaussian transform of the same order: i.e.
184 156CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES F {U(x, y, z)} = U(x, y, z)e jpx e jqy dxdy. (5.14) and the inverse Fourier transform is then written as: F 1 {U(p, q, z)} = ( ) 2 1 2π U(p, q, z)e jpx e jqy dpdq. (5.15) Once the beam has been propagated by multiplying point to point both the field matrix and the propagation matrix we can transform the field back to the time domain using the two dimensional inverse FFT. If z 0 is the propagation starting point then the final field corresponds to: U(x, y, z L ) = F 1 {F {U(x, y, z 0 )} A(p, q, z L )}. (5.16) Thus, this method basically consists of transforming the input field into the Fourier domain in two dimensions and propagation of this field along the z axis. Then the resulting field is transformed back to the time domain using the inverse FFT. Based on this principle we developed our own code in Matlab R, which allowed us to propagate the field and to reflect it off the mirrors surface. Two independent codes were developed, one at the University of Western Australia (UWA) and the other at the California Institute of Technology (Caltech), for the purpose of verification of results. For the simulation we assume a perfect surface for the test masses, but any imperfection can easily be added. Starting from the waist of the cavity we propagate the beam down to the ETM where it is reflected from the mirror surface and propagate it back through the waist to the ITM. Here part of the beam is transmitted out of the cavity through the substrate, and the rest is reflected back to the waist of the cavity completing the round trip. This is then iterated until the power inside the cavity has reached the steady state. In order to calculate the diffraction losses for different higher order modes it was necessary first to make the modes resonant inside the cavity. Starting from a nominal value of 4 km for the cavity length we move the ETM away from the ITM up to a maximum of half the laser wavelength (λ = µm) until we find the cavity length that maximises the circulating power for a particular mode. For each small step that the ETM is moved several round trips are done to calculate the power builtup inside
185 5.5. RESULTS 157 the cavity. Once the cavity is set at the resonance length we propagate the mode inside the cavity for several round trips until the circulating power reaches the steady state, which corresponds to the maximum circulating power. Most of the calculations presented here were done using a elements grid. This proved to be good for the calculations of the parameters we are interested in, for the different modes. However, it was observed that for higher order modes (order higher than 6) a grid of is not enough when using mirrors of infinite size due to aliasing. In those cases we increased the size of the grid to a , using also finer elements. Another option could have been the use of antialiasing filters or an adaptive grid. 5.5 Results In order to determine the parametric instability gain we need to know more than just the diffraction losses of the higher order modes. Therefore we also calculated the optical gain, the frequency, and the optical Qfactor of each mode. We also examined the mode shapes, and how they change with the size of the mirrors. In each case the calculations were done separately for each mode. We explored the variation of the size of the mirrors that form the cavity, while keeping the mirror radius of curvature and losses constant. The results were crossed checked between UWA and Caltech finding very good agreement between both simulations. A cavity formed by two mirrors with the losses previously mentioned and a constant radius of curvature of 2076 m at a nominal length of 4000 m was assumed for these simulations. On each simulation a pair of mirrors of the same diameter was used. For all our simulations it was assumed that the substrate of the test masses will have two flat sides proportional to the substrate diameter for suspension attachment as shown in figure 5.1. We also assumed that the mirror coating covers the whole front surface of the test mass (surface 1 in figure 5.1).
186 158CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES Figure 5.1: Advanced LIGO substrate dimensions for ITM and ETM test masses showing the flat sides of the test mass for suspension attachment. From LIGO drawing D B Diffraction losses The diffraction losses for different modes for varying mirror size are presented in figure 5.2. As expected the diffraction losses are the same for modes of order 1 like HG01, HG10 and LG01. For HG modes the order number is given by (m + n) while for LG modes the order number is given by (2l + m). It is clear (as previously showed by Fox and Li in ref [11]) that the higher the order of the mode the higher is the loss, especially for smaller mirrors (smaller Fresnel number). Diffraction losses can be separately analysed by the order of the mode, but when the order of the mode increases we need to take in to account the symmetry of the modes. For example mode HG60 is mainly distributed along one axis compared to HG33, which is of the same order but is evenly distributed on the mirror s surface. In such case and due to the energy distribution of each mode the losses for mode HG33 are much smaller than mode HG60. From figure 5.2 we can deduce that when the order of the mode is increased the grouping of the diffraction losses spreads out due to the different symmetries of the modes. If we compare the diffraction losses of modes of order 6 and 7 (figures 5.2 (c) and 5.2 (d) we notice that there is an overlap of some modes. Therefore the low loss modes of order 7 like LG07 have lower diffraction losses than the high loss modes of order 6, namely LG22 and LG30. Even though it is difficult to distinguish in figure 5.2, modes HG11 and LG02 have
187 5.5. RESULTS 159 (a) (b) Diffraction losses [ppm] 10 6 Diffraction losses for modes up to order HG00 HG01 HG10 LG01 LG02 HG11 HG20 LG10 LG03 HG12 HG30 LG11 LG04 HG22 HG13 HG40 LG12 LG20 Diffraction losses [ppm] Diffraction losses for 5 th order LG05 HG23 HG14 LG13 HG50 LG Mirror radius size [cm] Mirror radius size [cm] (c) (d) Diffraction losses [ppm] Diffraction Losses for 6 th order LG06 HG33 HG24 HG15 LG14 HG60 LG22 LG30 Diffraction losses [ppm] Diffraction Losses for 7 th order LG07 HG43 HG52 LG15 HG16 LG23 HG70 LG Mirror radius size [cm] Mirror radius size [cm] Figure 5.2: Diffraction losses for different higher order modes. Starting from the top left corner we present modes up to order 4 in ascendant orders from HG00 up to LG20. For clarity we separately present modes of order 5, 6 and 7. HG and LG modes are plotted together for comparison. almost the same losses. This is easy to explain since we can see in figure 5.3 that LG02 corresponds to HG11 twisted by 45 o (or viceversa) and therefore orthogonal to each other. Therefore the difference in diffraction losses comes from the flat sides of the test mass affecting differently each mode. It is also interesting to notice that the highest loss is in mode LG10, which is a more symmetric one, but has more energy at the edge of the mirror, compared with other 2 nd order modes. We have compared our results with calculations of the diffraction losses using the cavity eigenvalues by Juri Agresti from Caltech [21]. He calculated the diffraction losses for several LG modes. His results are in very close agreement with the results
188 160CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES Figure 5.3: Intensity profile at the ITM in a mirror of diameter 34 cm. On the left hand side we can see mode HG11 compared to mode LG02 in the middle. Also for comparison the energy distribution of mode LG10 is presented on the right hand side. here presented. The average difference between the FFT simulation and the eigenvalue approximation is of 1%, while the biggest difference is less than 5% for lower order modes. However his calculations were done for a previous design of Advanced LIGO test masses with smaller mirrors of 31.4 cm in diameter, which we also use for comparison. In parallel with the FFT simulations and the eigenvalues calculations done at Caltech we did our own eigenvalues calculations based in an eigenvector method proposed by C. Yuanying et al [22]. The results obtained through this method showed that in a perfect aligned cavity with cylindrical test masses (circular mirrors) only LG modes and their rotated orthogonal modes will resonate. However the need of suspend the mirrors requires the test masses to have two flat sides as can be seen in figure 5.1 (LIGO technical document D B). This breaks the symmetry. As the circular symmetry is broken when solving with the eigenvector method for this cavity it shows that HG modes are now part of the eigenvectors solution of this cavity. Therefore HG modes are partially supported by the cavity even if it is perfectly aligned. However these modes are mainly distributed along the horizontal axis aligned with the flat sides of the test mass. The symmetry break not only changes the eigenvectors and eigenvalues solution for this system, but also induces a particular orientation of the higher order mode that minimises the diffraction losses. The main difference with the eigenvector method is that in the FFT simulation
189 5.5. RESULTS 161 Diffraction losses comparison between FFT and Eigenvalues Diffraction losses (ppm) 1.E+06 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 HG01 LG01 HG11 HG02 LG02 LG10 HG12 HG03 LG03 LG11 HG22 HG13 HG04 LG04 LG12 LG20 HG14 HG05 HG23 LG05 LG13 LG21 HG15 HG24 HG33 HG06 LG06 LG14 LG22 LG30 HG16 HG43 HG52 HG07 LG07 LG15 LG23 LG31 HG08 HG17 HG26 HG35 HG44 LG08 LG16 LG24 LG32 LG40 1.E+00 HG00 1.E Optical modes FFT Results Eigenvalues Figure 5.4: Comparison of diffraction losses obtain with FFT simulations and the eigenvector method for the proposed Advanced LIGO type cavity. we can choose which mode we are going to propagate inside the cavity. The FFT simulation allows the mode to change its shape while it propagates inside the cavity. In order to determine when the mode shape is stable we calculate the nonorthogonality between the input and the circulating beam, this subject to the finesse of the mode and the power to build up in the cavity. Therefore the mode labels in the graphs correspond to the input mode used in the FFT simulation and not necessarily correspond to that of the final mode shape. Figure 5.4 show the diffraction losses results obtained with both methods. The results are in close agreement, but we noticed that some of the modes that we injected are not supported by the cavity. These modes in fact do not appear in the eigenvector method; moreover we can see that those modes are the ones that their mode shape changes in to a lower loss mode of the same order. This explains why modes HG15, HG24, HG33 have all similar losses as mode LG06. We further analyse the mode shape changes in section
190 162CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES Optical gain Considering the parameters here presented for Advanced LIGO type optical cavities, the optical gain of the main arms is about 800. Assuming infinite size mirrors and the losses already presented for this cavity, the gain that we obtain from the simulations is close to 793. Using mirrors of finite size increases the diffraction losses, thus reducing the gain. This can also be deduced from equation 5.6. Gain 10 0 Optical Gain for higher order modes HG00 HG01 LG02 HG20 LG10 LG03 HG30 LG04 HG40 LG05 LG20 LG06 HG50 LG07 HG60 LG30 HG Mirror size radius [cm] Figure 5.5: Cavity optical gain for some HG modes of different orders. The modes have been plotted in descendant order, starting from the top with HG00 to finish with HG70 at the bottom. HG and LG modes are plotted together for comparison. For a given finite mirror size the higher the order of the mode, the higher the diffraction losses. This in turn means lower optical gain and lower finesse for the higher order mode, which as a consequence implies a reduction of the circulating power. Figure 5.5 shows the optical gain for some of the HG and LG modes. The figure shows how the gain of each order changes with the mirror size (or Fresnel number). The fundamental mode doesn t change much, but as the order of the mode increases the diffraction losses increase reducing the gain. With infinite size mirrors all the modes have the same gain since there are no diffraction losses. Using the nominal Advanced LIGO mirror size (34 cm diameter) we can plot the
191 5.5. RESULTS 163 Higher Order Modes Optical Gain 1.E+03 1.E+02 LG23 LG31 HG00 HG01 LG01 HG11 HG20 LG02 LG10 HG12 HG30 LG03 LG11 HG22 HG13 HG40 LG04 LG12 LG20 Gain LG13 LG21 HG15 HG24 HG33 HG60 LG06 LG14 HG16 HG43 HG52 LG07 LG22 LG30 HG70 LG15 HG08 HG17 HG26 HG35 HG44 LG08 LG16 LG24 LG32 LG40 HG14 HG50 HG23 LG05 1.E+01 1.E+00 1.E Optical Modes Figure 5.6: Optical gain variation for higher order modes for a mirror diameter of 34 cm TEM 40 intensity profile for different mirror size Inf 36 cm 34 cm 32 cm 31.4 cm 30 cm 28 cm Intensity Mirror size [m] Figure 5.7: Intensity profile variation of mode HG40 due to the different mirror size. optical gain versus mode number as shown in figure 5.6. Here we notice how the gain is reduced for a particular mirror size when the order of the mode increases. Again we can see the dependency on the energy distribution and symmetry of the mode.
192 164CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES The gain reduction due to the finite size of the mirrors can also be appreciated in figure 5.7, which shows the effect in the intensity profile of mode HG40. We can see how the optical gain is reduced with the size of the mirrors. The use of smaller mirrors increases the diffraction losses leading to smaller gain as shown in table 5.1. This effect is even stronger in higher order modes Mode frequency As previously mentioned the calculation of the resonance length was done by moving the ETM away from the ITM until the circulating power is maximised. The resonance length is different for each mode, but when the mirror size was changed a minor variation in resonance length was noticed. This suggests that mirrors of different size will also alter the resonance conditions of the cavity, thus changing the mode frequency. It is well known that the frequency shift for higher order modes is given by [23]: υ 0 π (m + n) arccos( g 1 g 2 ) for HG modes, (5.17) υ 0 π (2l + m) arccos( g 1 g 2 ) for LG modes. (5.18) Here υ 0 corresponds to the Free Spectral Range (FSR) in Hz, g 1 and g 2 correspond to the stability factor of each mirror, define as g = (1 L/R), L being the cavity Mirror Diameter Diffraction Losses Optical Gain (cm) (ppm) Table 5.1: size. Diffraction losses and cavity optical gain of mode HG40 for different mirror
193 5.5. RESULTS 165 length, and R the radius of curvature of the mirror. Thus to calculate the frequency separation of a higher order mode with respect to the fundamental mode we use the following relation: f l = 1 L f Y AG + N π c 1. (5.19) 2L 2 (2LR L 2 1/2 ) Where f corresponds to the frequency variation of the mode, l to the cavity length variation and f Y AG to the Nd : Y AG laser fundamental mode frequency. Here N = (m + n + 1) for HG modes and N = (2l + m + 1) for LG modes, c corresponds to the speed of light in vacuum and R the radius of curvature of the mirrors. We note that in reality the laser is locked to the T EM 00 cavity mode. Equation (5.19) is special case for cavities with two mirrors of the same radius of curvature and as a consequence with g 1 = g 2. Using equations (5.17) and (5.18) the mode separation per mode order using the proposed Advanced LIGO parameters is khz. According to our simulations (a) (b) (c) (d) (e) (f) (g) (h) Figure 5.8: Modes of order 7 as they would appear in an infinite sized mirror. Therefore no diffraction losses or mode shape changes affect these modes. Figures (a), (b), (c) and (d) correspond to modes HG07, HG16, HG25 and HG34. Figures (e), (f) (g) and (h) correspond to modes LG07, LG15, LG23 and LG31.
194 166CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES this is true only when using mirrors of infinite size. The use of mirrors of finite size introduces diffraction losses which not only reduce the circulating power but also shift the frequency of the higher order modes. First we consider the fundamental mode where the frequency deviations are negligible, of the order of millihertz. The smallest mirror in our simulations is of 28 cm in diameter. In this case the frequency variation from the infinite mirror case is about Hz, which compared to the laser fundamental mode frequency, is a variation of For a mirror of 34 cm in diameter the frequency variation is reduced to Hz. These results show a clear agreement with the diffraction losses for modes of order 7 shown in figure 5.2 (d). The highest diffraction losses are from mode LG31, while the lowest losses are from mode LG07, which also has the smallest frequency variations from the infinite mirror case. These results show that the frequency not only depends on the mode order, but also on the symmetry and energy distribution of the mode subject to the size of the mirror. The frequency variation will also have an impact on the possible parametric instabilities calculations. Figure 5.9 show the frequency variations for the higher order modes in the proposed cavity with mirrors of 34 cm in diameter. Cavity losses define the optical Qfactor, which for any given mode is given by: Mirror f req f req f req f req f req f req f req f req Diameter HG70 HG61 HG52 HG43 LG07 LG15 LG23 LG31 (cm) (Hz) (Hz) (Hz) (Hz) (Hz) (Hz) (Hz) (Hz) Table 5.2: Frequency shift of modes of order 7 for different size mirrors compared to the frequency of the same mode when using infinite sized mirrors.
195 5.5. RESULTS 167 Higher Order Modes Frequency Shift Delta Frequency (Hz) 1.E+04 1.E+03 1.E+02 1.E+01 1.E+00 1.E01 1.E02 HG01 LG01 HG11 HG20 LG02 LG10 HG12 HG30 LG03 LG11 HG22 HG13 HG40 LG04 LG12 LG20 HG14 HG50 HG23 LG05 LG13 LG21 HG15 HG24 HG33 HG60 LG06 LG14 LG22 LG30 HG16 HG43 HG52 HG70 LG07 LG15 LG23 LG31 HG08 HG17 HG26 HG35 HG44 LG08 LG16 LG24 LG32 LG40 1.E03 HG00 1.E Optical Modes Figure 5.9: Frequency variations from the theoretical value (infinite size mirror) for each higher order mode when using finite size mirrors of diameter 31.4 cm. Q = w 2δ T. (5.20) Here w corresponds to the frequency of the mode and δ T to the relaxation rate of that particular mode [1]. Therefore it is expected that the higher the diffraction losses the lower the optical Q of that mode. The Qfactor of the optical modes has a direct effect on the parametric gain R 0 therefore a reduction of this factor will also reduce the parametric gain. The Qfactor of the optical modes will also have a dependency on the size of the mirrors, since both the frequency and the losses depend on the size of the mirror as well. Figure 5.10 shows the optical Qfactor for the higher order modes for the 34 cm diameter mirrors. Here we can see that the Qfactor follows the same trend as in the optical gain, which also is inversely proportional to the total losses of the cavity Mode shape Higher order modes have a more spread intensity profile. As a consequence depending on the symmetry of the mode, it will cover a larger area of the mirror s surface. This not only causes high diffraction losses due to the energy loss per round trip, but in
196 168CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES Optical Q factor for higher order modes 1.E+13 Q factor 1.E+12 1.E+11 HG00 HG01 LG01 HG11 HG20 LG02 LG10 HG12 HG30 LG03 LG11 HG22 HG13 HG40 LG04 LG12 LG20 HG14 HG50 HG23 LG05 LG13 LG21 HG15 HG24 HG33 HG60 LG06 LG14 LG22 LG30 HG16 HG43 HG52 HG70 LG07 LG15 LG23 LG31 HG08 HG17 HG26 HG35 HG44 LG08 LG16 LG24 LG32 LG40 1.E Optical Modes Figure 5.10: Optical Qfactor for the higher order modes in the proposed cavity. some cases it also distorts the mode. The overlap parameter Λ used to calculate the parametric gain R 0 depends on the mode distribution over the mirror s surface [3]. An interesting case is mode HG33. This mode resonates and keeps its shape inside the cavity if the mirrors are of infinite size (figure 5.11 (a)), but when using finite size mirrors the mode is completely distorted and after a few round trips it doesn t look like a mode HG33 anymore, but more like mode LG06, although twisted by 30 o and thus orthogonal to LG06 as can be seen in figure The distorted mode HG33 still show some features from the original mode like the small energy distribution at the centre of the intensity profile. However it is easy to see the similarity and why the diffraction losses are higher for mode HG33 due to the energy loss at the edge of the mirror. The interpretation is that even if we forced mode HG33 in to a perfectly aligned cavity it will not resonate inside the cavity and will give rise to a twisted LG06 mode (figure 5.11 (b)). This mode will also be one of the eigenvectors of the cavity since it is orthogonal to the original LG06 shown in figure 5.11 (c). Similar effects were observed for modes HG43 and HG52.
197 5.6. CONCLUSIONS 169 (a) (b) (c) Figure 5.11: (a) Intensity profile of mode HG33 in a infinite sized mirrors. The next two intensity profiles are for finite sized mirrors of diameter 34 cm. (b) Shows the intensity profile of mode HG33 on the ITM at end of the simulations. (c) Shows the intensity profile of mode LG06 also for finite size mirrors. 5.6 Conclusions We have shown how the diffraction losses of various modes in large optical cavities depend on the diameter of the mirrors. Moreover they also depend on the shape of the mirror. We show that the predicted mode frequencies are also offset from the infinite mirror case by up to a few khz. The diffraction losses are needed to determine the optical Qfactor of each mode, and this combined with mode frequency data is necessary to estimate the possibility of parametric instabilities, through the calculation of the parametric gain. We have shown that finite size mirrors significantly alters the shape of the higher order modes, and due to high diffraction losses on each round trip also the optical gain is reduced. The mode shape variations affect the overlap integral calculation which determines the optoacoustic coupling in the parametric instabilities calculations. We wish to point out that in a power recycled interferometer, the design of the power recycling cavity can vary the coupling losses so as to increase the high order mode losses. The high order losses can never be less than the diffraction losses predicted here. However it is also true in the case of coupled cavities the mode shape could be significantly altered (compared with the single cavity modes considered here), and this could vary the diffraction losses. The presence of a signal recycling cavity
198 170CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES can also affect the parametric instability. Acknowledgements The authors would like to thanks Erika D Ambrosio, Bill Kells and Hiro Yamamoto for useful discussions. Also thanks to Jérôme Degallaix for his help in developing the FFT code at UWA, and Juri Agresti for his eigenvalues calculations. David G. Blair and Li Ju would like to thanks the LIGO Laboratory for their hospitality. Biplab Bhawal is supported by National Science Foundation under Cooperative Agreement PHY This work was supported by the Australian Research Council, and is part of the research program of the Australian Consortium for Interferometric Gravitational Astronomy.
199 5.7. APODISING COATING Apodising coating Study of the possible reduction of parametric instability gain using apodising coating in test masses P. Barriga 1 and R. DeSalvo 2 1 School of Physics, University of Western Australia, Crawley, WA6009, Australia 2 LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA Introduction As part of the study of the possibility of the occurrence of parametric instabilities in advanced gravitational wave interferometers, it was suggested the use of an apodising coating for the test masses. The main goal is to reduce the parametric gain R 0 by increasing the diffraction losses of the high order optical modes keeping the fundamental mode diffraction loss below 1 ppm. This section concentrates on the effect of an apodising coating over the diffraction losses for an Advanced LIGO type of cavity. Diffraction losses changes will affect the frequency of the mode, the Qfactor and the total losses, also affecting the mode shape and the overlapping parameter as a consequence. Therefore it is not straight forward to determine the effect of diffraction losses changes over the parametric gain, but in general we need higher diffraction losses in order to reduce the parametric instability gain. The diffraction losses for an Advanced LIGO type of cavity have been previously calculated [16, 24]. The Fast Fourier Transform (FFT) method developed at The University of Western Australia allows us to inject any optical mode in to the cavity. Inside the cavity the mode is propagated using a FFT and thus the beam is free to change according to the resonance conditions imposed by the cavity parameters. Previous simulations show that a mode which is not supported by the cavity will morph in to a different mode of the same order but with lower diffraction losses. A change in
200 172CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES the mode shape is accompanied by the corresponding frequency shift. In such cases it is not possible to say that the nominal mode resonates inside the cavity. As a result the simulations presented here show the high order modes diffraction losses of the injected mode. Figure 5.12 shows a comparison of the diffraction losses obtained using a FFT simulation and the results obtained using the eigenvalues calculations for this cavity using an eigenvector method [22]. For these calculations a test mass of 34 cm in diameter was assumed according to LIGO drawing D B. A homogeneous coating with 50 ppm losses was also assumed. Diffraction losses comparison between FFT and Eigenvalues Diffraction losses (ppm) 1.E+06 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 HG01 LG01 HG11 HG02 LG02 LG10 HG12 HG03 LG03 LG11 HG22 HG13 HG04 LG04 LG12 LG20 HG14 HG05 HG23 LG05 LG13 LG21 HG15 HG24 HG33 HG06 LG06 LG14 LG22 LG30 HG16 HG43 HG52 HG07 LG07 LG15 LG23 LG31 1.E+00 HG00 1.E Optical modes FFT Results Eigenvalues Figure 5.12: Diffraction losses for an Advanced LIGO type cavity, comparison between FFT and eigenvalues calculations Apodising coating In order to compare the effect that an apodising coating will have on the diffraction losses the designs presented in figure 5.13 were tested using different values for L1, L2 and L3. Several simulations were done in order to determine the best coating absorption combination by comparing results based on the diffraction losses of the fundamental mode. In all simulations the same substrate was used according to the LIGO document E , which corresponds to a 34 cm diameter test mass including
201 5.7. APODISING COATING 173 the chamfer and the flat sides for suspension attachment. Let T i = 5000 ppm and L i = 15 ppm be respectively the transmission and dielectric losses for the Input Test Mass (ITM) and T e = 1 ppm and L e = 15 ppm the corresponding values for the End Test Mass (ETM) [18]. In this case it is clear that the major loss contribution comes from the transmission losses of the ITM. The results presented in the next section correspond to the more relevant ones. The simulations also show that the minimum coating size for the fundamental mode to have diffraction losses of 1 ppm corresponds to a circular coating with a diameter of 33.1 cm. This is assuming that outside the coating all photons will be loss. However by reducing the losses outside the central coating we are able to reduce the size of this central coating proportional to the reduction of the outer ring losses. For these simulations the same coating is assumed for both ITM and ETM. L1 L1 L cm L cm 17 cm 17 cm Figure 5.13: Proposed apodising coatings design for the ITM and ETM of an Advance LIGO type cavity. The main purpose is to simulate their influence on diffraction losses and ultimately their effect in the parametric gain R Results Figure 5.14 shows the diffraction losses for a standard homogeneous coating with L = 50 ppm, also other configurations with homogeneous coatings with losses of 1000 ppm and ppm have been considered. The figure also include two apodising coatings, one with losses given by L1 = 50 ppm and L2 = ppm (coating mean value of ppm) and one with losses given by L1 = 50 ppm and L3 = ppm
202 174CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES (coating mean value ppm). However due to the different energy distribution of the higher order modes the absorption losses of the apodising coatings will be different for each mode. However the simulation results presented in figure 5.14 show no big difference in terms of diffraction losses when using different coatings. Normal DiffLoss DS DiffLoss DiffLoss DiffLoss DS2 (L = 50) (L1 = 50 (L1 = 1000) (L = 10000) (L1 = 50 L2 = 25000) L3 = 10 5 ) mean L = mean L = Table 5.3: Diffraction losses for the fundamental mode HG 00 using the different coatings. All values in ppm. Table 5.3 shows the diffraction losses for different coating losses on the test masses. We can infer from the table that when using a homogeneous coating the higher the losses the lower the diffraction losses. When using an apodising coating the diffraction losses for the fundamental mode are increased, but always keeping them below 1 ppm. Diffraction Losses Coating Comparison (mirror = 34 cm) Diffraction losses (ppm) 1.E+06 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 1.E+00 HG00 HG01 LG01 HG20 HG11 LG02 LG10 HG30 HG12 LG03 LG11 HG40 HG13 HG22 LG04 LG12 LG20 HG50 HG14 HG23 LG05 LG13 LG21 HG60 HG15 HG24 HG33 LG06 LG14 LG22 LG30 HG70 HG16 HG52 HG43 LG07 LG15 LG23 LG31 1.E Optical modes (L = 50) (L1 = 50, L2 = 25000) (L = 1000) (L = 10000) (L1 = 50 L3 = ) Figure 5.14: Diffraction losses of higher order modes for an Advanced LIGO type of cavity using different combinations of homogeneous and apodising coatings. For higher order modes the diffraction losses comparison between the different
203 5.7. APODISING COATING 175 coatings will depend of the energy distribution on the beam profile. Therefore modes with higher energy distribution closer to the edge of the test mass will be more affected by the apodising coating. A similar effect can be seen in the shape of the mode which is also affected by the coating losses, which in consequence will affect the overlapping parameter and therefore the parametric gain R 0. Also due to the higher losses the gain of the cavity is reduced, thus the circulating power inside the cavity also drops. Diffraction Losses Ratio (mirror = 34 cm) Ratio HG00 HG01 LG01 HG20 HG11 LG02 LG10 HG30 HG12 LG03 LG11 HG40 HG13 HG22 LG04 LG12 LG20 HG50 HG14 HG23 LG05 LG13 LG21 HG60 HG15 HG24 HG33 LG06 LG14 LG22 LG30 HG70 HG16 HG52 HG43 LG07 LG15 LG23 LG Optical modes AP1/Normal (L=1000)/Normal (L=10000)/Normal AP2/Normal Figure 5.15: Ratio between the different coatings and a normal coating, assuming that the normal coating has homogeneous losses of 50 ppm. Figure 5.15 shows the ratio between the different coatings when compared to the standard coating, which is assumed to have homogeneous losses of 50 ppm. It is interesting to see that the biggest effect of the differential coatings is on the fundamental mode, with an increase of losses of 27%, which is reduced for the higher order modes. Most of the higher order modes have more or less similar diffraction losses except for the more symmetric HG modes. Figure 5.16 shows the comparison between the three different coatings, where DiffLoss 50 ppm corresponds to the original calculations with a homogeneous coating with dielectric losses of 50 ppm. DiffLoss AP1 corresponds to the first test with a central area of 50 ppm and an external ring of 5.7 cm with a loss of ppm. DiffLoss AP2 corresponds to a central area of 50 ppm as well, but with an external ring of 4.1 cm with a loss of a ppm. As we can see from figure 5.16 there is
204 176CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES Diffraction Losses Comparison for Different Coatings 1.E+06 Diffraction Losses (ppm) 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 HG01 LG01 HG11 HG20 LG02 LG10 HG12 HG30 LG03 LG11 HG22 HG13 HG40 LG04 LG12 LG20 HG14 HG50 HG23 LG05 LG13 LG21 HG15 HG24 HG33 HG60 LG06 LG14 LG22 LG30 HG16 HG43 HG52 HG70 LG07 LG15 LG23 LG31 1.E+00 HG00 1.E Optical Modes DiffLoss 50ppm DiffLoss AP1 DiffLoss AP2 Figure 5.16: Diffraction losses comparison between the original homogeneous coating and the two apodising coatings. not much difference between the different coatings, however we can still see that the biggest difference in terms of diffraction losses is for the lower modes, in particular the fundamental mode HG 00. The main difference however is in the coating absorption for each mode, while having a homogeneous coating shows homogeneous absorption it is not the case for the apodising coatings. In the case of an apodising coating the absorption will depend on the energy distribution of the mode, therefore it is also important the change in shape of the mode since it will also affect its absorption. This calculation was done by normalising the circulating power inside the cavity and integrating the field of each mode over the absorption map over the test mass surface. Figure 5.17 shows the results for the two different apodising coatings. Not surprisingly the coating absorption goes up with the mode order. As in the diffraction losses case this is cause by the mode shape changing inside the cavity. This is caused by the circular symmetry, which favours the resonance of LG modes more than HG modes. We have to remember that in this case no external means of exciting higher order modes have been considered, no mirror tilt, no mechanical resonance and no suspension residual noise for example. We can also notice in figure 5.17 that for the lower modes there is not much
205 5.7. APODISING COATING 177 Comparison of coating absorption 1.E+05 Coating Absorption (ppm) 1.E+04 1.E+03 HG11 HG20 LG02 LG10 HG12 HG30 LG03 LG11 HG22 HG13 HG40 LG04 LG12 LG20 HG14 HG50 HG23 LG05 LG13 LG21 HG15 HG24 HG33 HG60 LG06 LG14 LG22 LG30 HG16 HG43 HG52 HG70 LG07 LG15 LG23 LG31 1.E+02 HG00 HG01 LG01 1.E Optical Modes Absorption AP1 Absorption AP2 50 ppm Figure 5.17: Shows the different dielectric absorption for higher order modes. We notice that there is not much difference between the two coatings when analysing the lower modes, but there is a clear difference for the higher order ones. difference between the two coatings absorptions. But for the higher order modes, starting from order 3, the two curves start to show some difference. This is due to the energy distribution of the higher order modes and the fact that the outer ring of the second coating has higher losses even though it is slightly narrower Conclusions Based on these results the use of an apodising coating will increase the diffraction losses for several modes. Somewhat unanticipated was to see that the effect is bigger in the lower order modes. The diffraction loss change comes with the corresponding frequency shift and a Qfactor change for the optical mode. Consequently there is also a change in the mode shape due to the different coating losses. In general these changes are too small and any effect on the overlapping parameter will be negligible. As a result the effect of an apodising coating on the parametric instabilities gain R 0 for an advanced gravitational wave interferometer it is also expected to be negligible.
206 178CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES Acknowledgements The authors would like to thank Phil Wilems and Erika D Ambrosio for useful discussions. Pablo Barriga would like to thank the LIGO Laboratory for their hospitality. This work was supported by the Australian Research Council, and is part of the research program of the Australian Consortium for Interferometric Gravitational Astronomy.
207 5.8. MIRROR TILT Mirror tilt Effect of mirror tilt in higher order optical modes Introduction The physical parameters of the cavity like distance between mirrors, their radius of curvature, size and losses (substrate and coating) define the characteristics of the cavity. These include the beam waist and spot size at the input and end test mass (ITM and ETM), the final circulating power, the free spectral range, diffraction losses and the higher optical modes frequency separation. All these parameters are necessary to determine the possibility of parametric instability in the main arms of advanced gravitational wave detectors by the calculation of the parametric gain R 0 (shown in equation 5.1). As seen in section 5.2 in order to estimate the parametric gain R 0 accurately it is very important to correctly estimate the frequency, the diffraction losses and the Qfactor of the higher optical modes. First we review the effects of the geometry of the mirror in the higher order modes and the frequency separation. This defines the reference for the analysis on the effect that the tilt of the mirror has on the frequency of the higher order modes for an Advanced LIGO type of cavity. In general a mirror tilt will increase the cavity diffraction losses, depending on the ratio between the size of the mirror and the laser spot size. The simulations show that the tilt of the mirrors increases the diffraction losses changing the frequency separation of the higher order modes. As a consequence the Qfactor and the gain of the optical modes is also affected, which ultimately affects the power inside the cavity Mirror geometry An important parameter that needs to be taken in to account is the geometry of the test mass. The frequency of the transverse modes depends on the radius of curvature of the mirror and the cavity length. Normally infinite size mirrors and therefore symmetric are considered for the calculations of the frequency mode separation. It has been shown that the size of the mirror introduces a frequency shift from the
208 180CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES theoretical value expected for the frequency of the transverse modes, and this shift increases with the order of the modes [24]. In laser interferometric gravitational wave detectors it is necessary to suspended the optics in order to isolate the test mass from seismic noise. The proposed solution is then to cut two flat sides in opposite sides of the substrate as shown in figure 5.1 which allows for the bonding of the fused silica suspension elements. Assuming that the coating covers all of surface 1 we calculate the eigenvalues for an Advanced LIGO type cavity using an eigenvector method proposed by C. Yuanying et al [22]. The results obtained through this method showed that in a perfectly aligned cavity with symmetric cylindrical test masses (circular mirrors) only LaguerreGaussian (LG) modes and their rotated orthogonal modes will form the eigensolution. Due to the cylindrical symmetry of this system they are able to freely rotate and diffraction losses will be independent of the mode orientation, hence no preferred orientation for the optical modes on the test mass. In reality, we know that due to misalignments of the optics HermiteGaussian (HG) modes will also appear, but these can still be represented as a linear combination of LG modes. The current design for the suspension of the test masses requires them to have two flat sides for suspension attachment as shown in figure 5.1 (LIGO technical document D B). This breaks the cylindrical symmetry of the system and therefore of the mirror under the assumption that the coating covers the whole front surface of the substrate. This break of symmetry affects the eigenvectors of this cavity. There is a small increase in diffraction losses due to the smaller area of the high reflective coating, but there is also a preferred orientation for the eigenvectors, which minimise the loss for that particular mode, but in turn affects the orthogonal optical mode as shown in figure 5.19 using mode LG21 as an example. A similar case with mode LG03 can be seen in figure Figures 5.18, 5.19 and 5.20 show the effect of the flat sides of the test masses on the resonant optical modes inside the cavity. Using mode LG21 as an example we notice that in circular mirrors the mode can freely rotate on the mirror surface with no effect in diffraction losses. With the flat sides needed for the attachment of the suspension the symmetry is broken introducing an orientation of the optical mode. This orientation corresponds to the one that minimises the diffraction losses for that
209 5.8. MIRROR TILT 181 LG21 LG21 Figure 5.18: Optical mode LG 21 in a perfectly aligned cavity with circular mirrors. LG21 HG05? Figure 5.19: Optical mode LG21 in a perfectly aligned cavity with flat sides. The shape of the mirrors induce an orientation on the mode which minimises the diffraction losses, but worsens the orthogonal mode. particular mode. However this strongly affects the corresponding orthogonal mode now resembling a mode HG05, which is the result of having a square area within the surface of the mirrors. This can only occur if the spot size is big enough so the higher order modes intensity pattern is constrained by the edge of the mirror. In advanced interferometric gravitational wave detectors large spot size are required in order to minimise the test mass thermal noise [25]. Table 5.4 shows the diffraction losses and the shift in frequency from the theoretical value of khz (or infinite sized mirror case) for an Advanced LIGO type cavity. The same effect can be seen in all higher order modes with the exception of the modes
210 182CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES (a) LG03 LG03 (b) LG03 HG21? Figure 5.20: The figure shows another example of the effects of the flat sides for suspension attachment in the cavity eigenmodes and their orientation. (a) shows the case of a perfectly aligned cavity with circular mirrors where mode LG03 and its orthogonal mode (also LG03) are part of the eigensolutions for this cavity, (b) shows the case for the mirror with flat sides for mirror suspension where mode LG03 also resonates, but with a different orientation and the orthogonal mode resembles an HG21 optical mode. with circular symmetry like LG10, LG20, LG30,...etc. With the break of symmetry these modes do not form part of the eigensolutions for this cavity, but they could still resonate in the cavity if the conditions are right. As seen in figures 5.18 and 5.20 (a) in a cavity with circular mirrors an optical mode and its orthogonal mode are coupled. As a consequence they will have the same diffraction losses, mode frequency separation, optical gain and Qfactor. The
211 5.8. MIRROR TILT 183 Diffraction losses Frequency shift (ppm) (Hz) LG21 circular LG21 flat sides HG05? flat sides Table 5.4: Comparison for LG21 mode between circular mirrors and mirrors with flat sides for suspension attachment. Only one of the circular mirror cases is presented since the orthogonal mode has similar values. flat sides in the mirrors break this symmetry as seen in figures 5.19 and 5.20 (b). With the break in symmetry an optical mode will not have the same diffraction losses as its orthogonal mode. Figure 5.21 shows that the higher the order of the mode the larger the diffraction losses difference between orthogonal modes. This difference in diffraction losses also implies different frequency separation, different optical gain and optical Qfactor. Therefore the higher the order of the optical modes the larger these differences will be, and as a consequence a decoupling of the higher order optical modes doublets. The mirror size also plays an important role in the diffraction losses and the frequency shift of the higher order modes as seen in sections and Figures 5.22 and 5.23 show the influence of the mirror size on the frequency separation between the fundamental mode and the higher order modes. We notice that the higher the order of the mode the stronger the effect of the mirror size. This occurs due to the more spread pattern of the higher order modes and as a consequence the larger area of the energy distribution which increases the diffraction losses for these modes. The figures show the results for simulations using the Advanced LIGO design. However the secondary horizontal axis shows the ratio between the spot size and the mirror size and therefore can be applied to any advanced interferometer design.
212 184CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES Mirror geometry and diffraction losses Diffraction losses (ppm) LG01 LG02 LG03 LG11 LG04 LG12 LG05 LG13 LG06 LG21 LG07 LG14 LG22 LG15 LG23 LG Optical modes Figure 5.21: The graph shows the difference between orthogonal modes in diffraction losses introduced by the mirror geometry. The difference appears due to the break in symmetry of the mirrors Mirror tilt We have defined the conditions of a perfectly aligned cavity with finite size mirrors. Therefore we can proceed to misalign the cavity by tilting one the mirrors. This will allow us to study the effect of the mirror tilt in higher order modes diffraction losses and frequency shift. For our simulation we use the FFT based code previously presented in section 5.4. This time it also includes a mirror tilt angle for the ETM as shown in figure It has been shown that the tilt perturbation shifts the mode pattern to one side of the cavity which results in additional coupling loss [26]. The increase in diffraction power loss due to misalignment is proportional to the square of the mirror tilt angle and therefore a quadratic increase of power loss with the tilt angle will occur. Figure 5.25 shows a simplification of the problem, where a small tilt in one test mass changes the optical path of the laser beam. Now the tilted optical path crosses the horizontal (or perfectly aligned) optical path at an angle θ. This path length difference ( L) can be calculated with some simple trigonometry. The optical path difference is define by L = L L, where L = L a + L b.
213 5.8. MIRROR TILT 185 Frequency change with mirror size Frequency [Hz] Frequency (Spot size/mirror Change with size) Mirror ratio Size Order Order Order Order Mirror size radius (m) HG16 HG43 HG52 HG70 LG07 LG15 LG23 LG31 HG15 HG33 HG24 HG60 LG06 LG14 LG22 LG30 HG14 HG23 HG50 LG05 LG13 LG21 HG13 HG22 HG40 LG04 Figure 5.22: The graph shows for higher order modes the variation in frequency from the fundamental mode TEM 00. For modes of order lower than 4 the variation is too small for the scale of this graph and are not shown. We notice that for a given spot size the smaller the mirror the higher the frequency shift and therefore the higher the divergence from the theoretical value. This can also be seen on the top secondary x axis which shows the ratio between the spot size and the mirror size. tan(θ) = x 1 L 1 = x 2 L 2. (5.21) The tangent of the angle θ can be determined by the displacement of the spot size ( x 1,2 ) and the distance between the mirror and the centre of rotation of the beam path (L 1,2 ) as shown in equation (5.21). L 1 = L a = L b = L ( 1 + x 2 ), (5.22) x 1 L x 2 2, (5.23) L x 2 1. (5.24) Equations (5.22), (5.23) and (5.24) show the relations between the beam path and the displacement of the spot on the mirrors in order to calculate the length difference
214 186CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES Frequency change with mirror size for modes order 7 Frequency Change (Spot with size/mirror Size size) for ratiomodes of order Frequency [Hz] Advanced LIGO Spot size: 60 mm Test Mass radius: 170 mm Mirror size radius (m) HG16 HG43 HG52 HG70 LG07 LG15 LG23 LG31 Figure 5.23: The graph shows for modes of order 7 the effect of the mirror size in the frequency gap with the fundamental mode. We use the Advanced LIGO design for our study, where the spot size has a diameter of 0.6 cm and the mirror a diameter of 17 cm. The simulation does include the flat sides for suspension attachment. R1 R2 L = 4000 m α Figure 5.24: Considering only one of the mirrors to be tilted there is a change in the optical path for the laser beam. As a consequence there is a change in the resonance conditions of the cavity. between the perfectly align situation and the tilt one. For these calculations also the Sagitta S has been considered, and is defined by:
215 5.8. MIRROR TILT 187 x 1 L a L 1 θ L = L + L 1 2 * L = L a + L b θ L b L 2 x 2 Figure 5.25: With some simple geometry it is possible to calculate the change in the path length of the laser beam and also the displacement of the spot from the centre of both test masses. S 1 = R R 2 x 2 1 (5.25) tan(θ) = tan(θ) = x 1 = x 1 R2 + x 2 1 Rα 2R L Rθ. (5.26) 1 + θ 2 Analogous calculations for the end test mass show that the spot displacement is given by: x 2 = R (θ α). (5.27) It is well known that the frequency variation ( f) of the fundamental modes for variations of the cavity length is given by: f = L L f Y AG. (5.28) Where L corresponds to the optical path length variation, L is the total optical path length and f Y AG is the laser frequency. In our case we are using an Nd:YAG laser with a wavelength of µm, which corresponds to a frequency of Hz. However for higher order modes this frequency variation has an extra component. For higher order modes the frequency is then given by: f = c ( q + N ) 2L π arccos( g 1 g 2 ). (5.29)
216 188CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES Where c corresponds to the speed of light in vacuum, q to the axial mode, g 1 and g 2 corresponds to the stability factor of each mirror. N corresponds to the mode number given by (m + n + 1) for HG modes or (2m + n + 1) for LG modes. In this case the radius of curvature of both mirrors is the same with a value of R, therefore g = g 1 = g 2. This means that the frequency variation for higher order modes due to a small change in the optical path length is given by: f = L ( ) N L f c 1 Y AG + L. (5.30) π 2L 2 2LR L 2 The extra term in this equation is comparatively small, only 11.4 mhz for high order optical modes of order 2. It has also been shown that cavities with stability gfactor close to 1 are more efficient [27]. However the resonator becomes very sensitive to misalignment of the mirrors and diffraction losses will increase rapidly with mirror tilt. The losses increase with the square of the tilt angle and can be characterised by the misalignment sensitivity D i as proposed by Hauck et al [28]: D 2 i = πl λ ( gj g i ) (1/2) 1 + g 1 g 2. (5.31) (3/2) (1 g 1 g 2 ) Where D i corresponds to the misalignment sensitivity of mirror i referred to mirror j, with L the cavity length, λ the laser wavelength and g 1 and g 2 the stability factors for mirrors 1 and 2. The reciprocal value of D i is the tilt angle for that mirror which increases the losses by 10%. If both mirrors are misaligned, the losses proportional to D 2 i are summed up. Therefore the misalignment of the cavity is defined as: D = D D 2 2. (5.32) We are analysing a case with a cavity length of 4 km and mirrors with 2076 m radius of curvature the stability factor is given by g = g 1 = g 2 = , and with λ = 1.064µm we then obtained a D i value of which in turn means that diffraction losses will increase by 10% for a tilt angle of radians. The use of Fast Fourier Transform (FFT) to simulate the propagation of the beam inside the cavity allows the mode shape to change due to the finite mirror size and hence enables a much better approximation of the diffraction losses, and therefore
217 5.8. MIRROR TILT 189 of the other parameters of interest [14]. A simulation program based on the beam propagation using FFT was developed at UWA which was previously used for the calculation of diffraction losses in advanced interferometers [24]. The same program allows us to introduce mirror tilts and estimate their effect in the diffraction losses and frequency of the higher order optical modes. Due to the tilt of the mirror there will be a displacement of the spot proportional to the mirror tilt angle. In a cavity with finite mirrors this mode pattern shift closer to the edge of the mirror results in an additional diffraction loss. The FFT code not only allowed us to estimate the diffraction losses and frequency changes for each mode, but also to check the spot displacement and therefore the change in geometry of the cavity. The increase in diffraction losses is proportional to the square of the tilt angle, but the constant of proportionality decrease with increasing mirror size (or Fresnel number). In the case of an infinite concave mirror the diffraction losses will always be zero for any small tilt angle. However even in the case of an infinite size mirror there will be an increase in power losses due to mismatch between the input beam and the circulating beam. The loss of circulating power is proportional to the square of the tilt angle and its constant decrease with mirror size. Figure 5.26 shows the shift in frequency when the ratio between the spot size and mirror size varies. The figure shows where the Advanced LIGO design is in terms of this ratio and shows that for higher order modes (which are more affected) a change of 1 % in the spot size/mirror size ratio produces a change between 7 Hz and 50 Hz in frequency for modes of order 6, depending on the mode energy distribution. For example, LG14 will have a frequency shift of 7 Hz for a 1 % ratio change, while LG30 will have a frequency change of 47 Hz. This variation is larger for higher high order modes. This however it also implies that thermal tuning can be use to advantage for frequency tuning of the higher order modes. With the mirror tilted, i.e. the cavity is misaligned, the mode is no longer concentric on the test mass and there is an asymmetric edge effect. This is due to the fact that the mode moves closer to one edge, and because we are on the exponential tail of the mode the differential effect is different on the near side leading to a quadratic tilt
218 190CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES Frequency change with spot/mirror size ratio for modes of order Frequency (Hz) Advanced LIGO Spot size/mirror size ratio LG06 LG14 LG22 LG30 Figure 5.26: The graph shows the variation in frequency of modes of order 6. Effects like thermal lensing will change the (spot size/mirror size) ratio changing the frequency gap between the fundamental mode and higher order modes. This frequency shift will change the resonant conditions for parametric instabilities. sensitivity. This explains why higher order modes are more affected. It also shows that the effects will depend on the orientation of the mode profile in relation to the tilt. For example, for a certain tilt direction, mode HG05 will be more affected than HG50 which comes as a consequence of the flat sides on the test mass substrate as shown in figure The frequency change will depend on the orientation of the mode profile in relation to the tilt. This can also be interpreted as the frequency change for a given higher order mode will depend on the tilt direction in relation with the energy distribution of a particular mode. Therefore the frequency gap between the fundamental mode and the higher order modes will also depend on this relative orientation Conclusions We have shown the effect of the mirror geometry and the mirror tilt in an align cavity in terms of diffraction losses and the frequency separation of the higher order modes.
219 5.8. MIRROR TILT 191 Frequency change with ETM mirror tilt for modes of order HG LG20 Frequency (Hz) LG12 HG31 HG22 LG Mirror tilt (nrad) LG04 LG12 LG20 HG40 HG31 HG22 Figure 5.27: The graph shows the variation of the frequency gap between the fundamental mode and the modes of order 4. We notice that the effects of the mirror tilt in the frequency gap depends on the energy distribution of the mode. As a consequence the frequency variation for a given tilt of the mirror not only depends on the mode order but also on the energy distribution of the optical mode. This study forms part of a broader understanding of the possibility of parametric instabilities in advanced interferometric gravitational wave detectors. Using an FFTbased code developed at UWA we have quantified the effects of mirror tilt in terms of the frequency shift. In reality the cavity length effect due to displaced spots is generally a common effect on the TEM 00 and higher order modes which should be removed by the control system. We have also shown that changes in the spot size/mirror size ratio can affect the frequency separation between higher order modes. This can be use to advantage by thermally tuning the spot size on the test masses as a way to control the diffraction losses. In turn it is possible to control the mode gain, optical Qfactor and frequency separation, ultimately controlling the parametric gain R 0.
220 192CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES Frequency change with mirror tilt HG50 60 F re q u e n cy (H z) HG40 10 HG05 0 HG Mirror tilt (nrad) HG04 HG40 HG05 HG50 Figure 5.28: The graph shows the frequency variation for a few modes of order 4 and 5. It shows that the frequency change will depend on the orientation of the mode profile. 5.9 Postscript The calculations made in the publication of Numerical calculations of diffraction losses in advanced interferometric gravitational wave detectors were based on the advanced LIGO design at that time. Since publication in June 2007, the design of Advanced LIGO has changed. In this particular case our main concern is the effect that the new arms configuration will have on the diffraction losses and therefore on the parametric gain. The changes to the radius of curvature of the test masses seek to reduce the spot size at the beamsplitter, effectively reducing the diffraction losses in the recycling cavities [29]. There is also a reduction of the reflectivities of the mirrors, reducing the finesse of the arm cavities. This reduction does not affect the quantum noise in an advanced dual recycling interferometer such as Advanced LIGO. The main reason for the reduction of finesse for the arm cavities is the use of fused silica as the substrate of choice for Advanced LIGO. By selecting fused silica as the substrate for the test masses, absorption is less of a problem when compared to the original sapphire option [30]. Table 5.5 shows the parameters of the arm cavities
221 5.9. POSTSCRIPT 193 comparing the previous design with the current design of Advanced LIGO. Parameter Previous AdvLIGO Current AdvLIGO ITM radius of curvature 2076 m 1971 m ETM radius of curvature 2076 m 2191 m ITM spot size radius 60 mm 55 mm ETM spot size radius 60 mm 62 mm Waist size radius 11.6 mm 11.8 mm Waist position from ITM 2000 m 1885 m Cavity gfactor Mode spacing khz khz Table 5.5: Comparison of design parameters between the previous Advanced LIGO design and the current one. Figure 5.29 shows the diffraction losses for both configurations. It shows that some modes are more affected by the change in radius of curvature of the test masses, but in general the difference in the round trip diffraction losses is very small. This was expected since the increase in spot size at the ETM is compensated by the reduction at the ITM, leaving the round trip total roughly the same. The most affected modes are those with more energy spread over the mirror coating. Leading to bigger spot size on the ETM which has a greater impact on roundtrip losses. Conversely the optical gain and Qfactor of the higher order modes depend on the total losses, including diffraction losses and coupling losses. As a consequence we can see a reduction of the gain in figure 5.30, in particular for the fundamental mode and the lower of the higher order modes. This was expected, since the reduction of cavity finesse implies less circulating power in the arm cavities. Interestingly the higher order modes have somewhat higher gain in the current design, even though it is only a small fraction. The gain for the fundamental mode was reduced from 790 to 285, which implies a considerable reduction in circulating power. The case of the Qfactor shown in figure 5.31 is slightly different. Here the higher order modes are always higher in the previous design, with a Qfactor of the fundamental mode of
222 194CHAPTER 5. DIFFRACTION LOSSES AND PARAMETRIC INSTABILITIES Comparison of diffraction losses between original and current Advanced LIGO design Diffraction losses (ppm) 1.E+06 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 1.E+00 HG00 HG01 LG01 HG11 HG02 LG02 LG10 HG12 HG03 LG03 LG11 HG22 HG13 HG04 LG04 LG12 LG20 HG14 HG05 HG23 LG05 LG13 LG21 HG15 HG24 HG33 HG06 LG06 LG14 LG22 LG30 HG16 HG43 HG52 HG07 LG07 LG15 LG23 LG31 HG08 HG17 HG26 HG35 HG44 LG08 LG16 LG24 LG32 LG40 1.E Optical modes Original AdvLIGO Current AdvLIGO Figure 5.29: Comparison of diffraction losses between the previous design for Advanced LIGO and the current design , which is reduced to in the current configuration. The reduction of the optical gain of the higher order modes and of Qfactor in particular for the lower of the higher order modes, points to a lower parametric gain and therefore a lower chance of parametric instabilities in the current Advanced LIGO design. Since this is only in comparison with the previous design it does not guarantee that parametric instabilities will not occur. High Order Modes Optical Gain 1.E+03 Gain 1.E+02 1.E+01 HG00 HG01 LG01 HG11 HG02 LG02 LG10 HG12 HG03 LG03 LG11 HG22 HG13 HG04 LG04 LG12 LG20 HG14 HG05 HG23 LG05 LG13 LG21 HG15 HG24 HG33 HG06 LG06 LG14 LG22 LG30 HG16 HG43 HG52 LG07 LG15 HG17 HG26 HG35 HG44 LG08 1.E+00 HG07 LG23 LG31 HG08 LG16 LG24 LG32 LG40 1.E Optical modes Original AdvLIGO Current AdvLIGO Figure 5.30: Comparison between the gain of the higher order modes in the previous Advanced LIGO design and the current design.
223 5.10. REFERENCES 195 Optical Q factor for higher order modes 1.E+13 Q factor 1.E+12 1.E+11 HG00 HG01 LG01 HG11 HG02 LG02 LG10 HG12 HG03 LG03 LG11 HG22 HG13 HG04 LG04 LG12 LG20 HG14 HG05 HG23 LG05 LG13 LG21 HG15 HG24 HG33 HG06 LG06 LG14 LG22 LG30 HG16 HG43 HG52 HG07 LG07 LG15 LG23 LG31 HG08 HG17 HG26 HG35 HG44 LG08 LG16 LG24 LG32 LG40 1.E Optical modes Original AdvLIGO Current AdvLIGO Figure 5.31: Comparison between the Qfactor of the higher order modes in the previous Advanced LIGO design and the current design. Frequency Comparison Delta Frequency (Hz) 1.E+04 1.E+03 1.E+02 1.E+01 1.E+00 1.E01 HG01 LG01 HG11 HG02 LG02 LG10 HG12 HG03 LG03 LG11 HG22 HG13 HG04 LG04 LG12 LG20 HG14 HG05 HG23 LG05 HG15 LG21 LG13 HG24 HG33 HG06 LG06 LG14 LG22 LG30 HG16 HG52 HG43 HG07 LG07 LG15 LG23 LG31 HG08 HG17 HG26 HG35 HG44 LG08 LG16 LG24 LG32 LG40 1.E02 1.E03 HG00 1.E Optical modes Original AdvLIGO New AdvLIGO Figure 5.32: Comparison between the frequency shift of higher order modes in the previous Advanced LIGO design and the current design References [1] V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, Parametric oscillatory instability in FabryPerot interferometer, Phys. Lett. A 287 (2001) [2] W. Kells and E. D Ambrosio, Considerations on parametric instabilities in Fabry Perot interferometer, Phys. Lett. A 299 (2002) [3] C. Zhao, L. Ju, J. Degallaix, et al, Parametric instabilities and their control
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227 Chapter 6 Stable Recycling Cavities 6.1 Preface The author presented a preliminary design for the Australian International Gravitational wave Observatory (AIGO) during an international workshop in Gingin, Western Australia. At this workshop Volker Quetschke from the University of Florida presented the concept study for a stable power recycling cavity proposed for the Advanced LIGO design. Following several discussions on the possible benefits of stable recycling cavities compared with marginally stable recycling cavities used by the operating interferometric GW detectors, the author started to research the possibility of incorporating a stable power recycling cavity in the AIGO design. The main design constraint was the requirement to use the available infrastructure to fit the extra mirrors and vibration isolation systems necessary for the stable recycling cavity. A stable signal recycling cavity was successfully added to the design and discussed with Guido Mueller (also from the University of Florida) during the Amaldi 7 conference on gravitational waves in Sydney. A more complete presentation by G. Mueller followed at a Parametric Instabilities workshop in Gingin. The original concept of incorporating a stable power recycling cavity was extended to include a stable signal recycling cavity. Following the selection of the RF sidebands for the length sensing control system the design of AIGO was then extended to form a complete optical design for an advanced dual recycled interferometric gravitational wave detector. This extended study was completed in collaboration with the University of Florida, in particular with the coauthors of the paper Optical design of the proposed Australian International Gravitational Observatory. The calculations were done by the author in consulta 199
228 200 CHAPTER 6. STABLE RECYCLING CAVITIES tion with the group at the University of Florida. Muzammil Arain did the mode matching calculations and the tolerances of the mirrors. Additional simulations were completed during the evolution of this design, some results of which are presented in the paper at the core of this chapter. A later simulation of the beamsplitter thermal effects was recorded as an internal technical report and added as a postscript to this chapter.
229 6.2. INTRODUCTION 201 Optical design of the proposed Australian International Gravitational Observatory Pablo Barriga 1, Muzammil A. Arain 2, Guido Mueller 2, Chunnong Zhao 1 and David G. Blair 1 1 School of Physics, The University of Western Australia, Crawley, WA Department of Physics, University of Florida, Gainesville, FL32611, USA Marginally stable power recycling cavities are being used by nearly all interferometric gravitational wave detectors. With stability factors very close to unity the frequency separation of the higher order optical modes is smaller than the cavity bandwidth. As a consequence these higher order modes will resonate inside the cavity distorting the spatial mode of the interferometer control sidebands. Without losing generality we study and compare two designs of stable power recycling cavities for the proposed 5 kilometre long Australian International Gravitational Observatory (AIGO), a high power advanced interferometric gravitational wave detector. The length of various optical cavities that form the interferometer and the modulation frequencies that generate the control sidebands are also selected. 6.2 Introduction The addition of a southern hemisphere interferometric gravitational wave (GW) detector in Gingin ( 80 km north of Perth in Western Australia) will greatly improve the angular resolution of the existing network of GW detectors [1, 2], while also providing resolution for both polarisations, thereby allowing measurement of the luminosity distance of sources. Thus the addition of a single southern hemisphere GW detector of sensitivity comparable to the proposed northern hemisphere detectors (Advanced Laser Interferometric Gravitationalwave Observatory (LIGO) [3], Largescale Cryogenic Gravitationalwave Telescope (LCGT) [4], and Advanced VIRGO [5]) allows the number of detectable sources to be doubled [6].
230 202 CHAPTER 6. STABLE RECYCLING CAVITIES Similar to the 4 km long LIGO detector or the 3 km VIRGO detector, the proposed Australian detector (AIGO) will be a Michelson interferometer with 5 km long optical cavities in each arm. The interferometer will be kept at a dark fringe such that all the light reflects back towards the laser. A mirror placed between the laser and the Michelson Interferometer (MI) is then used to form the power recycling cavity (PRC) that further increases the circulating power in the arm cavities [7]. Future detectors such as Advanced LIGO and AIGO will also employ signal recycling where an additional mirror at the output port of the MI is used to buildup the signal in a tuneable frequency band. The spatial eigenmodes of both recycling cavities have to match the spatial eigenmode of the arm cavities. This ensures an efficient extraction of the signal or GW induced sidebands and a good mode matching between the spatial modes of the carrier and the spatial modes of phase modulation radio frequency (RF) sidebands used to control the longitudinal and angular degrees of freedom. It also improves the coupling of the carrier field into the arm cavities. The first generation of large scale interferometric GW detectors, LIGO and VIRGO, used marginally stable power recycling cavities in an essentially flat/flat configuration. These cavities did not confine the spatial modes of the RF sidebands which led to significant spatial modemismatch between them and the carrier [8]. A sophisticated thermal correction system was necessary in order to overcome these problems during the commissioning phase of the interferometer [9]. Stable PRC for Advanced LIGO to confine the spatial modes of the RF sidebands have first been proposed by the University of Florida [10, 11] as part of their input optics work [12]. A group at the California Institute of Technology discovered that marginally stable signal recycling cavities will also reduce the amplitude of the GWsidebands by resonantly enhancing the scattering of light into higher order spatial modes [13]. Since then the concept of stable recycling cavities has been intensively discussed for all next generation interferometric gravitational wave detectors and they are now part of the baseline design for Advanced LIGO [14, 15]. This paper analyses the concept of stable recycling cavities within the design scenario of the proposed AIGO interferometer. In Section 6.3 we will review dual
231 6.2. INTRODUCTION 203 recycled Michelson interferometer and define key parameters of the proposed AIGO interferometer. The suppression of higher order modes (HOM) as a function of the Gouy phase will be reviewed in the Section 6.4. In Section 6.5 we will discuss and compare possible designs of stable power recycling cavities given the current vacuum envelope of the Gingin test facility. The current vacuum envelope and the specific design of the PRC will then be used to select the RF used to generate the control sidebands. In the remainder of the paper we will apply the same design principles discussed in Section 6.5 to design the signal recycling cavity (SRC). ETM 1 Degrees of freedom Input Mode Cleaner M1 M2 L IMC L 1 East FabryPerot Cavity ITM 1 L + =(L 2 +L 1 )/2 L=(L 2 L 1 )/2 l + =l pr +(l 2 +l 1 )/2 l=(l 2 l 1 )/2 l src =l sr +(l 2 +l 1 )/2 l 1 M3 PRM BS ITM 2 L 2 ETM 2 100W PSL Nd:YAG laser λ = 1064nm l pr l sr l 2 South FabryPerot Cavity SRM To Detector Bench Figure 6.1: Configuration of the proposed AIGO interferometer as a dual recycling interferometer with marginally stable recycling cavities. The figure shows a prestabilized 100 W Nd:YAG laser and the input mode cleaner (IMC). The power recycling mirror (PRM), beamsplitter (BS), both input test masses (ITM 1 and ITM 2 ), and both end test masses (ETM 1 and ETM 2 ). It shows the power recycling cavity (PRC) formed by the power recycling mirror (PRM) and both input test masses (ITM 1 and ITM 2 ) passing through the BS. Also the signal recycling cavity (SRC) formed by the signal recycling mirror (SRM) and both ITMs. It also shows the different degrees of freedom that need to be controlled.
232 204 CHAPTER 6. STABLE RECYCLING CAVITIES 6.3 Dual recycling interferometers An interferometric GW detector consists of a set of coupled optical cavities in a MI as shown in figure 6.1. Long kmscale cavities are used to enhance the displacement sensitivity of the MI arms. The power recycling mirror (PRM) creates a composite cavity (PRC) with the common mode of the arm cavities, while the signal recycling mirror (SRM) creates another composite cavity (SRC) with the interferometer differential mode. This configuration operated in signal recycling mode was first proposed by the Glasgow group [16] while the resonant sideband extraction (RSE) mode was later described by Mizuno et al [17]. The lengths of the two long FabryPerot cavities that form the arms of the interferometric GW detector need to be controlled. In addition, the MI has to be kept dark such that all the light is send back towards the laser. The position of the PRM has to be controlled to keep the PRC on resonance for the laser field. The SRM is controlled to resonantly enhance or extract the signal fields. Using the notation described in figure 6.1, it involves controlling five length degrees of freedom (L +, L, l +, l, l src ) of the seven suspended optics PRM, BS, ITM 1, ETM 1, ITM 2, ETM 2 and SRM. Length sensing and control schemes for this interferometer configuration were first developed and tested by [18, 19, 20, 21]. These schemes are still evolving as the interferometer designs evolve [22]. All utilise phase modulation sidebands and/or phase locked lasers to generate the necessary control signals. However, it is currently assumed that two pairs of phase modulation sidebands are sufficient to control all longitudinal degrees of freedom. The lengths of both recycling cavities are directly linked to the RF used to generate the sidebands. This will be discussed in more detail in Section 6.6. In order to calculate the parameters for the stable PRC we need to define the FabryPerot arms of the interferometer. A large spot size is needed in order to maximise the averaging over the thermal fluctuations in the test masses and thereby reduce the test mass thermal noise [23]. At the same time it is necessary to keep the fundamental mode diffraction losses sufficiently low (normally below 1 ppm per round trip). AIGO current design uses sapphire as the substrate of choice for the test masses. The parameters of the arm cavities for the proposed AIGO interferometer are presented in table 6.1.
233 6.4. HIGHER ORDER MODES SUPPRESSION 205 Cavity Length 5000 m ITM (ETM) radius of curvature 2734 m ITM (ETM) diameter 32 cm Cavity gfactor Waist size radius mm Spot size radius mm Free spectral range khz High order modes separation khz Cavity finesse 1220 Cavity pole Hz Table 6.1: Parameters of the arm cavities for the AIGO interferometer. The proposed design assumes sapphire test masses with a diameter of 32 cm and a spot size radius of 55 mm in order to keep diffraction losses of the fundamental mode below 1 ppm. 6.4 Higher order modes suppression The transversal spatial eigenmodes of optical cavities formed between mirrors with spherical radii of curvatures can be approximated by a set of HermiteGaussian eigenmodes. For a given wavelength λ this set is defined by a waist size w 0 and its location z 0 along the optical axis [24]. Each eigenmode is described by a pair of mode numbers m, n. Each number is associated with one of the transversal directions defined by our coordinate system and corresponds to the order of the Hermite mode which is used to describe the field distribution along this coordinate axis. Current interferometric GW detectors operate with the 00mode being resonant in the arm cavities and the PRC. Imperfections in the mirrors, such as radii of curvature mismatches, misalignments, or other spatial variations of the mirror surfaces will cause scatter between the eigenmodes. This will reduce the power in the 00mode, increase the stray light, and create spurious error signals. The scatter is resonantly enhanced into specific HOM if they are close to resonance and encounter only very small diffraction losses inside the interferometer. Modes with large mode numbers have larger effective cross sections and finite apertures will increase their interferometer internal losses. In addition, the
234 206 CHAPTER 6. STABLE RECYCLING CAVITIES scatter efficiency between the 00mode and HOM with large mode numbers is usually much lower than the scatter efficiency into low order HOM. Consequently, the resonant enhancement of scattered light into HOM is mainly a problem for the lowest order HOM. The goal of the stable recycling cavities is to avoid incidental resonances of any of the low order HOM. Transversal eigenmodes are resonant inside a cavity if their roundtrip phase shift is a multiple of 2π. The roundtrip phase shift of each HermiteGaussian mode has two contributions. The main contribution is the phase shift associated with planewave propagation: φ = 2kL (6.1) where L is the length of the cavity. This phase shift is common to all modes. The second contribution is associated with the Gouy phase [25], which can be calculated from the generalised cavity gfactor: with Ψ G = arccos ±g (6.2) g = A + D (6.3) where A and D are the diagonal matrix elements of the ABCD matrix which describes the roundtrip through the cavity. Each HermiteGaussian mode is acquiring an additional phase which depends on the mode number: Ψ mn = (m + n + 1)Ψ G (6.4) This additional phase breaks the degeneracy between the modes and causes the HOM to have different resonance frequencies or length [24]. Figure 6.2 shows the normalised buildup of the first ten HOM (mode number = m + n) as a function of the Gouy phase Ψ G of the cavity. Any significant buildup of a HOM could significantly reduce the stability of the optical interferometer with the
235 6.4. HIGHER ORDER MODES SUPPRESSION 207 Transmission [a.u.] 10 0 High Order Modes Transmission π 0.1π 0.15π 0.2π 0.25π 0.3π 0.35π 0.4π 0.45π 0.5π Ψ: Gouy phase shift (rad) Figure 6.2: Transmission of HOM as function of the Gouy phase shift. All HOM resonate at Ψ G = 0, however only even modes resonate at Ψ G = 0.5π. By selecting a Gouy phase around 0.18π the highest transmission is for modes of order 5 and 6, while at Ψ G close to 0.15π it will be orders 6 and 7, but with a higher transmission for order 1. From 0.5π to π the transmission of HOM mirrors the transmission here presented. lower modes being the most critical ones. The graph shows which of the lower order modes are the least critical ones and can be used to optimise the Gouy phase inside the cavity. For example a Gouy phase of Ψ G 0.18πrad or a cavity gfactor of 0.7 for the recycling cavities would allow the modes with mode numbers 5 and 6 to have some buildup while all other modes are well suppressed. Figure 6.3 shows the suppression of HOM compared to the 00mode for different cavity designs with different stability gfactors. All shown designs have a finesse of 95. The first three designs do not use any focusing elements inside the recycling cavities. The first design (L = 12 m) corresponds to a marginally stable recycling cavity; this is about the maximum length for a nonfolded recycling cavity which would fit into the Gingin central building. The second (L = 500 m) and third designs (L = 1 km) would require to fold the recycling cavity through one of the long vacuum
236 208 CHAPTER 6. STABLE RECYCLING CAVITIES tubes which would also house the arm cavities. Although the suppression of HOM increases for longer recycling cavities, it is still rather small compared to a recycling cavity with a g 0.7. In the remainder of the paper we will use this gfactor for the recycling cavities. This gfactor can be obtained with two different Gouy phases: Ψ G = 0.18π rad = 0.58 rad and Ψ G = 0.82π rad = 2.57 rad. At these Gouy phase values the first HOM (TEM 01 and TEM 10 ) have lower buildup when compared to a marginally stable design. However the lower buildup of these modes in the recycling cavity is somewhat compensated by the higher overlap between the carrier and the RF sidebands in the stable PRC. This allows for error signals for alignment controls. Moreover, the alignment sensing matrix for the stable cavity design allows for a better decoupling of various alignment signals. This is a tradeoff which will require further studies. 40 High order modes suppression for recycling cavities Magnitude (db) Mode order Marginally stable (g = ) 500 m stable (g = ) 1 Km stable (g = ) Short stable (g = ) Figure 6.3: Comparison of the intensity suppression of HOM between a 12 m long marginally stable recycling cavity, a 500 m long inline PRC, a 1 km long inline PRC, and the proposed design for a stable PRC. The graph shows HOM up to order 10.
237 6.5. POSSIBLE SOLUTIONS FOR STABLE RECYCLING CAVITIES Possible solutions for stable recycling cavities In the previous section we showed that without any additional focusing elements very long recycling cavities are needed to gain enough Gouy phase in order to achieve a moderately good suppression of HOM. Since this option is impractical we evaluate some alternative solutions that allow us to obtain the required Gouy phase and thus the necessary HOM suppression without extending the recycling cavities by a large amount. To be more specific we use the current vacuum envelope at the Gingin test facility to constrain the parameter space. The vertex of the two arms restricts the distance between the BS and the recycling mirrors to about 3 m while the dimensions of the central building restricts the distance between the ITMs and the recycling mirror to about 12 m. Currently the end stations are 80 m from the central tank. For a future 5 km detector, the ITMs could be relocated into these end stations while the BS, the recycling cavity mirrors, and the auxiliary optical components could all be left in the central station. The ABCD matrix method [26] is used to simulate the propagation of the laser beam inside the interferometer. The spatial mode inside the recycling cavity has to match the circulating beam of the main arm cavities. Therefore we start our analysis from the waist of the arm cavities moving backwards towards the recycling cavity input mirror. It is assumed that the test mass substrates are made of sapphire which has an index of refraction of Consequently, the ITM has an effective focal length of 3645 m for both recycling cavities. In the following two sections we will discuss two designs for stable recycling cavities for AIGO. Both designs would fit into the current vacuum envelope at Gingin. They will be compared in the last section of this chapter Straight stable recycling cavity A first option for a stable recycling cavity will be the addition of a lens inside the common arm of the MI. As the distance between the PRM and the BS is limited to 3 m, the focusing element would have to have a focal length in the order of or
238 210 CHAPTER 6. STABLE RECYCLING CAVITIES ITM East PR2 East From MC PR1 BS PR2 South ITM South Figure 6.4: Schematic diagram for the proposed stable PRC design for AIGO advanced interferometer (figure not to scale). This solution includes a lens inside the recycling cavity in order to achieve the required Gouy phase. smaller than 3 m to accumulate any appreciable Gouy phase. This will create a beam size of well below 100 µm on the PRM. This small beam size has multiple disadvantages such as an intensity of at least 10 MW/cm 2 for typical input powers and power recycling gains and very stringent requirements on the focal length of the lens and the radius of curvature (ROC) of the recycling mirror [10]. These problems can be significantly reduced when moving the ITMs to the current end stations and place one lens (PR2) in each arm of the MI as shown in figure 6.4. This lens could be combined with the compensation plate which will be installed in most high power interferometers to compensate thermal lenses generated in the ITM substrates [27, 28] or it could even be polished into the backside of the ITM substrate. The focal length of this lens would be roughly similar to the distance between the ITM and the central building. This creates a waist close to PR1, which then would be placed at the position where the acquired Gouy phase is equal to the design Gouy phase. Its ROC needs to match the ROC of the Gaussian eigenmode at that location in order to modematch the recycling cavity to the arm cavities. Here and throughout this document the modematching refers to modematching in power.
239 6.5. POSSIBLE SOLUTIONS FOR STABLE RECYCLING CAVITIES 211 (a) x Mode matching as function of the optics tolerances PR2 focal length normalized error PR1 radius of curvature normalized error 0.02 (b) 1 x 103 Mode matching as function of the optics tolerances PR2 focal length normalized error PR1 radius of curvature normalized error Figure 6.5: (a) Modematching drop as a function of PR1 radius of curvature and PR2 focal length. Note that modematching drops mainly due to PR2 radius of curvature error. (b) The optimised modematching after repositioning PR1. An important consideration for the design of the stable recycling cavity is the tolerance of the mirrors and lenses. Here we consider modemismatch, manufacturing tolerances based on the current glassmanufacturing technology and the ability to correct them by adjusting the distance between PR1 and PR2. This will be discussed using a specific set of parameters (see tables 6.2 and 6.3) for the PR mirrors which
240 212 CHAPTER 6. STABLE RECYCLING CAVITIES fix the length and generate a Gouy phase of 0.58 rad. The nominal ROC of PR1 in this design is m while the focal length of PR2 is m. Any changes from the design values of PR1 and PR2 will decrease the modematching into the arm cavity. Figure 6.5(a) shows the modematching drop as a function of the normalised error in ROC of the mirrors. Figure 6.5(a) shows that the modematching depends mainly on the normalised error in PR2. This is also easy to understand since the modematching in a simple two element telescope depends on the absolute errors in the focal lengths. However, as shown in figure 6.5(b) this can be compensated by changing the distance between PR2 and PR1 to match up with the asbuild focal lengths. Note that the Gouy phase after reoptimising the distance is again very close to its design value. For example, with these tolerances the Gouy phase change is about 0.07 rad in the worse case. This can be recovered by distance reoptimisation to about ±0.02 rad Folded stable recycling cavity A different approach is the possibility of a folded recycling cavity allowing a stable PRC to be constructed within the existing facilities. In this case we add two mirrors to the recycling cavities to attain the necessary Gouy phase for the HOM suppression. Figure 6.6 shows a schematic of the proposed design for a folded stable PRC. The original PRM from figure 6.1 is replaced by three power recycling mirrors PR1, PR2 and PR3. This creates a modematching telescope that also works as a PRC. By carefully choosing the ROC and the distance between these mirrors it is possible to obtain any gfactor. As a consequence the distance l pr shown in figure 6.1 now corresponds to the distance between PR1 to the BS passing by the secondary and tertiary mirrors PR2 and PR3. As seen before the distance between the input mirror of the recycling cavity and the BS is very short. This restriction requires us to install the PRC mirrors before the BS (to the left of the BS in figure 6.6). Because of the strong focusing power needed to achieve the required Gouy phase the power density at PR1 is around 3 MW/cm 2. It is a similar case to the one presented in the previous section where a short distance cavity will have a very steep change in Gouy phase (again due to a very short Raleigh
241 6.5. POSSIBLE SOLUTIONS FOR STABLE RECYCLING CAVITIES 213 From MC PR1 ITM East PR2 PR3 BS ITM South Figure 6.6: Schematic diagram for the proposed stable PRC design for AIGO advanced interferometer (figure not to scale). range) making it very difficult to control and with very high power density on the input mirror. As a consequence we need to put PR2 behind the BS as in figure 6.6. This set some constraints to the geometry of the recycling cavity. First PR1 and PR2 will need to be offset from the PR3 BS ITM line. The larger the offset the larger the angle at which the beam will impinge PR2 and PR3 increasing the astigmatism in the PRC. However a larger offset will allow for larger distances between the mirrors which will ease the mirror tolerance for the Gouy phase control. It is also necessary to keep a minimum clearance between the BS and the PR2 PR3 circulating beam which sets a maximum for the distance between PR1 and PR2 and the angle of incidence at PR3. The parameters for the proposed designs which take all these constraints into account are listed in tables 6.2 and 6.3. The smallest spot size radius is mm at PR2 sustaining a power density of 17.9 kw/cm 2 ; well below the damage threshold. A potential problem with the folded stable cavity design is the astigmatism due to the angles at which the beam will impinge the mirrors. Given the constraints discussed above the angles at which the beam will impinge PR2 and PR3 are very similar and close to 2.38 o. The combination of a relatively short distance between
242 214 CHAPTER 6. STABLE RECYCLING CAVITIES the mirrors and the even shorter focal length creates some astigmatism. The worse case is at PR1 with a vertical spot radius of mm and a horizontal spot radius of mm, corresponding to an ellipticity of 0.8. This is interesting when compared to the case with a Gouy phase of 2.57 radians, where the ellipticity is the same, but the horizontal spot size radius is reduced to mm, hence changing the ellipse major axis and the power density. In both cases the difference in focal length between the horizontal and vertical axis is close to 1.5 m. The difference in Gouy phase between the two axes makes the suppression of the HOM by means of a stable recycling cavity ineffective without the addition of extra optical components. Using offaxis parabolic mirrors it is possible to compensate for the astigmatism due to the large angles of incidence. Assuming the use of offaxis parabolic mirrors, next we present an optical layout for the modematching telescope for an intermediate optical mode between the horizontal and the vertical directions. Following a similar criterion as in the straight PRC we assigned the tolerances for the ROC of the mirrors that form the folded stable PRC. The designed value of PR2 ROC is m, while the designed value of PR3 ROC is 12.4 m. Figure 6.7(a) shows the decrease in modematching due to ROC errors. It shows the modematching as a function of the normalised error in ROC of PR2 and PR3. As seen in the straight cavity case we can improve the modematching by optimising the distance between PR2 and PR3. After optimising the distance the modematching improves to better than 99.9%. Note that moving PR2 will require repositioning PR1 to keep the length of the recycling cavity constant. Adaptive heating on PR2 and PR3 can modematch to a significant modal space in the arm cavity. The modematching can drop by as much as 0.5% over the range of expected values for the arm cavities test masses. This by itself shows that the proposed cavity is quite tolerant to errors in ROC. However, it is desirable to improve the modematching because of errors in the ROC of the ITM can change the Gouy phase of the recycling cavity. Since this static error is a one time only process, the modematching can be improved by repositioning the PR2 mirror also restoring the Gouy phase. The improved modematching is shown in figure 6.7(b).
243 6.5. POSSIBLE SOLUTIONS FOR STABLE RECYCLING CAVITIES 215 (a) 1 x 103 Mode matching as function of mirror tolerances PR3 radius of curvature normalized error PR2 radius of curvature normalized error (b) Mode matching as function of the ITM and ETM ETM radius of curvature (m) ITM radius of curvature (m) Figure 6.7: (a) Modematching as a function of proposed radius of curvature tolerance limits on PR2 and PR3 for a fixed folded PRC design. The contour lines are lines of constant modematching. (b) Improved modematching as a function of expected values of ITM and ETM radius of curvature after optimising PR2 position Comparison between designs Due to the long distance between PR1 and the lens PR2 in the straight design, there is a smooth transition of Gouy phase from 0 radians near the ITM to 0.58 radians
244 216 CHAPTER 6. STABLE RECYCLING CAVITIES at PR1 (stability gfactor of 0.7). In the folded design there is a Gouy phase shift of only radians from ITM to PR3. From PR3 to PR2 the Gouy phase shift is 0.58 radians (stability gfactor of 0.7). Since PR3 is in the far field of the arm cavities a very small Gouy phase shift is induced between ITM and PR3. As a consequence the distance between them has a very small effect in this stable PRC design. For this simulation we assumed that the ITM is at the end station 80 m from the BS. These results show that for the folded design the ITM could also be inside the main lab. The small change in Gouy phase can be compensated by adjusting the position or the ROC of PR2. As a consequence the effectiveness of the stable PRC will be mostly affected by errors in the ROC of PR2. Figure 6.8(a) shows the tolerance of the PR2 lens in the straight cavity design while figure 6.8(b) shows the tolerance of the PR2 mirror in the folded cavity design. Both figures show the accumulated Gouy phase variation and the spot size radius at PR1 as a function of the PR2 normalised error. We notice that for both designs similar percentage errors in PR2 have a similar effect in the spot size variation and the accumulated Gouy phase shift at PR1. The spot size has a steeper variation with the normalised error in the straight cavity design, with the accumulated Gouy phase also showing a steeper slope around the design value. This makes the tolerances for manufacturing the optics very demanding, but we have shown that these errors can be overcome by repositioning the optics. The major contributor to the thermal effects in both designs is the substrate thermal lens in the ITM. With high circulating power inside the arm cavities this substrate thermal lens is mainly generated by the power absorbed by the coating of the ITMs. For almost 1 MW of circulating power, the absorbed power is about 0.5 W. The required amount of compensation depends upon the material chosen. For a lens made of fused silica, the required compensation will be in the shape of an annulus pattern and the required compensating power would be about ten times the absorbed power. A CO 2 laser can be used to create the required pattern, in such case about 5 W would be sufficient for the compensation [29]. Note that two optical components would be necessary to compensate the thermal effects inside the interferometer. For the folded design, this could be the substrate side of the ITMs
245 6.5. POSSIBLE SOLUTIONS FOR STABLE RECYCLING CAVITIES 217 (a) (b) Gouy phase and spot size at PR1 as function of PR2 Gouy phase and PR1 spot size as function of PR Gouy phase Spot size Gouy phase Spot size One way Gouy phase (rad) Spot size at PR1 (mm) One way Gouy phase (rad) Spot size at PR1 (mm) PR2 ROC Normalized Error PR2 Normalized Error Figure 6.8: (a) Variation of the accumulated Gouy phase in the straight PRC design and spot size radius at PR1 as a function of the PR2 lens normalised focal length. (b) Variation of the accumulated Gouy phase in the folded PRC design and spot size radius at PR1 as a function of the normalised PR2 ROC. or independent compensation plates. In the straight design PR2 can be used for compensation. With one optical element in each arm, it is possible to compensate both common mode as well as differential mode distortions in the recycling cavities. A major concern is the spot size at the BS. With a spot size radius of mm in the folded design the temperature rise due to the circulating beam is less than 0.01 o K from room temperature. This assumes a substrate like Suprasil 3001 for the BS, with substrate absorption of 0.25 ppm/cm [30]. As a consequence the thermal effects are quite small and the modemismatch due to these effects negligible. In the straight cavity design the spot size radius at the BS is only 2.30 mm and therefore the thermal effects stronger. The temperature rise at the substrate is close to 0.01 o K, while at the coating almost 0.06 o K. As a consequence the strongest modemismatch will occur in the inline arm, caused by the beam crossing the substrate at 45 o inducing some astigmatism. The modemismatch can be reduce to less than 0.1 % by repositioning PR2 also reducing the astigmatism. By using the PR2 lens as a compensation plate it is also possible to make some corrections through variations in the lens focal length. Since each MI arm of the PRC has its own lens it is possible to tune the modematching more effectively since each PRC arm will have a different mismatch with
246 218 CHAPTER 6. STABLE RECYCLING CAVITIES the main arm cavities. The bigger spot size in the folded cavity design will also increase the diffraction losses by at least an order of magnitude. This however can be overcome by reducing the spot size at the vertex using different ROC for the ITM and ETM [15]. While in the case of the straight cavity design the much smaller spot size radius at the BS means that its contribution to diffraction losses is negligible. A smaller spot size generates tighter alignment restrictions. Any offcentring of the beam at the BS will increase the leakage of power from the PRC into the dark port. The amount of power that leaks into the dark port depends on this contrast defect, the circulating power in the PRC and the transmissivity of the SRM. The restrictions are achievable with the advanced vibration isolation system under development for AIGO [31], but will require further revision. In the straight cavity design we have two optical elements, one inside each arm, which can be used for thermal compensation. The thermal effects on the BS can be compensated by operating on these two optical elements. The repositioning of the PR2 elements and the control of their ROCs via thermal compemsation provides the four degrees of freedom required for mode matching into four cavities; namely PRM and inline arm, PRM and offline arm, SRM and inline arm and SRM and offline arm. The smaller beam size at the BS will also increase the coating noise contribution of the BS to the overall noise of the interferometer. Thermooptic noise can be a main source of noise in advanced interferometers, but it has been suggested that it has been overestimated in the past [32]. It is one of the main fields of research for the next generation of interferometric GW detectors. Therefore the entire design would have to be further refined looking into other effects such as thermooptic noise. Table 6.2 shows the distances between the different optical elements of each stable recycling cavity design. The focal length of the optical elements selected for these simulations are presented in table 6.3. In both designs PR1 is a mirror. PR2 is a lens in the straight cavity solution while for the folded case is a mirror. PR3 is a mirror in the folded cavity solution and it is not required in the straight design. Table 6.4 shows the spot size radius on the different optical elements that form each of the
247 6.5. POSSIBLE SOLUTIONS FOR STABLE RECYCLING CAVITIES 219 Straight + lens PRC Folded stable PRC Elements Distance (m) Elements Distance (m) ITM  PR ITM  BS PR2  BS BS  PR BS  PR PR3  PR PR2  PR Total Total Table 6.2: designs. Distance between the different optical components in both cavity recycling Straight + lens PRC Folded stable PRC Optics Focal Length (m) Focal Length (m) PR PR PR36.2 ITM Table 6.3: Focal length of the different components that form the inline PRC with a lens to obtain the necessary Gouy phase and the folded PRC, which uses three mirrors to obtain the necessary Gouy phase.
248 220 CHAPTER 6. STABLE RECYCLING CAVITIES Straight + lens PRC Folded stable PRC Position Spot size (mm) Position Spot size (mm) Arm waist Arm waist ITM ITM PR BS PR BS PR PR PR Table 6.4: Development of the spot size radius on different optical components starting from the arm waist through the ITM back to the PRC input mirror PR1. In the inline design the lens PR2 is between the ITM and the BS, while in the folded design the mirrors are between the input mirror PR1 and the BS. Note the difference in spot size radius at the BS. stable PRC proposed designs. Based on the results of our analysis a straight stable PRC seems a better choice. The modemismatch due to ROC errors and the Gouy phase variations are lower than for the folded stable PRC. The modemismatch due to the thermal gradient at the BS can be corrected in both designs. A folded stable PRC will introduce some degree of astigmatism. However this can be greatly reduced by properly selecting the angle of incidence at two offaxis parabolic mirrors, but this will further complicate the optical design and the control system of the interferometer. 6.6 Sidebands and the stable recycling cavities Modulation frequencies calculations In the previous section we have selected a Gouy phase for the suppression of HOM in the PRC. Therefore we have a known length for a stable PRC. The resonance condition of this cavity will determine the modulation frequencies (f m ) for the RF sidebands. In this case the carrier sees an overcoupled arm reflectivity. Therefore if
249 6.6. SIDEBANDS AND THE STABLE RECYCLING CAVITIES 221 the carrier is resonant in the PRC the sidebands need an extra phase shift, thus the following relation applies to the modulation frequencies of the sidebands. f m = Here n 1 is an integer and L P RC l pr + (l 1 + l 2 )/2. ( n ) c. (6.5) 2 2L P RC the average PRC length, which is defined by With l 1 and l 2 the distance from the BS to the ITM, l 1 on the perpendicular arm and l 2 on the inline arm as seen in figure 6.1, and c the speed of light in vacuum. The selected modulation frequencies and the carrier frequency will all have to go through the IMC. Therefore the length of the IMC (L IMC ) has to be such that its FSR, defined as F SR IMC = c/2l IMC, allows the transmission of the carrier and both modulation frequency sidebands. Therefore the sideband frequencies need to satisfy the following relation as well: c f m = n 2. (6.6) 2L IMC Here n 2 is an integer equal or bigger than 1. Therefore the two sideband modulation frequencies will be an integer multiple of the PRC FSR (F SR P RC ). In the previous section we have seen that the length of the straight PRC is about 82.2 m, while for the folded cavity solution the length is 94.7 m. This means that each stable cavity solution has a different FSR; 1.82 MHz for the straight cavity and 1.58 MHz for the folded cavity solution. As a consequence each configuration will require different modulation frequencies. In both designs the modulation frequencies must not resonate inside the arms of the interferometer. Using equations (6.5) and (6.6) we can then calculate the maximum length of the IMC subject to the vacuum constraints. At the Gingin main lab in the current configuration the maximum IMC length is 10 m. This implies that for a straight cavity the IMC length will be 9.67 m with a FSR of 15.5 MHz. For the folded design the IMC length is 9.97 m with a FSR of 15.0 MHz. The maximum modulation frequency is determined by the bandwidth of the high efficiency photodiodes used in the readout, thus we choose to demodulate at f m1 + f m2 < 100 MHz. By introducing a difference in the arm lengths in the recycling cavities, we can
250 222 CHAPTER 6. STABLE RECYCLING CAVITIES arrange for the output of the interferometer to be dark for the carrier but not dark for one of the sidebands [33]. This asymmetry is required in order to have some signal in the SRC that allows for the control of that degree of freedom. This Michelson arm length difference known as Schnupp asymmetry [34] is defined as l l 1 l 2 which corresponds to the difference between BS ITM SOUT H (inline) and BS ITM EAST (perpendicular) lengths. We can define this asymmetry as l = c/(4f m2 ) since we assume that the reflectivities of the mirrors in the PRC and SRC are the same. Therefore the asymmetry will be 967 mm for the straight PRC and 997 mm for the folded cavity. The lower modulation frequency (f m1 ) needs to be resonant in the PRC and nonresonant in the arm cavities. This allows the control of the common mode signal and the PRC length. The higher modulation frequency (f m2 ) needs to be resonant in the PRC and in the SRC as well in order to be able to control the SRC length and the Michelson arms. Under these conditions the higher modulation frequency will be 77.5 MHz for the straight PRC, with f m1 + f m2 = 93.0 MHz, and 75.2 MHz for the folded cavity solution with f m1 + f m2 = 90.2 MHz. There is however an ongoing investigation for a different selection criteria for the modulation frequencies that could allow for a lower high frequency to be selected without extending the Schnupp asymmetry [35] Signal recycling cavity The length for the SRC will depend on the operation scheme selected for the interferometer. For example for a narrow band detection of black hole  black hole inspiral or neutron star  neutron star inspiral we can select a peak frequency around 300 Hz [36]. This has been extensively studied and it is not the purpose of this paper to analyse in detail the theory behind the detuning of the SRC in order to obtain the desired resonance peak frequency [37, 38]. We will study the case where no detuning (φ s = 0) is required, which corresponds to a RSE configuration for the interferometer. Both modulation frequencies need to fulfil equations (6.4) and (6.5), but only the higher frequency (f m2 ) will be use to determine the length of the SRC as per the following relation [20]:
251 6.6. SIDEBANDS AND THE STABLE RECYCLING CAVITIES 223 L SRC + L SRC = c 2πf m2 (n 3 π + φ s ). (6.7) Here L SRC + L SRC corresponds to the length of the SRC plus its detuning, φ s corresponds to the signal recycling detuning in radians and n 3 an integer. In order to operate the interferometer in a broadband RSE configuration we need to select n 3 carefully. Since the modulation frequencies are multiples of each other and only one of the sideband frequencies (f m2 ) needs to be resonant in the SRC. When operating in a detuned configuration all frequencies will be offresonance. The peak frequency response of the interferometer will depend on the SRC detuning. However the peak frequency not only depends on the length of the SRC. It also depends on the coupling of the cavities, which in turn depends on the transmission and losses of the mirrors that shape the interferometer. How these cavities are coupled (undercoupled, overcoupled or matched) will play an important role in the interferometer frequency response. At the same time the coupling combined with the length of the cavities and in particular the ratio between the lengths of the SRC and the arm cavities will define the slope of the detuning. The bandwidth of the frequency response and the highest possible frequency at which will be possible to tune the interferometer will be defined by the transmission and losses of the mirrors. This is not a simple matter and has been studied in greater detail in references [37, 38]. With a folded PRC a folded SRC of m is needed in order to obtain a broadband RSE interferometer. This implies a distance of 16.2 m between the BS and SR1, which can be easily accommodated at the Gingin test facility (there is a possible maximum of 7 m between the BS and the SRM). This corresponds to a folded stable SRC that follows a similar design as the PRC. The ROC of the SR1 mirror will need to be of 1.52 m. The SRC will then have a Gouy phase of 2.65 radians or a gfactor of For simplicity we have assumed that the mirrors for the SRC are of the same ROC as the one used in the PRC, but this will not necessary be the case. A folded SRC allows us to select the ROC of the mirrors to obtain an optimum Gouy phase. This is still under intense investigation [13]. If required, a marginally stable SRC could be easily accommodated, instead of the SRC extra mirrors (SR1, SR2 and SR3), one SRM at the output of the interferometer can be installed in order to obtain
252 224 CHAPTER 6. STABLE RECYCLING CAVITIES a marginally stable SRC. However special attention will be needed to select the right reflectivity for the mirror according to the combined reflectivities of the PRC mirrors. The straight PRC solution needs a slightly different approach since the lens PR2 is sitting inside the MI. To obtain a stable SRC we still need to make the higher modulation frequency (f m2 ) resonant in the SRC. This means a SRC of m long with a SR1 of 1.5 m ROC at 4.13 m from the BS. With the mirror at this position the Gouy phase shift for the SRC cavity will be 2.4 radians (gfactor 0.52). However if a marginally stable SRC is required it will be necessary to increase the cavity Gouy phase to π radians. Since the Gouy phase transition is rather slow under this configuration it is not possible to obtain the necessary extra Gouy phase shift within the available space inside the Gingin main lab. As a result it will be necessary to add another lens in front of SR1 to obtain the extra Gouy phase. We can install this lens 1.9 m in front of the SR1 and with a focal length of 1.88 m obtain a Gouy phase of nearly π radians at the SR1 making the SRC a marginally stable cavity. In such case the SRC will have the same length as the previous solution, but the SR1 will need to be replaced by a 1.2 m ROC mirror. This extra lens could also be used for tuning the SRC in order to obtain a more suitable Gouy phase Summary A summary of the calculations for the different cavity lengths is presented in table 6.5 for the proposed AIGO interferometer. Table 6.6 shows the details of the recycling cavities for both proposed solutions. The results are for a double recycling interferometer with stable PRC and stable SRC to be operated as a broadband RSE interferometer. 6.7 Discussion We studied two alternative solutions for stable recycling cavities included in the optical design of an advanced interferometric GW detector. The operation as a dual recycled interferometer with a broadband RSE scheme was also presented. This can be extended to a detuned narrow band interferometer by selecting the appropriate
253 6.7. DISCUSSION 225 Parameter Straight PRC Folded PRC L IMC m m L P RC m m L Arms 5000 m 5000 m L SRC m m Asymmetry m m f m MHz MHz f m MHz MHz Table 6.5: Optical length of the different cavities proposed for the AIGO dual recycled interferometer. Parameter Straight PRC Folded PRC PR1  BS BS  IT M Inline BS  IT M P erp L PRC Inline L PRC P erp SR1  BS Table 6.6: Distance (in meters) between the different mirrors that form the proposed stable recycling cavities for the AIGO interferometer.
254 226 CHAPTER 6. STABLE RECYCLING CAVITIES detuning for the SRC. The proposed design includes arm cavities of 5 km long and stable recycling cavities. This will further increase the sensitivity of the interferometer by increasing the power in the recycling cavities and in the main FabryPerot arms. We have shown that, without the addition of extra optical elements, kilometre long inline recycling cavities are not efficient in terms of HOM suppression. However, the addition of a lens in the recycling cavity allows for the adjustment of the accumulated Gouy phase within the PRC and thus to select a suitable level of suppression of the HOM. The study of an alternative solution with a folded stable PRC was also presented. This can be accommodated in probably every GW design depending on the constraints on each particular case. The latter design introduces some level of astigmatism in the stable recycling cavities which will depend on the geometry of the cavity. This is especially adverse in the SRC since it will contain the information of the GW signal coming from the arm cavities. We have also shown that the straight design is less susceptible to errors in the accumulated Gouy phase due to errors in the focal length of the optics which can be corrected by small changes in the position of the lens also recovering the modematching into the main arm cavities. The modemismatch introduced by the thermal gradient induced in the BS as well as any mismatch due to ROC errors can be compensated by reoptimising the optics position and/or the use of the PR2 lens as a thermal compensation plate. These are the main reasons to favour a straight recycling cavity with the addition of a lens in order to obtain the required Gouy phase and HOM suppression. Even though, this design will impose more stringent alignment requirement for the optical elements. It is expected that a reduction of HOM of about 20 db will help to reduce the possibility of parametric instabilities in advanced interferometers to a level where they can be further reduced using passive techniques [39]. Future work will address in more detail the effect of stable recycling cavities in the possibility of parametric instabilities. A further refined look into the noise sources, alignment issues, vibration isolation, and thermooptic noise among others will also be necessary. This will help to outline the final design of future advanced interferometric GW detectors.
255 6.7. DISCUSSION 227 Acknowledgments The authors would like to thanks Slawomir Gras from The University of Western Australia, Jérôme Degallaix from the Albert Einstein Institute Hannover and Volker Quetschke from the University of Florida for useful discussions. This work is supported by grant of the National Science Foundation, the Australian Research Council, and is part of the research program of the Australian Consortium for Gravitational Astronomy.
256 228 CHAPTER 6. STABLE RECYCLING CAVITIES 6.8 Beamsplitter thermal effects Beamsplitter thermal effects in the proposed AIGO stable recycling cavity Introduction As part of the ongoing design of AIGO, the future Australian advanced interferometric gravitational wave detector we include the study of two options for a stable power recycling cavity (PRC). The aim is to use the current location of the East and South end stations 80 m away from the central tank at the vertex of the two arms, which will contain the beamsplitter (BS). The two possible solutions under study are a folded modematching telescope cavity, similar to the current design for Advanced LIGO, and a straight cavity which instead uses an additional lens to obtain the necessary Gouy phase that will allow for a higher order mode frequency gap larger than the cavity bandwidth. Both proposed designs consider the installation of the ITM at the location of the current end stations. From MC PR1 ITM East PR2 ITM East PR2 East From MC ITM South PR3 BS ITM South PR1 BS PR2 South Figure 6.9: Schematics showing the two possible solutions for the AIGO stable recycling cavity, a folded stable recycling cavity on the left and a straight recycling cavity with the addition of a lens on the right.
257 6.8. BEAMSPLITTER THERMAL EFFECTS Astigmatism in folded design One of the main problems of the folded cavity solution is the astigmatism created by the angles at which the beam impinges the mirrors that form the folded recycling cavity. The elliptical spots at the mirrors due to the beam impinging at an angle means different sagitta values for the horizontal and vertical planes. With a circulating power of 2 kw the PRC astigmatism is enhanced by the small thermal effects that this circulating power will induce in the mirrors. A similar situation occurs in the current Advanced LIGO design. Even though it is a milder effect due to the smaller angles at which the beams hit the mirrors that form the folded PRC. The distance between the vacuum tanks in the current Advanced LIGO design allows for the mirrors PR1 and PR2 to be placed before the BS. Distances of more than 16 m between the mirrors that form the PRC allows them to achieve small angles. In the proposed AIGO design we can only achieve 12 m, but with PR2 behind the BS and only 3 m from PR3 to the BS. PR2 is then behind the BS as shown in figure 6.9 the clearance between the BS and the PR2 PR3 circulating beam creates an extra constraint to the minimum angle we can achieve in this design. Figure 6.10 shows the astigmatism on both proposed folded designs (Advanced LIGO and AIGO). The figure shows the different beam size in the horizontal and vertical axis as a result of the beam impinging the PRC mirrors at an angle. A measure of this is shown as the ellipticity of the beam at the PR1 mirror. Table 6.7 presents a summary of these values. Interferometer AIGO Folded PRC Advanced LIGO Folded PRC Axis X Y X Y Waist position (m) Waist size radius (m) Table 6.7: Comparison of the astigmatism induced in the stable folded cavity designs for AIGO and Advanced LIGO. The table shows the waist spot size radius and its position in both stable folded PRC as a way of comparison.
258 230 CHAPTER 6. STABLE RECYCLING CAVITIES 70 Gaussian Beam in Power Recyling Cavity 70 Gaussian Beam in Power Recyling Cavity beam size (mm) Spot size radius X = mm Y = mm e ~ 0.80 beam size (mm) Spot size radius X = mm Y = mm e ~ x y 10 x y distance (m) distance (m) Figure 6.10: Astigmatism comparison between the AIGO design (left) and latest Advanced LIGO PRC design (right). Both designs are folded stable recycling cavities confined within their own vacuum constraints. They have different parameters and thus different spot size at PR1, but it can be seen that the ellipticity at PR1 for AIGO is 0.8 while for the Advanced LIGO design is only Thermal effects The main problem in the current straight cavity design is the smaller spot size at the BS. In the folded cavity design the BS spot size radius is mm (similar to the mm of the current Advanced LIGO design). Since the BS transmits 50% of the incoming beam it only has a few layers of coating, hence low coating absorption. For these simulations a coating absorption of 0.1 ppm was assumed. The substrate absorption of the proposed substrate (Suprasil 3001) [30] for the main optics is only 0.25 ppm/cm, and thus the thermal effects induced by a large spot size are negligible. In the straight cavity design the spot size radius is reduced to 2.29 mm at the BS, which means higher power density. Thermal lensing is mainly caused by the power absorbed by the coating and the power absorbed by the substrate. The power absorbed by the coating will cause a mechanical deformation of the optics changing its sagitta, while the temperature gradient induced by the beam going through the substrate will induce a path difference between the central and the external part of the transmitted beam. This is mainly caused by the refractive index change with temperature. In the case of the BS the thermal effect due to substrate absorption will only affect the inline beam, since the
259 6.8. BEAMSPLITTER THERMAL EFFECTS 231 perpendicular beam doesn t cross the substrate of the BS. An ideal BS will transmit only 50% of the beam reflecting the other 50%. However the beam is reflected back by the arm cavities. Therefore the beam circulating through the substrate and reflected in the coating will be close to the full circulating power in the PRC. Therefore both arms will have slightly different beam profiles at each ITM, which in turn will have different mode mismatch at the waist of the cavity. For all the simulations presented here a circulating power in the PRC of 2 kw was assumed. Figure 6.11 shows the different temperature gradient at the BS under both configurations. The simulations include the effects of coating absorption and bulk absorption. In the folded configuration the maximum temperature rise is only o K above room temperature (300 o K) for 2 kw of circulating power. In the straight configuration the highest temperature is o K degrees above room temperature mainly due to the coating absorption, which can be seen in figure 6.11 (d). This difference is caused by the different spot size at the BS, with a radius of 57.9 mm in the folded configuration and 2.3 mm in the straight configuration. Figure 6.12 shows the thermal lensing effect in the position and size of the waist in the inline arm cavity. In these graphs only the thermal effects of the BS are considered, the thermal effects at the ITM and recycling cavity optics have been omitted in order to compare only the contribution of the BS to the mode mismatch. The right hand side plot in the figure corresponds to the waist size and position after the lens PR2 has been repositioned in order to compensate for a much larger error. Originally the waist radius spot size is around 8.4 mm and the position almost 1.1 km closer to the ITM! A similar effect could be seen in the perpendicular arm, which didn t come as a surprise since the major temperature effect comes from the power absorbed at the coating. Figure 6.13 shows the waist size radius and its position in the perpendicular arm cavity. Even though the mode mismatch seems to be quite large the total effect in terms of power loss it is not as bad as originally thought. Based on the calculations of the couplings of the fundamental mode into higher order modes proposed by D. Z. Anderson [40] we obtained that the power loss to higher order modes due to the mode mismatch is given by:
260 232 CHAPTER 6. STABLE RECYCLING CAVITIES (a) (b) (c) (d) Figure 6.11: Comparison of the temperature profile in the BS due to the PRC circulating power in both proposed AIGO designs. The left hand side shows the temperature gradient induced in the folded PRC, while the right hand side shows the case with a straight PRC. All figures are set to the same temperature scale. We notice the high peak in temperature due to the small spot size at the BS under the straight stable cavity design caused by coating absorption. P P = ( ) 2 ( ) 2 w0 λ ( z) +. (6.8) w 0 2πw0 2 Where, w 0 corresponds to the difference between the size of the intrinsic waist of the cavity and the actual size of the waist of the incoming beam, with w 0 the intrinsic waist of the cavity. z corresponds to the difference in the position of the waist of the cavity; this is between the intrinsic position and the actual position of the waist, with λ the wavelength of the laser, in our case µm.
261 6.8. BEAMSPLITTER THERMAL EFFECTS Spot size and waist position variation with BS thermal lensing 18 Spot size and waist position variation with BS thermal lensing Beam radius (mm) x y Waist (mm) Pos (m) W X Y Beam radius (mm) x y Waist (mm) Pos (m) W X Y Distance (m) Distance (m) Figure 6.12: Comparison of the waist position and size between the two PRC designs. The graphs only show the effect of the BS thermal lensing in the inline arm cavity. It does not include thermal effects in the ITM or PRC mirrors. 18 Spot size and waist position variation with BS thermal lensing 18 Spot size and waist position variation with BS thermal lensing Beam radius (mm) x y Waist (mm) Pos (m) W X Y Beam radius (mm) x y Waist (mm) Pos (m) W X Y Distance (m) Distance (m) Figure 6.13: Comparison of the waist position and size between the two PRC designs. The graphs only show the effect of the BS thermal lensing in the perpendicular arm cavity. It does not include thermal effects in the ITM or PRC mirrors.
262 234 CHAPTER 6. STABLE RECYCLING CAVITIES Inline Arm Perpendicular Arm X Y X Y Waist size radius (mm) Waist position (m) Power loss(%) Table 6.8: Comparison between thermal effects in both arms of the interferometer. Therefore we have four different values for the power loss, since we have different values for the x and y axis position and size of the waist on each arm. The results are presented in table Discussion and conclusions After this brief study of the thermal effects at the BS due to the circulating power and some of its consequences, the conclusion would be to favour the straight cavity design over the folded cavity design. The thermal lensing at the BS will make the mode mismatch between the PRC and arm cavities larger for the straight cavity design even after repositioning the lens. However the astigmatism induced in the folded design outweighs the mode mismatch from the straight cavity. The ideal solution will be to increase the distance between the BS and the recycling mirrors, but also increasing the distance between the recycling mirrors in order to obtain a gentler slope for the Gouy phase transition. A major concern is the spot size at the BS. With a spot size radius of mm in the folded design the temperature rise due to the circulating beam is less than 0.01 o K from room temperature assuming a substrate like Suprasil 3001 with substrate absorption of 0.25 ppm/cm. As a consequence the thermal effects are quite small and the mode mismatch due to these effects is almost negligible. However in the straight cavity design the spot size radius at the BS is only 2.29 mm and therefore the thermal effects much stronger. In this case the main temperature rise is at the coating of the BS. The temperature rise at the substrate is close to 0.02 o K, while at the coating almost 0.03 o K. The strongest mode mismatch will occur in the inline arm, caused by
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266 238 CHAPTER 6. STABLE RECYCLING CAVITIES [29] P. Willems, Thermal Compensation Experience in LIGO, LIGO Presentation, G Z, (2007). [30] Suprasil 3001 and 3002 Data Sheet, [31] E. Chin, J. C. Dumas, C. Zhao, D. G. Blair, AIGO High Performance Compact Vibration Isolation System, J. Phys. Conf. Ser. 32 (2006) [32] M. Evans, S. Ballmer, M. Fejer, et al, Thermooptic noise in coated mirrors for highprecision optical measurements, arxiv: v1 [grqc], (2008). [33] B. J. Meers and K. Strain, Modulation, signal, and quantum noise in interferometers, Phys. Rev. A 44 (1991) [34] L. Schnupp, Internal modulation schemes, presented at the European Collaboration Meeting on Interferometric Detection of Gravitational Waves, Sorrento (1988). [35] K. Somiya, P. Beyersdorf, K. Arai, et al, Development of a frequencydetuned interferometer as a prototype experiment for nextgeneration gravitationalwave detectors, Appl. Opt. 44 (2005) [36] C. Cutler and K. Thorne, An overview of gravitationalwave sources, arxiv: v1 [grqc], (2002). [37] J. Mizuno, Comparison of optical configurations for laserinterferometric gravitationalwave detectors, PhD Thesis, Max Planck Institut für Quantenoptik, (1995). [38] J. E. Mason, Signal Extraction and Optical Design for an Advanced Gravitational Wave Interferometer, PhD Thesis, California Institute of Technology, (2001). [39] S. Gras, D. G. Blair, L. Ju, Test mass ring dampers with minimum thermal noise, Phys. Lett. A 372 (2008) [40] D. Z. Anderson, Alignment of resonant optical cavities, Appl. Opt. 23 (1984)
267 Chapter 7 Summary and conclusions This thesis has presented the author s contributions to the field of interferometric gravitational wave detection including: vibration isolation system design for auxiliary optics and test masses; advanced vibration isolation performance and local control system design; high power modecleaner optical design; and advanced interferometric gravitational wave detector arm cavities and stable recycling cavities modelling and design, which included the selection of RF sidebands for length sensing and control. The compact prototype vibration isolation system design for the modecleaner optics uses a four leg inverse pendulum and a pyramidal Roberts linkage as a double stage horizontal preisolation and includes Euler springs and cantilever spring blades for vertical isolation. A gimbal between the vertical stages provides a high moment of inertia rocking mass, including permanent magnets and copper plates for self damping of the pendulum modes. This design has shown good performance which will further improve once the described modifications are in place. The auxiliary vibration isolator is not only designed to be used for the modecleaner optics, but also for the optics of the recycling cavities. The development and testing of control electronics for the modecleaner vibration isolator (as part of the original design) was started in parallel with the mechanical design. The electronics were tested with the advanced vibration isolator soon after testing with the modecleaner vibration isolator was completed. A more complex, advanced vibration isolator for the test masses was developed at UWA. The combination of an inverse pendulum and Roberts linkage provide a double stage of horizontal preisolation, while a LaCoste spring configuration is used for vertical preisolation. Ultralow frequency control was provided for the LaCoste stage and the Roberts linkage stage through an ohmic thermal position control which, when 239
268 240 CHAPTER 7. SUMMARY AND CONCLUSIONS combined with large dynamic range magnetic actuators, provides complete control of the positioning and damping of the different stages of vibration isolation. The position sensing was provided by shadow sensors with large dynamic range. The addition of magnetic actuators to the Roberts linkage stage in combination with the local control system upgrade will allow for a superspring configuration in the near future, improving the low frequency performance of the vibration isolator. A passive multistage pendulum combines Euler springs for vertical isolation with rocker masses. The masses were combined with copper plates and permanent magnets to create passive viscous damping, using eddy currents to reduce the Qfactor of the pendulum normal modes. A control mass at the end of the pendulum chain provides the last stage of control and an intermediate stage for the test mass suspension made of Niobium ribbons. Since the installed electrostatic control system was not ready for use, the test mass was controlled through the control mass stage rather than directly. Due to the low signal to noise ratio of the control mass signal, the sensing of the test mass angular displacement was arranged through an optical lever placed outside the vacuum tank. The readout signal from a quad photodetector allowed for the test mass readout signal to be used in a control loop for damping of the high Q pendulum modes of the test mass suspension. The local control system is based on an offtheshelf DSP board by Sheldon Instruments. The remainder of the control electronics were developed by the author with the support of C. Zhao at UWA. This included the development of dual channel electronic boards and filter boards. All the necessary boards were mounted on a purposely developed backplane required for the distribution of the I/O signals for the local control system. The control electronics for the electrostatic board will be mounted in a separate chassis (installation pending at time of writing). The control electronics are connected to the vibration isolator through DB 25 cables which are fed into the vacuum tank through vacuum compatible connectors. Since all components inside the vacuum tank have to be ultra high vacuum compatible, thin polyimide copper wires are used for signal distribution. It was necessary to bundle the signals together in DB 25 connectors in order to pass through a sixway cross feedthrough and reach inside the vacuum tank. Once inside the vacuum
269 241 tank the signals first reach an intermediate board, from where they are distributed around the vibration isolator to the different sensors and actuators. This excludes the high current signals used for the ohmic thermal position control, which are fed into the vacuum tank through a separate high current connector with thicker vacuum compatible wiring. Long term stability of the vibration isolator will improve with the addition of an autoalignment system. Even though the cavity could be locked and operated for long periods of time, an autoalignment system will dramatically improve the duty cycle. The autoalignment system forms part of the current development of a hierarchical global control system, which will allow for the operation of longer cavities with higher finesse. The local DSP architecture will be replaced with a centralised system, where a single multicpu digital controller running realtime Linux OS will take over the control of the vibration isolators. Multiple PCIePCIX extension chassis with shared memory at each isolator will be connected to the controller using fibre optics, facilitating data exchange between each local control and the central processing unit. An optical design for the modecleaner was started in parallel with the auxiliary optics vibration isolator. The main objective of this design was to suppress the optical higher order modes and stabilise the laser beam entering the main cavities of the interferometer. Due to the high laser power required in advanced interferometric gravitational wave detectors, this design also considered the thermal effects of the high circulating power inside the modecleaner. These effects will change the cavity gfactor and as a consequence the suppression level of the higher order modes and more importantly will induce astigmatism in the outgoing beam. A new design, which could substantially reduce the astigmatism, was proposed taking into consideration the thermal effects induced by the high circulating power inside the modecleaner cavity. The analyses of optical mode suppression and thermal effects in the modecleaner cavity were then translated into the main arm cavities. The simulations undertaken for the behaviour of the higher order modes in the arm cavities were based on an FFT code developed by the author, with the main purpose being to analyse the different
270 242 CHAPTER 7. SUMMARY AND CONCLUSIONS effects of the mirrors on the higher order optical modes. Of particular interest were the effects with direct influence on parametric instabilities and the parametric gain R 0, such as diffraction losses and the frequency separation of the higher order modes from the fundamental mode. Analysis of a standalone cavity showed that the optical mode parameters were usually underestimated and that degrees of freedom such as mirror size, mirror tilt, optical mode orientation and energy distribution can have a large effect on the estimation of the parametric gain. Further analysis went on to add stable recycling cavities, with the initial intent of converting the marginally stable power recycling cavity into a stable power recycling cavity for dual recycled interferometric gravitational wave detectors. A stable power recycling cavity will help to improve the control signals and the power buildup in advanced interferometers. The stable design has also been extended to the signal recycling cavity. Following recent Advanced LIGO approval of the addition of extra auxiliary suspensions for the extra mirrors necessary for a stable recycling cavities, it will be necessary to analyse the impact of these cavities on parametric instabilities and the parametric gain. This will help to improve the design of advanced interferometric gravitational wave detectors in general and AIGO in particular.
271 Appendices 243
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273 Appendix A Science Benefits of AIGO The Science benefits and Preliminary Design of the Southern hemisphere Gravitational Wave Detector AIGO D. G. Blair 1, P. Barriga 1 A. F. Brooks 2, P. Charlton 7, D. Coward 1, JC. Dumas 1, Y. Fan 1, D. Galloway 6, S. Gras 1, D. J. Hosken 2, E. Howell 1, S. Hughes 8, L. Ju 1, D. E. McClelland 3, A. Melatos 4, H. Miao 1, J. Munch 2, S. M. Scott 3, B. J. J. Slagmolen 3, P. J. Veitch 2, L. Wen 1, J. K. Webb 5, A. Wolley 1, Z. Yan 1, C. Zhao 1 1 School of Physics, University of Western Australia, Crawley, Perth, WA 6009, Australia 2 Department of Physics, The University of Adelaide, Adelaide, SA, 5005 Australia 3 Department of Physics, Australian National University, Canberra, ACT 0200, Australia 4 School of Physics University of Melbourne, Parkville, Vic 3010, Australia 5 School of Physics, The University of New South Wales, Sydney 2052, Australia 6 School of Mathematical Sciences, Monash University, Vic 3800, Australia 7 School of Computing and Mathematics, Charles Sturt University, NSW 2678, Australia 8 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA , USA The proposed southern hemisphere gravitational wave detector AIGO increases the projected average baseline of the global array of ground based gravitational wave detectors by a factor 4. This allows the world array to be substantially improved. The orientation of AIGO allows much better resolution of both wave polarisations. This enables better distance 245
274 246 APPENDIX A. SCIENCE BENEFITS OF AIGO estimates for inspiral events, allowing unambiguous optical identification of host galaxies for about 25% of neutron star binary inspiral events. This can allow Hubble Law estimation without optical identification of an outburst, and can also allow deep exposure imaging with electromagnetic telescopes to search for weak afterglows. This allows independent estimates of cosmological acceleration and dark energy as well as improved understanding of the physics of neutron star and black hole coalescence. This paper reviews and summarises the science benefits of AIGO and presents a preliminary conceptual design. A.1 Introduction Currently there are four kilometre scale gravitational wave detectors in the world (3 LIGO detectors in the USL1 in Livingston, Louisiana, H1 and H2 colocated at Hanford, Washington) and the VIRGO detector in Pisa, Italy. There are also smaller detectors in Europe and Asia: GEO600 in Hannover, Germany, and TAMA in Tokyo, Japan. In the coming decade, advanced detectors will be built, either as upgrades to existing facilities (Advanced LIGO and Advanced VIRGO), or as new detectors: LCGT in Japan and AIGO in Australia. The advanced detectors are designed to have improved low frequency performance and lower shot noise, leading to an amplitude sensitivity about 10 times better than existing detectors, enabling them to monitor a volume of the universe 1000 times larger than current detectors. Frequent neutron star inspiral events should be detectable as well as more distant binary black hole coalescences. Coalescing neutron star binary systems will be able to be observed to about 200 Mpc, while black hole binaries will be able to be observed to distances 1 Gpc (see for example E E Flanagan and S A Hughes [1]). Future improvements using third generation detectors will improve this capability even further. The correlation of electromagnetic events with gravitational wave signals provides enormous science benefits. First it allows the velocity of gravitational waves to be estimated. Second, if the source is a binary inspiral, it allows the luminosity distance to be determined from the gravitational wave inspiral event itself, independent of
275 A.2. SCIENTIFIC BENEFITS OF THE AIGO OBSERVATORY 247 the red shift determined from observation of the host galaxy. This allows a powerful independent probe of the Hubble law, cosmological acceleration and the equation of state of dark energy [2]. However, this requires the identification of the gravitational waves source locations to find the electromagnetic counterpart of the event. Individual gravitational wave detectors have poor angular resolution with a beam width of 120 degrees, so they are good all sky monitors but are completely inadequate for directional searches. This situation is greatly altered if an array of detectors is used. Then the coherent analysis of signals from the array allows the network to have diffraction limited resolution, where, as with VLBI radio astronomy, the angular resolution is set by the ratio of the signal wavelength to the product of the projected detector spacing and the signal to noise ratio. A world wide array of detectors can achieve an angular resolution of 10 arc minutes for signals in the audio frequency terrestrial detection band as discussed further below. However, the two dimensional projected detector spacing can only be large for all directions in the sky if the array contains a southern hemisphere detector. Here we summarise the scientific benefits of the AIGO detector and the then go on to summarise a preliminary conceptual design for this detector. A.2 Scientific benefits of the AIGO observatory It has long been recognised that an Australian detector disproportionately improves the science return of the existing international network of gravitational wave detectors. This disproportionate impact comes about for several reasons. First, an Australian detector would greatly improve our ability to determine, from gravitational waves alone, the location a gravitational wave event on the sky. Gravitational wave detectors largely determine position by triangulation using time of arrival information between different detectorsphase fronts of an incident gravitational wave interact with different detectors at different times. Coherent network analysis effectively resolves these differing times of arrival, enabling the detector array to be an all sky monitor with good angular resolution over all source directions Detailed calculations by Wen et al [3, 4, 5] indicate that inclusion of AIGO would on average improve
276 248 APPENDIX A. SCIENCE BENEFITS OF AIGO the international network s ability to localise sources from about 12 square degrees to a fraction of a square degree as shown in figure A.1. Without AIGO, the error ellipses are typically about 1.5 deg 8 deg. With AIGO, they are significantly smaller than 1 deg 1 deg. The error ellipse would then be well matched to the field of view of most sensitive optical, Xray and radio telescopes, so that it becomes possible to conduct very long exposure searches for electromagnetic signatures of gravitational wave events. The second major impact of an Australian detector would be to improve greatly our ability to measure both polarisations of a signal. At least two detectors with substantial different orientations are needed to fully reconstruct both wave polarisations from the data. The LIGO detectors in the USA are oriented in such a way that they do not provide information about both polarisation components. This was a deliberate choiceby orienting both detectors such that they each measure the same wave polarisation, the statistical confidence in any given detection is greatly increased. For the initial goalunambiguous first detection of gravitational wavesthis is a natural and appropriate choice of detector orientations. Unfortunately, this is not such a good choice of detector orientations once direct detection has occurred. The primary goal of developing the science of gravitationalwave astronomy requires the measurement of the polarisations, as this greatly increases our ability to infer astronomically important information. Consider, for example, waves from binary coalescence, of which advanced detectors expect to detect more than 20 events per year. In this case, the amplitude ratio of the two gravitational wave polarisations encodes the inclination of the plane of the binary orbit with respect to the line of sight from the Earth. Once the orbital inclination is defined, the frequency evolution of the gravity wave signal contains a complete description of the system, and the observed amplitude therefore encodes the distance of the source. This remarkable property of gravitational wave signals enables them to be very powerful cosmological probes. Adding an Australian detector to the network immensely augments its capability to measure polarisations simply due to its orientation on the nearly spherical surface of the Earth. Thus with AIGO the network can obtain quite precise distance information to sources [2].
277 A.2. SCIENTIFIC BENEFITS OF THE AIGO OBSERVATORY 249 The third benefit from the Australian detector relates to the noise performance of the network. For broadband stationary noise, the noise of a network of detectors is reduced as the square root of the number of detectors. While this factor is not large (only 25%), it has a much larger effect on the number of detectable sources, since the number of detectable sources depends on the volume of the accessible universe, which increases as the cube of the detector strain sensitivity. Thus the global array can be expected to detect almost double the number of signals with the addition of a single southern hemisphere detector of sensitivity comparable to the northern hemisphere detectors. For nonstationary noise, a larger network has the benefit of being much better at rejecting spurious signals. Such signals must mimic a gravitational wave passing through the network by arriving at each detector at a time and with appropriate amplitude to be consistent with a real gravitational wave signal. The probability of such a signal reduces as the power of the number of detectors, so the addition of a single detector greatly reduces this probability. This reduction can be by a factor of depending on the types of signal. The above three improvements provided by adding AIGO to the world array vastly enhances the knowledge we can gain about the gravitational wave sources. For some sources, such as core collapse supernovae, the waves are likely to be poorly understood prior to gravitational wave observations. For others such as waves from black hole binary mergers, the signals are likely to be only moderately well understood, while waves from coalescing binaries prior to merger are well understood. In the not well understood regime one must use the observed waves to solve an inverse problem and obtain an understanding of the dynamics of the source. Gursel and Tinto [6] developed methods for performing the inverse problem, and examined how well it could be implemented using detectors located in North America, Germany, and Australia. Their work demonstrates that the addition of AIGO to the network greatly improves the reconstruction of such waves. In the other extreme of a well understood system, the signal can be used to define the source distance and location on the sky. Figure A.1 demonstrates the advantage of increasing the number of detectors in the array and also of obtaining maximum outofplane volume in the array by placing one detector in the southern hemisphere. AIGO significantly improves the angular
278 250 APPENDIX A. SCIENCE BENEFITS OF AIGO resolution and also eliminates the ambiguity problem which arises if all the detectors are close to a common plane. The outofplane response also increases the maximum baseline significantly thereby obtaining good angular resolution in almost all sky directions. The array is even further improved if LCGT is added. (a) (b) Figure A.1: Angular area maps for world array. The angular uncertainty for each geocentric sky direction is indicated as an ellipse on the sky. These are normally highly elongated. (a) LIGO and VIRGO. (b) LIGO, VIRGO and AIGO. A further improvement is obtained if LCGT is added to the array as shown in figure A.2. To quantify the problem of host galaxy determination, we need an estimate of the number of galaxies within the detector array angular resolution. Figure A.2 shows the average number of galaxies per 1 σ error ellipse for different gravitational wave detector arrays, as reported in Wen et al in 2007 [5]. We see that the average number of galaxies at 200 Mpc varies from in excess of 200 for LIGO VIRGO array (LHV) to about 4 if AIGO an LCGT are added to the array. Taking the galaxy distribution into account, Wen et al showed that about 25% of sources can be unambiguously identified with a galaxy. This allows the Hubble Law to be tested without actual identification of an optical outburst.
279 A.3. PRELIMINARY CONCEPTUAL DESIGN FOR AIGO number of galaxies within a 1 σ error circle distance (Mpc) Figure A.2: The average number of galaxies expected within a 1 σ error ellipse for different gravitational wave detector arrays, based on the angular resolution of each array for each sky direction. The figure shows that the number of galaxies per error ellipse is reduced from almost 200 for the LIGO VIRGO array, to less than 10 for LIGO VIRGO AIGO. For a single additional detector, AIGO gives the greatest benefit, but the best array contains AIGO and LCGT. The symbols in the figures are: C LCGT, A AIGO, V VIRGO, LH LIGO. A.3 Preliminary conceptual design for AIGO A preliminary conceptual design for AIGO utilises the maximum arm length possible on our site of about 5 km. The extra arm length has the advantage of diluting local noise sources such as thermal noise and control system noise, allowing slightly less demanding specification for coating acoustic loss and control systems. With Advanced LIGO (AdvLIGO) test mass and control specifications, the maximum inspiral range is increased from about 200 Mpc to 250 Mpc, corresponding to a doubling of the accessible volume of the universe. Our design uses slightly smaller beam spots than AdvLIGO, (55 mm) to maintain 1 ppm arm cavity diffraction loss. This slightly increases the test mass thermal noise but reduces the overlap factor for parametric instability. The vacuum arms are chosen to be the same diameter as LIGOs. The vacuum design [7] has been shown to allow
280 252 APPENDIX A. SCIENCE BENEFITS OF AIGO LIGO vacuum specifications to be exceeded. A passive solar thermal bakeout system has been demonstrated experimentally [8]. The vacuum system will include future provision for ion pumps. Vibration isolation will use multistage passive isolators developed at UWA [9], subject to successful evaluation. Two are currently being evaluated on an 80 m optical cavity. The isolators have an extra preisolation stage compared with the VIRGO design. Otherwise they are conceptually similar to those of VIRGO but are much more compact, occupying about 10% of the volume (and are correspondingly cheaper). We propose to use sapphire for the test masses, because this material has a low acoustic mode density compared with fused silica, which would reduce the problem of parametric instability [10]. Sapphire also has the advantage of fast thermal response, which means that the system comes into thermal quasisteady state much more rapidly than fused silica, further aiding parametric instability control [11]. The absorption requirement of the two inner test masses is 50 ppm/cm. This is typical of good quality material currently available. The end test masses can use sapphire of lower optical quality. Auxiliary optics will be made from fused silica: mirrors for a quasistable power recycling cavity, the 10 m input modecleaner and 1 m output mode cleaner. The test mass suspension system follows the design developed by Lee at UWA [12], utilising four thin niobium ribbons with microcantilever suspension from equatorial holes in the test masses. The test masses are controlled electrostatically using annular bifilar comb capacitors incorporating an RF local control readout which serves as an auxiliary local sensor. They are supported by a control mass which plays the role of a VIRGO marionetta. Main test mass control is by actuators on the control mass. The laser, injection locked to the successful Adelaide 10 W laser now in use [13], will initially have a power of 100 W. This may need to be augmented to 200 W, subject to performance of the vacuum squeezing. Vacuum squeezing is currently not part of the AIGO baseline design but could be implemented if the technology is available. Two prototype auxiliary optics suspensions have already been developed [10]. These use two stages of horizontal preisolation (one inverse pendulum, one Roberts linkage), one Euler spring and one pair of blade springs for vertical isolation. These
281 A.4. DISCUSSION AND CONCLUSIONS 253 will be used to support the mode cleaner mirrors as well as the power recycling and signal recycling mirrors. Hartmann sensor technology for monitoring wavefront distortion has been developed at Adelaide and recently demonstrated in Gingin [15]. The AIGO design includes Hartmann sensors with CO 2 laser thermal control to monitor and compensate wavefront distortion in all test masses as well as the beamsplitter and compensation plates. The control specification is to better than 1 nm. Closed loop thermal compensation control has been successfully implemented at the Gingin facility. The proposed interferometer configuration will be detuned Resonant Sideband Extraction. The control strategy is likely to be based on an ANU concept in which control signals are injected into both beamsplitter ports. We propose to use an 80 m stable power recycling cavity containing a thermal compensator lens. The input modecleaner design for AIGO will use a triangular 10 m cavity with apex mirror output for reduced astigmatism [16]. An output mode cleaner will be employed to reject light in higher order modes and control modulation sidebands from reaching the photodetectors. This is likely to be based on the AdvLIGO concept of a 4 mirror ring cavity silicate bonded onto a glass breadboard. A.4 Discussion and conclusions We have shown that a global array of gravitational wave detectors that contains AIGO is substantially improved. Better polarisation resolution allows improved distance estimates for inspiral events. Roughly 25% of detected inspiral events can be identified with a particular galaxy. This enables independent measurements of the Hubble constant with or without detection of an optical outburst. The addition of AIGO doubles the number of detectable sources and reduces nonstationary noise by more than an order of magnitude. A 5 km interferometer has significant advantages, particularly in reducing thermal noise, and in principle can detect double the number of sources compared with a single AdvLIGO interferometer.
282 254 APPENDIX A. SCIENCE BENEFITS OF AIGO A.5 References [1] E. E. Flanagan and S. A. Hughes, Measuring gravitational waves from binary black hole coalescences. I. Signal to noise for inspiral, merger, and ringdown, Phys. Rev. D 57 (1998) [2] N. Dalal, D. E. Holz, S. A. Hughes, and B. Jain, Short GRB and binary black hole standard sirens as a probe of dark energy, Phys. Rev. D 74 (2006) (9pp). [3] L. Wen and B. Schutz, Coherent data analysis strategies using a network of gravitational wave detectors, Technical Report, G , LIGO, (2005). [4] L. Wen, Network analysis of gravitational waves, Technical Report, G , LIGO, (2006). [5] L. Wen, E. Howell, D. Coward, and D. Blair, Host galaxy discrimination using world network of gravitational wave detectors, in Proceedings of the XLI Ind Rencontres de Moriond on Gravitational Waves and Experimental Gravity (J. Dumarchez and J. T. T. Van, eds.), , The Gioi Publishers, [6] Y. Gürsel and M. Tinto, Near optimal solution to the inverse problem for gravitationalwave bursts, Phys. Rev. D 40 (1989) [7] S. Sunil and D. G. Blair, Investigation of vacuum system requirements for a 5 km baseline gravitationalwave detector, J. Vac. Sci. Technol. A 25 (2007) [8] D. Berinson, D. Blair, P. Turner, and T. Simaile, Test of a model solar bakeout system for laser interferometer gravitational wave detectors, Vacuum 44 (1993) [9] E. J. Chin, J. C. Dumas, C. Zhao, et al, AIGO high performance compact vibration isolation system, J. Phys. Conf. Ser. 32 (2006) [10] L. Ju, S. Gras, C. Zhao, J. Degallaix, and D. Blair, Multiple modes contributions
283 A.5. REFERENCES 255 to parametric instabilities in advanced laser interferometer gravitational wave detectors, Phys. Lett. A 354 (2006) [11] J. Degallaix, C. Zhao, L. Ju, and D. Blair, Thermal tuning of optical cavities for parametric instability control, J. Opt. Soc. Am. B 24 (2007) [12] B. H. Lee, Advanced Test Mass Suspensions and Electgronstatic Control for AIGO. PhD Thesis, School of Physics, The University of Western Australia, [13] D. Mudge, M. Ostermeyer, D. J. Ottaway, et al, Highpower Nd:YAG lasers using stable unstable resonators, Class. Quantum Grav. 19 (2002) [14] P. Barriga, A. Woolley, C. Zhao, and D. G. Blair, Application of new preisolation techniques to mode cleaner design, Class. Quantum Grav. 21 (2004) S951 S958. [15] A. F. Brooks, T.L. Kelly, P. J. Veitch, and J. Munch, Ultrasensitive wavefront measurement using a Hartmann sensor, Opt. Express 15 (2007) [16] P. J. Barriga, C. Zhao, and D. G. Blair, Astigmatism compensation in modecleaner cavities for the next generation of gravitational wave interferometric detectors, Phys. Lett. A 340 (2005) 1 6.
284 256 APPENDIX A. SCIENCE BENEFITS OF AIGO
285 Appendix B Control of Parametric Instabilities Strategies for the control of parametric instability in advanced gravitational wave detectors L. Ju, D. G. Blair, C. Zhao, S. Gras, Z. Zhang, P. Barriga, H. X. Miao, Y. Fan and L. Merrill School of Physics, University of Western Australia, Crawley, Perth, WA 6009, Australia Parametric instabilities have been predicted to occur in all advanced high optical power gravitational wave detectors. In this paper we review the problem of parametric instabilities, summarise latest findings, and assess various schemes proposed for their control. We show that nonresonant passive damping of test masses reduces parametric instability but has a noise penalty, and fails to suppress the Qfactor of many modes. Resonant passive damping is shown to have significant advantages but requires detailed modelling. An optical feedback mode suppression interferometer is proposed which is capable of suppressing of all instabilities but requires experimental development. B.1 Introduction Groundbased gravitational wave detectors are designed to make extremely high precision measurements of the motion of test masses with perturbations limited by quantum measurement theory. To obtain high sensitivity, high laser power is required. If 257
286 258 APPENDIX B. CONTROL OF PARAMETRIC INSTABILITIES there is any mechanism that a small amount of this power can be coupled directly into oscillatory vibration of the test masses, this can lead to their uncontrolled mechanical oscillation. A particular means by which this can occur is called parametric instability (PI). The possibility of PI in advanced laser interferometer gravitational wave detectors was first shown to be a potential problem by Braginsky et al [1, 2]. Later, threedimensional modelling showed that parametric instability was likely for current proposed interferometer configurations [3]. If the high frequency acoustic modes in test masses have significant spatial overlap with high order optical cavity modes that satisfy a resonance condition, then parametric instability could occur. It appears that parametric instability is likely to be a potential threat to any interferometer which uses high optical power and low acoustic loss test masses. Instability causes test mass acoustic modes to ring up to a large amplitude in a time that could be in the range 50 ms to hundreds of seconds. The large amplitude destroys the interferometer fringe contrast. The phenomenon cannot be filtered or reduced through any postprocessing of data. Since the original predictions, parametric instability has been extensively modelled. It has been undertaken by groups at The University of Western Australia (UWA), Moscow State University, The Japanese Large Scale Cryogenic Gravitational Wave Telescope (LCGT) project and members of the Laser Interferometer Gravitational Observatory (LIGO) laboratory at the California Institute of Technology [3, 4, 5, 6, 7, 8]. Modelling requires detailed knowledge of optical modes and acoustic modes in test masses. Results are dependent on small changes in the test mass geometry (which change the acoustic mode spectrum) and on the mirror diameter and shape, which affect the optical mode spectrum. Small features such as small wedge angles in the test masses and the placing of flats on the test mass sides have very strong effects because they break both the acoustic mode and optical mode degeneracy, acting in general to increase the number of potentially unstable modes. There is no significant disagreement between estimates of instability. However, the precise detail of instability is extremely sensitive to system parameters. For example, a change in mirror radius of curvature of 1 part in 10 4 is sufficient to modulate individual mode parametric gain by a large factor.
287 B.1. INTRODUCTION 259 Such changes are within the uncertainties of test mass material parameters and 1 2 orders of magnitude larger than the mirror radius of curvature tolerance. Because of the high sensitivity of the resonant interactions to small parameter changes, only models which use the same finite element modelling (FEM) test mass meshing, the same acoustic losses of test masses including the contributions from coatings, the same model for optical cavity diffraction losses and the same material parameters will give strictly identical results. Modelling results also contain significant uncertainties due to the limited precision of FEM [9]. Thus modelling is unlikely to be able to predict the detailed instability spectrum of a real largescale interferometer. However, modelling results are likely to present an accurate statistical picture. Best estimates currently predict an average of 10 unstable modes per test mass in an Advanced LIGO 1 type configuration. The most realistic modelling to date has been applied to a proposed Advanced LIGO configuration. Modelling takes into account the test mass shape and acoustic losses due to mirror coatings. Results predict that between 0.2% and 1% of acoustic modes in the frequency range khz are likely to be unstable depending on the precise instantaneous value of the effective mirror radius of curvature. The parametric gain for unstable modes varies between 1 and Over a radius of curvature range of 30 m (a range chosen to take into account thermally induced changes as well as manufacturing tolerances), current estimates indicate a positive parametric gain greater than unity for modes spread across four test masses. The number of modes and their gain fluctuate as the radius of curvature is thermally tuned. Over the 30 m radius of curvature thermal tuning range, more than 700 acoustic modes are parametrically excited in each test mass (but at any one radius of curvature, the number is only about 5 10 per test mass). Parametric instability estimates were first obtained for interferometer arm cavities alone [1]. However, the real gravitational wave detector configuration is more complicated with nested cavities of power recycling and signal recycling cavities [10]. The effects of degenerate power and signal recycling cavities were then considered [2, 5], including the realistic case of nonmatched arm cavities [11]. By selectively suppress 1
288 260 APPENDIX B. CONTROL OF PARAMETRIC INSTABILITIES ing arm cavity modes, the effect of using stable (nondegenerate) power recycling cavities can be estimated. To date, all configurations analysed show instability. The magnitude of parametric instability gain scales as the product of three Q factors: those of the optical cavity main mode, the relevant cavity high order mode and the test mass acoustic mode. Unstable modes are generally in the frequency range of khz. Most suggestions of methods for controlling instability have focused on changing the value of the Qfactors of the acoustic modes. In any room temperature interferometer in which thermal lensing is significant, parametric instability will be tuned by timevarying thermal lensing. Thus, changes will occur over a thermal lensing timescale, which is seconds to minutes in single crystal test masses (such as sapphire or silicon) and 1 h for fused silica test masses. In 2007, the UWA group observed threemode parametric interactions for the first time [3]. The parametric gain was shown to be tunable through variation in the test mass radius of curvature in agreement with predictions. In a cryogenic interferometer, parametric instability is likely to be frozen into a particular configuration since the temperature coefficient for thermal lensing falls effectively to zero at cryogenic temperatures. However, the precise configuration is unlikely to be able to be predicted in advance. Since advanced gravitational wave detectors are already under construction, it is critical to focus on strategies for the control of parametric instability, which is the focus of this paper. Instability control should be achievable without noise penalty because the instabilities are all narrow band and all occur outside the measurement band. However for many reasons, noise free PI control is not simple. Section B.2 summaries the theory of parametric instability and the status of modelling results and explains why PI has not already been seen in LIGO and VIRGO. Section B.3 discusses various approaches to PI control and shows that simple methods have noise penalties, disadvantages and risks. Optical feedback control is also discussed, which although complex, can be implemented as a fully automatic external suppression system.
289 B.2. PARAMETRIC INSTABILITIES THEORY AND MODELLING 261 B.2 Parametric instabilities theory and modelling B.2.1 Summary of theory Parametric interactions can be considered as classical ponderomotive interactions of optical and acoustic fields or as simple scattering processes [13], as indicated in figure B.1. In (a), a photon of frequency ω 0 is scattered, creating a lower frequency (Stokes) photon of frequency ω s and a phonon of frequency ω m, which increases the occupation number of the acoustic mode. In (b), a photon of frequency ω 0 is scattered from a phonon creating a higher frequency (antistokes) photon of frequency ωa, which requires that the acoustic mode is a source of phonons, thus reducing its occupation number. The scattering could create entangled pairs of phonons and photons [14]. (a) (b) ω 1 = ω 0  ω m ω 0 ω 0 ω a = ω 0 + ω m ω m ω m Figure B.1: Parametric scattering of a photon of frequency ω 0 (a) into a lower frequency Stokes photon, ω s, and a phonon of frequency ω m, and (b) into a higher frequency anti Stokes photon ω a, which require destruction of a phonon. Figure B.2 illustrates threemode parametric interactions in an optical cavity from a classical viewpoint. In this example, stored energy in the form of a TEM 00 mode is shown scattering into a TEM 11 mode by a particular acoustic mode. The interactions can only occur strongly if two conditions are met simultaneously. First, the optical cavity must support eigenmodes that have a frequency difference approximately equal to the acoustic frequency ω 0 ω 1 ω m. Here, ω 0 is the cavity fundamental mode frequency while ω 1 represents either Stokes or antistokes high order mode. Second,
290 262 APPENDIX B. CONTROL OF PARAMETRIC INSTABILITIES the optical and acoustic modes must have a suitable spatial overlap. Parametric interaction effect can be expressed by parametric gain R. If R > 1, instability will occur. Radiation pressure force Input frequency ω o Acoustic mode ω m Cavity fundamental mode ω o (Stored energy) Stimulated high order optical mode ω 1 Figure B.2: A cartoon of threemode interactions in an optical cavity from a classical point of view, showing an acoustic mode scattering stored energy from the cavity fundamental mode into a high order mode while acting back on the test mass by radiation pressure. The parametric gain in a simple cavity is given by [2] R = ± 4P 0Q 1 Q m LMcω 2 m Λ 1 + ( ω/δ 1 ) 2 (B.1) Here, P 0 is the power stored in the TEM 00 mode, which is the fundamental mode in the cavity; M is the mass of the acoustic resonator; L is the length of the cavity; ω m is the acoustic mode frequency; ω = ω 0 ω 1 ω m ; δ 1 = ω 1 /2Q 1 is the half linewidth (or damping rate) of the high order optical mode; ω 1 is the frequencies of the Stokes (antistokes) high order optical modes; and Q 1 and Q m are the quality factors of the high order optical mode and the acoustic mode respectively. The factor Λ measures the spatial overlap between the electromagnetic field pattern and the acoustic displacement pattern defined by [2]
291 B.2. PARAMETRIC INSTABILITIES THEORY AND MODELLING 263 Λ = ( V ψ0 ( r ) ψ 1 ( ) 2 r ) u zdr ψ0 2 dr ψ1 2 dr u 2 dv, (B.2) where ψ 0 and ψ 1 describe the optical field distribution over the mirror surface for the TEM 00 mode and higherorder modes respectively, u is the spatial displacement vector for the mechanical mode and u z is the component of u normal to the mirror surface. The integrals dr and dv correspond to integration over the mirror surface and the mirror volume V, respectively. It should be pointed out that there are often multiplemode interactions, and the above equation should include a summation over all the possible modes (both Stokes and antistokes modes) [15]. Stokes and antistokes modes, respectively. The positive and negative signs in R correspond to From equation (B.1), it can be seen that parametric gain is a product of three parameters: P 0, Q m and Q 1, corresponding to the arm cavity power, acoustic mode Q factor, and high order optical mode Qfactor respectively. Compared with the baseline parameters for Advanced LIGO, the initial LIGO arm cavity power (P 0 20 kw) is 40 times lower, the acoustic Q ( 10 6 ) is 10 times lower and optical Q 1 (finesse of 200) is 2 times lower than those of Advanced LIGO (P kw, Q m 10 7, finesse of 450). In addition, the test masses are smaller so that the acoustic mode density is lower at high frequency. The parametric gain in initial LIGO should be more than 800 times lower than Advanced LIGO, and the risk of instability is very low. Hence, it is no surprise that LIGO has not observed PI. It is important to point out that the derivation of equation (B.1) is based on a model that does not consider diffraction losses of the optical cavities. We found that in the case of a simple cavity, the addition of diffraction loss does not change the formulation and equation (B.1) is still valid. To incorporate diffraction losses in the analysis of parametric instabilities in interferometers with recycling cavities is mathematically difficult and has not yet been accomplished. However, in practice for optical cavities with relatively low finesse, the diffraction losses of the low spatial order modes are small compared with the coupling loss. Only very high order modes have strong diffraction losses. The very high order mode contribution to PI is usually small; thus, diffraction losses can often be ignored. So, the results presented here do
292 264 APPENDIX B. CONTROL OF PARAMETRIC INSTABILITIES not strongly depend on this issue. B.2.2 Modelling approach We have conducted many simulations of various interferometer configurations. In general, we use finite element modelling to simulate several thousand acoustic modes of the test masses and their diffraction losses [3]. We have used both modal analysis and FFT codes to model cavity optical modes. We have simulated interferometers with both signal recycling mirror (SRM) and power recycling mirror (PRM) using Advanced LIGO parameters. This involves an elaborated form of equation (B.1) which takes into account both power and signal recycling [11]. We normally assume identical main cavities of 4 km length and a circulating power of 830 kw. We also assume that a power recycling cavity operates in a marginally stable scheme. For stable power recycling cavities, one expects that the finesse of some high order modes will increase, thereby increasing Q 1 and the associated parametric gain. Unfortunately, we do not have the formalism available at present to analyse PI in an interferometer with stable power recycling cavities. Stokes and antistokes modes are constructed using a set of transverse optical modes up to eleventh order. In a typical analysis, about 60 optical higher order modes (HOM) and 5500 elastic modes of the test mass are taken into account. The HOM mode shapes and resonant frequency are obtained numerically using the eigenvalue method whereas elastic modes are calculated using finite element modelling. The test mass model takes into account most of the detailed structure. For our most precise model (see results below), test masses are modelled as 20 cm thick 17 cm radius cylinders with flats on the circumference, including chamfers and back face wedge of 0.5 o. In addition, the optical modes frequency detuning due to the finite mirror geometry [3] is also taken into account. The quality factor of optical modes is determined by coupling losses, while the Qfactor of elastic modes is based on substrate losses [17] and coating losses [18]. Diffraction losses of the high order modes are not significant for low finesse design presented below. Our analysis usually includes up to the fifth longitudinal mode number from the main cavity mode, which ensures that all possible interactions of optical modes with
293 B.2. PARAMETRIC INSTABILITIES THEORY AND MODELLING 265 an elastic mode are taken into account. However, usually only the first three mode numbers contribute significantly to the parametric gain. The PI analysis is carried out for different radii of curvature (RoC) of the end test mass (ETM) mirror. In the analysis presented below, the ETM RoC is allowed to vary from km to km with 0.1 m steps. The radius of curvature for input test mass (ITM) is set as a constant of km. For each ETM RoC data point, the resonance condition ω and the overlapping parameter Λ are calculated. Estimating PI for different RoC of the ETM enables us to simulate the thermal tuning of the interferometer and thus probe changes of the resonant conditions. B.2.3 Modelling results Our simulations of the Advanced LIGO detector with power and signal recycling cavities reveal that the instability can occur over the whole RoC range considered. For this configuration, there is a very strong dependence of the mirror radius of curvature on the parametric gain R of unstable modes. Figure B.3 shows the number of unstable modes for different RoC. At certain values of RoC, there is up to 20 unstable modes while for other values, only a few unstable modes are present. The number of unstable mode is on average 8 ± 3 for each test mass. However, the total number of unstable modes over the whole range is as high as 777. Figure B.4 shows the highest parametric gain of the unstable modes at different radius of curvature. It can be seen that parametric gain can vary from 2 up to It should be noted that the instability condition for certain mechanical modes is very sensitive to the change of radius of curvature, in the range 0.1 m. Therefore, very small changes of the effective mirror curvature result in strong changing of the instability condition from one mode to another. This can be visualised by examining the movies located at Movies.htm. These movies show parametric gain for all modes during a sweep through the test mass radius of curvature. We recommend playing the sweep manually by dragging the tracker button, to enable careful examination of the gainfrequency structure. Figure B.5 shows a statistical result of gain values of unstable modes. It can be seen that the majority of unstable modes have R < 5. However there are a substantial
294 266 APPENDIX B. CONTROL OF PARAMETRIC INSTABILITIES number of modes with gain R > 100. Figure B.3: Number of unstable modes as a function of ETM radius of curvature for an interferometer with advanced detector parameters. Both substrate and coating acoustic losses were taken into account for the elastic modes Qfactor. The transmissivity of the SRM, PRM, ITM, ETM are set to proposed Advanced LIGO values 20%, 2.5%, 1.4%, and 5 ppm, respectively. The detuning of the signal recycling cavity was δ=20 deg. B.3 Possible approaches to PI control B.3.1 Power reduction and thermal radius of curvature control The simplest approach to PI control is to reduce the input power by a factor equal to the peak instability gain that might be encountered. To ensure stable operation in a reasonable range of mirror radii of curvature, the power would have to be reduced, say, to 1% of nominal power, thereby increasing the detector shot noise by an order of magnitude.
295 B.3. POSSIBLE APPROACHES TO PI CONTROL 267 Figure B.4: Maximum parametric gain for an interferometer with different ETM radius of curvature using advanced detector parameters. Figure B.5: The parametric gain distribution of unstable modes, for all of the unstable modes in a 30 meter radius of curvature range. The majority unstable modes have R value< 5.
296 268 APPENDIX B. CONTROL OF PARAMETRIC INSTABILITIES A much better option would be to use CO 2 laser heating to stabilise the interferometer test masses at radii of curvature where the peak parametric gain is minimum as indicated in figure B.4. In the absence of any other damping scheme, a power level of 10% of the proposed peak power could be achievable. Since power causes thermal radius of curvature tuning, the process of powering up sweeps the cavity through different parametric instability gain regimes on a test mass thermal timescale ( 1000 s for fused silica test masses). The ringup time constant of an unstable mode can be written as [15] τ = 2Q ω m (R 1). (B.3) For a test mass to ringup from thermal amplitude of m to 10 9 m (the assumed breaking lock amplitude), the break lock time will be t B = 23Q ω m (R 1). (B.4) For example, a 30 khz mode with Q = 10 7 and R = 20 would cause the interferometer to break lock in about 1 min. While thermal actuation with CO 2 lasers may compensate for this, the process is complicated by the long time constants in fused silica test masses. It is not clear to us how to simultaneously control the instantaneous radius of curvature (which determines the parametric instability) and the thermal lensing environment (which defines the optical mode matching) with a single CO 2 laser actuation system. Because of the long thermal time constants, interferometer lock acquisition would need to use thermallensdependent power level control to navigate past instabilities without losing lock. Once operation is stabilised to a radius of curvature where the PI gain is low, the power level could be ramped up to just below the PI threshold. The ramping up of power would need to be slow enough so that the radius of curvature stayed within a tight range. It could be possible to operate proposed interferometers at between 10% and 30% of the proposed power in the absence of other control schemes. After losing lock, it might be a very slow process to regain stable operation because of the long thermal memory of the test masses.
297 B.3. POSSIBLE APPROACHES TO PI CONTROL 269 B.3.2 PI control by using ring dampers or resonant acoustic dampers Ring dampers The idea of ring dampers is to apply lossy strips on the circumference of test masses to suppress the Qfactors of the acoustic modes. This Qfactor reduction will not greatly increase the thermal noise of the test mass if the lossy parts are far from the laser spot at the test mass [19, 20]. By carefully choosing the position of the lossy strips, it is possible to reduced the test mass mode Qfactor without greatly degrading the thermal noise performance of the interferometer. The lossy strips could be an optical coating or a layer of Al 2 O 3, Au or Cu applied by conventional ionassisted deposition techniques. We have analysed the use of ring dampers. This work has been published elsewhere [6] and here we summarise their performance. The ring damper method can effectively suppress many acoustic mode Qfactors by a factor of 50. As shown in figure B.5, most of the unstable modes have R values < 5. Thus, by applying ring dampers to the test masses most of the unstable modes can be suppressed. Unfortunately the effect of the ring damper is not uniform: some modes are only weakly damped and few unstable modes are very difficult to suppress. Figure B.6 shows the reduction of unstable modes as a function of RoC for a typical ring damper design. Here we assume a simple cavity interferometer without recycling cavities but with Advanced LIGO parameters. The ring damper is a 2 cm wide, 20 m thick strip with loss angle of The exact number of unstable modes at certain RoC may differ from those when recycling cavities are considered but the statistical results of the simulation are not altered. There is always a thermal noise penalty by applying lossy ring dampers on the test masses. The model shown here contributes a Brownian thermal noise penalty of 5% [6]. If the noise penalty is allowed to increase to 20%, substantial instability free windows appear in the RoC tuning range. Resonant dampers The use of resonant dampers, widely used in vibration isolation systems, was proposed by Evans et al [21], and DeSalvo [22]. Preliminary analysis shows that in principle,
298 270 APPENDIX B. CONTROL OF PARAMETRIC INSTABILITIES (a) (b) Figure B.6: Example of unstable mode suppression using a 2 cm wide, 20 m thick rind damper on the test masses in a simple cavity interferometer with Advanced LIGO parameters. (a) Unstable mode number without ring damper on the test masses. (b) Unstable mode number with ring dampers on the test masses. There is substantial reduction of unstable modes due to the applied lossy ring damper. it is possible to attach a small lossy springmass resonator to a test mass to damp the resonant modes. If we choose the mass of the damper to be 1 g and the resonant frequency of the damper to be 20 khz, it is easy to show that the Qfactor of the resonances of a 10 kg effective mass mirror, in the range of khz, can be reduced
299 B.3. POSSIBLE APPROACHES TO PI CONTROL 271 from 10 7 to < as shown in figure B.7. This result assumes perfect coupling to the test mass modes. To be usable in practice, there are several factors that must be considered. For the damping to be effective, the damper should be placed sufficiently close to the antinodes of the resonant mode to be damped. Therefore, many dampers may be necessary to obtain good coupling to the large number of potentially unstable modes. Also from the thermal noise point of view, it is desirable that the dampers should be placed far away from the laser spot. This limits the effectiveness of the damping for some modes that have high amplitude near the laser spot. Detailed modelling is required to determine an optical configuration of dampers and their thermal noise contribution Damped mode Q Frequency (Hz) Figure B.7: Internal modes Qfactor of a test mass with a small resonant damper attached. Parameter used: effective mode mass of the test mass 10 kg; original Qfactors of the internal modes 10 7 ; small resonator damper mass 1 g; damper Qfactor 1 and the frequency of the damper 20 khz. Here we assume perfect coupling between the damper resonance and the test mass internal modes. However, it is clear from figure B.6 that strong damping can be achieved in the khz band where instabilities occur, with weak damping within the gravitational wave signal band. Thus, this method is particularly worthy of further investigation.
300 272 APPENDIX B. CONTROL OF PARAMETRIC INSTABILITIES B.3.3 Local control of PI by acoustic excitation sensing and feedback Local feedback is a potentially simple solution to PI. The idea is to sense the acoustic excitation of a test mass and provide a derivative feedback force using standard control theory. Motion sensing could use optical, radiofrequency or electrostatic sensing, while feedback forces could be applied electrostatically. However, there are several issues with this method. a) First, the sensor must have sensitivity sufficient to maintain amplitudes small compared with the position bandwidth of the interferometer (position bandwidth is the linewidth expressed in displacements units). The position bandwidth of Advanced LIGO is 1 nm. The sensing noise must be small compared with this distance or else the feedback controller will excite all mechanical modes within its bandwidth (say khz) to an amplitude equal to the sensor noise floor. As long as the sensor noise floor is less than about m/ Hz, this will not represent a severe degradation of fringe contrast. However, if the sensor noise was m/ Hz (typical of current local control sensors), this would represent a significant loss of fringe contrast. It is also worth pointing out that the requirement of test mass motion < m/ Hz in the interferometer detection band (below a few khz) means that the force noise from feedback actuation on the test mass should be sufficiently low to avoid injecting extra noise in the detection band. This imposes even tighter requirements on the sensor noise floor. b) The second problem is the large number of test mass acoustic modes that are potentially excited, as shown in section B.2.3. Many of the acoustic modes have a complex mode structure. The overlap integral between each acoustic mode and an electrostatic sensor must be large to be able to get good signal coupling. It is often very difficult to excite test mass modes using electrostatic exciters because the exciter applies forces across a node so that the positive and negative displacements partially cancel, leading to small electromechanical coupling. In practice, we have found it very difficult to achieve strong coupling to high Q modes with high frequencies in our high Q test masses in an 80 m cavity. To accommodate high coupling to a large set of modes, the sensor and actuator would have to be broken up into a set of separately
301 B.3. POSSIBLE APPROACHES TO PI CONTROL 273 addressable elements. Each mode would require a different combination of actuator elements. Sensing suffers from the same problem, so that high signaltonoise sensing will require sensing at different locations on the test mass. c) The next problem is one of gain. The parametric gain we wish to suppress is typically 10, sometimes 100 and if unlucky more than Let us assume that we choose a gain of 100 as the maximum we want to control. This means that the actuator must be able to excite the test mass mode at a rate 100 times faster than the ringdown time. For example, if the acoustic mode ringdown time is 10 s, the ringup time will be 0.1 s. Such strong excitation is possible for small gap spacings and high excitation voltages, but in our experience such strong coupling is difficult to achieve. It would require a very high voltage small gap spacing exciter, and it would have to be designed so that it did not supply residual inband noise above the detector noise floor. B.3.4 PI control using global optical sensing and electrostatic actuation PI will be very easily detected in the dark fringe signal from the interferometer, which is a much better probe of instability than any local sensor. The signal could be applied to all test masses simultaneously even though the unstable acoustic modes will be shared across four test masses. There should be no problem with driving test masses at frequencies where there is no instability, except under the unlucky circumstance that the frequency coincided with another test mass acoustic mode. If this occurred, the system should also be able to automatically damp such a selfinduced instability. Using the global PI signal overcomes the signaltonoise ratio (SNR) problem associated with local sensing, but there still remains the need to apply quite large forces to the test masses in the case of high gain instabilities. This is still limited by the problem of achieving a large overlap integral with the electrostatic driver as discussed in section B.3.2. Control of the end test masses will need to take account of the time delay phase shift. To evaluate the active acoustic damping solution requires the testing of suitable actuators to confirm adequate coupling to all of the predicted unstable test mass acoustic modes.
302 274 APPENDIX B. CONTROL OF PARAMETRIC INSTABILITIES B.3.5 PI control using global optical sensing and direct radiation pressure The force required to reduce the Qfactor of an acoustic mode is directly proportional to its amplitude. If a suppression system is required to act on large acoustic amplitudes (say m), the force requirements are too large to use direct radiation pressure actuation. However, direct radiation pressure actuation is practical if instability is caught before the amplitude has grown too large. If we apply direct radiation pressure feedback control when the oscillation increases to ten times the equilibrium thermal amplitude, the maximum force F 0 required to reduce the acoustic mode Q factor is corresponding to a laser power of F 0 = 10 2mω2 k B T Q f, (B.5) P = cf 0 2 = 5c 2mω2 k B T, (B.6) Q f where k B is Boltzmann s constant, c is the speed of light and Q f is the desired Q factor. Assuming an effective mass m = 10 kg, ω m = 30 khz, Q f = 10 5, T = 300 K, the required power is P = 0.8 W. Here, we have used point mass approximation which assumes 100% overlap between the acoustic mode and the actuation force. For nonideal overlap, corresponding to actuation far from an antinode of the acoustic mode, more power will be required. In 2005, we used a laser walkoff delay line as a radiation pressure actuator [23] and proposed to use direct radiation pressure as a means of PI control. Multiple reflections can reduce the power requirements or increase the force by 30 times. In all cases, the main problem is to obtain large overlap with all acoustic modes. The actuation phase depends on the location of the actuator and must be well defined for each relevant mode. If instability was not suppressed early enough, such an actuator would be unable to achieve suppression. This threshold effect imposes strong requirements on the reliability and completeness of the suppression system. Braginsky and Vyatchanin [24] suggested using an external short optical tran
303 B.3. POSSIBLE APPROACHES TO PI CONTROL 275 quiliser cavity to control instability. This method is also limited by the requirement of overlap between the actuator and the test mass acoustic modes. In [25], the method was shown to be technically difficult but viable in principle. Again, in practice several such systems would be required for each test mass to obtain adequate overlap with the acoustic modes, greatly increasing the complexity of the interferometer system. B.3.6 PI control by global optical sensing and optical feedback The parametric interaction provides forces to the test mass via the high order optical modes which are excited in the cavity. It should be possible to suppress the high order mode in the cavity by introducing an antiphase high order mode. Here, we summarise the analysis of such a high order mode interference system. It has the advantage that it could act very rapidly and even suppress very high gain instabilities. If it was practical and robust, it could eliminate the need for any other instability suppression scheme. To model the system with optical feedback, Zhang et al [26] used a classical model of a cavity with fields as shown in figure B.8 including an injected field f 0 (t). The fundamental mode (E i with frequency ω 0 ) and the high order mode (f i with frequency ω 1 ) contribute to create the radiation pressure force at the differential frequency of ω 0 ω 1. This force acts back on the test masses which vibrates at its internal mode frequency ω m. The parametric instability comes from the interaction between the radiation pressure force and the mechanical mode vibration. If no other external field is injected into the cavity apart from the fundamental mode, the high order mode field f i is produced due to the scattering of the fundamental mode into higher order modes and their resonant buildup inside the cavity. The back action force is determined by the product of f i and E i. As E i is a constant depending on the input power and cavity parameters, the strength of the instability is solely determined by f i and consequently by the parameters in equation (B.1)the frequency difference between ω 0 ω 1 and ω m, the overlap between the high order mode and the test mass internal mode, and the quality factors of the cavity and the test mass internal mode. If we inject another high order mode optical field f 0 out
304 276 APPENDIX B. CONTROL OF PARAMETRIC INSTABILITIES f 0 (t) f 1 (t) f 2 (t) E Laser E 1 E in E 4 E 3 f 4 (t) L f 3 (t) x(t) Figure B.8: The fields of the fundamental mode, high order mode and the feedback of the cavity. The fundamental mode fields with frequency ω 0 is denoted E i (i = 1, 2, 3, 4), and the high order mode with frequency ω 1 is denoted f i (i = 1, 2, 3, 4). The feedback field is f 0, L is the length of the cavity and x(t) is the perturbation of the cavity length due to mirror vibration. of phase with f 1 into the cavity, it will destructively interfere with f 1 to suppress the instability. By detecting the optical signal at the interferometer dark port, it is possible to determine the amplitude, frequency and phase of f 1. One can then inject field f 0 with an appropriate phase and amplitude to suppress the instability. Figure B.9 is a simplified schematic diagram for this optical feedback control system. The beam pickoff mirror (BS1) diverts a small part of the main laser beam to a PZT mirror which is used for locking a beam injection MachZehnder interferometer. After passing through two phase modulators and a phase mask, the pickoff beam recombines with the main beam at BS2 and then is injected to the interferometer. The phase modulator EOM1 together with photodetector PD1, mixer and the amplifier are used to phase lock the pickoff beam to the main beam. The modulator EOM2 is used to create the sideband at the test mass internal mode frequency ω m. The phase mask converts the fundamental mode to the high order mode. The photodetector PD2 detects the high order mode amplitude and frequency that is fed back to control the driver of EOM2 to create an appropriate level sideband signal. The sideband signal is injected into the cavity to suppress the high order mode created by the test mass internal mode scattering. This prevents the buildup of the frequency ω 1, thereby suppressing the instability.
305 B.4. CONCLUSIONS 277 Figure B.9: Schematic diagram of the PI optical feedback control setup. The beam pickoff mirror (BS1) diverts a small part of the main laser beam to the PZT mirror. After passing through two phase modulators and a phase mask the pickoff beam recombines with the main beam at BS2 and then is injected to the interferometer. Detailed analysis and numerical results [26] show that in principle, it should be possible to suppress parametric instability in the next generation detector such as Advanced LIGO. This would only require external optical and electronic components at the corner station and would not require modification of the test masses or the local control systems. It has the advantage that it could act very rapidly and even suppress very high gain instabilities. The disadvantage of this method is that each high order mode to be suppressed required a MachZehnder arrangement and a phase mask for adding the high order mode. This will increase the complexity if many modes were to be suppressed. However, it would be possible to use this technique combined with other methods for PI suppression. If the injection optics in advanced interferometers included the basic beam splitters required, particular problematic instabilities could be suppressed without internal modification of the interferometer. An experiment at the high optical power facility [9] at Gingin is planned to investigate this solution. B.4 Conclusions We have shown that without taking active control measures, advanced gravitational wave interferometers may suffer from parametric instabilities. These will render the
306 278 APPENDIX B. CONTROL OF PARAMETRIC INSTABILITIES devices inoperable at high power. The stochastic nature of instability has been emphasised, with instability sometimes occurring suddenly, as the thermal conditions of test masses vary with time. The difficulty of making precise predictions has also been emphasised. Control of instability using simple active damping is shown to be difficult. We have shown that passive ring dampers can significantly reduce parametric gain but not eliminate instability. Simple modelling shows that a number of passive resonant dampers could be attached to test masses to reduce instability and probably control it, but this requires detailed modelling. A less invasive technique based on optical feedback has been proposed. This technique allows instability control by selectively suppressing all large amplitude high order modes. This technique enables all instabilities to be controlled but requires knowledge of the high order mode, and a separate phase mask for each high order mode that requires suppression. This is a most attractive backup solution which could be implemented during commissioning if required for particular high order modes without requiring mechanical changes to the core optics. Acknowledgment We would like to thank the LIGO laboratory and the International Advisory Committee of the Gingin Facility for their encouragement and advice. This research was supported by the Australian Research Council and the Department of Education, Science and Training. This paper has been assigned LIGO Document Number LIGO P B.5 References [1] V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, Parametric oscillatory instability in fabryperot interferometer, Phys. Lett. A 287 (2001) [2] V. B. Braginsky and S. P. Vyatchanin, Low quantum noise tranquilizer for FabryPerot interferometer, Phys. Lett. A 293 (2002)
307 B.5. REFERENCES 279 [3] C. Zhao, L. Ju, J. Degallaix, S. Gras, and D. G. Blair, Parametric instabilities and their control in advanced interferometer gravitationalwave detectors, Phys. Rev. Lett. 94 (2005) [4] L. Ju, S. Gras, C. Zhao, et al, Multiple modes contributions to parametric instabilities in advanced laser interferometer gravitational wave detectors, Phys. Lett. A 354 (2006) [5] A. Gurkovsky, S. Strigin, and S. Vyatchanin, Analysis of parametric oscillatory instability in signal recycled ligo interferometer, Phys. Lett. A 362 (2007) [6] S. Gras, D. Blair, and L. Ju, Test mass ring dampers with minimum thermal noise, Phys. Lett. A 372 (2008) [7] H. S. Bantilan and W. P. Kells, Investigating a Parametric Instability in the LIGO Test Masses, Technical Report, T , LIGO, [8] K. Yamamoto, T. Uchiyama, S. Miyoki, et al, Parametric instabilities in the LCGT arm cavity, J. Phys. Conf. Ser. 122 (2008) (6pp). [9] S. Strigin, D. Blair, S. Gras, and S. Vyatchanin, Numerical calculations of elastic modes frequencies for parametric oscillatory instability in advanced ligo interferometer, Phys. Lett. A 372 (2008) [10] B. J. Meers, Recycling in laserinterferometric gravitationalwave detectors, Phys. Rev. D 38 (1988) [11] S. Strigin and S. Vyatchanin, Analysis of parametric oscillatory instability in signal recycled ligo interferometer with different arms, Phys. Lett. A 365 (2007) [12] C. Zhao, L. Ju, Y. Fan, et al, Observation of threemode parametric interactions in long optical cavities, Phys. Rev. A 78 (2008) [13] J. Manley and H. Rowe, Some General Properties of Nonlinear ElementsPart I. General Energy Relations, Proc. of the IRE 44 (1956)
308 280 APPENDIX B. CONTROL OF PARAMETRIC INSTABILITIES [14] S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, Continuousvariable entanglement and quantumstate teleportation between optical and macroscopic vibrational modes through radiation pressure, Phys. Rev. A 68 (2003) [15] L. Ju, C. Zhao, S. Gras, et al, Comparison of parametric instabilities for different test mass materials in advanced gravitational wave interferometers, Phys. Lett. A 355 (2006) [16] P. Barriga, B. Bhawal, L. Ju, and D. G. Blair, Numerical calculations of diffraction losses in advanced interferometric gravitational wave detectors, J. Opt. Soc. Am. A 24 (2007) [17] S. D. Penn, A. Ageev, D. Busby, et al, Frequency and surface dependence of the mechanical loss in fused silica, Phys. Lett. A 352 (2006) 3 6. [18] I. Martin, H. Armandula, C. Comtet, et al, Measurements of a lowtemperature mechanical dissipation peak in a single layer of ta2o5 doped with tio2, Class. Quantum Grav. 25 (2008) (8pp). [19] Y. Levin, Internal thermal noise in the LIGO test masses: A direct approach, Phys. Rev. D 57 (1998) [20] K. Yamamoto, M. Ando, K. Kawabe, and K. Tsubono, Thermal noise caused by an inhomogeneous loss in the mirrors used in the gravitational wave detector, Phys. Lett. A 305 (2002) [21] M. Evans, J. S. Gaviard, D. Coyne, and P. Fritschel, Mechanical Mode Damping for Parametric Instability Control, Technical Report, G , LIGO, [22] R. DeSalvo, 2008, LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA, (private communication, 2005). [23] M. Feat, C. Zhao, L. Ju, and D. G. Blair, Demonstration of low power radiation pressure actuation for control of test masses, Rev. Sci. Instrum. 76 (2005) (3pp).
309 B.5. REFERENCES 281 [24] V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, Analysis of parametric oscillatory instability in power recycled ligo interferometer, Phys. Lett. A 305 (2002) [25] S. W. Schediwy, C. Zhao, L. Ju, et al, Observation of enhanced optical spring damping in a macroscopic mechanical resonator and application for parametric instability control in advanced gravitationalwave detectors, Phys. Rev. A 77 (2008) (5pp). [26] Z. Zhang, C. Zhao, L. Ju, D. G. Blair, and P. Willems, Enhancement and suppression of optoacoustic parametric interactions using optical feedback, Phys. Rev. A, 2009, submitted. [27] L. Ju, M. Aoun, P. Barriga, et al, ACIGA s high optical power test facility, Class. Quantum Grav. 21 (2004) S887 S893.
310 282 APPENDIX B. CONTROL OF PARAMETRIC INSTABILITIES
311 Appendix C Vibration Isolator Control Electronics C.1 Introduction The local control system is a multiple inputmultiple output (MIMO) system, which is then converted via a sensing matrix into a single inputsingle output (SISO) system. Each input passes through a dedicated digital filter and PID control loop before being converted by a driving matrix into signals to drive each of the actuators. As a consequence different matrices, filters and control loops have been created for the different stages depending on their resonant frequencies, the number of actuators, and their distribution on that particular stage. The control electronics provides means to interconnect actuators and sensors to the DSP on which the local control system is based. The user control and interface was written in LabView R. The builtin libraries provided by the DSP manufacturer allow for the control loops to run on the DSP board isolated from the local operating system. Through the interface the operator can remotely monitor the performance of the different stages of the isolator, and adjust the various loop gains and settings if necessary. The operator interface is a series of graphical screens showing the current status of each stage of the isolators. Figure C.1 shows a block diagram of the local control system and figure C.2 shows a general schematic for its wiring. C.2 Control electronics The control electronics were developed in standard 6U height Eurocard format at UWA. Each board in the control electronics contains two channels. Each channel 283
312 284 APPENDIX C. VIBRATION ISOLATOR CONTROL ELECTRONICS Inverse Pendulum 4ACT + 4SS 4SS 4ACT La Coste Vertical Stage 1HC + 2ACT + 1SS Roberts Linkage 4HC + 2SS 1SS 2ACT 1HC 2SS 4HC High Current Power Supply High High Current PreIsolator High Current High Current Power Current Power Supply Power Supply Power Supply Supply Multipendulum stage 1CS 4CS DSP Sheldon MOD6800 Control Mass 5ACT + 5SS Optical Lever 1QPD Test Mass Control 5ACT 5SS 1OL 1QPD Laser λ = 1064 nm Figure C.1: Block diagram of the local control system for the advanced vibration isolator. provides an infrared LED control circuit, a dual photodetector circuit, and a control signal circuit. The LED control circuit (shown in figure C.8) provides power to the infrared LED and monitors its connectivity, which can be seen in the frontpanel of the board. The photodetector circuit (shown in figure C.9) filters and amplifies the signal from both photodetectors that comprise the shadow sensor. Both signals are independently fed to the DSP board, and can also be accessed from the front panel for external measurements or monitoring. The photodetectors differential signal can also be accessed from the front panel. The control signal from the DSP is fed to the corresponding control channel circuit (shown in figure C.10) through the backplane. The control signal goes through a filter and a transimpedance circuit before going through a high speed amplifier. This circuit converts the voltage control signal into a corresponding current level that drives the magnetic field produced by the coils which in turn acts on the permanent magnet and thus on the corresponding stage of the vibration isolator. The control signal circuit includes a potentiometer (POT2 in figure C.10) which allows for the adjustment of the gain in order to maximise the dynamic range of the actuators. Small differences on each actuator impedance due to distance and/or wiring, including smaller actuators for the control mass stage, makes it necessary to tune each circuit in order to maximise the control signal dynamic
313 C.2. CONTROL ELECTRONICS 285 Current Power Supplies Intermediate Board LaCoste Heating Roberts Linkage Heating Inverse Pendulum PC/DSP Control Electronics Vacuum 106 mbar Roberts Linkage LaCoste Control Mass Horizontal Control Mass Vertical Figure C.2: Control electronics and wiring strategy. A 100 pin Sub D connector is used to connect the PC/DSP which host the local control software and processor to the control electronics chassis through its backplane. From here the I/O signals are distributed to the high current power supplies or to the magnetic actuators. While the shadow sensors provide the sensing, magnetic actuators control the positioning of each stage. The signals are distributed through an intermediate board which distributes them to each axis on the isolator. Low frequency position control is also obtained using current power supplies. These signals are distributed separately through high current wires to each stage. range. An external signal can be connected to this circuit through a connector in the front panel. The external signal is added to the control signal from the local control system. The filter board (shown in figures C.13 and C.14) contains five independent filter circuits connecting the control signals to the current power supplies that drive each of the degrees of freedom of the ohmic position control. Four are used for each Roberts linkage axis and one for the LaCoste stage which connects all the springs of this stage in series. Figure C.3 shows the distribution of the different boards and channels that populate the control electronics. Figure C.4 shows the location of the different connection points in the isolator frame, using the east arm ITM vibration isolator as an example. The connectors with only shadow sensors (ss) are installed for the Roberts linkage sensing, and the horizontal actuators and shadow sensors (as) are for the inverse pendulum stage. High current connections (h) are used for the ohmic position control, which provides ultra low frequency position control for the Roberts linkage and LaCoste stages, while
314 286 APPENDIX C. VIBRATION ISOLATOR CONTROL ELECTRONICS (a) Optical Lever Input Auxiliary I/O Control Mass Tilt B Control Mass Tilt D Control Mass Horizontal BC Control Mass Horizontal CD Control Mass Spare Control Mass Horizontal A High Current Filters 0 High Current Filters 1 Roberts linkage D Roberts linkage A LaCoste Vertical C LaCoste Vertical A Inverse Pendulum B Inverse Pendulum D Inverse Pendulum C Inverse Pendulum A (b) Figure C.3: Control electronics board distribution. Each board has two channels which filter and amplify the signals from shadow sensors and magnetic actuators. The heating positioning control signal is connected directly from the DSP to the high current power supply through the filters board. vertical actuation (va) is only for the LaCoste stage. In addition, five shadow sensors and actuators pairs are necessary for the control mass stage. These connectors are not visible from the top view of the vibration isolator. The central tube connections (c) are for the electrostatic control and its wiring is fed through the central tube of the multistage pendulum in order to reach the suspension cage at the bottom where the electrostatic board is installed next to the test mass. Since each vibration isolator is orientated according to the test mass orientation, and therefore with respect to the incoming laser beam, they will have different geographical orientation. For this reason A, B, C and D are defined as the cardinal
315 C.2. CONTROL ELECTRONICS 287 c2 op2 as3 va2 h3 h2 ss shadow sensors as horz. actuator + shad. sens. c central tube h high current/heating op options/auxiliary va vertical actuator A +V c3 ss1 North as2 D op1 z y x as4 B High current board c1 h4 East ITM Isolator h1 va1 h1 C ss2 c4 Intermediate Board Figure C.4: The figure shows the distribution of the different connection points in the isolator. It also shows the distribution actuators for the inverse pendulum and LaCoste stages and the direction of the actuators response to a positive control signal. points for the vibration isolators. Each vibration isolator will have different geographical orientation but the same relative orientation within its components. This facilitates the local control system setup since it does not need to be tailored for each isolator and as a consequence the wiring and channel distribution remains the same. Inside the vacuum tank each shadow sensor and actuator is ultra high vacuum compatible and each has to be wired to a connection port requiring ultra high vacuum compatible boards, connectors, pins and wires. Polyimide insulated copper wires of 0.25 mm diameter were used in order to connect the shadow sensors and actuators around the vibration isolator. The wiring was arranged so as to minimise the transmission of vibration. It was also designed to be modular, with several identical extensions and connectors fabricated in parallel. This strategy simplified the replacement of faulty pieces. The intermediate board shown in figure C.4 distributes the I/O signals within the vibration isolator. This board was developed in order to collect the multiple signals
316 288 APPENDIX C. VIBRATION ISOLATOR CONTROL ELECTRONICS from the shadow sensors and actuators in the isolator and group them in a smaller number of DB 25 terminals that could be easily connected through a sixway cross, used as feedthrough connection port to the outside world as shown in figure C.5. From the sixway cross, 25 vias cables with standard DB 25 connectors were used to connect the signals to the control electronics. The east board in figure C.4 distributes the high current signals for the heating stages of the LaCoste and Roberts linkage. A picture of the intermediate board as installed on the ITM vibration isolator are shown in figure C.6, while the diagrams for the signal distribution are shown in figures C.15, C.16, C.17, C.18, and C.19. (a) (b) Figure C.5: The picture shows a 6way cross used as wires feedthrough. Each of the remaining sides has a vacuum compatible DB 25 that connects the shadow sensors and magnetic actuators signals. A backplane was developed in order to distribute the I/O signals between the different control boards, the vibration isolator, and the DSP. A block diagram of the backplane is shown in figure C.20 which is followed by the signal distribution through the backplane connectors. Figure C.21 shows the different I/O signals for the horizontal preisolator which correspond to the inverse pendulum actuators and shadow sensors. Figure C.22 shows the I/O signals for the vertical axis which correspond to the LaCoste actuators and shadow sensors. Figure C.23 shows the Roberts linkage
317 C.3. CONCLUSIONS 289 shadow sensors. Figure C.24 shows the signals for the ohmic position control. Figure C.25 shows the horizontal control mass I/O signals. Figure C.26 shows the tilt (or pitch) control mass I/O signals. Figure C.27 shows the distribution of the I/O signals of the different connectors at the backplane. Figure C.28 shows the signal distribution of the 100 pin SubD connector which connects the DSP to the backplane. (a) (b) Figure C.6: (a) Intermediate board installed at the ITM vibration isolator. (b) shows the intermediate board and to the right the high current board. C.3 Conclusions The current DSP based local control system will be replaced with a multicpu digital controller server running realtime Linux. The server will be configured to allow separate control loops to run on each of the CPU cores. A PCIExpress expander chassis will be used to allow fibreoptic cables to be run to each of the isolators (ITM and ETM) on each arm of the interferometer. Analog to Digital and Digital to Analog converters, as well as AntiAliasing and AntiImaging filters will be installed at each station. This will connect to the control electronics currently in use for the signal distribution on each of the vibration isolators.
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